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A Timeline Of Circles
2009-Apr-12, 07:51 AM
Moved from thread about speed of Earth rotation...

I might be stealing this thread but I don't want to start a new one, but another quick question; Isn't the universe expanding due to the force of the Big Bang?

Jens
2009-Apr-12, 08:10 AM
I might be stealing this thread but I don't want to start a new one, but another quick question; Isn't the universe expanding due to the force of the Big Bang?

The mainstream explanation is no. If that were so, the expansion would be slowing down, but in fact it appears to be accelerating, so it is hypothesized that the expansion is caused by something called "dark energy."

A Timeline Of Circles
2009-Apr-12, 08:44 AM
The mainstream explanation is no. If that were so, the expansion would be slowing down, but in fact it appears to be accelerating, so it is hypothesized that the expansion is caused by something called "dark energy."

Interesting, I might start a new thread:) But wouldn't something of the that magnitude, Big Bang, have so much force to literally drive dark energy/matter out to? I mean do not we see the same thing in star explosions? That they leave trails of force or matter?

loglo
2009-Apr-13, 11:10 AM
The Big Bang wasn't an explosion. It didn't drive matter anywhere, it just expanded space which dragged matter along with it. A subtle but important difference.

matt.o
2009-Apr-13, 12:58 PM
The mainstream explanation is no. If that were so, the expansion would be slowing down, but in fact it appears to be accelerating, so it is hypothesized that the expansion is caused by something called "dark energy."

This is not correct. Dark energy is not wholly responsible for the expansion. At the current epoch, the expansion is due to both inertia and dark energy in roughly equal parts, however at early times the expansion was due mainly to inertia. The current theory is that inflation gave the initial "kick" for the expansion, and things just keep uniformly expanding unless they have some other action operating on them such as gravity, or dark energy.

Cougar
2009-Apr-13, 01:17 PM
Isn't the universe expanding due to the force of the Big Bang?

The "big bang" is poorly named. As loglo said, it was not an explosion or "bang." The big bang is essentially the initial expansion. So your question converts to:


Isn't the universe expanding due to the inertia of the initial expansion?

Here, I think the answer is yes, but as Jens points out, there is something else going on. Until 10 or 12 years ago, scientists always figured the expansion must be slowing due to the gravitational effect of all the mass in the universe. But careful, independent observations have shown this does not appear to be the case. The expansion is very, very slowly accelerating. The mechanism causing the acceleration is as yet unknown.

Kwalish Kid
2009-Apr-13, 02:24 PM
This is not correct. Dark energy is not wholly responsible for the expansion. At the current epoch, the expansion is due to both inertia and dark energy in roughly equal parts, however at early times the expansion was due mainly to inertia. The current theory is that inflation gave the initial "kick" for the expansion, and things just keep uniformly expanding unless they have some other action operating on them such as gravity, or dark energy.
This is more correct, but not completely correct.

The expansion is something completely separate from a "big bang" event. The universe could be infinitely old and we would still have expansion. That the average distance of ideal galaxies in space increases (or decreases) is a feature of the geometry of the universe which is essentially set by the initial conditions of the universe and is modified by physical properties of the universe.

So, if there was a big bang event, it set the expansion rate. This rate was immediately slowed down by the energy density of the very early universe at a known relationship. Then, perhaps, there was a sudden increase and then sudden decrease in expansion that we call inflation. Then the expansion went back to being slowed by the influence of the energy density and, later, the mass density of the universe.

Along with the mass-energy densities that slow expansion, there is also the potential for influences to increase expansion. It may be that expansion simply increases in rate over time because of the equation that governs the relationship between spacetime and gravity. It might be the the energy of the vacuum contributes to increasing the rate of expansion. So far, it is not sure which is contributing to increasing the rate of expansion, but one of them is.

However, all of this expansion is purely geometrical, it is not due to inertia or to impressed force.

Ken G
2009-Apr-13, 04:59 PM
However, all of this expansion is purely geometrical, it is not due to inertia or to impressed force.Exactly, it seems one of the most pervasive of all Big Bang myths, even among those who know a lot about it, that the Big Bang is "caused" by something. At present, there are no falsifiable theories that can attribute a cause to the Big Bang, so it would seem more appropriate to label it an "initial condition". Physics has never really known what to do with initial conditions, the normal expectation is that they are in turn caused by something else, but the something else always also involves initial conditions, so it is really no solution to the "initial condition problem." I'd call it a fundamental part of the structure of physics, so we should not be terribly surprised it is also reflected in our theories of origin of the universe.

01101001
2009-Apr-13, 05:08 PM
Big Bang questions

Please study Scientific American: Misconceptions about the Big Bang (http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03) by Lineweaver and Davis.

For instance:


Brooklyn is not expanding. People often assume that as space expands, everything in it expands as well. But this is not true.

A Timeline Of Circles
2009-Apr-13, 09:03 PM
Ok, so too an extant..will the universe ever end? Stop expanding after all the inertia is withered away? Do we have any theories out that could give some back round on that?:think:

Bearded One
2009-Apr-13, 11:25 PM
Ok, so too an extant..will the universe ever end? Stop expanding after all the inertia is withered away? Do we have any theories out that could give some back round on that?:think:Inertia can't be withered away, it's an inherent property of all matter. Inertia is an objects resistance to a change of motion. I suspect what you are referring to would be more closely defined by the term kinetic energy. It's related to, but not the same as inertia.

Think about it this way, what direction is this initial "inertia" directed?

Jeff Root
2009-Apr-14, 12:33 AM
Think about it this way, what direction is this initial "inertia" directed?
Sheesh! That's a question I've thought about many times, and I
haven't yet figured out what the question is, much less an answer.
You'll make smoke come out of a newbie's ears with that one!

-- Jeff, in Minneapolis

A Timeline Of Circles
2009-Apr-14, 12:37 AM
Inertia can't be withered away, it's an inherent property of all matter. Inertia is an objects resistance to a change of motion. I suspect what you are referring to would be more closely defined by the term kinetic energy. It's related to, but not the same as inertia.

Think about it this way, what direction is this initial "inertia" directed?

I thought it went in all directions like a sphere from the center point? So in a sense (tell me if I am wrong) the middle of the universe.

Can someone describe how the big bang could relate to something in today's tech, as in a bang that never ceases to loose force?

pzkpfw
2009-Apr-14, 01:06 AM
There is no centre of the Universe.

The "Big Bang" was not some explosion that occured "somewhere" that flung stuff outwards from that point.

Use search - there's been lot's of threads.


(In brief - look at a balloon. The surface of that balloon.

There is no inside, there is no outside. Just the balloon itself.

Now imagine that balloon expanding. Which point on the balloon is the "centre"?)

A Timeline Of Circles
2009-Apr-14, 01:48 AM
(In brief - look at a balloon. The surface of that balloon.

There is no inside, there is no outside. Just the balloon itself.

Now imagine that balloon expanding. Which point on the balloon is the "centre"?)
Yes but how can we relate the expansion of the balloon with the expansion of the Universe? The balloon has a point in which we ourselves put the oxygen in the balloon. That would be inferring that the expansion of the universe has an outside object choosing how fast the expansion is.

01101001
2009-Apr-14, 01:53 AM
Yes but how can we relate the expansion of the balloon with the expansion of the Universe


Please study Scientific American: Misconceptions about the Big Bang (http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03) by Lineweaver and Davis.

Please.


This balloon analogy should not be stretched too far. From our point of view outside the balloon, the expansion of the curved two-dimensional rubber is possible only because it is embedded in three-dimensional space. Within the third dimension, the balloon has a center, and its surface expands into the surrounding air as it inflates.

Bearded One
2009-Apr-14, 02:03 AM
I thought it went in all directions like a sphere from the center point? So in a sense (tell me if I am wrong) the middle of the universe.

Can someone describe how the big bang could relate to something in today's tech, as in a bang that never ceases to loose force?To answer your second question first, no. Best you will get is poor analogies that break down so quickly that they may do more harm than good. You almost have to take multiple, sometimes conflicting, analogies together to try to get a real grasp of it. You just have to deal with the conflicting elements in whatever way suits your fancy. To truly understand, you have to go beyond the analogies and just learn to grasp it directly. You need to be able to "see" it in the equations and the models without referring to everyday situations.

That leads into the answer to your first question. You are envisioning a center to the universe, a common issue when trying to relate to everyday experiences. That leads you astray. The balloon analogy that is often used tends to leave the center question open. A balloon has a center. You will be told to just think of the surface of the balloon, but your mind will keep drawing you to that center that you know is there! It's just an analogy, don't take it to far.

Now, what was I blabbing about? Oh yeah, that direction stuff.

What direction are we moving in? You are inquiring as to whether we still have some of the initial kinetic energy that we should have got from the big bang. What direction would that be expressed in? Everything (galactic scale) is moving away from us. We must have an extreme case of galactic body odor, everything hates us and is trying to get away. Not only that, they seem to be accelerating their desire to avoid us. :(

The sad thing is that everybody in the Universe has the same impression, we are all getting further and further away from each other. The Universe is anti-social. :( So which way are we going? This way, that way or maybe the other way? It really depends on who you ask. Ask someone in a galaxy a billion light years to the left and they will say we're going this aways, the person in the galaxy a billion light years on the right says we're going the otherways. So how is this initial "inertia" represented?

It wasn't an explosion and the explosion analogy falls apart very quickly. You can determine the center of an explosion from the blast and it's remnants. I can't think of an everyday analogy that comes close. It's a bit like an expanding balloon, a bit like an explosion and a bit like an expanding gas cloud, yet different from all of them. It is what it is. Don't try to force it into something it isn't.

I'm not trying to give any actual answers here, if anything I'm expounding on the questions. The Universe is a strange place, stranger than we could ever imagine. We developed to exist on this planet and to deal with the situations we encounter on this planet. We cannot expect that we could readily comprehend the realities that exist beyond.

Cougar
2009-Apr-14, 02:48 AM
The expansion is something completely separate from a "big bang" event.

Well, yes, of course, but isn't the expansion the apparent result of whatever that big bang event was?


The universe could be infinitely old and we would still have expansion.

I don't follow you there. If the universe is infinitely old, why aren't all the stars dead?


That the average distance of ideal galaxies in space increases (or decreases) is a feature of the geometry of the universe which is essentially set by the initial conditions of the universe and is modified by physical properties of the universe.

That's an excellent point! Especially if you have some appreciation for what the "geometry of the universe" might be.


So, if there was a big bang event, it set the expansion rate.

Ah, so you're saying this is a feature of the space between particles, as opposed to any inertia imparted to the particles?! Yes, that must be right, because other than local gravitational perturbations, all the superclusters are essentially suspended within vast volumes of space that are apparently, from widely independent, confirming observations, slowly but inexorably expanding in all directions. Understanding Space - The Final Frontier. :)

Tarkus
2009-Apr-14, 10:29 AM
Space is expanding.... taking matter with it?

Reminds me of refrigeration.

No wonder it's getting colder.

Kwalish Kid
2009-Apr-14, 03:25 PM
I don't follow you there. If the universe is infinitely old, why aren't all the stars dead?
From the point of view of a toy model with expansion, this isn't really a problem. There are a number of scenarios whith infinite histories and stars. As long as there was a period in the past where the scale factor was relatively small compared to the present, there could be stars, as all the available matter and energy would be squeezed down to a usable state. Or there could be a matter/energy creation field of some sort. These sorts of models have, I believe, been almost ruled out be various observations, but they remain on the perifery of possibility. Hoyle, Narlikar and Burbidge really did some amazing work in trying to get their alternatives to the standard model off the ground. They showed that certain models were possible, even if they didn't fit the available evidence (and even if they seem far too interested in the possibility of intensional or unintensional conspiracy against their work).

You seem to be getting the rest of my post. This is not easy stuff to write in short bursts. Or long ones, I'm finding.

Ken G
2009-Apr-14, 03:44 PM
Indeed, as I often like to point out, the whole idea that "space is expanding and carrying matter with it" is certainly just a useful picture, rather than any kind of statement of verifiable physical truth. There is no model for how space "carries" things, space is, as yet, just an invention to support a concept of a ratio between a distance and a measuring standard. It also gives us some place to put dark energy, but we have no idea what that is anyway. To really see why "space is expanding" is not a physically testable statement, I point out that "matter is shrinking" can be made to be completely indistinguishable in every way from expanding space. It requires suspension of disbelief that matter can really do that, but so does expanding space. At the end of the day, all we do is model and invent pictures to support the models.

George
2009-Apr-14, 03:47 PM
At present, there are no falsifiable theories that can attribute a cause to the Big Bang, so it would seem more appropriate to label it an "initial condition". Yes, they are just, as Cougar coined, theorinos (something to take somewhat seriously, but short of a true theory), right ? :)

Not that I read many, but the more I read books addressing pre-BBT views the more I appreciate your elevated stance in properly pointing-out that they are not true scientific theories, contrary to their claims. It is surprising that some cosmologists seem to have no problem in claiming genuine theoretical status of how to address t=0, as well as, that which was before and beyond.

Cougar
2009-Apr-14, 04:14 PM
However, all of this expansion is purely geometrical, it is not due to inertia or to impressed force.
Exactly, it seems one of the most pervasive of all Big Bang myths, even among those who know a lot about it, that the Big Bang is "caused" by something.

I think "myth" is a bit harsh, as if someone fell for a fully imaginary story, when in actuality it was not too long ago that introductory texts and other publications put forward the "inertia" explanation for the ongoing expansion... or else I completely misunderstood this same explanation on numerous past occasions, in which case, it must not have been explained very precisely...

Timothy Ferris's observation is relevant here:




"We live in a changing universe, and few things are changing faster than our conception of it."

Kwalish Kid
2009-Apr-14, 05:01 PM
Can you find one of these texts with a discussion of "inertia" in this context? I would be interested to see one.

Jeff Root
2009-Apr-14, 05:19 PM
Ignoring the acceleration, how is the observed expansion different
from inertial motion? I don't see a difference.

-- Jeff, in Minneapolis

speedfreek
2009-Apr-14, 06:13 PM
Ignoring the acceleration, how is the observed expansion different
from inertial motion? I don't see a difference.

-- Jeff, in Minneapolis

Ignoring the acceleration of the rate of expansion, we find that the further away a distant galaxy is the faster it is apparently receding. How could that be explained through inertia? Why would inertia cause galaxies to accelerate as they move away from us?

If we then add the cosmological principle and assume that we are not at the centre of the universe, we find that the expansion of the background metric would explain our observations and so we think that wherever you were in the universe you would see the distant galaxies apparently receding from you in the same way. How could inertia work like a changing background metric?

speedfreek
2009-Apr-14, 06:24 PM
Please study Scientific American: Misconceptions about the Big Bang (http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03) by Lineweaver and Davis.


Here is a reformatted pdf file of that article, with all the diagrams intact, from the authors website:

Misconceptions about the Big Bang (http://www.mso.anu.edu.au/%7Echarley/papers/LineweaverDavisSciAm.pdf).

"Baffled by the expansion of the universe? You’re not alone. Even astronomers frequently get it wrong".

Jeff Root
2009-Apr-14, 09:29 PM
Ignoring the acceleration, how is the observed expansion different
from inertial motion? I don't see a difference.
Ignoring the acceleration of the rate of expansion, we find that the
further away a distant galaxy is the faster it is apparently receding.
How could that be explained through inertia?
The galaxies are just coasting away from each other. That is all
"inertial motion" means. I didn't ask about what caused the motion.
I asked how the observed motion, ignoring the recently-discovered
acceleration, is different from inertial motion.



Why would inertia cause galaxies to accelerate as they move away
from us?
Already you are ignoring my request to ignore the acceleration!
Ignore it already! This is 1997. Nobody has a clue yet that the
expansion might be accelerating.



If we then add the cosmological principle and assume that we are
not at the centre of the universe, we find that the expansion of the
background metric would explain our observations and so we think
that wherever you were in the universe you would see the distant
galaxies apparently receding from you in the same way. How could
inertia work like a changing background metric?
If the galaxies are just coasting away from each other, it should
look exactly like what we see now, in 1997, as you described:
The farther away a distant galaxy is, the faster it is receding.

-- Jeff, in Minneapolis

speedfreek
2009-Apr-14, 11:33 PM
I think I need to go back to 1929 in order to understand this myself! :)

Why does redshift increase with distance?

Why aren't distant galaxies coasting towards the galaxies beyond them? Why are the more distant galaxies coasting away faster than the closer ones? They all seem to have different speeds relative to us. The further away a galaxy is, the faster it recedes. It is as if they are are accelerating away from us. It is as if we are at the epicentre of an explosion and everything is flying away from us, gaining speed as it goes.

I often see it said that inertia can describe the metric expansion of the universe, but I cannot understand how the idea that "an object in motion stays in motion" applies when things seem to be receding from us faster, the further we look. I can understand model explanations like "they are lodged in the fabric of expanding space" or "the metric that defines distance is changing, over time", but how does coasting away from each other work across increasing distance?

Grey
2009-Apr-14, 11:42 PM
I often see it said that inertia can describe the metric expansion of the universe, but I cannot understand how the idea that "an object in motion stays in motion" applies when things seem to be receding from us faster, the further we look. I can understand model explanations like "they are lodged in the fabric of expanding space" or "the metric that defines distance is changing, over time", but how does coasting away from each other work across increasing distance?Look at it the other way around. Say a bunch of galaxies all start at the same spot, and begin moving away from each other at random speeds. Wait 13 billion years. Now, if galaxy A was moving away from us twice as fast as galaxy B, in those 13 billion years it will have moved twice as far, so when we look at it now, it will be twice as far away. There's your redshift/distance relationship. Notice that in this scenario, the Hubble constant is constantly decreasing over time. That is, if you wait another 13 billion years, galaxy A won't have doubled in speed somehow. It will have remained at the same speed (lower actually, due to gravitational attraction, but let's not worry about that part), so now the Hubble parameter is half what it was. But galaxy B, still moving at half the speed of galaxy A, is still half as far away as galaxy A, so the redshift to distance relationship still holds, just with a new proportionality constant. Now, that model doesn't work at all with an accelerating rate of expansion, but it does work with a decreasing rate of expansion. Does that make sense?

matt.o
2009-Apr-14, 11:58 PM
Can you find one of these texts with a discussion of "inertia" in this context? I would be interested to see one.

The question "why is the Universe expanding" or "why are things receding" often comes up and a common answer is "because it was yesterday!" (well not quite that flippant, but you get the gist). In fact, when discussing the nature of the expansion in Cosmological Physics, Peacock says "If we understand that objects separate now only because they have done so in the past, there need be no confusion."

To describe inertia, my high school physics teacher said "Inertia is when something keeps on doing what it's already doing". Thus, I think the use of inertia in answering why the Universe is expanding is justified.

speedfreek
2009-Apr-15, 12:05 AM
Now, that model doesn't work at all with an accelerating rate of expansion, but it does work with a decreasing rate of expansion. Does that make sense?

Yes it does, thank you. I did know that, somewhere in the back of my mind (sorry Jeff!), but my problem is that I have only started looking into cosmology over the past few years, so my limited knowledge is more biased towards all the recent discoveries. I have come at the whole thing backwards.

So, well before any observations of an accelerating expansion, when we thought that the rate of expansion was still decelerating due to gravity, what happened when we found high redshift objects apparently receding faster than light?

Tarkus
2009-Apr-15, 03:45 AM
They're being sucked out - it's a vacuum up there you know..

Running now

Grey
2009-Apr-15, 12:51 PM
So, well before any observations of an accelerating expansion, when we thought that the rate of expansion was still decelerating due to gravity, what happened when we found high redshift objects apparently receding faster than light?Well, long before we saw evidence of the expansion accelerating, we had abandoned the view of the expansion as galaxies moving through space, and adopted the idea of an expansion of space carrying the galaxies along, as it were. So that already solved the issue of superluminal recession before any such galaxies were observed (that is, there isn't any restriction in general relativity of a net increase in distance between two distant objects that is greater than the speed of light due to the expansion of space; you still won't see one object go zipping right past another object faster than light). But, we could continue our path of thought, and cast ourselves back in time to the 1920's, and imagine what they would think of extremely high redshift galaxies. At that point, the thought was that those redshifts were Doppler redshifts from recession velocity. The relativistic Doppler formula can be written as z = sqrt[(c + v) / (c - v)] - 1. If you look at that equation, you'll see that as v approaches c, z increases without bound. So no matter how large a Doppler redshift is, it can be produced by an object receding at less than the speed of light.

01101001
2009-Apr-15, 04:51 PM
Please study Scientific American: Misconceptions about the Big Bang (http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03) by Lineweaver and Davis.Here is a reformatted pdf file of that article, with all the diagrams intact, from the authors website:

Misconceptions about the Big Bang (http://www.mso.anu.edu.au/%7Echarley/papers/LineweaverDavisSciAm.pdf).

Thank you for that. I keep forgetting Scientific American castrates images in old articles as a way of enticing visitors to subscribe. I condemn that as rude, but certainly within their rights, and understandable as a way for the old media to try to make a buck on this new Internet thing nobody understands. Scientific American, please find a better way.

The PDF link comes with the pictures intact, so has more value to the reader. And, I'm mentioning both here, just so I have half a chance to find the link to the unabridged PDF when next I search for the article's location.

And, did I mention: those who have Big-Bang questions, be that reader. Please read it first to see if your own questions have already been answered. And enjoy the pictures.

Smoke Ring
2009-Apr-15, 04:58 PM
If you have ever blown up a balloon, when you first start blowing you must blow hard to start the balloon expanding and as you keep blowing it becomes easier and easier. Is it possible that we are seeing a phenomena similar to this?

speedfreek
2009-Apr-15, 05:18 PM
Well, long before we saw evidence of the expansion accelerating, we had abandoned the view of the expansion as galaxies moving through space, and adopted the idea of an expansion of space carrying the galaxies along, as it were. So that already solved the issue of superluminal recession before any such galaxies were observed (that is, there isn't any restriction in general relativity of a net increase in distance between two distant objects that is greater than the speed of light due to the expansion of space; you still won't see one object go zipping right past another object faster than light).

And here is where my earlier confusion lies. If we had abandoned the view of expansion as galaxies moving through space well before the acceleration of the expansion was discovered, how could their movement have been considered inertial?

Jeff Root
2009-Apr-15, 05:25 PM
long before we saw evidence of the expansion accelerating, we
had abandoned the view of the expansion as galaxies moving through
space, and adopted the idea of an expansion of space carrying the
galaxies along, as it were.
Is there any difference between the two? Wouldn't galaxies
coasting through space look exactly the same as galaxies moving
along with expanding space? If there is a difference, what is it?

-- Jeff, in Minneapolis

rommel543
2009-Apr-15, 05:29 PM
speedfreek did an amazing explanation of expansion here (http://www.bautforum.com/space-astronomy-questions-answers/84823-location-big-bang.html#post1447613). Really helped me understand why the expansion is accelerating.

speedfreek
2009-Apr-15, 06:07 PM
Is there any difference between the two? Wouldn't galaxies
coasting through space look exactly the same as galaxies moving
along with expanding space? If there is a difference, what is it?

-- Jeff, in Minneapolis

I'm thinking the difference is the difference between the Copernican and cosmological principles. Greys earlier explanation doesn't seem to me like it would adhere to the cosmological principle (my bold):


Say a bunch of galaxies all start at the same spot, and begin moving away from each other at random speeds. Wait 13 billion years. Now, if galaxy A was moving away from us twice as fast as galaxy B, in those 13 billion years it will have moved twice as far, so when we look at it now, it will be twice as far away.

And how could galaxies apparently coast through space faster than light?

Ken G
2009-Apr-15, 08:08 PM
Wouldn't galaxies
coasting through space look exactly the same as galaxies moving
along with expanding space? If there is a difference, what is it?
There might be a confusion here between two senses of the term "inertial". This is not the way you mean it, I think, but in one sense of the term, the motion of the galaxies is purely inertial, because there are no forces on them (in GR gravity is not a force, it is the participation of spacetime in the dynamics). However, this type of inertial action begs the question of why the universe is expanding, as that addresses the issue of what is happening to the spacetime metric. One cannot call something an "explanation" if it only talks about what is not present (forces).

But what I think you mean by a purely "inertial" explanation is different from the above, if I correctly take your meaning that spacetime may be viewed as globally static and Minkoskian (a la special relativity), and the galaxies may be viewed as moving in a fixed and constant way (or even decelerated/accelerated by various gravity terms) through that Minkowskian spacetime. That is the "Milne" model, but that description simply does not succeed in matching the cosmological data (even before the acceleration business), and prompts the need to use the equations of general relativity-- not those of special relativity with some quasi-Newtonian deceleration tacked on. The key reason is that the Minkoswki metric, coupled with some velocity evolution of the galaxies, cannot fit the data in a way such that the velocity evolution also obeys the laws of dynamics. In other words, any effort to preserve Minkowskian spacetime requires throwing out general relativity, which might not be a wise thing to do given its great success.

Jeff Root
2009-Apr-15, 08:12 PM
Wouldn't galaxies coasting through space look exactly the same as
galaxies moving along with expanding space? If there is a difference,
what is it?
I'm thinking the difference is the difference between the Copernican
and cosmological principles.
The cosmological principle is essentially that the Universe looks
pretty much the same nomatter where you are in it: Galaxies all
around, moving away from you at a speed proportional to their
distance from you; Cosmic background radiation at a temperature
of about three kelvins in all directions. Would that be different if
galaxies are moving through space rather than with space?

The cosmological principle is based on what we can see from here.
It would be strictly true if the Universe were infinite in extent, or if
the Universe were "closed" in such a way that it curves in on itself
like the surface of a planet. The part of the Universe that we can
see appears to be "flat", so either it is not curved, or the Universe
is so immensely humongous that the curvature doesn't show up
even over the enormous distances we can see. I reject the idea
that the part of the Universe that is participating in the cosmic
expansion could be infinite, because everything participating in the
cosmic expansion is causally-connected, and that would not be
possible if the matter involved were infinite in extent. Space itself
might be infinite, but the matter and energy participating in the
expansion cannot be.

So it seems to me that the cosmological principle is rather weak.
It can't be extended very far with any degree of certainty. It might
actually apply everywhere, but logically, that seems very unlikely.
The fact that we see no edge to the Universe could just be the
result of the Universe being absurdly enormous, and our happening
to not be particularly close to the "edge".



Greys earlier explanation doesn't seem to me like it would adhere to
the cosmological principle (my bold):


Say a bunch of galaxies all start at the same spot, and begin moving
away from each other at random speeds. Wait 13 billion years. Now, if
galaxy A was moving away from us twice as fast as galaxy B, in those
13 billion years it will have moved twice as far, so when we look at it
now, it will be twice as far away.
So we need to bring in the Inquisition to determine whether he's a
closet Copernican.



And how could galaxies apparently coast through space faster than light?
Do they apparently coast through space faster than light?

-- Jeff, in Minneapolis

DrRocket
2009-Apr-15, 08:18 PM
Is there any difference between the two? Wouldn't galaxies
coasting through space look exactly the same as galaxies moving
along with expanding space? If there is a difference, what is it?

-- Jeff, in Minneapolis

But the galaxies, on the largest scales(neglecting local groups and such) are ALL moving appart from one another, directly away from the perspective of EACH galaxy. You can't do that with a "dift" through space. So, not they would not look exactly the same.

In fact, a way to think about it is that the galaxies are stationary, and the space between then is opening up. This gets back to the balloon analogy. Put ink dots on the balloon and then inflate it. The dots move out radially (a direction that has no meaning for the manifold for which this is just an analogy so can be considered "stationary" in some sense) while they move apart on surface as the surface expands.

Jeff Root
2009-Apr-15, 09:19 PM
Both Ken's posts and DrRocket's are often difficult to reply to.
In this case, DrRocket's is easier, so DrRocket first.




Is there any difference between the two? Wouldn't galaxies
coasting through space look exactly the same as galaxies moving
along with expanding space? If there is a difference, what is it?
But the galaxies, on the largest scales (neglecting local groups and such)
are ALL moving apart from one another, directly away from the perspective
of EACH galaxy. You can't do that with a "drift" through space. So, no
they would not look exactly the same.
That is just an assertion of what I'm questioning. There is no
obvious difference between galaxies coasting through space and
galaxies moving with expanding space. I don't doubt that your
abbreviated description of the expansion is accurate. But I see
no reason to think that it can't be accounted for by galaxies
moving through space inertially.

-- Jeff, in Minneapolis

pzkpfw
2009-Apr-15, 09:35 PM
The speed of light limit?

Jeff Root
2009-Apr-15, 11:12 PM
Wouldn't galaxies coasting through space look exactly the same as
galaxies moving along with expanding space? If there is a difference,
what is it?
There might be a confusion here between two senses of the term
"inertial". This is not the way you mean it, I think, but in one sense
of the term, the motion of the galaxies is purely inertial, because
there are no forces on them ...
It seems to me that the second sense is an extension of the first.
So I definitely mean the first, but may also mean the second.

However, I question whether there really are two senses of "inertial".
The first is in some way comparable to "A body at reast remains at
rest as long as no net force is applied to it." The second may be
compared to "A body in motion remains in motion as long as no net
force is applied to it." Those seem different but are really one.

I suspect that the descriptions of moving galaxies are similarly
equivalent. What does it mean to say that "galaxies are moving
through space"? It means only that the distances between them
are changing. What does it mean to say that "galaxies are at rest
in space, but the space they are in is expanding"? It means only
that the distances between galaxies are increasing.

If there were some way to detect the motion of a body through
space, as opposed to detecting changing distances between
bodies, then I would readily agree that the two are different.



But what I think you mean by a purely "inertial" explanation is
different from the above, if I correctly take your meaning that
spacetime may be viewed as globally static and Minkowskian
(a la special relativity), and the galaxies may be viewed as moving
in a fixed and constant way (or even decelerated/accelerated by
various gravity terms) through that Minkowskian spacetime. That
is the "Milne" model, but that description simply does not succeed
in matching the cosmological data (even before the acceleration
business), and prompts the need to use the equations of general
relativity-- not those of special relativity with some quasi-Newtonian
deceleration tacked on. The key reason is that the Minkoswki metric,
coupled with some velocity evolution of the galaxies, cannot fit the
data in a way such that the velocity evolution also obeys the laws
of dynamics. In other words, any effort to preserve Minkowskian
spacetime requires throwing out general relativity, which might not
be a wise thing to do given its great success.
I think you are probably right in understanding what I mean by
"inertial", but I will have to ask you to explain what you mean by
your basic terms: "globally static" and "Minkowskian". I'm just
ignorant of their meaning, I'm not questioning your useage. :)

What I'm questioning here are the "data" you refer to, and
interpretations of that data. I'm just not aware of observations
which cannot be interpreted as simple inertial motion of galaxies
away from each other, aside from gravitational slowing and the
recently-discovered acceleration.

-- Jeff, in Minneapolis

Jeff Root
2009-Apr-15, 11:18 PM
The speed of light limit?
Was that in response the the last line of my post immediately
preceding yours?


But I see no reason to think that it can't be accounted for by
galaxies moving through space inertially.
If so, what is the conflict between the speed of light limit and an
interpretation of cosmic redshift as the motion of galaxies through
space inertially?

-- Jeff, in Minneapolis

DrRocket
2009-Apr-15, 11:21 PM
Both Ken's posts and DrRocket's are often difficult to reply to.
In this case, DrRocket's is easier, so DrRocket first.


That is just an assertion of what I'm questioning. There is no
obvious difference between galaxies coasting through space and
galaxies moving with expanding space. I don't doubt that your
abbreviated description of the expansion is accurate. But I see
no reason to think that it can't be accounted for by galaxies
moving through space inertially.

-- Jeff, in Minneapolis

Then explain how, simply in terms of the kinematics, you can have the galaxies moving through space in a manner in which, no matter which point you choose to measure from, all the galaxies appear to moving directly away from you. You simply cannot do that with motions through 3-space.

Let us take a simple example. You are standing at a point with a bow and arrow. Your friend is standing 100 feet away. He is not a good friend, so you shoot your arrow/galaxy at him. A better friend is standing halfway between, slightly to the side. You see the arrow going away from you. But he sees it coming directly at him. The guy in the middle sees it coming toward him for a while and then going away. In the situation that we see with galaxies, all three of you should see the arrow going directly away from you.

Now, if instead you are both standing on a rubber sheet, and someone stretches the sheet, then all of you see all of the others receding.

After that you might try to address the issue raised by pzkfpw as to how one attains superluminal recession rates without violating special relativity.

pzkpfw
2009-Apr-15, 11:31 PM
Was that in response the the last line of my post immediately
preceding yours?

Yes. I'd have quoted if it didn't seem clear - sorry it wasn't.


If so, what is the conflict between the speed of light limit and an
interpretation of cosmic redshift as the motion of galaxies through
space inertially?

-- Jeff, in Minneapolis

I thought that (mainstreamly thinking) the speed of Galaxies is "too high" to be simple motion through space, and had to be accounted for by expansion of the Universe itself. (DrRocket confirms this).

(Breaking the speed of light limit is one of the things ATMers raise as "evidence" against the expansion of the Universe.)

DrRocket
2009-Apr-15, 11:37 PM
If so, what is the conflict between the speed of light limit and an
interpretation of cosmic redshift as the motion of galaxies through
space inertially?

-- Jeff, in Minneapolis

Very simply the Hubble constant derived from the redshift data shows and essentially linear relationship between recession speed and distance. Objects sufficiently far away have a recession velocity that is greater than c. As an ordinary speed of a body moving through space that would violate relativity. But it does not violate relativity if the perceived recession speed is due to an expansion of space itself.

Jeff Root
2009-Apr-15, 11:44 PM
DrRocket,

I can't make sense of your analogy. What do I represent? What
do the other people represent? What does the arrow represent?
At what era in the history of the Universe does your analogy apply?
I strongly suspect that the analogy is faulty, in addition to being
unclear. The point you want to make may be absolutely correct,
but the analogy doesn't work for me. I like analogies though, so
please try again.



After that you might try to address the issue raised by pzkfpw as
to how one attains superluminal recession rates without violating
special relativity.
Are superluminal recession rates seen? I don't think so. I think
they are only inferred from theory. But if they are seen, then
they do violate special relativity.

-- Jeff, in Minneapolis

Jeff Root
2009-Apr-16, 12:03 AM
Very simply the Hubble constant derived from the redshift data shows
an essentially linear relationship between recession speed and distance.
Objects sufficiently far away have a recession velocity that is greater
than c.
Is that velocity observed? What is the redshift of a galaxy with a
recession velocity greater than c?



As an ordinary speed of a body moving through space that would
violate relativity. But it does not violate relativity if the perceived
recession speed is due to an expansion of space itself.
It still violates special relativity, even then.

-- Jeff, in Minneapolis

pzkpfw
2009-Apr-16, 12:12 AM
It still violates special relativity, even then.

-- Jeff, in Minneapolis

No.

An ant walking across the floor of my car (towards the front) is not walking faster than "maximum ant speed".

If you are going to say a Galaxy can actually move through space faster than c - that might need to be taken out of this Q&A thread and into a new ATM thread.

Bearded One
2009-Apr-16, 12:19 AM
It still violates special relativity, even thenIt's more a matter of being out of the scope of special relativity, much as Newtonian mechanics are out of scope for high speed interactions.

I tend to view SR as an intermediary theory, much like Newton's laws. It's fine within it's scope, but don't take it to far. SR is good for local interactions, trying to apply it to the whole Universe is beyond it's scope just as trying to use Newton's equations to model higher speed interactions.

Jeff Root
2009-Apr-16, 12:31 AM
As an ordinary speed of a body moving through space that would
violate relativity. But it does not violate relativity if the perceived
recession speed is due to an expansion of space itself.
It still violates special relativity, even then.
No.

An ant walking across the floor of my car (towards the front) is
not walking faster than "maximum ant speed".
DrRocket is saying that a galaxy can have a recession speed
greater than c. He is also saying that that recession speed
does not violate relativity. I replied that it does violate special
relativity. I'm not the least bit sure that that comment supports
my position, but I'm pretty sure that it is true, and mainstream.
Special relativity does not contain a concept of anything similar
to a car floor in your analogy. If the ant represents a pulse of
light, then your car is an impossibility under special relativity--
or at least, driving at close to or faster than ant speed.

-- Jeff, in Minneapolis

speedfreek
2009-Apr-16, 12:43 AM
Is that velocity observed? What is the redshift of a galaxy with a
recession velocity greater than c?

-- Jeff, in Minneapolis

I spent ages attempting to answer your earlier post, but ultimately failed to be able to put my argument into words. Then I saw how the thread had moved on since!

In reference to the above quote, the velocity is inferred from current theory, according to which, all galaxies with a redshift of more than z~1.5 were apparently receding faster than light when they emitted the light we are seeing, and they are apparently still doing so. This roughly equates to all light-travel times of over 9 billion years or so. In 1990 I think we had seen redshifts of z=4, at least.

I seem to remember seeing an estimate of the redshift required for a recession speed of c based on relativistic doppler effect instead of cosmological redshift and I think it was something around z=1.8.

Bearded One
2009-Apr-16, 12:46 AM
What I'm questioning here are the "data" you refer to, and interpretations of that data. I'm just not aware of observations
which cannot be interpreted as simple inertial motion of galaxies
away from each other, aside from gravitational slowing and the
recently-discovered acceleration.I think the cosmological principle is critical here. If we assume, and it is an assumption, that every observer in the Universe sees the same thing then simple kinetic effects don't work. I believe an earlier post of mine asked the question of direction. If we are moving apart from our neighbors in the Universe due to an initial force impacted on us by the creation event, then what direction is that force (kinetic motion) expressed in? Force itself may be scalar, but to meaningful to us here it must be a vector. As Dr. pointed out, try to model such expansion in 3d, or even 2d, without expanding the space. Simply applying a kinetic impulse to the dots doesn't work. Remember also that you can only give an initial impulse, you can't be tweaking it constantly.

pzkpfw
2009-Apr-16, 12:46 AM
DrRocket is saying that a galaxy can have a recession speed
greater than c. He is also saying that that recession speed
does not violate relativity. I replied that it does violate special
relativity.

That recession speed is not the speed of that Galaxy through space, so it doesn't violate special relativity. Space itself is expanding.



Special relativity does not contain a concept of anything similar
to a car floor in your analogy.

My car is (kind of, but not really) the Universe expanding. But don't take that analogy too far. It was really just an illustration of how apparent speed is different from actual speed.
- Nobody watching my car drive past would say the ant is able to walk at 100.000001 km/hr.
- We apparently "see" galaxies receeding faster than c, but they are not thought to be actually moving faster than c through space.



If the ant represents a pulse of
light, then your car is an impossibility under special relativity--
or at least, driving at close to or faster than ant speed.

Use ants sitting on an expanding balloon if you like, where the balloon represents the expanding Universe. The balloon expands and the ants appear to each other to be receeding from each other. The speed of that recession may appear to be faster than an ant can actually walk - but that's fine, none of the ants are walking.

(An ant might decide to get up and walk around a bit. That's fine. Some Galaxies are getting closer to each other too - even colliding (local effects of gravity).)

speedfreek
2009-Apr-16, 12:53 AM
DrRocket is saying that a galaxy can have a recession speed
greater than c. He is also saying that that recession speed
does not violate relativity. I replied that it does violate special
relativity. I'm not the least bit sure that that comment supports
my position, but I'm pretty sure that it is true, and mainstream.
Special relativity does not contain a concept of anything similar
to a car floor in your analogy. If the ant represents a pulse of
light, then your car is an impossibility under special relativity--
or at least, driving at close to or faster than ant speed.

-- Jeff, in Minneapolis

I can only post the original paper on which those earlier SciAm articles were based on.

Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe (http://arxiv.org/abs/astro-ph/0310808)

It definitely does not ignore the acceleration of the expansion, but we had observed galaxies with apparently superluminal recession speeds before we found the evidence for the acceleration and the two are not directly related - we would see galaxies apparently receding faster than light even if the universe had continued to decelerate.


We use standard general relativity to illustrate and clarify several common misconceptions about the expansion of the Universe. To show the abundance of these misconceptions we cite numerous misleading, or easily misinterpreted, statements in the literature. In the context of the new standard Lambda-CDM cosmology we point out confusions regarding the particle horizon, the event horizon, the ``observable universe'' and the Hubble sphere (distance at which recession velocity = c). We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light. We explain why this does not violate special relativity and we link these concepts to observational tests. Attempts to restrict recession velocities to less than the speed of light require a special relativistic interpretation of cosmological redshifts. We analyze apparent magnitudes of supernovae and observationally rule out the special relativistic Doppler interpretation of cosmological redshifts at a confidence level of 23 sigma.

Jeff Root
2009-Apr-16, 01:40 AM
What I'm questioning here are the "data" you refer to, and
interpretations of that data. I'm just not aware of observations
which cannot be interpreted as simple inertial motion of galaxies
away from each other, aside from gravitational slowing and the
recently-discovered acceleration.
I think the cosmological principle is critical here. If we assume, and it
is an assumption, that every observer in the Universe sees the same
thing then simple kinetic effects don't work. I believe an earlier post
of mine asked the question of direction. If we are moving apart from
our neighbors in the Universe due to an initial force impacted on us by
the creation event, then what direction is that force (kinetic motion)
expressed in? Force itself may be scalar, but to meaningful to us here
it must be a vector. As Dr. pointed out, try to model such expansion in
3d, or even 2d, without expanding the space. Simply applying a kinetic
impulse to the dots doesn't work. Remember also that you can only
give an initial impulse, you can't be tweaking it constantly.
What do you mean it doesn't work? How would a bunch of galaxies
moving away from each other through space look different from those
same galaxies motionless in expanding space?

I gather that the expansion can be modelled in 1d without expanding
the space. Is that correct? The problem shows up in going to 2d?
Can you describe this problem?

-- Jeff, in Minneapolis

DrRocket
2009-Apr-16, 01:48 AM
Is that velocity observed? What is the redshift of a galaxy with a
recession velocity greater than c?


It still violates special relativity, even then.

-- Jeff, in Minneapolis

The allowability of superluminal recession speeds comes with general relativity, not special relativity. And it is allowable because in general relativity the prohibition against superlumninal speed applies against a local speed faster than c -- that is special relativity applied in a local coordinate patch of the entire space-time manifold. The superluminal recession speed is the result of a phenomena that is not described within special relativity.

What you have is a situation where in local coordinates the galaxies are essentially stationary, which is what would be seen by a nearby observer. But because the manifolditself is expanding, the rate of separation between two points can exceed c even though there is no motion relative to the local frame of the manifold itself.

DrRocket
2009-Apr-16, 01:55 AM
DrRocket,

I can't make sense of your analogy. What do I represent? What
do the other people represent? What does the arrow represent?
At what era in the history of the Universe does your analogy apply?
I strongly suspect that the analogy is faulty, in addition to being
unclear. The point you want to make may be absolutely correct,
but the analogy doesn't work for me. I like analogies though, so
please try again.

-- Jeff, in Minneapolis

In the analogy you are an observer and so are the other two people. The arrow represent somthing moving through space. It is intended to show you that it is impossible for an object to move through space in a manner in which all observers everywhere in space see as moving directly away from them. It is simply kinematically impossible.

If an object is moving through space there will always be some location at which an observer would see it as moving towards him -- basically being in front of the arrow. It is simply not the case that everywhere can be behind a given arrow.

Yes, it is that simple. The pointy end of an arrow and the nock end are not the same end.

The only alternative is to reject the cosmological principle and assume that by some miracle we are located at the origin of an explosion in space, and therefore have a unique vantage point in the universe. You can do that, but that puts you rather on the fringe of cosmology.

Bearded One
2009-Apr-16, 02:21 AM
What do you mean it doesn't work? How would a bunch of galaxies
moving away from each other through space look different from those
same galaxies motionless in expanding space?

I gather that the expansion can be modeled in 1d without expanding
the space. Is that correct? The problem shows up in going to 2d?
Can you describe this problem?I almost said 1D in my post but didn't feel confident about it. It bugged me enough that I started playing with things on my desk. It seems I can make it work for a 1 dimensional Universe. It leads into all sort of other issues, but that's another discussion. For a 2 dimensional Universe it seems impossible. I haven't played with the math but the angles seem to cause problems. Angles don't exist in a 1D Universe.

Jerry
2009-Apr-16, 02:56 AM
It is important to remember that the expanding universe is a mathematical abstraction based upon two primary assumptions: 1) Most of the observed redshift in distant galaxies is due to relative motion - the space between clusters of galaxies is getting greater. 2) The Cosmic Microwave Background is a relic of a time when all of the known universe was primal - before matter as we know it became matter.

The current iteration of the theory also assumes there was a period of very rapid expansion, then a slowing and now a 'slow acceleration' of the expansion. We are observing from a 'saddle' between the rapid and the accelerating rate. This is not very Copernicun (assuming now is a special time); but this is the current model based upon this set of assumptions and observations.

DrRocket
2009-Apr-16, 02:59 AM
I almost said 1D in my post but didn't feel confident about it. It bugged me enough that I started playing with things on my desk. It seems I can make it work for a 1 dimensional Universe. It leads into all sort of other issues, but that's another discussion. For a 2 dimensional Universe it seems impossible. I haven't played with the math but the angles seem to cause problems. Angles don't exist in a 1D Universe.

It won't work in 1D either. In 1D a moving object is either moving left or right, and that will be independent of a selection of origin or of the location of an observer. If from the point of view of an observer at the origin, everything is moving away from him that means that things are the positive x-axis are moving to the right and thiings on the negative x-axis are moving to the left. But to someone far out on the x-axis, sat at x=1000 the objects between him and 0 will be coming right at him. Therefore not everyone sees the same thing.

It is not a problem of angles. It is a problem of arrows. You can formuate this more formally, I am sure, in terms of vector fields, but it comes down to the fact that arrows have a clear pointy end.

pzkpfw
2009-Apr-16, 03:01 AM
Take

1. G1-G2-G3-G4 ...

then

2. G1---G2---G3---G4 ...

then

3. G1-----G2-----G3-----G4 ...

then

4. G1-------G2-------G3------G4 ...

G1 sees G2 moving away at speed "--". (And vice versa)
G2 sees G3 moving away at speed "--". (And vice versa)
G3 sees G4 moving away at speed "--". (And vice versa)

So:

G1 sees G3 moving away at speed 2 x "--". (And vice versa)
G1 sees G4 moving away at speed 3 x "--". (And vice versa)


Eventually the speed of Gn (from the point of view of G1) appears greater than c.


If the apparent speed of the Galaxies did not behave this way, it would be similar to the expansion of the Universe not being "even" throughout the Universe (part of the balloon made of thicker-less-stretchy rubber). We'd see that in the way the distribution goes...

e.g. step 4 might be:

4'. G1-------G2------G3-----G4 ...

(G2, G3 and G4 all got further from G1; but the G2 to G3 distance increased less than G1 to G2).

We would then no longer have the symmetry implied by the cosmological principle. The view of G2 to G3 is not the same as from G2 to G1 etc.

...and that's not what is observed.

Ken G
2009-Apr-16, 03:04 AM
It seems to me that the second sense is an extension of the first.
So I definitely mean the first, but may also mean the second.If you mean the first, you are not saying anything-- this is perfectly "standard canon" to assert that the motion of objects under pure gravity is inertial. The real question of the thread is, why is the universe expanding. That does not in any way single out inertial motion, as all motion under gravity is inertial. The real question is, why is that motion that of expansion? The preseence of inertia has nothing at all to say on the matter, as inertia never has anything to say about motion under gravity (it doesn't even appear in that situation).


However, I question whether there really are two senses of "inertial".
The first is in some way comparable to "A body at reast remains at
rest as long as no net force is applied to it." The second may be
compared to "A body in motion remains in motion as long as no net
force is applied to it." Those seem different but are really one.
True they don't seem different, but neither is the issue here. The issue here is whether or not inertia has anything to say about motion under gravity. It does not. (That is the "principle of equivalence" in a nutshell.) So if inertia has nothing to add to the issue of motion under gravity, then saying "it's inertia" has no useful meaning in the context of expansion under gravity. It isn't wrong, it simply explains nothing.


What does it mean to say that "galaxies are moving
through space"? Nothing at all, I imagine. That's why no one who is careful says it. Instead, they say that a purely Minkowskian description of the global spacetime simply does not work with the machinery of general relativity, which does work.


What does it mean to say that "galaxies are at rest
in space, but the space they are in is expanding"? It means only
that the distances between galaxies are increasing.Again, the issue of the thread is simply not "are galaxies really moving or not", it is, what is the appropriate geometry of spacetime. The answer is, not globally Minkowskian. That is the real flaw in your position, not who is moving and who isn't (which is a coordinate issue, not an invariant like the geometry).


I think you are probably right in understanding what I mean by
"inertial", but I will have to ask you to explain what you mean by
your basic terms: "globally static" and "Minkowskian". The usual comparison between dynamical space, and static space with dynamical motion of galaxies, which seems to be the contrast you are asking about, is actually a difference in the invariant aspects of the spacetime metric. It comes back to that key issue of what is an inescapable aspect of an observation, and what is simply a coordinatization that we choose to express our story about the observation. "Are galaxies moving or not" is of the latter type, "there's no difference between a dynamical spacetime and dynamical galaxies in a static spacetime" is false because it runs afoul of the former type. The machinery of general relativity acts against the backdrop of a spacetime geometry, and it simply won't work against the backdrop of a static local metric that is extrapolated globally (which is all it can really mean to say the whole story is the motions of galaxies and not the dynamics of spacetime).


What I'm questioning here are the "data" you refer to, and
interpretations of that data. Well, it is certainly always valid to question data, and with dark matter and dark energy afoot, there's no telling where the problem might lie. But you were wondering if there is any difference between dynamical spacetimes and static ones with dynamical galaxies, and that sounds to me like a distinction between a globally evolving spacetime and a locally static (Minkowskian, like special relativity) extended globally. There is such a difference, and the latter doesn't work at all, even if the former still has its bugbears.


I'm just not aware of observations
which cannot be interpreted as simple inertial motion of galaxies
away from each other, aside from gravitational slowing and the
recently-discovered acceleration.The observations that support general relativity are all of that nature. If you demand that we subtract all observations that support general relativity, then yes, there's no difference.

Ken G
2009-Apr-16, 03:23 AM
It won't work in 1D either. In 1D a moving object is either moving left or right, and that will be independent of a selection of origin or of the location of an observer.But you are forgetting that the motion of the origin is also arbitrary. Hence we can judiciously combine different locations of origins, with different motions of origins, to return to exactly the same picture. None of the things you are talking about (like leftward or rightward motion) are observable invariants, they are all just reflections of the coordinates chosen (including the motion of the coordinates relative to other coordinates). The invariants of motion are all framed as relative motions between objects, not relative motions between an object and an origin, and relative motions between objects are not leftward or rightward, they are expansive or compressive (or rotational, but that gets tricky, Machian and so forth). The origin can be anywhere and doing anything you like, it cannot change any of the physics. That's the principle that GR is built around (to the best of my nonexpert understanding).

Kwalish Kid
2009-Apr-16, 03:28 AM
We are observing from a 'saddle' between the rapid and the accelerating rate. This is not very Copernicun (assuming now is a special time); but this is the current model based upon this set of assumptions and observations.
That the universe has recently begun a period of acceleration of expansion after a period of deceleration of expansion is is not an assumption, it is a result. One does not have to assume that there was a past of greater density, nor does one have to assume general relativity. (See, for example, the discussion in Riess et al. The Astrophysical Journal, 607:665–687, 2004 June 1, and Turner, M., & Riess, A. G. 2002, ApJ, 569, 18.)

One could, of course, deny these results on the basis that one wanted to preserve the assumption that we are not at a special time, but this doesn't seem reasonable.

Cougar
2009-Apr-16, 03:50 AM
I think the cosmological principle is critical here. If we assume, and it is an assumption, that every observer in the Universe sees the same thing then simple kinetic effects don't work.

That's what I thought, and I was going to prove it geometrically to show Jeff the difference between the expanding universe model and the kinetic model that Grey first introduced. But I ran into a little problem: my proof kept pointing to the conclusion that the kinetic model preserves the cosmological principle! The kinetic model has a center, but I kept finding that even a galaxy far from the center would see a redshift in proportion to distance. Now, my proof was not particularly rigorous, and I'm still not convinced, but, well, I'm kinda stuck at this point...

The above is only considering the redshift/distance relation. I don't know how the kinetic model can address the smoothness of the CMB at all. Well, I don't think it can.

publius
2009-Apr-16, 03:57 AM
Let me go off on a little pedantic excursion here. :)

First rather than fretting about "galaxies receding faster than c", forget about the constant 'c', and just speak about null paths, the paths light takes in space-time. Nothing every moves faster than null. The speed of null paths happens to be the constant 'c' globally in inertial frames in flat space-time. While it always 'c' locally, it is not so globally in non-inertial, and curved space-times. What you mean by distance and what you mean by speed in curved space-times is a coordinate thing anyway.

And then "static", "dynamic", etc. These terms have a precise meaning in GR (and a meaning I get confused about as it's sort of backwards to the way I'd think).

A *static* space-time is actually an invariant property, along with its less restrictive cousin, *stationary* space-times. A stationary space-time, in simple terms (the rigorous defintion is a doosie, involving time like Killing vectors), is one where coordinates can be found where the metric is not changing with time. A dynamic space-time is one where no such coordinates exist. In an invariant sense, one can say the geometry is "changing with time" truly. Well, I don't know if that's exactly kosher or not because it does involve separating space and time. But "dynamic" is nonetheless the proper term. :)

A static space-time involves the further restriction that coordinates can be chosen where there are no "space/time cross terms" in the metric; that is, no frame dragging terms. For example, Schwarzchild is static. Kerr is just stationary; there is an invariant notion of frame dragging there that cannot be transformed away.

These are invariant properties of the space-time geometry. The easiest way to picture it is to use the "coordinate can be found" descriptions, but the complex rigorous definitions do involve invariant language.

The thing is one can choose coordinates where the metric has terms that vary with time, but the space-time is still stationary or static. DeSitter, a simple "expanding universe" model is such an example. It is completely static. You can write in a nice FLRW form with a scale factor, but you can transform to static coordinates.

So just because the metric in some coordinates has terms that vary with time doesn't mean the space-time is not stationary or static.

A static space-time is the most restrictive, and a lot of things can be defined there that are problematic in the general case. Energy can be conserved globally trivially there and stuff like that. There is also a well defined rest frame for the mass distribution.



-Richard

Cougar
2009-Apr-16, 04:16 AM
It won't work in 1D either. In 1D a moving object is either moving left or right, and that will be independent of a selection of origin or of the location of an observer. If from the point of view of an observer at the origin, everything is moving away from him that means that things are the positive x-axis are moving to the right and thiings on the negative x-axis are moving to the left. But to someone far out on the x-axis, sat at x=1000 the objects between him and 0 will be coming right at him. Therefore not everyone sees the same thing.

Nope, because the guy at x=1000 is moving a little faster than somebody at x=900, and a lot faster than somebody at x=100. They appear to moving away from him with a redshift/distance relation. It's a little trickier in 2D, but even there, I can't convince myself it violates the cosmological principle.

pzkpfw
2009-Apr-16, 04:46 AM
Nope, because the guy at x=1000 is moving a little faster than somebody at x=900, and a lot faster than somebody at x=100. They appear to moving away from him with a redshift/distance relation. It's a little trickier in 2D, but even there, I can't convince myself it violates the cosmological principle.

I think the difference is that if it's plain intertial movement, then something has to actually be moving faster than something else. (e.g. that guy at x=1000 is moving faster than the guy at x=900 - it's not just the perception of the guy at x=0).

So while everyone could think they are at the non-moving centre of the Universe and everything else is (on average) moving away, because that's the way it looks to all of them, there'd be an actual non-moving centre.

Doesn't that violate the cosmological principle? (...as well as meaning something actually is breaking the speed of light limit?)


Part 2:

I think I just realised what DrRocket specifically means by 'sat at x=1000'. If the Universe is not expanding, a Galaxy at x=1000 could stay there, and some other Galaxies, as they are moving through space, would pass by.

Whereas, if the Universe is expanding, the guy (Galaxy) at x=1000 moves onwards and outwards along with the other Galaxies, and doesn't see them approach.

Sorry if I misunderstand, Dr.

DrRocket
2009-Apr-16, 06:11 AM
Both Ken's posts and DrRocket's are often difficult to reply to.
In this case, DrRocket's is easier, so DrRocket first.


That is just an assertion of what I'm questioning. There is no
obvious difference between galaxies coasting through space and
galaxies moving with expanding space. I don't doubt that your
abbreviated description of the expansion is accurate. But I see
no reason to think that it can't be accounted for by galaxies
moving through space inertially.

-- Jeff, in Minneapolis

OK , I think I see in part at least what you are getting at.

I think it is something like this. Suppose you set off a bomb. The pieces fly apart. It is pretty clear to a stationary observer that the process is not homogeneous and isotropic. Some pieces fly at him and some fly away. But suppose you select a distinguished class of observers, those observers attached to one of the flying pieces. From the perspective of those observers does it not appear that all the pieces are flying directly away from them and that the recession velocity is proportional to distance ? No tricks, just plain old Newtonian mechanics.

The answer to that question is yes. It is not hard to show it either. It follows rather simply from just having the position being proportionala to velocity. Then, selecting a reference frame attached to any of the pieces of the bomb the distance speed proportionality follows immediately, with the ratio of recessioin speed to distance being just time.

It is in fact possible to build a cosmological theory out of Newtonian mechanics that adheres to a somewhat restricted cosmological principle and produces results that are formally the same as a relativistic cosmology. Apparently Macrae and Milne did just this in 1934 and you can find a discussion in Theoretical Cosmology by A.K. Raychaudhuri. The development there is more sophisticated and covers a bit more than the simple ideas noted above.

Note that nothing that I have said here is intended to indicate that Newtonian cosmologies are satisfactory in the face of all of the data that is available now, and it certainly is not meant to in any way suggest that general relativity is not the proper setting for cosmological theories. It is simply to note that the single issue of an isotropic picture of recession can be handled in that manner. Just because it can be done does mean that it should be done.

DrRocket
2009-Apr-16, 06:31 AM
But you are forgetting that the motion of the origin is also arbitrary. Hence we can judiciously combine different locations of origins, with different motions of origins, to return to exactly the same picture. None of the things you are talking about (like leftward or rightward motion) are observable invariants, they are all just reflections of the coordinates chosen (including the motion of the coordinates relative to other coordinates). The invariants of motion are all framed as relative motions between objects, not relative motions between an object and an origin, and relative motions between objects are not leftward or rightward, they are expansive or compressive (or rotational, but that gets tricky, Machian and so forth). The origin can be anywhere and doing anything you like, it cannot change any of the physics. That's the principle that GR is built around (to the best of my nonexpert understanding).

I was simply looking at the problem from a kinematic, Newtonian perspective, which is the way that I interpreted the OP. I was not even attempting to address the issue from the perspective of GR.

As far as I know GR does not require you to attached the origin to any physical object, and even then "origin" doesn't mean anything except in a coordinate patch. There are no global coordinates. That is the nature of manifolds.

DrRocket
2009-Apr-16, 06:37 AM
Nope, because the guy at x=1000 is moving a little faster than somebody at x=900, and a lot faster than somebody at x=100. They appear to moving away from him with a redshift/distance relation. It's a little trickier in 2D, but even there, I can't convince myself it violates the cosmological principle.

Your observation is correct. See my earlier note to Jeff. If you attach the coordinate system to one of the moving particles then you will observe all of the other particles moving away with a speed proportional to distance. The proportionality constant is pretty simple -- time. And dimension doesn't matter. (This is all Newtonian).

But you do have to select a distinguished class of reference frames -- those attached to the moving bodies which started at the origin in the "rest" frame and move at constant velocity.

publius
2009-Apr-16, 06:50 AM
From the perspective of those observers does it not appear that all the pieces are flying directly away from them and that the recession velocity is proportional to distance ? No tricks, just plain old Newtonian mechanics.

The answer to that question is yes. It is not hard to show it either. It follows rather simply from just having the position being proportionala to velocity. Then, selecting a reference frame attached to any of the pieces of the bomb the distance speed proportionality follows immediately, with the ratio of recessioin speed to distance being just time.



I actually went through such a Newtonian "expanding universe" model in thread here a while back, but I had a little trouble with a disgruntled heckler who took the fun out of it. :)

Anyway, construct a spherical cloud of uniform density, and give all the particles a radial initial velocity profile proportional to radius. From a "comoving" frame, you get a nice Hubble relationship, relative velocity proportional to distance in all directions. Deep inside, it looks like everything is flying away from you. Now, with the self gravity of the mass, the expansion slows down, but everywhere the "Hubble profile" is maintained.

If the escape velocity profile is greater than an "escape velocity" which is proportional to the initial density, the sphere expands forever. If less, it stops and collapses back on itself.

The basic FLRW universe is the GR version of that. In Newton, you have to have a boundary somewhere, otherwise you get infinite mass, and other infinities. But in GR, with appropriate boundary conditions, the system can be "closed" with a finite mass, and there is no boundary anywhere. A real expert would have to explain how that works, but I think we can all sort of appreciate it. Other than, it's the same darn thing as that simple Newtonian model.

You can even add the Newtonian approximation of "dark energy"/cosmological constant which behaves like an additional "repulsive" negative mass density and get the accelerating expansion behavior. You'll see there is a point where Lambda can take over, and make the total gravity repulsive. If you give the system enough initial velocity, it will reach that point and expand forever at an accelerating rate. If below that, it will still collapse. And there is a equilibrium where Lamdba just balances the normal gravity and you can have a static universe, which was Einstein's original intent.

All the basic features of FLRW can be seen from such a simple Newtonian model. It just doesn't give you the coupling of space-time to that mass and all the wild and crazy things that does.

-Richard

Jeff Root
2009-Apr-16, 10:09 AM
There is a lot here to reply to! I especially want to thank speedfreek
for his extensive efforts. Thank you, speedfreek! Thanks to all the
rest of you who have replied, too. Even you two curmudgeons, Richard
and Ken.

It appears that DrRocket has changed his mind about the kinematic
description of the expansion. Either you were swayed by Ken's and
Cougar's comments, or you just thought about it some more on your
own, but I gather that you would agree that at least on a primitive
level, ignoring gravitational slowing and later acceleration, it is similar
to these two animations I made. The first is 1-dimensional, the 2nd
is 2-dimensional:

Uniform 1-D expansion (http://www.freemars.org/jeff2/expand3e.htm)
Uniform 2-D expansion (http://www.freemars.org/jeff2/expand5a.htm)

The notable features of the expansion in these animations are that it
looks the same from any location and there is no unique location that
can be called a center, so it appears to preserve the cosmological
principle. There is no way to say whether the animation shows dots
moving apart through space or non-moving dots in expanding space.
Those two descriptions are entirely equivalent as far as the animation
is concerned.

So if there is a difference between galaxies moving apart through space
and non-moving galaxies in expanding space, it is not detectable at the
level of simple kinematics.

Are we agreed on that, or have I jumped to a conclusion too far?

Bearded One, do you agree? Cougar, does this match what you found?

-- Jeff, in Minneapolis

gzhpcu
2009-Apr-16, 11:19 AM
So if there is a difference between galaxies moving apart through space
and non-moving galaxies in expanding space, it is not detectable at the
level of simple kinematics.

Having just read through this entire thread, the assumption seems reasonable to me.

What baffles me though, is the fact that the expansion of spacetime is not limited by c. I am not questioning the fact, this seems to be the consensus. Am just trying to understand why.

This because the effects of gravity, the warping of space, caused by an object, is limited to c. This deformation of space seems to me to be a similar type of distortion of the fabric of spacetime as expansion is.

So what causes spacetime to expand? Dark energy? Why does the limit of c not apply?

slang
2009-Apr-16, 11:23 AM
I actually went through such a Newtonian "expanding universe" model in thread here a while back, but I had a little trouble with a disgruntled heckler who took the fun out of it. :)

This one (http://www.bautforum.com/space-astronomy-questions-answers/77359-recent-expansion-space-papers-implications.html)? That was an enjoyable explanation, I even understood some of it :)

malaidas
2009-Apr-16, 11:28 AM
It still doesn't work Jeff, Simple pythag serves to illustrate

In a 1D reference frame yes reletive speed differences would Appear to imitate the cosmological principle.

P1 - P2 -P3
P1 -- P2 --P3

so if P2 is moving twice as fast as P2 and P3 is moving twice as fast as P2 then yes each would appear to be moving away from the other such that the cosmological principle holds.


However Put it into 2D and I can't do the diagram here. You have to ask about lines in different angles and do so pythag on the points on the different lines.

Consider 2 lines of the above travelling from a centre point along different radial lines. Let us call The Second Line R1,R2,R3

would the distance seen between P2 and P3 be the same as P2 and R2?

Well lets assume an origin at 0,0 and then move P1,R1 at rate 1 and P2,R2 at rate 2 and P3,R3 at rate 4. This obviously gives each line the illusion of coordinate centre for each point

We will fire line P along X=1 Y = 0 therefore At Time t=1
P1 = [1,0]
P2 = [3,0]
P3 = [5,0]

For simplicity lets fire R at 90' in first instance

therefore
R1 = [0,1]
R2 = [0,3]
R3 = [0,5]

At t=1 therefore distance between P2 ->P1 = 2 whilst the distance between P2 -> R2 = root 18 = ~4.24

Therefore the fact that there is a centre from which things are moving, implies that there would be a directionality to the expansion

Ok this is extreme, but even at a lesser angle an observed difference would be shown unless we are at a special point in the universe (which of course cannot be ruled out), either at the centre of the expansion or the expansion just happens to be that each possible point is moving at a rate to fool us that we are at the centre. This would involve a huge coindicence and implies that the bigbang was very directional.

I don't see any other possibilities.

DrRocket
2009-Apr-16, 01:15 PM
The
It appears that DrRocket has changed his mind about the kinematic
description of the expansion. Either you were swayed by Ken's and
Cougar's comments, or you just thought about it some more on your
own, but I gather that you would agree that at least on a primitive
level, ignoring gravitational slowing and later acceleration, it is similar
to these two animations I made. The first is 1-dimensional, the 2nd
is 2-dimensional:

Uniform 1-D expansion (http://www.freemars.org/jeff2/expand3e.htm)
Uniform 2-D expansion (http://www.freemars.org/jeff2/expand5a.htm)

The notable features of the expansion in these animations are that it
looks the same from any location and there is no unique location that
can be called a center, so it appears to preserve the cosmological
principle. There is no way to say whether the animation shows dots
moving apart through space or non-moving dots in expanding space.
Those two descriptions are entirely equivalent as far as the animation
is concerned.

So if there is a difference between galaxies moving apart through space
and non-moving galaxies in expanding space, it is not detectable at the
level of simple kinematics.

Are we agreed on that, or have I jumped to a conclusion too far?


-- Jeff, in Minneapolis

What I agreed with is that if you select as a distinguished class observers those observers "riding" on one of the flying pieces that the movement of the collection of pieces will appear to be uniformly away from the piece on which the observer is "riding" independent of which piece you chose as the observing location.


Here is how the derivation goes. We are using ordinary Newtonian kinematics, and as you said earlier ignoring gravitational interactons.

Fix an interial coordinate system. Suppose that now you set off a bomb, and eject a number of pieces P_i each moving at a constant velocity vector V_i. The position or P_i at time t , call it X_i will be simply tV_i. Now choose a piece at random, fix it, and arrange the labels so that it is P_1. Consider the motion of P_i for and arbitrary i relative to P_1. The position of P_i relative to P_1 is just the difference in the position vectors relative to our original fixed inertial coordinate system, or
X_i - X_1 = t V_i - t V_1 = t (V_i -V_1) .

Similarly the velocity of P_i relative to P_1 is just V_i - V_1
From this you can see that the relative position is just the scalar t times the relative velocity. So the relative velocity vector is parallel to the relative position vector -- P_i is moving directly away from P_1 and the velocity has the required linear dependence on positionn the proportionality constant being simply time.

So, from the perspective of an observer riding on any piece the expansion would be isotropic, as required.

Note that this does NOT say that the expansion is isotropic with respect to any arbitrarily chosen abstract observer (for instance one fixed at some point other thant the origin of the original inertial reference frame), but it does apply to any observer who is riding one of the pieces that participated in the initial event. So for the cosmological expansion question, an observer riding a body that resulted from the initial "explosion" would see an isotropic expansioin. I don't think that fits the strong cosmological principle, but it is perhaps an acceptable substitute.

I have not worked out the details, but I suspect that this would also work if you threw in special relativity.

This also does NOT say that this theory is a viable competitor to the standard general relativistic model for the big bang, in which the expansion is an expansion of space itself. It does not, for instance, explain superluminal recession rates, and it does not explain the uniformity of the cosmic background radiation. Above all it is not consistent with the observed data that shows that general relativity more accurately describes physics than does Newtonian mechanics.

Cougar
2009-Apr-16, 01:15 PM
At t=1 therefore distance between P2 ->P1 = 2 whilst the distance between P2 -> R2 = root 18 = ~4.24

Sorry, but I don't get how this shows anything. :confused: The distance between P2 and R2 is greater than that between P2 and P1, and they are moving away from each other.

malaidas
2009-Apr-16, 01:36 PM
Sorry, but I don't get how this shows anything. :confused: The distance between P2 and R2 is greater than that between P2 and P1, and they are moving away from each other.

The Point is that the expansion wouldn;t seem to be like you were in the centre as looking in different directions you would see different rates. Depending on the direction you were looking reletive to the centre of exansion
Whereas if you expand space every direction looks identical.

malaidas
2009-Apr-16, 01:52 PM
Let me clarify a little here,

What we are considering is whether or not the cosmological principle can hold for any Point within the universe if the expansion is due to kinetics. If the universe is 2D (and this can of course be readilly expanded to any nuimber of D). Then conmsider the spread of matter coming from an 'explosion'.

WHat my example shows is that for a given t, looking in one direction gives a movement of 1 value for the expansion rate, and looking in another shows another expansion rate. To really see it you have to have multiple radial lines and consider what it would look like in terms of relative movements looking across the coordinate space. In general you will see variance whichever way you look. Unless things are very carefully configured (and I'm not sure its even possible to do over time) the only place where the CP holds is at the origin.

Theerfore if we observe (and we do) that the CP holds for our position, then it must be the case that either we are at the origin, or things are very carefully setup (as above) or that the kinetic theorum is wrong.

The expansion of space theorum allows the CP to hold from any point in the coordinate system.

malaidas
2009-Apr-16, 02:12 PM
What I agreed with is that if you select as a distinguished class observers those observers "riding" on one of the flying pieces that the movement of the collection of pieces will appear to be uniformly away from the piece on which the observer is "riding" independent of which piece you chose as the observing location.


Here is how the derivation goes. We are using ordinary Newtonian kinematics, and as you said earlier ignoring gravitational interactons.

Fix an interial coordinate system. Suppose that now you set off a bomb, and eject a number of pieces P_i each moving at a constant velocity vector V_i. The position or P_i at time t , call it X_i will be simply tV_i. Now choose a piece at random, fix it, and arrange the labels so that it is P_1. Consider the motion of P_i for and arbitrary i relative to P_1. The position of P_i relative to P_1 is just the difference in the position vectors relative to our original fixed inertial coordinate system, or
X_i - X_1 = t V_i - t V_1 = t (V_i -V_1) .

Similarly the velocity of P_i relative to P_1 is just V_i - V_1
From this you can see that the relative position is just the scalar t times the relative velocity. So the relative velocity vector is parallel to the relative position vector -- P_i is moving directly away from P_1 and the velocity has the required linear dependence on positionn the proportionality constant being simply time.

So, from the perspective of an observer riding on any piece the expansion would be isotropic, as required.

Note that this does NOT say that the expansion is isotropic with respect to any arbitrarily chosen abstract observer (for instance one fixed at some point other thant the origin of the original inertial reference frame), but it does apply to any observer who is riding one of the pieces that participated in the initial event. So for the cosmological expansion question, an observer riding a body that resulted from the initial "explosion" would see an isotropic expansioin. I don't think that fits the strong cosmological principle, but it is perhaps an acceptable substitute.

I have not worked out the details, but I suspect that this would also work if you threw in special relativity.

This also does NOT say that this theory is a viable competitor to the standard general relativistic model for the big bang, in which the expansion is an expansion of space itself. It does not, for instance, explain superluminal recession rates, and it does not explain the uniformity of the cosmic background radiation. Above all it is not consistent with the observed data that shows that general relativity more accurately describes physics than does Newtonian mechanics.

I cannot agree here. Unless something very odd is going on. As you look across space you will see varying rates of expansion depending on the angle you look, For example an object of equal velocity at a low arc from ourselves may appear to have a lower rate of expansion than along our own radial line. Others much greater, it would depend entirely on the direction one looks and what direction they are travelling relative to us. There is no way it would look as if everything is moving away from us at a rate proportional to the distance edit: (in general).

Cougar
2009-Apr-16, 02:13 PM
To really see it you have to have multiple radial lines and consider what it would look like in terms of relative movements looking across the coordinate space.

That is exactly what I thought and what I did when I set out to geometrically prove to Jeff that any point away from the center in the explosion model would view the distance/redshift relation differently. But it didn't turn out that way! Remember - any point away from the center is in motion away from the center, which must be considered in its redshift measurement.

Grey
2009-Apr-16, 02:15 PM
So we need to bring in the Inquisition to determine whether he's a closet Copernican.Hey, now, no need for that. I was just describing the original view of the expansion, as galaxies moving normally through "normal" space. As I pointed out, that idea fit the data they had back when the idea of a "big bang" was first suggested, but we've known for a long time that it doesn't fit all the observations we have now. Even long before we'd seen evidence that the expansion was accelerating, it was clear that we needed a model that incorporates general relativity.

malaidas
2009-Apr-16, 02:24 PM
I'm not sure that its motion from the origin is important. Surely its meerly a factor of their increasing distance with regards to us.

Grey
2009-Apr-16, 02:28 PM
That is just an assertion of what I'm questioning. There is no obvious difference between galaxies coasting through space and galaxies moving with expanding space. I don't doubt that your abbreviated description of the expansion is accurate. But I see no reason to think that it can't be accounted for by galaxies moving through space inertially.Actually, Jeff, I think you're correct, sort of. That is, if you give a set of galaxies some random set of initial velocities starting from a central point, let them explode outward, and wait over time, you'll get something like the redshift-distance relationship that Hubble observed. As long as you're far enough from the edge of the expansion, it will look sort of like what we see. However, the details of what we see (for example, how the different types of distance measurement, like angular size distance and luminosity distance, relate to each other and to the redshift observed) do not match this simple model. So we know that model isn't a good one, not because there's no way to get something like a redshift-distance relationship out of it, but because the detailed prediction of that kind of model can't be made to match what we observe.

Ken G
2009-Apr-16, 02:41 PM
So if there is a difference between galaxies moving apart through space
and non-moving galaxies in expanding space, it is not detectable at the
level of simple kinematics.
This is true, but the point that publius and I have been making is that GR is not a theory of kinematics, it is a theory of dynamics, and furthermore, the OP question is about dynamics. The difference in those words has to do with self-consistency: kinematics is the way motion happens after you've already analyzed the effects that govern that motion, but dynamics is the self-consistent study of the effects that govern motion. Since the OP is about explaining the expansion, it is about the things that govern motion. It is a dynamical question, so "simple kinematics" is not responsive to the question.

So in summary:
1) It is true that the simple kinematic argument is not wrong from the point of view of simple kinematics, but it is not the answer to why the universe continues to expand the way it does, because the question is dynamical in nature, kinematics is insufficient.
2) The answer to why the universe started expanding, and why it continues to expand, cannot center on inertia, because inertia plays no role in motion purely under gravity-- and as yet no player other than gravity has been identified.

And thanks to Richard for clarifying some of the stickier issues about what is dynamical and what is static, as usual it is not as simple as one might imagine.

malaidas
2009-Apr-16, 02:48 PM
I'm not sure that its motion from the origin is important. Surely its meerly a factor of their increasing distance with regards to us.

Appologies I wasn't thinking straight. Of course it matters.

DrRocket
2009-Apr-16, 04:06 PM
I cannot agree here. Unless something very odd is going on. As you look across space you will see varying rates of expansion depending on the angle you look, For example an object of equal velocity at a low arc from ourselves may appear to have a lower rate of expansion than along our own radial line. Others much greater, it would depend entirely on the direction one looks and what direction they are travelling relative to us. There is no way it would look as if everything is moving away from us at a rate proportional to the distance edit: (in general).

The mathematics (see my post above) would seem to say something quite different.

Note that the angle effect is taken care of in the vector calculation. It is done in a coordinate-free manner, so you don't see trig functions coming in explicitly. That calculation is also independent of the dimension.

publius
2009-Apr-16, 05:01 PM
This one (http://www.bautforum.com/space-astronomy-questions-answers/77359-recent-expansion-space-papers-implications.html)? That was an enjoyable explanation, I even understood some of it :)


Yes, that was the thread. Thanks for finding it! In there, I tried to convey a picture of "comoving observers", those riding with the particles making up the spherical mass distribution.

You can see there are other observers, which can be completely inertial, which are not comoving. Give some observer an initial velocity at some high speed relative to a dust particle, and let him follow an inertial path. He will see something very different than the "cosmological principle" view.

And there are easily such frames and coordinates in the GR FLRW models. Imagine an inertial frame moving towards Andromeda at 0.9999c. That frame will certainly see some galaxies highly blue shifted, and those behind highly red shifted. It will go with the flow of its geodesic, but see something very different than isotropic.

Such frames can be seen in the simplified deSitter space-time as world lines on the hyperboloid very different from the co-moving world lines. They are just as valid as any other coordinates, but very different and don't expose the symmetry and other nice properties of the comovers.

-Richard

DrRocket
2009-Apr-16, 05:08 PM
Yes, that was the thread. Thanks for finding it! In there, I tried to convey a picture of "comoving observers", those riding with the particles making up the spherical mass distribution.

You can see there are other observers, which can be completely inertial, which are not comoving. Give some observer an initial velocity at some high speed relative to a dust particle, and let him follow an inertial path. He will see something very different than the "cosmological principle" view.

And there are easily such frames and coordinates in the GR FLRW models. Imagine an inertial frame moving towards Andromeda at 0.9999c. That frame will certainly see some galaxies highly blue shifted, and those behind highly red shifted. It will go with the flow of its geodesic, but see something very different than isotropic.

Such frames can be seen in the simplified deSitter space-time as world lines on the hyperboloid very different from the co-moving world lines. They are just as valid as any other coordinates, but very different and don't expose the symmetry and other nice properties of the comovers.

-Richard

Nice explanation.

BTW I found a reference -- General Relativity by Wald -- that makes the point that in general, there will be only one observer at each point for whom space-time is homogeneous and isotropic. There are some exceptions apparently, such as flat space-time, but for a general curved situation, only one per point. I had not realized that it was that restrictive.

Ken G
2009-Apr-16, 06:56 PM
Yes that is interesting, though of course you meant to say at most one-- the cosmological principle is the statement that there is indeed one.

pzkpfw
2009-Apr-16, 09:45 PM
yelram posts moved to new thread:
http://www.bautforum.com/against-mainstream/87256-yelram-claims-expansion-universe.html

DrRocket
2009-Apr-16, 09:58 PM
Yes that is interesting, though of course you meant to say at most one-- the cosmological principle is the statement that there is indeed one.

Perhaps a better way to make the statement that I made is that in a homogeneous and isotropic space-time there will in general (there are some exceptions as for instance in the case of flat space) be only one observer at each point for whom the isotropy is realized. The assumption of homogeneity and isotropy is the cosmological principle. It actually takes a bit of work to make that notion precise within the framework of GR, and the way it is done is to postulate the existence of a one-parameter family of spacelike hypersurfaces foliating space-time having a certain isometry property and with additional conditions that assure isotropy.

The really interesting things are that once that is done you have a global parameter that is a sort of "time" and something of a definition of "space". They are not clear unique concepts, but this foliation answers a question that I have had regarding what is meant when people talk about the curvature of "space". Space in this setting is represented by the hypersurfaces.

Those hypersurfaces turn out to be spaces of constant curvature, which then means that there exists a known topological classification. This is the source of the often cited dependence of the closedness or openness of space on curvature. The one kicker, to my mind, is that the case of negative curvature is not so clean as is usually stated. There are closed 3-manifolds of constant negative curvature, which have apparently been disallowed on the basis of being "unnatural", and I don't buy that without some further explanation. Bill Thurston did some ground-breaking (and Fields Medal winning) work on hyperbolic manifolds a few years ago that ought to figure into this somehow. http://en.wikipedia.org/wiki/Hyperbolic_3-manifold

It is also interesting to me that this foliation is apparently dependent on homogeneity and isotropy, which apply only to large-scale approximations to the real space-time. The universe at a detailed level is clearly not homogeneous and isotropic, so there is no such foliation, and therefore as one usually understands in GR there is no useful global defintion of space or of time, but only of space-time.

publius
2009-Apr-16, 11:37 PM
The really interesting things are that once that is done you have a global parameter that is a sort of "time" and something of a definition of "space". They are not clear unique concepts, but this foliation answers a question that I have had regarding what is meant when people talk about the curvature of "space". Space in this setting is represented by the hypersurfaces.



Yes, yes. This is very important to appreciate. In the co-moving coordinates, the proper clocks attached to the particles making up the mass distribution form a common "cosmological time". This is just a coordinate time like any other, but special significance can be attached to it. It in the strict sense of Relativity, one shouldn't do that, but one tends to anyway. :) And that's what cosmologists do.

Consider a standard FLRW metric:

ds^2 = dt^2 + a(t)*(dR^2), where dR^2 represents the spatial line element.

R = constant represents co-movers. Now, what is the proper clock rate of any comover? ds/dt = 1! The spatial hyperslices are chosen (well, work out to be) such that all co-moving clocks read the same thing "now". (This is very different from SR of course!) Distance between co-movers is increasing with time, yet there is no "time dilation" in the sense of "now" defined by these coordinates.

This is something that can be very confusing. The "stretching of signals" between comovers in these coordinates is entirely due to a(t), "expanding space". In static coordinates, which can be adopted in a simple deSitter model, things are quite different, and there is more familiar "time dilation", both gravitational and kinematic. Anyway, this co-moving notion of simultaneity can be very tricky to appreciate, especially for those with just an SR background. The spatial hyperslices, about which we fret if they are "flat" are not, are those arbitrary surfaces defined about those co-moving world lines. Big deal in the covariant, coordinate dependent framework of GR.

There are a lot of "stories" about the universe's expansion works that are told that are simply misleading, and based on coordinates, and worse, misunderstandings about how those coordinates themselves work. Many things are asserted that have no real meaning in GR, and come from attaching "too much reality" to the comoving coordinate choice. And worse, there are then misconceptions there as well.

-Richard

Ken G
2009-Apr-17, 01:21 AM
Yes, these are quite interesting posts, and I think they point to a still-open question: GR does not favor any particular coordinate choice, but the mass distribution of the universe seems to, so we come back to Mach's principle-- GR doesn't need Mach's principle because it doesn't need to favor comoving coordinates, but it may also be incomplete in some way (especially in regard to The Beginning). Perhaps there is something behind Mach's principle after all, that does select that frame, and establishes some kind of descriptive underpinning of the Big Bang other than just "it's all the initial condition".

publius
2009-Apr-17, 01:35 AM
.... so we come back to Mach's principle-- GR doesn't need Mach's principle because it doesn't need to favor comoving coordinates, but it may also be incomplete in some way (especially in regard to The Beginning). Perhaps there is something behind Mach's principle after all, that does select that frame, and establishes some kind of descriptive underpinning of the Big Bang other than just "it's all the initial condition".

Ah, Mach. But consider this. What is the "distant cosmic mass" doing relative to our *inertial*, near comoving vantage here in the Milky Way? It is not rotating, but it is accelerating! Our own universe does not obey Mach's Principle, if somehow inertia is define relative to the whole mass. We think of Mach as about rotation at first blush, but it has to be about translational acceleration as well.

This would be the case even in straight FLRW model without Lambda. Lamdba appears to have taken over and is accelerating the expansion (meaning the tide is now "ripping"), but without it, the expansion would be slowing (whether or not it would expand forever or stop and recollapse). Any inertial dust particle sees the others as deccelerating, and a compressive cosmic tide.

-Richard

Ken G
2009-Apr-17, 02:20 PM
But to me, that doesn't violate Mach-- I think of Mach as the statement that all acceleration must be defined in regard to other mass, it is never an innate property of a particle (which is a GR-like thing to say), and furthermore, the mechanism by which matter "tells" other matter what its acceleration is referenced to is gravity. So the inertia of one mass does not just create its gravity, the very meaning of ts inertia comes from the gravity of other masses. For example, a universe with only a single particle could not have any meaningful notion of inertia for that particle, and an accelerometer attached to that particle could never change readings.

Now, that's easy to say, as a universe of only one particle can have no forces on it. But at least we can say that's consistent with Mach. Now let's consider a universe of two identical masses, what then? In the Machian view, I would say this means the two masses can never under any circumstances have different accelerometer readings, as they can only reference their acceleration to each other, since acceleration relates to inertia and they get their inertia from each other. Again, this doesn't say much, because even in Newton the two identical particles must acclerate the same because their interaction must be an action/reaction pair. But again we can say it is consistent with Mach's principle.

Now let's have one mass comprise of twice the number of identical particles as the other. The force between them is still an action/reaction pair, but why does the mass with more particles accelerate less? Because it has a greater "vote" in the very meaning of what acceleration is, by virtue of its greater mass. So the relative inertia of the two masses comes from their gravity, it's consistent with Mach.

If we now go to a whole universe of comoving masses, they can expand or compress, uniformly, and still be consistent with Mach-- because all of the matter in the prevailing distibution is doing the same thing, they can be getting their inertia from the whole distribution (rotation would spoil that property). Indeed, we don't even get a sense of inertia at all, until we look at the outliers that have special forces on them, and when we ask what their response is to those forces, we need to reference their acceleration to something else-- the comoving distribution.

So I'm certainly no expert on Mach's principle, but it seems to me that the cosmological principle is quintessentially Machian, albeit in a kind of generalized sense where the motion of matter can show no less symmetry than does the matter distribution.

Kwalish Kid
2009-Apr-17, 04:58 PM
But what does Mach's Principle actually do? It doesn't tell us anything about how to determine inertia. It seems to me to replace something for which we have no explanation for something else equally mysterious.

publius
2009-Apr-18, 06:55 AM
Ken,

I've been meaning to respond, but it's taking a lot of thought. :lol: This gets deep. I see what you're saying, but I disagree. Now, as you note, there is no way to test this because we don't have any empty test universes.

In the Wheeler formulation of GR, matter tells space-time how to curve, and space-time tells matter how to move. The latter should be clarified as telling matter what it's inertial, force free way to move is -- the geodesics. Inertia, *as viewed by current theory*, is entirely local. Shaping the geodesics, which is what the EFE is about, is not the same thing as resisting deviation from them. One might suspect a connection with this "mass" stuff that both shapes geodesics and governs resistance from them, but they seem independent effects.

Imagine a pure mathematical reference in empty space-time. Let that frame deviate from the geodesic, and be accelerated. Said frame will experience inertial forces, terms in the equations. Those terms, the "Christoffel symbols" are a local thing, referenced to the local geodesic. Any mass we throw in globally will change those geodesics, but it will not change this business about proper acceleration being a local deviation from the geodesic. This is an invariant, not something relative/coordinat dependent.

In a black box, a local accelerometer does not care what the geodesic is, it only cares about deviation from it. Now you can argue that any real accelerometer means we've got to have mass, and real forces to effect acceleration, but I say we can imagine a massless limit of pure reference frames.

In our current body of theory, that's how it works. Granted, we can't empircally verify it but it fits everything we can verify. That is, a theory that has inertia as a local property relative to the geodesic is consistent with what can verify.

One can modify Mach into something more GR-ish, but that seems to me to be the tail wagging the dog. As I understand what Mach originally intended, things just didn't pan out.

-Richard

Ken G
2009-Apr-18, 09:20 AM
But what does Mach's Principle actually do? It doesn't tell us anything about how to determine inertia. It seems to me to replace something for which we have no explanation for something else equally mysterious.
By "Mach's principle", I don't necessarily mean what Mach said prior to the invention of GR, but rather, I mean possible metaphysical constraints that GR might be expected to respect, that are inspired by what Mach said prior to the invention of GR. The key issue is, if you have an observer standing in the center of a round pool, and the observer spins, nothing happens to the shape of the water. But if the pool spins, whether the observer does or not, the shape of the water does change. What are the possible ways that can observable fact can emerge from GR, and can it appear in non-Machian ways?

Newton argued that this observation means that inertial frames are absolute, which in some sense means, space exists independently of the pool and the observer. In Richard's carefully thought out response above, he takes a similar tack, though makes the GR-appropriate substitution of the set of geodesics for Newton's more liberal concept of absolute space. Richard is saying that gravity determines the geodesics, but motion relative to those geodesics is something entirely independent, and that independence is what affords it a kind of "absolute" quality. The shape of the water changes when forces cause it to not follow an allowed geodesic set out by gravity.

But I'm going to argue that is the Machian perspective, as soon as we stipulate that the prevailing mass distribution determines the geodesics. It seems pretty inescapable that the Einstein field equations require a boundary condition, and that will be set in some way by the prevailing mass distribution. Mass determines gravity, and gravity determines the geodesics, and that all seems very Machian to me.

I think the problem here is that I was not precise enough in how I was defining what I am calling Mach's principle. I said it is the principle that motion cannot break a symmetry that the prevailing mass distribution does not break, but I should have said that inertial motion does not break that symmetry. I agree with Richard that real forces can do what they like, and cause arbitrary deviations from geodesics, without any GR consequences because the mass involved can be so tiny, and the forces so weak, that we are in effect talking about a "bare reference frame", as Richard alluded to.

So getting back to the pool and the observer, what would Mach say is happening there, versus Newton? My impression is, Newton would say that even a universe with only a pool and an observer would still allow the surface to bow if the pool rotated, and not if the observer rotated. Mach might say that in such a universe, with no third party to give meaning to the question of "which rotated", one could not have one situation where the water bows, and another where it does not-- presumably either the water never bows, always bows, or the observer is simply not able to experience relative rotation with respect to the pool. The geodesics simply don't know what to anchor themselves in, they just kind of stream off wantonly, and all is up in the air.

Now personally, I think that if there were no prevailing mass distribution, some other effect involving the history or origin of the universe would still have to dictate a boundary condition, and so you'd still have geodesics. So on that score I agree with Newton and Richard. But I agree with Mach that if you do have a prevailing mass distribution, and it does exhibit some symmetry, then the geodesics will exhibit that symmetry too, and so will inertial motion. So getting back to the cosmological principle, I would argue that the cosmological principle is a manifestation of Mach's principle, when you throw in a kind of "chicken-and-egg" problem-- inertial motion in the universe can break no symmetry that the mass distribution doesn't break, but the mass distribution, following as it does inertial motion, cannot break the symmetry either. We live in a universe that exhibits a cosmological principle because purely inertial motion must always give you that, in light of Mach's principle that the mass distribution itself defines what inertial motion is.

Ken G
2009-Apr-18, 09:27 AM
In a black box, a local accelerometer does not care what the geodesic is, it only cares about deviation from it. Now you can argue that any real accelerometer means we've got to have mass, and real forces to effect acceleration, but I say we can imagine a massless limit of pure reference frames. Yes, I'm actually fine with this, and I'm fine with your separation of purely inertial motion from the effects of real forces. Everything I'm talking about is actually in the absence of any deviations from geodesics, so there are no real forces on the scales I have in mind (those of the universe as a whole). Now you may be saying that Mach's principle breaks down when real forces enter the problem, and I don't disagree with that, nor do I know what Mach himself intended. I'm just saying that to me, an important principle does survive here, in the context of purely inertial motion, and it is the principle I clarified in my response to Kwalish Kid.

One can modify Mach into something more GR-ish, but that seems to me to be the tail wagging the dog. As I understand what Mach originally intended, things just didn't pan out.I'm not really sure what Mach originally intended, frankly, he may have been saying that geodesics are simply impossible in an empty universe. That's really an irrelevant question because it is so different from our universe that it's not even science any more, but I would tend to think that an empty universe would still have some kind of history that would still have some way to generate geodesics, and I think the idea that geodesics are generated in some concrete or invariant way is the core of what you are saying. I'm just still calling that a Machian perspective-- to me, a non-Machian perspective is that the geodesics are inherent to spacetime and don't need to know about the mass distributions within, or history of, or boundaries at the edges of, that spacetime, whereas the self-consistency requirement of GR between spacetime and matter seems to me to still be quite Machian. GR seems very Machian to me, and so does the cosmological principle, because at its core Mach's principle is a chicken-and-egg principle, that's my point.

Or put differently, the only way to know what inertial motion means for the universe as a whole is to look at what the mass distribution of the universe as a whole is doing. That seems like Mach in a nutshell. It then has to exhibit a cosmological princple because there is no outside source or reference point from which to generate anything else, except of course for smaller-scale instabilities.

loglo
2009-Apr-18, 12:49 PM
Mach's Principle has to be the most insidious idea in cosmology that no physicist really trusts. In response to Richard's remark that Mach didn't pan out for Einstein, as noted by Barbour, there appears to be as many different definitions of Mach's Principle as there are physicists. Einstein wrote this of it around 1911:-

This assumption of the exact physical equivalence [of a uniform gravitational field and uniform acceleration] makes it impossible for us to speak of absolute acceleration of the system of reference, just as the usual theory of relativity makes it impossible for to talk of the absolute velocity of a system.

In itself this result is of great interest. It shows that the presence of inertial shell K
increases the inertial mass of a material point P within it. This makes it plausible that the entire inertia of a mass point is the effect of the presence of all other masses, resulting from a kind of interaction between them. This is exactly the standpoint E. Mach has argued persuasively for in his penetrating investigation of this question. - Mach's Principle: From Newton's Bucket to Quantum Gravity (Einstein Studies No 6)

Einstein goes on to describe the idea of a "relativity of inertia" as a goal of Mach's and a logical conclusion of his Principle. Barbour has cited this as a prime source of confusion of the meaning of Mach's Principle as Mach was "not concerned with inertial resistance but with the law of inertia. He was concerned solely with what is often called kinematical relativity."

and
"Whereas the logic of Mach's comment called for the explicit derivation of the distinguished local frames reference from a relational law of the cosmos as a whole, Einstein is working towards the elimination of the distinguished frame by asserting they are not really distinguished after all."

Einstein conceded two years after publishing GR that he was "led to introduce a quite different Mach's Principle". Barbour argues that when Einstein considered the cosmological implications of GR his version without boundary conditions was an attempt at completing the implementation of Mach's Principle within it. This is an interesting contrast to Ken's argument above that Machian boundary conditions are required of GR for a realistic cosmology. (If that was what he was saying!)

Interestingly Mach's Principle was a prime motivator for the introduction of the cosmological constant:-

The aim of this term, which introduced an effective repulsion of matter, was to ensure that the equations had no reasonable cosmological solution without the presence of matter. The necessary presence of matter would then ensure, in Einstein's opinion, that inertia was not merely influenced by matter but completely determined by it. "Mach's Principle and Relativistic Cosmology (http://www.platonia.com/barbour_modern_cosmology.pdf)"

De Sitter apparently never trusted the idea of a relativity of inertia and the famous model he produced shortly after Einstein deliberately avoided it, producing a model that was stable without the presence of matter. Friedmann then came along and showed both DeSitter's and Einstein's models were special cases of the same equations. The rest is history. Where that leaves Mach though I'm really not sure.

DrRocket
2009-Apr-18, 03:55 PM
So I'm certainly no expert on Mach's principle, but it seems to me that the cosmological principle is quintessentially Machian, albeit in a kind of generalized sense where the motion of matter can show no less symmetry than does the matter distribution.

It seems to me that you can argue the implications of General Relativity intelligently, because it is rather well defined, and subject to experimental verification and analysis via mathematics that is understood.

But to clain that a concept is "Machian" is not really debatable since "Mach's Principle" is sufficiently poorly defined so as to mean different things to different people.

http://en.wikipedia.org/wiki/Mach's_principle

Why try to argue the correspondence with some vague principle, which will lead to typical philosophical word games, when Albert formulated a nice clean and elegant theory that can be debated with perfect clarity as to what you are talking about?

Ken G
2009-Apr-18, 03:59 PM
Where that leaves Mach though I'm really not sure.I'd say that sums it up pretty well! No doubt "Mach's Principle" is a moving target, but to me, it is the simple idea that mass defines geodesics, so the geometry of spacetime is not something independent of the mass. The basic idea is, if you ask me "what are the natural geodesics that spacetime supports," I have to first ask you, "where's the mass and what is it doing"? Maybe that's Einstein's Mach principle, not Mach's, but there you have it. In this thread, it started with the issue of whether or not inertial motion can "explain" the motion of the universe, and I'm saying Mach's principle would assert that is backward logic-- the motion of the universe explains inerial motion. Nothing but some kind of boundary condition "explains" the motion of the universe, which of course is no explanation at all.

DrRocket
2009-Apr-18, 04:28 PM
I'd say that sums it up pretty well! No doubt "Mach's Principle" is a moving target, ...

As noted earlier, I'm not sure that I can say that the cosmological principle is the same as or implied by Mach's principle, since the latter is not well-defined.

But I can say that the "cosmological principle" is not a principle, and it is not even correct. It is quite clear that the real universe is neither homogeneous nor isotropic. It woud be a pretty boring place if it were.

The "cosmmological principle" is really the "cosmological idealization". It results in a fictitious universe that one hopes has some global resemblance to the real one and that is amenable to being modeled in manner sufficiently simple so as to admit analysis in manner that a human might comprehend.

One would hope that a real philosophical principle would be more universally applicable than an idealization.

Ken G
2009-Apr-18, 04:31 PM
The "cosmmological principle" is really the "cosmological idealization". It results in a fictitious universe that one hopes has some global resemblance to the real one and that is amenable to being modeled in manner sufficiently simple so as to admit analysis in manner that a human might comprehend.
Too true, yet I would argue that so it is for all "principles", that's just exactly what a principle is and we should stop asking it to be something different!

One would hope that a real philosophical principle would be more universally applicable than an idealization.Did I just hear you put "real" and "philosophical" right next to each other in the same sentence? You are making progress.... ;)

Ken G
2009-Apr-18, 04:47 PM
It occurs to me that we might do well to come to some kind of agreement about what we mean by "Mach's Principle", and that to do that, we might need a "weak" and "strong" version. I would say the weak version is simply the statement that the mass distribution (the stress-energy, really) defines the geometry of the geodesics, which in turn defines inertial motion. Real forces that cause real acceleration are something added to that, and Newton figured out that part long ago. So that all seems perfectly GR-ready.

As for a "stronger" version, I think one might go all the way back to the Greek philosopher (how often that's true) Parmenides, who claimed that both change and motion are impossible. That's about as Machian as you can get, and does not separate real acceleration from inertial motion because they are both some kind of illusion: "[What exists] is now, all at once, one and continuous... Nor is it divisible, since it is all alike; nor is there any more or less of it in one place which might prevent it from holding together, but all is full of what is." This was 2500 years before spacetime or field theories. Parmenides was quite interesting, a very ancient philosopher and few of his writings survive. My reaction to him is that he is obviously wrong, but probably right anyway.

Interestingly, if Parmenides was right, then grand unification will come not by making gravity a force, but by recognizing that forces, being illusory, are gravity.

Ken G
2009-Apr-18, 04:55 PM
Why try to argue the correspondence with some vague principle, which will lead to typical philosophical word games, when Albert formulated a nice clean and elegant theory that can be debated with perfect clarity as to what you are talking about?For the same reason that always merits conjoining philosophy and physics-- the search for the next thing. Don't forget the unification problem that GR faces, not to mention a few more sticky cosmological mysteries like the Initial Condition and dark energy, etc. As long as there are mysteries, we will need new insights, and philosophy has often been the place to search for old insights reborn as the new ones we need.

DrRocket
2009-Apr-18, 08:36 PM
Too true, yet I would argue that so it is for all "principles", that's just exactly what a principle is and we should stop asking it to be something different!Did I just hear you put "real" and "philosophical" right next to each other in the same sentence? You are making progress.... ;)

Maybe I should have said "surreal",or at least made a clear distinction between a real philosophical principle and a real principle.:) .

I think there are real principles that are not just idealizations. Conservation of momentum leaps to mind. That seems to apply in all situations -- let us not start a debate here on the reality of physical law in general as it not my intent to open that can of worms.

The situation here is that if one accepts general relativity as being accurate, then the cosmological principle is not true except as an idealization and that idealization is easily refuted. In fact you don't even have to accept general relativity in order to refute the cosmological principle at human scales -- all you need is a pulse. Eyesight would help but is not essential. Heck, if the cosmological principle of homogeneity wre true, then you and I would agree on the merits of philosophy.

DrRocket
2009-Apr-18, 08:45 PM
"[What exists] is now, all at once, one and continuous... Nor is it divisible, since it is all alike; nor is there any more or less of it in one place which might prevent it from holding together, but all is full of what is." .... My reaction to him is that he is obviously wrong, but probably right anyway.

Interestingly, if Parmenides was right, then grand unification will come not by making gravity a force, but by recognizing that forces, being illusory, are gravity.

How could you possibly decide of Parmenides was right or wrong ? It seems to me that to make that decision you first require a meaningful sentence.

"That's not right. It's not even wrong." -- Wolfgang Pauli

Ken G
2009-Apr-18, 11:19 PM
I think there are real principles that are not just idealizations. Conservation of momentum leaps to mind. That seems to apply in all situations --No, it is an idealization. It only applies, even in principle, to closed systems, and of course there are none of those. So to say the cosmological principle is not a "real" principle because it only applies to the universe as a whole, and not to pockets of it, is no different from saying conservation of momentum is not a "real" principle for exactly the same reason. It is all a question of scale-- much smaller systems are effectively closed than are effectively cosmological. Quantitative difference yes, qualitative, no. All science starts with idealizations, that's part of the point of doing it.

In fact you don't even have to accept general relativity in order to refute the cosmological principle at human scales -- all you need is a pulse. It is not the least bit unusual for a physical principle to come equipped with a scale of interest. Indeed, the contrary is more unusual.

Nereid
2009-Apr-19, 12:39 AM
Is there any experiment or observation, or finite set of experiments and observations, that could, even if only in principle, distinguish a description of the (observable) universe (or some part of it) based on Machian ideas from a description with comparable scope that is not so based?

If so, what?

If not, have we just determined that Machian stuff is not, by definition, part of science?

DrRocket
2009-Apr-19, 12:52 AM
No, it is an idealization. It only applies, even in principle, to closed systems, and of course there are none of those. So to say the cosmological principle is not a "real" principle because it only applies to the universe as a whole, and not to pockets of it, is no different from saying conservation of momentum is not a "real" principle for exactly the same reason. It is all a question of scale-- much smaller systems are effectively closed than are effectively cosmological. Quantitative difference yes, qualitative, no. All science starts with idealizations, that's part of the point of doing it.
It is not the least bit unusual for a physical principle to come equipped with a scale of interest. Indeed, the contrary is more unusual.

I certainly agree that conservation of momentum applies only in a closed system. But such do exist, either as reasonable approximations, or in the extreme to the universe as a whole.

I did NOT say that the cosmological principle applies only to the universe as a whole. It does not. The cosmological principle is a statement that basically says that at each point in the universe the universe is the same and that there are no preferred directions. I said that is patently false. The universe is clearly not homogeneous, different points see different aspects --the Sahara desert is not the same as the Caribean. And it not isotropic, thngs do no look the same in all directions -- Phyllis Dyller is quite different in appearance from Kim Basinger. The cosmological principle is nothing more or less than an idealization that one hopes is a good approximation to reality on large scales, and that provides a model that is analytically tractable.

Note that on a very small scale and for sufficiently short period of time there are real closed systems. Particles in small and isolated systems are true closed systems on time scales sufficiently small that they cannot interact with any particles outside of the system. And conservation of energy applies at a fixed point in time, so it is valid to limit the time interval under consideration. Take a collision problem between elementary particles. That is a situation in which conservation of energy applies, and not as just and idealization, at least in the quantum sense which includes uncertainty. Qualitative yes.

And while it is appropriate for physical models to come with a scale of interest, the notion of homogeneity, certainly in a classical sense as is the case with relativity, does not make a distinction basec on scale. So there is clear distinction between and idealization and a principle here.

This is not different from the idealization of a rigid body. There ia no such thing, and it violates basic physical laws -- lots of them. But it is a useful model, and one that is applicable only on relative small scales, and not to every small scale problem. It is one of those things, like the cosmological principle, that we know for a fact is wrong, but that provides a useful approximation anyway.

Ken G
2009-Apr-19, 01:00 AM
The cosmological principle is a statement that basically says that at each point in the universe the universe is the same and that there are no preferred directions. I said that is patently false. It's a question of scale, that's the point. No one who uses the cosmological principle thinks it does, or should, apply to the Sahara desert, just as no one who applies conservation of momentum thinks it does, or should, apply on distance scales shorter than h/p, or for long enough times that systems cannot remain closed. It's all a matter of scale, there is no other difference between such "real principles".


This is not different from the idealization of a rigid body. There ia no such thing, and it violates basic physical laws -- lots of them. But it is a useful model, and one that is applicable only on relative small scales, and not to every small scale problem.Right, and that is true of all physical principles, including the cosmological principle. Again, that's exactly what a physical principle is.


It is one of those things, like the cosmological principle, that we know for a fact is wrong, but that provides a useful approximation anyway.Yes, and like all principles of physics. It's all a matter of how precise one needs to be in order to find usefulness, and on what scale that precision appears.

Ken G
2009-Apr-19, 01:04 AM
Is there any experiment or observation, or finite set of experiments and observations, that could, even if only in principle, distinguish a description of the (observable) universe (or some part of it) based on Machian ideas from a description with comparable scope that is not so based?The classic example is geodesics that show a net rotation relative to the prevailing mass distribution of the universe, say frame-dragging for a Kerr black hole. In Machian terms, one could not have a Kerr solution to a nonrotating (relative to the prevailing mass distribution) black hole, though there is no other reason to rule out that possibility. We could observe one tomorrow, for all we know, it is only Machians who would doubt that. I also doubt it, but accept that my doubt carries no predictive power that could replace the act of observation. But it is certainly scientific to make the prediction that a black hole spacetime that breaks the symmetry of the distant masses must exhibit a history of relative motion in regard to those masses, whereas an unconstrained spacetime (quasi-Newtonian) would be held to no such standard.

DrRocket
2009-Apr-19, 01:30 AM
It's a question of scale, that's the point. No one who uses the cosmological principle thinks it does, or should, apply to the Sahara desert, just as no one who applies conservation of momentum thinks it does, or should, apply on distance scales shorter than h/p, or for long enough times that systems cannot remain closed. It's all a matter of scale, there is no other difference between such "real principles".

Right, and that is true of all physical principles, including the cosmological principle. Again, that's exactly what a physical principle is.
Yes, and like all principles of physics. It's all a matter of how precise one needs to be in order to find usefulness, and on what scale that precision appears.

OK, you hve stated that conservation of energy does not apply at scales shorter than h/p, and that is something that can be demonstrated to be valid with a basis in basic theory. I accept that.

Now, explicity provide the scale on which the cosmological principle holds, if you want me to accept it as principle on par with conservation of energy. Not only provide that scale but show why the features of the universe that are manifestly not homogeneous, not isotropic, and strongly curved do not upset the apple cart at the scales at which you claim this to be a valid "principle". Without that level of understanding and quantification of the limitations of the principle, I can accept it only as idealization that is useful but not amenable to rigorous justification.

This discussin is raising some interesting points in my mind. It don't think they are really in line with the OP however. But this might make a very interesting thread in and of itself. Let me ponder a moment and maybe pose a couple of questions elsewhere that are intriguing me at the moment.

Nereid
2009-Apr-19, 02:08 AM
Is there any experiment or observation, or finite set of experiments and observations, that could, even if only in principle, distinguish a description of the (observable) universe (or some part of it) based on Machian ideas from a description with comparable scope that is not so based?The classic example is geodesics that show a net rotation relative to the prevailing mass distribution of the universe, say frame-dragging for a Kerr black hole. In Machian terms, one could not have a Kerr solution to a nonrotating (relative to the prevailing mass distribution) black hole, though there is no other reason to rule out that possibility. We could observe one tomorrow, for all we know, it is only Machians who would doubt that. I also doubt it, but accept that my doubt carries no predictive power that could replace the act of observation. But it is certainly scientific to make the prediction that a black hole spacetime that breaks the symmetry of the distant masses must exhibit a history of relative motion in regard to those masses, whereas an unconstrained spacetime (quasi-Newtonian) would be held to no such standard.
Thanks! :)

I didn't know that (or, if I had once known it, have forgotten).

I'm not sure I really follow what you're saying, and I'm pretty sure that all readers of this post - other than publius and DrRocket - follow at least as poorly as I ... so would you mind expanding on this a bit please?

Ken G
2009-Apr-19, 02:51 AM
Certainly, though I must at the outset admit that it is very difficult to speak lucidly in the realm of Machian issues, and GR has so many subtle points (publius speaks in terms of -1/2 marks for imprecise language) that my answer might introduce its own flaws and misconceptions. But in my view, Newton's view is that spacetime (well, space for him) is absolute, completely independent of any other issue like what masses are doing anywhere. So the appearance of inertial forces (say centrifugal forces) come when you think you are moving inertially but you really aren't. You can then say "I see inertial forces, I must be moving non-inertially, relative to some absolute inertial spacetime that is invisible to me".

Mach would argue that if something is invisible to you, there's a reason for that-- it doesn't exist. Instead, "absolute space" is merely a kind of stand-in or proxy for a mass distribution that is somewhere else, the prevailing mass of the universe. You might not see that mass, but you could in principle, and then instead of saying you are moving non-inertially with respect to some absolute invisible spacetime, instead you say you have left a geodesic that is ruled by that distant mass you could in principle actually observe and understand why the geodesic connects to that mass in that way.

So when we talk about "rotating black holes", in the Newtonian approach, the rotation would be relative to its own local spacetime, and could in principle have nothing to do with any mass anywhere else. Hence, we could find a rotating (Kerr) solution to a black hole that showed no other tendency to rotate with regard to our universal (Machian) standards, it's just a little local whorl in the absolute spacetime in which it is embedded. But in the Machian approach, the local spacetime is not independent of the global mass distribution, so if the local spacetime shows rotation (say, frame-dragging), then it must be because that black hole had some kind of history of rotation relative to the distant masses, not just some arbitrary rotation relative to its own spacetime whorl.

In this sense, Mach's principle does not add something to the behavior of spacetime, it restricts what spacetime is allowed to do-- in ways that we have so far not seen any counterexamples (to my limited knowledge). As such, it seems consistent with not only GR, but also, simple Occam's razor. You may officially call me a Machian (until further notice).

Ken G
2009-Apr-19, 03:07 AM
Now, explicity provide the scale on which the cosmological principle holds, if you want me to accept it as principle on par with conservation of energy. Not only provide that scale but show why the features of the universe that are manifestly not homogeneous, not isotropic, and strongly curved do not upset the apple cart at the scales at which you claim this to be a valid "principle".It depends on what you mean by "on a par". If your standard for "par" can be measured in decimal places of accuracy, then the cosmological principle, even on the scale of the observable universe (its intended scale), will not compete with the raw accuracy of conservation laws on their preferred scales. However, if your standard of "par" is how powerfully the principle simplifies the calculations, and how deep of a creek you'd be up if you did not have that principle, then it is not only on par with conservation laws, it is virtually in a class by itself. So why do we have physical principles-- for their precision, or for their power? Note that physics in action is a constant tradeoff between power and precision; all that is really important is to understand the level of each that is in play for any given principle.


Without that level of understanding and quantification of the limitations of the principle, I can accept it only as idealization that is useful but not amenable to rigorous justification.I believe I understand what you are saying, you see deep (even philosophical) importance vested in conservation laws (as evidenced by that "law" word), but mere idealizations should not rise to the level of a "principle" in your view. So you would say that a star obeys a "law" of gravity, but if we stipulate that the star is a sphere, we are merely making an idealization. You would probably not assert a "principle" that stars are spherical, expressly because they are not. However, life is not really so cut-and-dried, because stars don't "obey" laws of gravity either. We already know that Newton's laws are false for stars (though they are great idealizations, especially if the star is spherical!), but we use them anyway, because we understand the errors involved and find them to be small in many contexts of interest-- and we can't use GR unless we make other idealizations as well because we can't solve the equations.

So it is with other principles though-- when the errors are small in the context of interest, it's a "principle". When the errors are downright minute, it's a "law", but these are all very slippery scales and I would argue there is not a fundamental difference anywhere in the lot. Whether we call it a "cosmological principle", or a "cosmological idealization", or a "cosmological law", it wouldn't alter in the least how it gets used, nor alter the contexts of the situations where observations demonstrate it cannot be used.

But this might make a very interesting thread in and of itself. Let me ponder a moment and maybe pose a couple of questions elsewhere that are intriguing me at the moment.I agree, we probably shouldn't say more here or we'll be off topic, but it is grist for a very interesting thread of its own.

DrRocket
2009-Apr-19, 03:14 AM
So it is with other principles though-- when the errors are small in the context of interest, it's a "principle". When the errors are downright minute, it's a "law", but these are all very slippery scales and I would argue there is not a fundamental difference anywhere in the lot.
I agree, we probably shouldn't say more here or we'll be off topic, but it is grist for a very interesting thread of its own.

And that thread now exists with questions posed here (http://www.bautforum.com/space-astronomy-questions-answers/87331-cosmological-principle-gr.html) I have no idea how to answer those questions. I hope you can help.

publius
2009-Apr-19, 03:59 AM
Actually you don't even need Kerr/frame dragging. Consider the "geodetic precession" of orbits in Schwarzschild, which I think is sometimes dubbed deSitter precession, as he was the first to explore it.

The inertial axes of a orbiting body actually rotate relative to a stationary observer, and to a free floting observer sufficiently far away to be practically stationary to the source mass. The effect is so small for weak fields that it's not noticeable (but can be measured with enough precision -- Gravity Probe B did confirm that, just had problems with the smaller frame dragging contribution). But Nordtvedt, et al, have confirmed it with the earth moon system via years of LLR data.

At any rate, an inertial observer in a circular orbit is thus "rotating with respect to the fixed stars". The water in the bucket is flat, yet the fixed stars are rotating around it.

This is not frame dragging, but is property of the local geodesics around the mass due to the space-time curvature. You can see the same effect on the 2D curved surface of a sphere. Construct an an equilateral triangle on that sphere, with one vertice at the pole and the others on the equator. Put a small 2D local cartesian axis at one vertice and let it make a round trip, keeping the locally axis "straight" relative to the the path confined on the curved surface(geodesic). It will rotate 90 degrees once per circuit.

And that same effect of 1time-3pace-D curvature causes local orbiting gyros to rotate (precess). Again, this has nothing to do with frame dragging or rotation of source masses, but is entirely a property of the geodesics of non-rotating, non-relatively moving sources.

What say Mach to this effect?

-Richard

Ken G
2009-Apr-19, 06:52 AM
What say Mach to this effect?
Beats me, that's a pretty subtle -1/2 there. But I don't think it's a fundamental problem for Mach, it's just a problem for too-naive descriptions of inertial axes. I think the effect you are talking about has to do with there being a nonzero mass of the bucket itself, but we can go back to your own "bare reference frame" approach to simplify the Machian perspective.

publius
2009-Apr-19, 08:13 PM
Beats me, that's a pretty subtle -1/2 there. But I don't think it's a fundamental problem for Mach, it's just a problem for too-naive descriptions of inertial axes. I think the effect you are talking about has to do with there being a nonzero mass of the bucket itself, but we can go back to your own "bare reference frame" approach to simplify the Machian perspective.

That's the thing -- it is a pure reference frame "feature" of the geodesics of Schwarzschild. The contribution of the orbiting mass itself would make things far more complex.

The (relatively!) simple way this is usually shown is to put a frame in a circular orbit geodesic, and keep the spatial axes aligned with what you think should be a non-rotating reference axes of the observer at infinity. You then calculate the Fermi-Walker derivatives, which define "proper rotation" and discover they are non-zero. And that is thus a "wow, something really strange is going on" moment. There is "coordinate rotation" between the observer at infinity and the orbiting observer. Both are inertial, yet see each other as rotating.

Again, the effect is vanishingly small for weak fields, but in a tight orbit around a black hole where local curvature is significant this rotation effect would be significant as well.

-Richard

Ken G
2009-Apr-20, 12:57 AM
I'm not disputing that, I'm saying that if you orbit a black hole, could you find some additional affect beyond the one you mention (say, some frame-dragging), even if the mass you are orbiting is not rotating with respect to the prevailing mass of the universe? It seems that the issue is, are the effects you get when a mass is rotating inherent in the relationship between the mass and the spacetime around it, or is the spacetime around it just some kind of proxy or stand-in for expressing a relationship with other masses, possibly quite far away? It strikes me as analogous to electric fields-- there is no "Mach's principle" for electric field, we simply expect all electric fields to be due to charges. No on asks, could there be an "inherent" field just kicking around, that charges modify? It seems to me that Mach is not saying anything much more than that, except in regard to inertial forces rather than electric forces.

Running with that analogy a bit, let's say we have a charge that is accelerating due to an electric field. Even in Newtonian mechanics, we could say that the charge is experiencing a balance between an inertial force and an electric force. Now, no one asks where the electric force comes from-- it is assumed that it comes from other charges. Why then is it so philosophically challenging to assert that the inertial forces comes from other masses? The equivalence principle would seem to be the embodiment of Machian thinking. One might imagine that spacetime is a field emanated by masses, much as the electric field is emanated by charges.

DrRocket
2009-Apr-20, 01:17 AM
I'm not disputing that, I'm saying that if you orbit a black hole, could you find some additional affect beyond the one you mention (say, some frame-dragging), even if the mass you are orbiting is not rotating with respect to the prevailing mass of the universe? It seems that the issue is, are the effects you get when a mass is rotating inherent in the relationship between the mass and the spacetime around it, or is the spacetime around it just some kind of proxy or stand-in for expressing a relationship with other masses, possibly quite far away? It strikes me as analogous to electric fields-- there is no "Mach's principle" for electric field, we simply expect all electric fields to be due to charges. No on asks, could there be an "inherent" field just kicking around, that charges modify? It seems to me that Mach is not saying anything much more than that, except in regard to inertial forces rather than electric forces.

Running with that analogy a bit, let's say we have a charge that is accelerating due to an electric field. Even in Newtonian mechanics, we could say that the charge is experiencing a balance between an inertial force and an electric force. Now, no one asks where the electric force comes from-- it is assumed that it comes from other charges. Why then is it so philosophically challenging to assert that the inertial forces comes from other masses? The equivalence principle would seem to be the embodiment of Machian thinking. One might imagine that spacetime is a field emanated by masses, much as the electric field is emanated by charges.

That inertial force comes, in part, from the electromagnetic field. The charge is acclelerating. Therefore it is radiating. That radiation is carrying away energy. That energy is coming from the kinetic energy of the particle which in turn is coming from the force that is accelerating and the energy of the force that is lost to radiation is realized as inertia. However, that calculation does not account for all of the inertia of even the electron, hence is clearly not enough to explain the inertia of the proton.

Ken G
2009-Apr-20, 02:14 AM
That inertial force comes, in part, from the electromagnetic field. The charge is acclelerating. Therefore it is radiating. The effect you describe would be small (and very small for the proton as you point out), but interestingly, it is far from clear that a uniformly accelerated charge does indeed radiate. For one thing, it would seem to violate the equivalence principle, for charges sitting on the surface of the Earth do not radiate. It's a very sticky issue no matter what model one uses to understand radiation, as you can get a flavor for here: http://www.mathpages.com/HOME/kmath528/kmath528.htm.

Still, we agree the issue is moot anyway-- we need a much larger inertial force in any case.

DrRocket
2009-Apr-20, 04:13 AM
The effect you describe would be small (and very small for the proton as you point out), but interestingly, it is far from clear that a uniformly accelerated charge does indeed radiate. For one thing, it would seem to violate the equivalence principle, for charges sitting on the surface of the Earth do not radiate. It's a very sticky issue no matter what model one uses to understand radiation, as you can get a flavor for here: http://www.mathpages.com/HOME/kmath528/kmath528.htm.

Still, we agree the issue is moot anyway-- we need a much larger inertial force in any case.

That is interesting and I will have to read through it in some detail.

But don't throw out the idea of a significant bit of mass coming from electromagnetics. The qualitiative explanation that I gave you is one that I could supply quickly and off the top of my head without a lot of detailed calculations. I think it may still work, and I am not quite ready to give up on the classical electrodynamics idea that accelerated charges radiate -- yet.

However, you can come at the issue from another direction, energy in the field of a moving charge. Feynman did just that and in The Feynman Lectures on Physics vol 2 you will find chapter 28 --"Electromagnetic Mass". I think you might enjoy reading that chapter.

He shows that 3/4 of the mass of the electron is related to this phenomena.

It seems to me that the point here is that mass and inertia are pretty subtle concepts and when one tries to determine their origin one may find that there is no single source. Mach may or may not be eventually shown to have a point, but I doubt it will be the whole story. I would be willing to make a Hawking-style bet on that, but chances of either side being able to collect within their lifetime is pretty small -- even if the LHC produces a Higgs particle.

DrRocket
2009-Apr-20, 05:11 AM
The effect you describe would be small (and very small for the proton as you point out), but interestingly, it is far from clear that a uniformly accelerated charge does indeed radiate. For one thing, it would seem to violate the equivalence principle, for charges sitting on the surface of the Earth do not radiate. It's a very sticky issue no matter what model one uses to understand radiation, as you can get a flavor for here: http://www.mathpages.com/HOME/kmath528/kmath528.htm.


OK, now I have at least given that essay the once-over. I am confused on a higher plane.

I am in the (long to say the least) process of trying to understand GR, and don't begin to understand the implications for electrodynamics. I find that in Misner, Thorne and Wheeler there is only one tiny section on electrodynamics. What they say is that a lot of things carry over from special relativity in relatively straightforward manner, but the wave equation is not one of those things.

Perhaps you or Publius, could elaborate as to how much of Maxwell carries over. That would be very interesting.

But I know what Maxwell implies in the classical setting. If you wiggle a charge, you get a time-varying current hence a time-varying B-field, and with that comes an electromagnetic wave. If that is wrong there are going to be a lot of very surprised electrical engineers. And just to check, I flipped the button on the TV set across the room that claims to be able to pick up signals from transmitters the design of which are based on just that weird idea -- and, voila',a talking head appeared. So I think there must be something to the idea. It can't be all wrong.


Aside: How credible do you find the site of that essay -- mathpages ? At first blush it seems OK, but the author is anonymous, and he seems to have taken some material from The Feynman Lectures on Gravitation. Now, Feynman is a personal hero. But those lectures (plus some that were not published in the book) were Feynman's attempt to develop a quantum theory of gravity. That attempt did not work. So we have somebody trying to understand Feynman's attempt to understand how to quantize gravity. Even a bright guy might have trouble gaining mastery of the subject in that situation. Are the people who write for that site at a sufficiently high level of bright ? In this particular case it is probably not terribly important since the author's final conclusion is "I dunno".

publius
2009-Apr-20, 07:05 AM
But I know what Maxwell implies in the classical setting. If you wiggle a charge, you get a time-varying current hence a time-varying B-field, and with that comes an electromagnetic wave. If that is wrong there are going to be a lot of very surprised electrical engineers. And just to check, I flipped the button on the TV set across the room that claims to be able to pick up signals from transmitters the design of which are based on just that weird idea -- and, voila',a talking head appeared. So I think there must be something to the idea. It can't be all wrong.


What Ken is talking about, the question of whether a uniformly accelerated charge radiates is a deep and subtle one in classical electrodynamics. The expression for radiation (see Jackson for a discussion) of an accelerated charge involves *endpoints*. The charge is accelerated for only a finite time, and the expression is "inconclusive" for a charge accelerated at a constant rate from -infinity to +infinty.

Feynman argued they didn't radiate.

An antenna, and any radiating system we can think of involves time vary acceleration. Wiggling charges is not constant acceleration.

Again, this gets very deep, and I've forgotten too much to discuss it, and I can only recommend reading a good reference. And as Ken mentions, there's all sorts of Equivalence Principle ramifications involved with this that get deep indeed. Here's a web reference that comes up with a Google on "Feynman accelerated charge":

http://www.mathpages.com/HOME/kmath528/kmath528.htm

(skimming over that, I'm not sure I agree with some of things he's saying about GR and absolute acceleration, but I just skimmed it).

The radiation reaction is known as the Abraham-Lorentz force, and this form applies only at low v/c (append a third hyphen, -Dirac for the relativistic version which is more complex), and looks something like this:

F_rad ~ q^2/c^3 * da/dt.

where da/dt is the "jerk", the time derivative of accleration, the third derivative of position.

Note that implies that a = constant means zero radiation force. But the thing to appreciate is the above formula comes from boundary conditions that themselves assume the motion is bounded/periodic! So the question is unresolved. You would think this should all be cut and dried, but this question is one of the unresolved issues of Classical Electrodynamics! You might hope QED resolved it, but it didn't, just "side stepped it". Don't ask me the details, I've just read that by experts on QED -- we'd need a real expert, a Schwinger or Feynman to properly explain all this :) )


Radiation reaction comes from a "self force", the particle *reacting with its own field*. This is very subtle and thorny issue. When you solve the equation of motion with a term that depends on its own acceleration, you get a term that depends on the *FUTURE* of applied force. You even get crazy solutions where a particle would spontaneously accelerate with no applied force! Those solutions are simply rejected because, well, that's nuts and nature shouldn't do such things. :lol: Again, I forgotten the details, but there was a paper that claimed to have resolved that problem. Whether it really did or not, I don't know. This mess is related to the "self-energy" problem of a point particle, and the "4/3" problem. IIRC, QED does better about that, but still has a problem (less than 4/3 but still > 1), and everyone hopes a TOE will resolve it.

This same thorny problem occurs in GR with gravitional radiation reaction, but one derivative higher (and adds more powers of 1/c). The quadrapole nature of gravitational radiation pushes everything up one time derivative.

-Richard

publius
2009-Apr-20, 07:07 AM
I'm a freakin' idiot, and didn't recognize that Ken posted that link and Dr. Rocket quoted the thing. It just didn't register.

-Richard

publius
2009-Apr-20, 07:30 AM
Oh, and by the same thorny problems occuring with gravitational radiation, I mean that gravitional radiation comes from a similiar "self interaction" of a mass with its own gravitational field. When a mass is radiating, it is being pushed off its geodesic by the effects of its own stress-energy on that geodesic.

From this, it would seem that we should "feel" gravitional radiation reaction force! I'm not completely sure about that (this is a very complex thing, believe me), but I think that is indeed the case. That is, if we're riding on some source mass in a gravitating system with no other forces acting, which is radiating, a local accelerometer should register the reaction force (which will be so vanishingly small in weak field systems that we couldn't hope to measure it, of course).

-Richard

DrRocket
2009-Apr-20, 08:36 AM
What Ken is talking about, the question of whether a uniformly accelerated charge radiates is a deep and subtle one in classical electrodynamics. The expression for radiation (see Jackson for a discussion) of an accelerated charge involves *endpoints*. The charge is accelerated for only a finite time, and the expression is "inconclusive" for a charge accelerated at a constant rate from -infinity to +infinty.

-Richard

This is fascinating. I had not heard of this issue before. But I am still having trouble understanding just what the problem really is.

I have the 2nd edition of Jackson. In that book, chapter 14 (Radiation by Moving Charges) section 14.2 seems to be the pertinent material. I have not checked Jackson's work, but his derivation (any of equations 14.27, 14.28 or 14.29) show, for a charge undergoing linear motion, a relationship between radiated power of an accelerated charge and the change of momentum with time or the change of energy with distance. It is pointed out, particularly in 14.29 that the radiated power is extremely low unless the energy gain per unit distance is humongous, but it is not zero. Those equations do not appear to require any endpoints. So I don't understand the statement that a uniformly accelerated charge might not radiate at all.

It would also seem that, at least in the context of special relativity, that a uniform acceleration from - infinity to + infinity would not be allowed.

Jeff Root
2009-Apr-20, 10:00 AM
This is way over my head, and doesn't have anything to do with
what I was discussing earlier in the thread, but let me see if my
intuition can jump to a correct conclusion...



It would also seem that, at least in the context of special relativity, that
a uniform acceleration from - infinity to + infinity would not be allowed.
I expect that Richard means the charge has always been accelerating
and always will be accelerating, in a constant direction. It's a little
hard to imagine a mechanism for that acceleration that avoids the
problems of special relativity, but it wouldn't conflict with relativity.

-- Jeff, in Minneapolis

Nereid
2009-Apr-20, 10:59 AM
Certainly, though I must at the outset admit that it is very difficult to speak lucidly in the realm of Machian issues, and GR has so many subtle points (publius speaks in terms of -1/2 marks for imprecise language) that my answer might introduce its own flaws and misconceptions. But in my view, Newton's view is that spacetime (well, space for him) is absolute, completely independent of any other issue like what masses are doing anywhere. So the appearance of inertial forces (say centrifugal forces) come when you think you are moving inertially but you really aren't. You can then say "I see inertial forces, I must be moving non-inertially, relative to some absolute inertial spacetime that is invisible to me".

Mach would argue that if something is invisible to you, there's a reason for that-- it doesn't exist. Instead, "absolute space" is merely a kind of stand-in or proxy for a mass distribution that is somewhere else, the prevailing mass of the universe. You might not see that mass, but you could in principle, and then instead of saying you are moving non-inertially with respect to some absolute invisible spacetime, instead you say you have left a geodesic that is ruled by that distant mass you could in principle actually observe and understand why the geodesic connects to that mass in that way.

So when we talk about "rotating black holes", in the Newtonian approach, the rotation would be relative to its own local spacetime, and could in principle have nothing to do with any mass anywhere else. Hence, we could find a rotating (Kerr) solution to a black hole that showed no other tendency to rotate with regard to our universal (Machian) standards, it's just a little local whorl in the absolute spacetime in which it is embedded. But in the Machian approach, the local spacetime is not independent of the global mass distribution, so if the local spacetime shows rotation (say, frame-dragging), then it must be because that black hole had some kind of history of rotation relative to the distant masses, not just some arbitrary rotation relative to its own spacetime whorl.

In this sense, Mach's principle does not add something to the behavior of spacetime, it restricts what spacetime is allowed to do-- in ways that we have so far not seen any counterexamples (to my limited knowledge). As such, it seems consistent with not only GR, but also, simple Occam's razor. You may officially call me a Machian (until further notice).
Thanks.

I guess my biggest area of ignorance is Machian ideas ...

... for starters, we don't live in a Newtonian universe, so how does this Machian approach incorporate (special) relativity? Specifically from where - in the Machian view - does 'relativistic mass' come from? And does inertia relate to mass-energy, in a particle-antiparticle annihilation, say?

I can see that a spinning black hole could, in principle, be used to test whether we live in a Machian or non-Machian universe, per your post, but then since any spinning black hole would have had a history before it became one, and its spin would be completely determined by its history (assuming nothing got added to it afterwards), there'd be no way to tell, would there?

Oh, and I also don't follow how frame dragging isn't inconsistent with Mach; nor, for that matter, gravitational wave radiation (in an inspiral event, where does all the inertia go?).

Like I said, I'm clearly ignorant of this Machian idea ...

DrRocket
2009-Apr-20, 02:46 PM
This is way over my head, and doesn't have anything to do with
what I was discussing earlier in the thread, but let me see if my
intuition can jump to a correct conclusion...


I expect that Richard means the charge has always been accelerating
and always will be accelerating, in a constant direction. It's a little
hard to imagine a mechanism for that acceleration that avoids the
problems of special relativity, but it wouldn't conflict with relativity.

-- Jeff, in Minneapolis

Acceleration in a constant direction would not violate special relativity.

But constant acceleration seems to me to mean a constant acceleration vectcor, magnitude and direction, and if the magnitude is constant, and the direction is constant, then in sufficient time you will have exceeded light speed.

publius
2009-Apr-20, 07:57 PM
What I meant was a charge under constant proper acceleration from time immemorial until eternity. :) This does not violate SR. One can feel a force of 1g forever. It's just that no reference frame will ever see that charge moving faster than light. Any local co-moving reference frame will see the coordinate acceleration at 1g always. Just give it enough time to that original co-moving frame and it will only asymptotically approach c.

And that's why the radiation expressions such as the Abraham-Lorentz force needs relativisitic modification (I forget what the actual expressions are, but they get complicated). If a charge is already moving close to 'c', and begins accelerating at a proper acceleration of 1g, it looks very different. But transform to a co-moving frame, and it looks normal, with Abraham-Lorentz applying there.

-Richard

publius
2009-Apr-20, 08:21 PM
This is fascinating. I had not heard of this issue before. But I am still having trouble understanding just what the problem really is.


Read over the relevant Jackson sections in detail when you get a chance. IIRC, the calculation in there for an accelerated charge involves an integral over a finite period of acceleration. Constant acceleration there thus means a particle starts accelerating at some rate 'a' at t = 0 and then stops at t = t1. You've got to have those endpoints there, and the calculation won't work from t = -infinity to +infinity (and wouldn't be relativistic anyway)and I think Jackson mentions that fact in the discussion.

That is not uniform acceleration. There are two large "jerks" at t = 0 and t = 1, where da/dt is a delta function, actually.

I don't remember if Jackson bothers to derive the relativistic version of the radiation expressions since they are so complex. But I believe Landau and Lifsh-itz (hyphen to avoid the profanity filter, which really ought to do better than this) do dabble with it in "The Classical Theory of Fields".

-Richard

publius
2009-Apr-20, 08:38 PM
Like I said, I'm clearly ignorant of this Machian idea ...

Mach is so a vague and nebulous that ignorance is sort of a built-in feature, I think. :) One just can't be sure exactly what one is arguing for or against.

-Richard

publius
2009-Apr-21, 01:51 AM
I think this is the paper that claimed to resolve the pathological run-away problems of the radiation reaction:

http://arxiv.org/abs/physics/0508031v3

Here's a page by the another author on this radiation problem which shows the relativistic case:

http://wwwphy.princeton.edu/~kirkmcd/examples/EM/rohrlich_ajp_65_1051_97.pdf

The relativistic formula for the radiation reaction, known as the Abraham-Lorentz-Dirac force, is best expressed in the more elegant
four vector form with the implied .

However, the radiated power expression can be expressed simply and gets a factor of gamma^6 in front of the usual expression. Yes, that's gamma to the sixth power.

According to the above, ol' Heaviside actually derived the relativistic version himself, but since that was before Einstein and SR, he didn't know what he had. Heaviside, my hero. :)

-Richard

DrRocket
2009-Apr-21, 03:38 AM
I think this is the paper that claimed to resolve the pathological run-away problems of the radiation reaction:

http://arxiv.org/abs/physics/0508031v3

Here's a page by the another author on this radiation problem which shows the relativistic case:

http://wwwphy.princeton.edu/~kirkmcd/examples/EM/rohrlich_ajp_65_1051_97.pdf

The relativistic formula for the radiation reaction, known as the Abraham-Lorentz-Dirac force, is best expressed in the more elegant
four vector form with the implied .

However, the radiated power expression can be expressed simply and gets a factor of gamma^6 in front of the usual expression. Yes, that's gamma to the sixth power.

According to the above, ol' Heaviside actually derived the relativistic version himself, but since that was before Einstein and SR, he didn't know what he had. Heaviside, my hero. :)

-Richard

I have been doing some reading on the classical case. But first things first, and your last phrase raises a question -- When will Oliver reclaim his position from helicopter Ben ? (I actually think highly of both of them, but Heaviside has greater class in a science forum, appropriate to he whom he represents).

On to classical electrodynamics.

Jackson seems to be as good a compact source as any, so for anyone with access to that text wishing to follow, the relevant chapters are 14 (Radiation by Moving Charges) and 17 (Radiation Damping), at least in the second edition. Relevant Wiki articles are
http://en.wikipedia.org/wiki/Larmor_formula
http://en.wikipedia.org/wiki/Abraham-Lorentz_force
and for the (special) relativistic case http://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz%E2%80%93Dirac_force

The Larmor radiation formula (equation 14.22) shows that any accelerating charge radiates and the power of the radiation is proportional to the square of the charge and the square of the magnitude of the acceleration. This formula does not make any assumptions regarding time intervals or periodic motion. So, at least in the classical case it seems that any accelerating charge does indeed radiate, even if the motion is linear.

From the Larmor formula you immediately can conclude on the basis of conservation of energy that acceleration of a charged particle encounters resistance, inertia, directly attributable to electromagnetic radiation. Thus there is an electromagnetic component to the mass of charged particles.

Based on that assessment, Newton's equation F=ma is written with two components (not vector components, but two vector forces of different origin) of force, one externally applied and one due to electromagnetic radiation.

One then writes an integral equation relating the dot product of the radiation force with velocity to the power from the Larmor equation over a time interval. At that point itis noted that if the motion is periodic or if the velocity and acceleration are orthogonal at the end points of the time interval then integration by parts gives a simple expression for the force due to radiation, and that is the radiation force that occurs in the Abraham-Lorentz equation of motion. I think this is the step that you recalled that depends on a finite time interval. This is where jerk comes into play.

That equation is then studied in more detail and a model developed to explain the force in terms of the interaction of a charged particle with its own field, the self-interaction problem. That work is relatively complicated and not entirely satisfactory from a theoretical perspective. There are problems with the assumed charge distribution (it is not just a point any longer) among other things and there is a discussion of the issues. These issues are not resolved, although quantum electrodynamics does provide procedures for performing the calculations.

There seems to be no particular issue with the basic question, in the classical case, of whether or not an accelerated charge radiates, whether the motion is linear or not. It is only when the model asks deeper questions that shortcomings in the theory are revealed.

The issue of a charge undergoing constant proper acceleration in a model based on general relativity is another kettle of fish. I don't know how that is handled or if there is any good explanation.

Jeff Root
2009-Apr-21, 07:57 AM
IIRC, the calculation in there for an accelerated charge involves an
integral over a finite period of acceleration. Constant acceleration
there thus means a particle starts accelerating at some rate 'a' at
t = 0 and then stops at t = t1. You've got to have those endpoints
there, and the calculation won't work from t = -infinity to +infinity ...

That is not uniform acceleration. There are two large "jerks" at
t = 0 and t = 1, where da/dt is a delta function, actually.
Again this is way over my head, but are you certain the jerks are
included in the math? I would expect that they would be ignored.
The motion of the particle before and after the period of interest
is simply unknown, unspecified, and unnecessary. There wouldn't
need to be any jerks at the endpoints because the acceleration
could have been in progress before t = 0 and continue after t = 1.

-- Jeff, in Minneapolis

DrRocket
2009-Apr-21, 02:54 PM
Again this is way over my head, but are you certain the jerks are
included in the math? I would expect that they would be ignored.
The motion of the particle before and after the period of interest
is simply unknown, unspecified, and unnecessary. There wouldn't
need to be any jerks at the endpoints because the acceleration
could have been in progress before t = 0 and continue after t = 1.

-- Jeff, in Minneapolis

Yes jerk is included in the math. You can see it in this Wiki article where it arises as a result of integration by parts, which results in the second derivative of velocity occuring in the integral that defines the Abraham-Lorentz force (for motions that cause the boundary term to vanish, such as periodic motion).

http://en.wikipedia.org/wiki/Radiation_reaction

You can find this same derivation in standard references on electrodynamics -- Classical Electrodynamics by J.D. Jackson for instance, as noted earlier in the thread.

Ken G
2009-Apr-21, 02:57 PM
... for starters, we don't live in a Newtonian universe, so how does this Machian approach incorporate (special) relativity? Specifically from where - in the Machian view - does 'relativistic mass' come from? And does inertia relate to mass-energy, in a particle-antiparticle annihilation, say?As publius points out, just what "Machian" means is not very precise, so we must attach our own meaning. To me, it is simply the idea that spacetime and mass are self-consistently determined, which is right at the heart of GR as per Wheeler's famous description of mass telling spacetime how to curve and spacetime telling mass how to move. That whole idea comes from Mach (and from where-ever he got it, no doubt several Greek philosophers said something like that). Newton's picture had mass and space completely disconnected, where space is just space whether there is mass in it or not, like a chessboard with or without any chess pieces on it. Mach's core idea is that the pieces make the board, even as the board constrains where the pieces can go. So if you ask "what is a chessboard" in the Machian analogy, you'd have to be ready to answer "what are chess pieces", and vice versa. I'd sum up: weak Mach: there is no board without the pieces, strong Mach: there is no board at all.

The reason Mach did not get much traction prior to GR is that Newton had a pretty good argument (long before) that if water knew to bow outward in a rotating bucket, then the relation between actual motion and a geodesic had to be a completely local relationship (publius spoke in similar terms above). But in my view, Newton's argument is not contradictory to Mach's requirement that noninertial motion be referenced globally-- Mach still allows noninertial motion to be a local comparison to the geodesic, he is just saying that the behavior of the geodesic has to be controlled nonlocally, geodesics are "boundary value" problems. I think that's just how GR treats them, which is the extent to which GR might be called Machian. However, there can be other meanings to that term, which GR does not conform to, ergo all the confusion about what is and what is not Machian.


I can see that a spinning black hole could, in principle, be used to test whether we live in a Machian or non-Machian universe, per your post, but then since any spinning black hole would have had a history before it became one, and its spin would be completely determined by its history (assuming nothing got added to it afterwards), there'd be no way to tell, would there?
I think it is likely a fairly technical issue as to how one could observationally tell if geodesics are really globally determined, rather than a local property of spacetime. publius talked about how the local mass, say a star you are orbiting, affects the spacetime and creates strange rotations of inertial axes as you go around (even without any rotation of that mass), but our question here is whether or not those local spacetime effects require a global boundary condition to become deducible. That's where I see the Machian issue coming in-- I see spacetime as a kind of fabric weaved by masses, and to get the local fabric you have to trace the threads all the way back to the loom.



Oh, and I also don't follow how frame dragging isn't inconsistent with Mach; nor, for that matter, gravitational wave radiation (in an inspiral event, where does all the inertia go?).These are likely even more technical questions, and I doubt that anything short of actual GR calculations could lend much insight to them. Ultimately, I don't trust any answer that does not stem directly from a calculation that solves the Einstein equations, especially my own.

Ken G
2009-Apr-21, 03:12 PM
However, you can come at the issue from another direction, energy in the field of a moving charge. Feynman did just that and in The Feynman Lectures on Physics vol 2 you will find chapter 28 --"Electromagnetic Mass". I think you might enjoy reading that chapter.

He shows that 3/4 of the mass of the electron is related to this phenomena.
That's quite interesting, the insights in those books are amazing. Note that real inertial comes from protons, not electrons, so we can agree that this is still a fairly moot issue on the Machian scale, but I agree that this does not make it uninteresting or unimportant. I don't know what publius' "4/3 problem" is, but I can't help wondering if that 4/3 isn't related to the 3/4 you are talking about here.

Mach may or may not be eventually shown to have a point, but I doubt it will be the whole story. I would be willing to make a Hawking-style bet on that, but chances of either side being able to collect within their lifetime is pretty small -- even if the LHC produces a Higgs particle.I'd never take a bet that says something is the "whole story", to be honest with you-- I believe the evidence, both rational and empirical, is all too clear that reality will forever be vastly more complex and rich with mystery than we will ever completely probe with our limited intelligence/technology.

DrRocket
2009-Apr-21, 03:48 PM
I believe I understand what you are saying, you see deep (even philosophical) importance vested in conservation laws (as evidenced by that "law" word), but mere idealizations should not rise to the level of a "principle" in your view. So you would say that a star obeys a "law" of gravity, but if we stipulate that the star is a sphere, we are merely making an idealization. You would probably not assert a "principle" that stars are spherical, expressly because they are not. However, life is not really so cut-and-dried, because stars don't "obey" laws of gravity either. We already know that Newton's laws are false for stars (though they are great idealizations, especially if the star is spherical!), but we use them anyway, because we understand the errors involved and find them to be small in many contexts of interest-- and we can't use GR unless we make other idealizations as well because we can't solve the equations.


Basically yes. But perhaps here is a more structured way to look the issues.

Let's take Newtonian mechanics (works the same way for relativity or quantum mechanics, or electrodynamics or ...)for instance. Setting aside the experimental basis, what you have at the level of the theory is a set of postulates that then serve to allow the application of the theory to a set of physical situations. Newton has his three laws. Those I see as principles. Items that can derived rigorously in the form of mathematical proofs, such as conservation of momentum, can also be viewed as principles, since they have precisely the same logical validity as do the original postulates. Everything else (like spherical stars or spherical cows) is just idealization and approximation that permit one to solve the equations that result from application of the principles to real problems of interest. Rigid bodies are an idealization, but certainly not a principle.

Cosmology is not a fundamental theory, but an application of existing theories, most notably general relativity. (Similarly solid state physics is not a fundamental theory, but an application of quantum mechanics.) The cosmological principle is not a principle in the sense that I have described, but an idealization used to simplify and make tractable the models that are based on general relativity. One might decide that the cosmological principle was flawed, in which case a lot of models and their implications would have to be abandoned, but cosmology would still go one. But if you decided that the equivalence principle of general relativity was untenable, then the theoretical structure would come crashing down. The equivalence principle is a real principle.

What I call principles are not so labeled because of "truth" in any absolute sense (Newton's laws are wrong for instance, as Einstein revealed) but because of their logical relation to the fundamental theories on which physics is built. In discussing physics, I find that first one needs to make clear the context, aka basic theory, within the confines of which the discussion is to be held, and that context then defines the set of "principles".

Principles are to physics as postulates are to mathematics. The only difference is that one attempts to support principles with experimental data, while postulates are accepted as merely acceptable on an intuitive basis and then serve as the basis for further development through pure logic. Euclidean geometry, for instance is built on a set of postulates that can be either accepted or rejected. Other geometries are needed for general relativity.

Newton's laws are rather like Euclid's axioms -- you can accept them and develop from that basis quite a useful model. You can adopt other principles and describe situations that are not amenable to description within Newtonian mechanics. But the basic principles, within the bounds of the theory, are inviolate. Most importantly, all that you need to develop the theory is those principles. You don't, in principle, need to know anything else. Anything that logically follows from the principles is also "true" within the theory, and can be used in combination with experiment to define the domain of validity of the theory from a "practical" viewpoint.

I think that defining the context is critical. We don't have any absolute knowledge of physics, all that we have are a few pretty good, but not perfect, theories. We know that those theories are not "true" in an absolute sense, wo we can only discuss approximations within the limitation imposed by those models -- the alternative is a free-for-all that would open the discussion to all sorts of "woo woo" ideas and magic.

So I am perfectly happy to discuss a problem in mechanics within the limitations of Newtonian mechanics, or special relativity or general relativity so long as it is clear from the discussion which model is being applied. If you use some other theory that is OK too, but I will then be rather skeptical as to the utility, since one has at that point abandoned the rather large experimental and theoretical basis supporting the established theories -- and when you get to the "Einstein was not too bright, but I have the truth" theories I will exit the discussion, and as I exit I will not that your theory has NO accepted principles on which to base a discussion.

When you start to talk about theories on the frontiers of research, then the issue becomes what the underlying principles really are, what the implications may be, and whether the predictions derived from the principles agree with experiment and whether they agree with the predictions of established theories within the known domains of validity.

Nereid
2009-Apr-21, 07:26 PM
As publius points out, just what "Machian" means is not very precise, so we must attach our own meaning. To me, it is simply the idea that spacetime and mass are self-consistently determined, which is right at the heart of GR as per Wheeler's famous description of mass telling spacetime how to curve and spacetime telling mass how to move. That whole idea comes from Mach (and from where-ever he got it, no doubt several Greek philosophers said something like that). Newton's picture had mass and space completely disconnected, where space is just space whether there is mass in it or not, like a chessboard with or without any chess pieces on it. Mach's core idea is that the pieces make the board, even as the board constrains where the pieces can go. So if you ask "what is a chessboard" in the Machian analogy, you'd have to be ready to answer "what are chess pieces", and vice versa. I'd sum up: weak Mach: there is no board without the pieces, strong Mach: there is no board at all.

The reason Mach did not get much traction prior to GR is that Newton had a pretty good argument (long before) that if water knew to bow outward in a rotating bucket, then the relation between actual motion and a geodesic had to be a completely local relationship (publius spoke in similar terms above). But in my view, Newton's argument is not contradictory to Mach's requirement that noninertial motion be referenced globally-- Mach still allows noninertial motion to be a local comparison to the geodesic, he is just saying that the behavior of the geodesic has to be controlled nonlocally, geodesics are "boundary value" problems. I think that's just how GR treats them, which is the extent to which GR might be called Machian. However, there can be other meanings to that term, which GR does not conform to, ergo all the confusion about what is and what is not Machian.

That makes sense, sorta.

Is it possible, then, to describe or define a version of Mach that is sufficiently precise and unambiguous as to allow an objective determination of whether falsifiable hypotheses can be derived from it?


I think it is likely a fairly technical issue as to how one could observationally tell if geodesics are really globally determined, rather than a local property of spacetime. publius talked about how the local mass, say a star you are orbiting, affects the spacetime and creates strange rotations of inertial axes as you go around (even without any rotation of that mass), but our question here is whether or not those local spacetime effects require a global boundary condition to become deducible. That's where I see the Machian issue coming in-- I see spacetime as a kind of fabric weaved by masses, and to get the local fabric you have to trace the threads all the way back to the loom.

These are likely even more technical questions, and I doubt that anything short of actual GR calculations could lend much insight to them. Ultimately, I don't trust any answer that does not stem directly from a calculation that solves the Einstein equations, especially my own.
OK ... accepting, for now, that that is what would be required, is it worth making the effort to find out? As in, per your earlier post, we could - in principle - show that the universe behaves as if a narrower rule were in play than GR.

And independent of any such effort, does taking a Machian view help to do astrophysics or cosmology? In the sense, perhaps, that it is easier to think about certain kinds of problems assuming Mach than not.

DrRocket
2009-Apr-21, 07:44 PM
That makes sense, sorta.

Is it possible, then, to describe or define a version of Mach that is sufficiently precise and unambiguous as to allow an objective determination of whether falsifiable hypotheses can be derived from it?

This was my earlier point. "Mach's Principle" seems to have no clear statement, nothing sufficiently precise so asl to allow quantification or falsification. Given that lack of precision, why not simply stipulate that Einstein may have been influenced by Mach and whatever might be meant by "Mach's Principle" that is clear enough for debate is incorporated into GR and then base the debate on GR. It seems to me that the alternative is either a clear formulation of Mach's principle in falsifiable terms that results in something different from GR or some sort of mushy argument that will not lead anywhere.

"There's no sense in being precise when you don't even know what you're talking about." -- John von Neumann

Ken G
2009-Apr-21, 07:50 PM
Items that can derived rigorously in the form of mathematical proofs, such as conservation of momentum, can also be viewed as principles, since they have precisely the same logical validity as do the original postulates. Yes, I agree that logical connections between postulates and theorems afford the theorem with the same status as the postulate, but what is that status? If someone says "let the Sun be a sphere, then..." I see that as not much different from "let this system be closed, then we have conservation laws...". We still pick the postulates we want, and any postulate can be labeled an arbitrary idealization. The accuracy of the assumptions can vary greatly, and we're still doing the same exercise, as long as the accuracy is tailored to some useful outcome. I think we just get so enamored of the fact that in certain situations we can get results that are so amazingly accurate, we simply forget that in most real situations, we cannot. So where is our justification for claiming that "nature obeys laws"? Rather, we obey laws when we study nature, and those are the laws of science.


Everything else (like spherical stars or spherical cows) is just idealization and approximation that permit one to solve the equations that result from application of the principles to real problems of interest. Rigid bodies are an idealization, but certainly not a principle.Yes, the idealization is not the same as the principle, it is the requirement for the principle. Yet all principles come with their associated idealizations. There is no principle that does not require an idealization, they are like two halves of the same coin.

The cosmological principle is not a principle in the sense that I have described, but an idealization used to simplify and make tractable the models that are based on general relativity. That is possible, it really depends on how the situation got the way it did. If we say that a star is a sphere, and this is not a principle but an idealization so that we can apply principles that require spheres (like point gravity and so forth), then we are begging the question of why stars are spheres in the first place. We can just say that this is an externally imposed condition that we observe, and call it an idealization rather than a principle, or we can say that there is a reason stars are spheres, and that reason is based in principles. You might then say that's fine, then identify the principle that makes them spheres. I would look to the same challenge for the "cosmological principle". So I agree that if all we want to do is say the universe looks isotropic and homogeneous so let's treat it as such, that should be the "cosmological idealization". But if we expect that no universe would look like that unless there was a reason, then the "cosmological principle" is whatever principle makes the cosmological idealizaton work. If we have not yet identified that principle, then you are right that we should admit we have not and not claim that we have (and "cosmological principle" suggests that we have). But I would only look to that term as a challenge to identify what that principle actually is-- and that's where I look to Machian ideas (to bring it back to the thread). The universe exhibits the symmetry it does because to not exhibit that symmetry would require a breaking of the symmetry by the mass distribution, but that is the very thing that requires the symmetry be broken to do. Space follows mass, mass follows space-- chicken and egg, cosmological principle Mach style.


But if you decided that the equivalence principle of general relativity was untenable, then the theoretical structure would come crashing down. More likely, it would just undergo violent readjustment, but so would cosmology if the "cosmological principle" did not hold. Still, I see value in distinguishing what is a principle from what is an idealization based purely on not knowing what else to do, so I think that's your main point here.

publius
2009-Apr-21, 07:54 PM
Ken,

The "4/3 problem" is a curious result of attempts at classically defining the electromagnetic mass. I forget the details, but you end with something like

E = 4/3 mc^2, (or the reverse E = 3/4 mc^2, I forget).

It is usually expressed as the 4/3 problem because it is typically written with the 4/3 factor rather than 3/4.

That is, you trying to define the mass/inertia of an electron or other fundamental "point particle" entirely in terms of classical electromagnetic field and soon discover it doesn't work.

Again, I forget all the details and we'd need someone up to speed to explain it, but QED does better, but still doesn't completely resolve it. "Self-energy" is a thorny problem.

-Richard

DrRocket
2009-Apr-21, 07:57 PM
That's quite interesting, the insights in those books are amazing. Note that real inertial comes from protons, not electrons, so we can agree that this is still a fairly moot issue on the Machian scale, but I agree that this does not make it uninteresting or unimportant. I don't know what publius' "4/3 problem" is, but I can't help wondering if that 4/3 isn't related to the 3/4 you are talking about here.
I'd never take a bet that says something is the "whole story", to be honest with you-- I believe the evidence, both rational and empirical, is all too clear that reality will forever be vastly more complex and rich with mystery than we will ever completely probe with our limited intelligence/technology.

The "4/3 problem" noted by publius is discussed on Jackson in connection with the Abraham-Lorentz model (it on page 795 following equation 17.48 in the second edition if you care to try to find it) and is associated with the relativistic model.

OOPS -- looks like publius supplied the answer while I was typing this.

Ken G
2009-Apr-21, 08:09 PM
Is it possible, then, to describe or define a version of Mach that is sufficiently precise and unambiguous as to allow an objective determination of whether falsifiable hypotheses can be derived from it?My answer to that would be that one need not leave general relativity to do it, it already appears in the form of the boundary conditions we choose. It all begins even in Newtonian gravity, where to do a problem involving the gravitational potential energy, you have to make an assumption about the potential at infinity (it's uniform). Make a different assumption, and you get a different solution to the same laws of gravity-- it's really just an ultraweak form of Machian thinking. I suspect general relativity must do similar things with the boundary conditions, and that's where one invokes the cosmological principle. So I think you are asking, do we invoke that principle because it is Machian, or just because it works and we haven't any necessary reason to do anything else? To that I'd say the latter, but it is nevertheless useful to have a philosophical basis behind the postulates that work. It might be entirely pedagogical, but pedagogies have an important place in understanding our own physics.



OK ... accepting, for now, that that is what would be required, is it worth making the effort to find out? As in, per your earlier post, we could - in principle - show that the universe behaves as if a narrower rule were in play than GR.The universe does behave as if a narrower rule were in play than GR, because GR is not a complete description. It always requires the manual imposition of boundary conditions in space and time, as did Newtonian physics. That's the main difference between a theory of dynamics, and more ancient styled theories based in concepts of things having a "proper state of being" and so forth. Dynamics begins with the recognition that we can never get the behavior entirely out of the laws, we need to append empirical or rational external constraints.


And independent of any such effort, does taking a Machian view help to do astrophysics or cosmology? In the sense, perhaps, that it is easier to think about certain kinds of problems assuming Mach than not.Yes, that is generally all one can really hope to get from philosophical perspectives, along with inspiration for where to turn for new ideas when something isn't working.

DrRocket
2009-Apr-21, 08:14 PM
Yes, I agree that logical connections between postulates and theorems afford the theorem with the same status as the postulate, but what is that status? If someone says "let the Sun be a sphere, then..." I see that as not much different from "let this system be closed, then we have conservation laws...". We still pick the postulates we want, and any postulate can be labeled an arbitrary idealization. The accuracy of the assumptions can vary greatly, and we're still doing the same exercise, as long as the accuracy is tailored to some useful outcome. I think we just get so enamored of the fact that in certain situations we can get results that are so amazingly accurate, we simply forget that in most real situations, we cannot. So where is our justification for claiming that "nature obeys laws"? Rather, we obey laws when we study nature, and those are the laws of science.

But it is not true that "in real situations we cannot". What is true is that in real situation we need sophisticated models with sophisticated approximations and numerical models to get very accurate answers.

Often the idealized models are pedagogical. We assume the earth is a uniform sphere to that we can apply Newtonian gravitation and treat the earth as a single point. That is done so that we can solve the equation with pencil and paper.

But when you are sending an ICBM to a target you don't model the earth as a uniform sphere, and the gravitational field is not spherically symmetric. So called gravitational "harmonics" are included in the detailed model to provide the required accuracy. We can send warheads to the lagoon at Kwajalein with astounding accuracy, in part because the gravitational field over that trajectory is rather well known and and the models are very accurate.

The accuracy of those models is the result of the use of what I am calling principles.


If we say that a star is a sphere, and this is not a principle but an idealization so that we can apply principles that require spheres (like point gravity and so forth), then we are begging the question of why stars are spheres in the first place. We can just say that this is an externally imposed condition that we observe, and call it an idealization rather than a principle, or we can say that there is a reason stars are spheres, and that reason is based in principles.

We actually have both. We know why stars are approximately spherical -- that geometry minimizes the system potential energy. An the minimization of the system energy is related to fundamental principles. That principle also finds application in finite-element models which permit the very accurate solution of the equations of elasticity and is the basis of modern structural analysis -- again difficult equations are solved as a result of application of fundamental principles and equally sophisticated approximations and numerical methods.

On the other hand in classroom problems the star is assumed to be spherical because the symmetry makes it possible to solve some illustrative example problems with simple techniques. And because stars really are reasonably close to spherical.

A spherical cow is somewhat different.



More likely, it would just undergo violent readjustment, but so would cosmology if the "cosmological principle" did not hold. Still, I see value in distinguishing what is a principle from what is an idealization based purely on not knowing what else to do, so I think that's your main point here.

Basically yes. And I think that it is not hard to reach agreement as a practical matter as to which is which.

Ken G
2009-Apr-21, 08:15 PM
It is usually expressed as the 4/3 problem because it is typically written with the 4/3 factor rather than 3/4.

That is, you trying to define the mass/inertia of an electron or other fundamental "point particle" entirely in terms of classical electromagnetic field and soon discover it doesn't work. OK, so it does sound related to DrRocket's reference to Feynman's electromagnetic mass of a point particle.

publius
2009-Apr-21, 08:19 PM
Actually, I remember reading something, generally well over my head (the author did lots of the fancy high powered tensor 'rithmetic) that one can let Mach live in the boundary conditions. But the objection to that is apparently that Mach in the boundary is too different from what Mach apparently intended, that what you're really doing is redefining Mach to match what you have, rather than what you have following from Mach. Or something. :)


-Richard

DrRocket
2009-Apr-21, 08:36 PM
OK, so it does sound related to DrRocket's reference to Feynman's electromagnetic mass of a point particle.

Yes, same thing, with more detail in Jackson, as noted (Jackson actually has an entire chapter on radiation damping).

It gets worse when you throw in considerations o self-energy, since the self-energy of a point charge is infinite (at least classically). I don't think electrons are quite THAT heavy.

Feynman wrote that piece in large part to give his freshmen an example of a problem of some import that is not settled -- giving them a flavor of research issues. As usual, his presentation is masterful. It is well worth the time required to read it.

DrRocket
2009-Apr-21, 08:44 PM
Actually, I remember reading something, generally well over my head (the author did lots of the fancy high powered tensor 'rithmetic) that one can let Mach live in the boundary conditions. But the objection to that is apparently that Mach in the boundary is too different from what Mach apparently intended, that what you're really doing is redefining Mach to match what you have, rather than what you have following from Mach. Or something. :)


-Richard

What boundary ? How can you do that without, at least, knowing the topology of the universe ? If the space-like slices are spheres, then finding the boundary would be a bit of a problem, no ?

In any case, my point in bringing up electromagnetic mass, is that whatever might be an eventual explanation for mass and intertia it will have more than a single aspect. There has to be more to is than just the relationship with mass distribution. Charge has to be in the picture somewhere.

publius
2009-Apr-22, 02:38 AM
What boundary ? How can you do that without, at least, knowing the topology of the universe ? If the space-like slices are spheres, then finding the boundary would be a bit of a problem, no ?


That's a good question. We don't know what the topology of the universe is. We know a bit about what we can observe, but what the whole thing is, well, we may never know.

What is meant by boundary conditions, in the context of the EFE, is pretty darn complex and goes beyond what you're thinking -- some notion of a space-like boundary. Space and time are mixed. When we think of a time constraint, we usually call that "initial conditions", and we think of spatial constraints, we call them boundary conditions. When dealing with space-time, you have both mixed together, so "boundary conditions" mean both, actually, and a mixture of both.

The EFE represent a humdinger of a set of coupled ellipitcal partial differential equations. And all such PDE's depend on boundary and initial conditions to determine a particular solution.

Consider the simple wave equation in one spatial dimension. Any functions f(x - ct) + g(x + ct) solves it. The initial and/or boundary conditions (which you use depends on the circumstance). Suppose you have fixed length string -- you use a combination of boundary conditions (must be zero at the endpoints) plus an initial condition of the shape of the string at
t = 0. There, you only have a series of allowed wavelengths/frequencies that fit on the string.

Without a boundary, an infinite length string, you specify initial conditions of the form (let u(x, t) be the desired wave function):

u(x, 0) = f(x)
du/dt(x, 0) = g(x) (d/dt represents the partial derivative here)

The solution is the then:

u(x, t) = 1/2[f(x -ct) + f(x + ct)] + 1/2c[ G(x + ct) - G(x - ct),

where G(y) is the integral of g(y). Add a source term, some s(x, t) for an inhomogenous equation and you get the above plus a double integral form of
s(x, t)dx*dt, which students of EM should immediately recognize as leading to the retarded (and advanced) potential formula. This is reminiscent of the solution to an inhomogenous ordinary differential equation, except rather than two arbitrary constants, we get two arbitrary whole functions.

The initial and boundary forms are somewhat different in style, but they are basically the same thing, constraints placed on the solution that allow us to choose one particular solution of many (infinite, in many cases) solutions that solve the equation. And besides those two forms, strict boundary vs strict initial, you can have a variety of constraints on the function and its derivatives at some points/regions of x or t.

The "laws" give us the equations to solve. But is the boundary/initial conditions that actually let us pin things down.

And likewise with the EFE. You have to impose some additional constraints to get a particular space-time solution.

Even the "empty space-time" Minkowksi solution of the EFE requires boundary-initial conditions on the EFE. Likewise the deSitter Lambda vacuum solution also requires boundary-initial conditions. And any FRLW solution requires them as well.

And in any event, you're solving equations to model what you can see. If that shoe fits, good, but you still have no clue about how well it fits what you can't see.


-Richard

DrRocket
2009-Apr-22, 05:36 AM
That's a good question. We don't know what the topology of the universe is. We know a bit about what we can observe, but what the whole thing is, well, we may never know.

What is meant by boundary conditions, in the context of the EFE, is pretty darn complex and goes beyond what you're thinking -- some notion of a space-like boundary. Space and time are mixed. When we think of a time constraint, we usually call that "initial conditions", and we think of spatial constraints, we call them boundary conditions. When dealing with space-time, you have both mixed together, so "boundary conditions" mean both, actually, and a mixture of both.

The EFE represent a humdinger of a set of coupled ellipitcal partial differential equations. And all such PDE's depend on boundary and initial conditions to determine a particular solution.

Consider the simple wave equation in one spatial dimension. Any functions f(x - ct) + g(x + ct) solves it. The initial and/or boundary conditions (which you use depends on the circumstance). Suppose you have fixed length string -- you use a combination of boundary conditions (must be zero at the endpoints) plus an initial condition of the shape of the string at
t = 0. There, you only have a series of allowed wavelengths/frequencies that fit on the string.

Without a boundary, an infinite length string, you specify initial conditions of the form (let u(x, t) be the desired wave function):

u(x, 0) = f(x)
du/dt(x, 0) = g(x) (d/dt represents the partial derivative here)

The solution is the then:

u(x, t) = 1/2[f(x -ct) + f(x + ct)] + 1/2c[ G(x + ct) - G(x - ct),

where G(y) is the integral of g(y). Add a source term, some s(x, t) for an inhomogenous equation and you get the above plus a double integral form of
s(x, t)dx*dt, which students of EM should immediately recognize as leading to the retarded (and advanced) potential formula. This is reminiscent of the solution to an inhomogenous ordinary differential equation, except rather than two arbitrary constants, we get two arbitrary whole functions.

The initial and boundary forms are somewhat different in style, but they are basically the same thing, constraints placed on the solution that allow us to choose one particular solution of many (infinite, in many cases) solutions that solve the equation. And besides those two forms, strict boundary vs strict initial, you can have a variety of constraints on the function and its derivatives at some points/regions of x or t.

The "laws" give us the equations to solve. But is the boundary/initial conditions that actually let us pin things down.

And likewise with the EFE. You have to impose some additional constraints to get a particular space-time solution.

Even the "empty space-time" Minkowksi solution of the EFE requires boundary-initial conditions on the EFE. Likewise the deSitter Lambda vacuum solution also requires boundary-initial conditions. And any FRLW solution requires them as well.

And in any event, you're solving equations to model what you can see. If that shoe fits, good, but you still have no clue about how well it fits what you can't see.


-Richard

I understand boundary conditions in general and I understand what you are saying in the generality in which you presented it as well.

However, if Mach's principle is taken to mean that what we perceive as mass or inertia of objects in space-time is the result of the entire mass distribution within space-time, then I don't see how boundary conditions can apply only to that bit of space-time that we can "see" -- which I suppose would limit things to what physicists call a coordinate sytem and what mathematicians would call a chart. It would have to apply to the whole enchilada. And that is where I am having trouble with burying Mach's principle in the boundary conditions, and the lack of any knowledge of a boundary. If the manifold turns out to be closed there is no boundary. If the manifold turns out to be open then it is still a manifold without boundary but you might make do with asymptotic conditoins that a physicist might call a boundary at "infinity".

On the other hand Mach's principle is so nebulous that perhaps there is some meaning that I am missing that would allow a treatment locally, and a chart could have a real boundary that might be put to use. Patching those charts together to make sense of a global Mach's principle might be a real trick.

publius
2009-Apr-22, 06:08 AM
But first things first, and your last phrase raises a question -- When will Oliver reclaim his position from helicopter Ben ?

I forgot about this. Oliver will be back when the current crisis is over, or I get tired of fussing over it, or when Helicopter gets fired/replaced (or the world comes to an end). Ben's term is up in 2010, and the rumor is he won't be reappointed and will be replaced by Larry Summers. I will not be putting any pictures of Summers up. Even if you paid me. :)


-Richard

Ken G
2009-Apr-22, 06:25 AM
But the objection to that is apparently that Mach in the boundary is too different from what Mach apparently intended, that what you're really doing is redefining Mach to match what you have, rather than what you have following from Mach. Yeah, that doesn't really bother me, it's not like Mach had some angel whispering in his ear or something. His idea evolved, that's not surprising-- what is so earth-shattering is how vastly different a picture it is to imagine that the geodesics are not served to us from on high, they (or what they represent to our feeble minds) are part of reality and so must connect with all the other real things that cavort across those geodesics. I think that is a very useful insight-- everything that is real comes together in the same package, there's never anything "first" and then something else gets "added later". When we "disentangle" reality we create something that fits better in our heads, but we also replace reality with something much different. I guess you can call me a monist as well as a Machian.

DrRocket
2009-Apr-22, 06:32 AM
I forgot about this. Oliver will be back when the current crisis is over, or I get tired of fussing over it, or when Helicopter gets fired/replaced (or the world comes to an end). Ben's term is up in 2010, and the rumor is he won't be reappointed and will be replaced by Larry Summers. I will not be putting any pictures of Summers up. Even if you paid me. :)
-Richard

That seems reasonable, and tasteful.

http://tbn2.google.com/images?q=tbn:RmCeIMoWCAcbUM:http://exiledonline.com/wp-content/uploads/2008/11/dr-lawrence-h-summers-82.jpg (http://images.google.com/imgres?imgurl=http://exiledonline.com/wp-content/uploads/2008/11/dr-lawrence-h-summers-82.jpg&imgrefurl=http://exiledonline.com/larry-summers-a-suicidal-choice/&usg=__LkZoZ_5izfFdU20jKgD9xXKmH0U=&h=768&w=511&sz=55&hl=en&start=2&um=1&tbnid=RmCeIMoWCAcbUM:&tbnh=142&tbnw=94&prev=/images%3Fq%3Dlarry%2Bsummers%2Bobama%26hl%3Den%26s a%3DN%26um%3D1)

It may be difficult to figure out precisely when the crisis is over though. If it ends when my net worth returns to 2007 levels, then I fear that I may not survive to see it.

And Oliver has so much class, that his absense is felt (although I actually do like Ben).

http://tbn0.google.com/images?q=tbn:do7MBLqi7g3vJM:http://www.oliverheaviside.com/oliver-portrait-selection.jpg (http://images.google.com/imgres?imgurl=http://www.oliverheaviside.com/oliver-portrait-selection.jpg&imgrefurl=http://www.oliverheaviside.com/&usg=__TJQOy0sUXKowFXpqco8BTaUzNv0=&h=312&w=246&sz=20&hl=en&start=6&um=1&tbnid=do7MBLqi7g3vJM:&tbnh=117&tbnw=92&prev=/images%3Fq%3DOliver%2BHeaviside%26hl%3Den%26sa%3DG %26um%3D1)

Now THERE'S a boundary condition.

DrRocket
2009-Apr-22, 06:39 AM
Yeah, that doesn't really bother me, it's not like Mach had some angel whispering in his ear or something. His idea evolved, that's not surprising-- what is so earth-shattering is how vastly different a picture it is to imagine that the geodesics are not served to us from on high, they (or what they represent to our feeble minds) are part of reality and so must connect with all the other real things that cavort across those geodesics. I think that is a very useful insight-- everything that is real comes together in the same package, there's never anything "first" and then something else gets "added later". When we "disentangle" reality we create something that fits better in our heads, but we also replace reality with something much different. I guess you can call me a monist as well as a Machian.

I have no idea what it is that you have said here.

Seriously, I read this three times and am clueless.

I don't even know what it is that I don't understand.

Ken G
2009-Apr-22, 06:40 AM
There seems to be a lot of worry about how "nebulous" Mach's principle is, but in my eyes, it's actually rather straightforward in principle (not in practice, that's GR boundary conditions, and as publius points out, that's some of the hardest stuff there is). Take Newton's bucket. If you only spin the bucket, we know what happens from experience-- the water bulges. A deep symmetry holds that only spinning the bucket must be the same as only spinning everything but the bucket in the opposite direction. Newton says that symmetry doesn't exist, it is broken by the "absolute reality" of space. Mach says that symmetry is not broken, because it is space that doesn't exist and so cannot break that symmetry. Ergo, spinning everything in the universe about the bucket, but not the bucket itself, should still cause the water in the bucket to bulge. That is a question for GR, and perhaps has an answer in GR, though I wouldn't know how to calculate it and perhaps it is still debated (based on appropriate boundary conditions and whatnot).

Another way to look at it is to now spin both the bucket and everything else in the universe around the same axis. Again Newton argues that space itself "knows" about that rotation, Mach says there is nothing that can know about that rotation, as space is nothing. Ergo, if you spin both the bucket and everything else, there's no bulging of the water. Those inclined to side with Mach simply ask, how would we ever know if the bucket and everything else "really are" spinning? It seems to me like one of those examples of reality being "less" than we imagine it is, expressly because there are whole equivalence classes of behaviors that we could never tell are happening so we erroneously imagine that we can tell they are not.

Ken G
2009-Apr-22, 06:46 AM
I don't even know what it is that I don't understand.Newton's idea was that space comes first. It's just there. Then you can imagine adding mass into space, and nothing happens to the space because it was already there. What I'm saying is, that is a technique we often use to help us picture things-- we "layer" our understanding, imagining that we can usher in the various aspects of reality one at a time, like painting a wall with multiple coats. But the "monist" view of reality is that reality is basically just one thing, and any effort to break it up into pieces will replace it with something it is not. It may have value to do it, and indeed it does have value to do it (it is almost the defining feature of all physics outside of GR), but something important about reality may be lost in translation when one does that. Specifically, if "monism" is right, it seems likely that grand unification will look more like GR (with its unification of space, time, and stress-energy) rather than like quantum mechanics (where time is a parameter, space is an observable, and energy is the generator of time evolution). String theory unifies the particles, but it takes the quantum mechanical approach of slicing time, space, and energy to ribbons (so far as I know about string theory).

DrRocket
2009-Apr-22, 07:10 AM
There seem to be a lot of worry about how "nebulous" Mach's principle is, but in my eyes, it's actually rather straightforward in principle (not in practice, that's GR boundary conditions, and as publius points out, that's some of the hardest stuff there is). Take Newton's bucket. If you only spin the bucket, we know what happens from experience-- the water bulges. A deep symmetry holds that only spinning the bucket must be the same as only spinning everything but the bucket in the opposite direction. Newton says that symmetry doesn't exist, it is broken by the "absolute reality" of space. Mach says that symmetry is not broken, because it is space that doesn't exist and so cannot break that symmetry. Ergo, spinning everything in the universe about the bucket, but not the bucket itself, should still cause the water in the bucket to bulge. That is a question for GR, and perhaps has an answer in GR, though I wouldn't know how to calculate it and perhaps it is still debated (based on appropriate boundary conditions and whatnot).

Another way to look at it is to now spin both the bucket and everything else in the universe around the same axis. Again Newton argues that space itself "knows" about that rotation, Mach says there is nothing that can know about that rotation, as space is nothing. Ergo, if you spin both the bucket and everything else, there's no bulging of the water. Those inclined to side with Mach simply ask, how would we ever know if the bucket and everything else "really are" spinning? It seems to me like one of those examples of reality being "less" than we imagine it is, expressly because there are whole equivalence classes of behaviors that we could never tell are happening so we erroneously imagine that we can tell they are not.

OK this I follow. And this is Mach as I understand Mach. If I understand Mach.

But I could never get myself too excited about it.

I agree that spinning a bucket in a universe that contains nothing but the bucket could be a surprising exercise. For two reasons. First, because I agree that you can't be just spinning, you have to be spinning with respect to some reasonable reference frame -- something else. Second, because just producing the experimental arrangement to conduct this excercise will be quite a trick. Even David Copperfield would be hard pressed to make everything, litterally everything, escept the bucket disappear. Spinning the universe about the bucket does not strike me as much easier. So it is a nice thought experiment, but not much more than that.

Newton's mechanics sidesteps the question neatly. The bucket buldges if it is spinning with respect to an inertial reference frame, which is the only reference frame in which Newton's mechanics holds. This is nice and circular because there is no objective means of identifying a Newtonian reference frame. I personally doubt that one exists, and not because of the transcendence of relativity.

Special relativity is not much better. The setting there is still a global intertial reference frame, and there is no more objective criteria for finding one than in the case of Newtonian mechanics.

The only salvation seems to be general relativity, which at least permits one to use local reference frames that might actually exist. But here I think, and perhaps Publius can confirm, that the theory is set up so as to conform to Mach's principle. But the experimental confirmation of general relativity only applies in THIS universe, which contains more than the bucket, so the reasoning is again a bit circular. What you do seem to have is a theory that explains our universe and also is consistent with Mach's thought experiment. So, it would be possible to design a universe in which Mach's experiment turns out as Mach supposed it might, I'll keep that in mind if I happen to become a god and need to desisgn a universe. I do know a lot of people who, for religious reasons, contemplate being in that situation, but I am not one of them.

The question then is whether general relativity is successful because of an influence from Mach's principle (in some form) or whether Mach's principle seems to be successful because of Mach's principle. Since Mach's principle is not one of the formal postulates from which general relativity is derived, I don't think it matters.

DrRocket
2009-Apr-22, 07:21 AM
Newton's idea was that space comes first. It's just there. Then you can imagine adding mass into space, and nothing happens to the space because it was already there. What I'm saying is, that is a technique we often use to help us picture things-- we "layer" our understanding, imagining that we can usher in the various aspects of reality one at a time, like painting a wall with multiple coats. But the "monist" view of reality is that reality is basically just one thing, and any effort to break it up into pieces will replace it with something it is not. It may have value to do it, and indeed it does have value to do it (it is almost the defining feature of all physics outside of GR), but something important about reality may be lost in translation when one does that. Specifically, if "monism" is right, it seems likely that grand unification will look more like GR (with its unification of space, time, and stress-energy) rather than like quantum mechanics (where time is a parameter, space is an observable, and energy is the generator of time evolution). String theory unifies the particles, but it takes the quantum mechanical approach of slicing time, space, and energy to ribbons (so far as I know about string theory).

OK this I understand. It seems to be written in the language that I have learned to speak.

Penrose would agree with you that general relativity is likely to stand up to unification better than the majority seems to think. He is in the minority of course, but I would be rather reluctant to bet against Penrose in a game involving intuition. If you have not read at least a little of his book The Road to Reality I suggest that you find a copy. I think you would enjoy it. It is an astounding book. As near as I can tell, Penrose tried to address, in some fashion and at varying levels, almost everything that he knows about. Penrose knows a lot about a lot of things. It is an astoundingly deep book.

publius
2009-Apr-22, 07:46 AM
This can really make your head hurt. Trouble is getting words out that make sense.

Start out with empty space-time (whatever that is). Plop a spherical mass in it. You get a different space-time, Schwarzschild. Now plop a rotating sphere in it. You get something different.

Now, rotate a reference frame in empty space-time. In that frame, plop sphere. You get the same thing as the rotating sphere above. Now plop a sphere rotating in the opposite direction, you get Schwarzschild (transformed to a rotating frame).

The Machian 64K question is what the devil is the mass or frame really rotating with respect to. If the mass is the only thing to reference against, why the two different solutions.

The answer is the boundary conditions, near as I can tell. "Empty space-time" is a solution to the EFE with zero source and certain boundary conditions. When we add the mass sphere, we're adding another "layer" on top, and that layer interacts and changes things a bit. Rotating or not rotating is relative to those boundary conditions.

Think of it this way. Go "outside" of space-time. Even infinite flat space-time. Just imagine being outside of it. Rotate it, translate it, accelerate it from that outside vantage. That doesn't matter, nothing inside would know the difference, and couldn't know. Your boundary conditions are on, well, the boundary between space-time and the "outside".

Hmmphhh. That all sounds as vague and nebulous as I said Mach was. :lol:

-Richard

Ken G
2009-Apr-22, 08:45 AM
If you have not read at least a little of his book The Road to Reality I suggest that you find a copy. I think you would enjoy it. It is an astounding book. As near as I can tell, Penrose tried to address, in some fashion and at varying levels, almost everything that he knows about.That's an excellent suggestion. I found "Emperor's New Mind" to be interesting but tough sledding, and quite speculative, but The Road to Reality I don't know anything about and it sounds good.

Ken G
2009-Apr-22, 08:57 AM
Go "outside" of space-time. Even infinite flat space-time. Just imagine being outside of it. Rotate it, translate it, accelerate it from that outside vantage. That doesn't matter, nothing inside would know the difference, and couldn't know. Your boundary conditions are on, well, the boundary between space-time and the "outside".
Yes, that's my take on it too, and it sounds all very Machian. Then throw in the Big Bang (for good measure), and there is ample opportunity for the "loom" of what exists in the universe (its mass/energy/stress) to have left its mark in the weave of spacetime. This all relates directly to the crucial question, what is real. I think GR, and Mach, have a lot to say about what is real, or more accurately, what isn't. That's what I mean by monism-- the story of what is real is entirely told by mass/energy/stress, and spacetime is nothing but a language for telling the story. That underpins Mach, it's the reason that space could not have been absolute, any more than a story can exist without the storyteller. Or to put the anthroporphisms where they belong, say that intelligent minds are the storytellers, mass/energy/stress is what the story is about, and spacetime is the prose that carries the meaning of the story. Literary and nebulous, certainly, but it has the ring of truth to me.

DrRocket
2009-Apr-22, 06:26 PM
Think of it this way. Go "outside" of space-time. Even infinite flat space-time. Just imagine being outside of it. Rotate it, translate it, accelerate it from that outside vantage. That doesn't matter, nothing inside would know the difference, and couldn't know. Your boundary conditions are on, well, the boundary between space-time and the "outside".

Hmmphhh. That all sounds as vague and nebulous as I said Mach was. :lol:

-Richard

I don't quite follow this last point. To have an "outside" you would need to have an isometric embedding of the space-time manifold in some larger space. I know that is possible for all Riemannian manifolds (Nash embedding theorem) but I don't know if there is a version for Lorentzian manifolds that guarantees any such manifold can be realized via an embedding in some simple space. You could always embed topologically without worrying about the metric (Whitney embedding theorem).

But suppose that somehow space-time is realized as an embedded manifold. The usual balloon analogy, realizes a sphere as an embedded manifold in 3-space for instance. Now, take the boundary of that embedded manifold, as a subspace of the space in which it is embedded. You may well, as for the balloon, find that the boundary is the entire manifold -- as is the case for the balloon in 3-space. That boundary has nothing to do with the notion of the boundary relative to the manifold structure, which may well be empty, as is the case for the balloon.

Simpler example: Consider an open line segment embedded in 2-space. For instance the open interval (0,1) of the x-axis in the plane. The boundary in that setting is the closed interval [0,1]. But viewed as a manifold the boundary would be only the end points. In this case the boundary is the entire manifold, plus two more points. This puts the idea of "boundary conditionis" in a rather strange light.

DrRocket
2009-Apr-22, 06:32 PM
That's an excellent suggestion. I found "Emperor's New Mind" to be interesting but tough sledding, and quite speculative, but The Road to Reality I don't know anything about and it sounds good.

It is a terrific book. But if you thought "Emperor" was tough sledding, you ain't seen nothing yet. The good news is that it can be read without attempting to understand all of the subtleties and details. I doubt anyone really understands all of them, likely including Penrose himself. A lot of the details simply are not there, but the outline of the concepts is. It is a tour de force from a brilliant and unconventional mind.

It is rather speculative, but it is responsible speculation. You might get a feel for the breadth and level of speculation from the full title:The Road to Reality, a Complete Guide to the Laws of the Universe. Needless to say, there are a few questions that remain unanswered.

publius
2009-Apr-22, 07:42 PM
I don't quite follow this last point. To have an "outside" you would need to have an isometric embedding of the space-time manifold in some larger space. I know that is possible for all Riemannian manifolds (Nash embedding theorem) but I don't know if there is a version for Lorentzian manifolds that guarantees any such manifold can be realized via an embedding in some simple space. You could always embed topologically without worrying about the metric (Whitney embedding theorem).


Embedding is a very complex subject and you'd need a real expert, a real high priest, to explain it. My understanding is you can do for psuedo-Riemannian manifolds, but you'll generally need more time-like and well as space-like dimensions, which sort of blows your mind. For example, simple Schwarzschild requires a (2, 4) embedding space, one more time and one more space dimension (but, curiously, if you supress the other two spatial dimensions and consider only the radial dimensions with time, you can embed that in (1, 2) flat space-time).

But any rate, I'm not even thinking of proper embedding spaces, although you could think of it that way. I'm just conceiving the inconceivable :), something "outside" of everything. No matter what you think of, I'll think of something being outside of it.

So construct a proper embedding space. I can imagine even higher dimensions than that, even letting them go to infinite number. An infinite number of both time and space dimensions. And even then, I can still conceive of something outside even that! :lol:

It's a land with perhaps magic unicorns and lollipops growing like daffodils which has whatever properties are required to contain as a subset whatever we can conceive of.

-Richard

DrRocket
2009-Apr-22, 08:18 PM
Embedding is a very complex subject and you'd need a real expert, a real high priest, to explain it. My understanding is you can do for psuedo-Riemannian manifolds, but you'll generally need more time-like and well as space-like dimensions, which sort of blows your mind. For example, simple Schwarzschild requires a (2, 4) embedding space, one more time and one more space dimension (but, curiously, if you supress the other two spatial dimensions and consider only the radial dimensions with time, you can embed that in (1, 2) flat space-time).

But any rate, I'm not even thinking of proper embedding spaces, although you could think of it that way. I'm just conceiving the inconceivable :), something "outside" of everything. No matter what you think of, I'll think of something being outside of it.

So construct a proper embedding space. I can imagine even higher dimensions than that, even letting them go to infinite number. An infinite number of both time and space dimensions. And even then, I can still conceive of something outside even that! :lol:

It's a land with perhaps magic unicorns and lollipops growing like daffodils which has whatever properties are required to contain as a subset whatever we can conceive of.

-Richard

I don't have a problem with high-dimensional spaces for the embedding. The Whitney and Nash embeddings generally take a space of minimum dimension quite a bit higher than that of the manifold being embedded. I am not surprised that in the pseudo-Riemannian case there is also the signature of the metric to be considered and that it has effect on the minimal embedding dimension. (I just found a reference, but don't have access to the full article http://rspa.royalsocietypublishing.org/content/314/1518/417.abstract). I don't even have a problem with infinite-dimensional spaces, although you have to be careful with the topologies -- they are not nearly as straightforward as for finite-dimensional spaces.

The problem as I see it is that when you embed a manifold in a higher-dimensional space you will have something that has empty interior when viewed as a subset of the higher dimensional space, and that puts the manifold in the closure of the host space. So when you take boundaries, you get back at least the whole manifold.

And yes, given an embedding you can quite easily come up with another one in a an evern higher space -- just go up one dimension by taking a product with the line for instance. But you still wind up with the entire manifold in the boundary, which makes "boundary conditions" rather problematic.

Ken G
2009-Apr-22, 09:54 PM
I believe you can think of the boundary conditions as whatever constraints you need to actually solve the Einstein field equations. So you always need them if you want to do G.R., they are not esoteric considerations. But what sets the boundary conditions is always the Big Unanswered Question thoughout physics, GR is by no means alone in that department. There is never a physics of boundary conditions-- they are whatever you care to make them, the physicist's fingerprints are always all over them. Sometimes it is more common sense than others, but it's always an entry point for philosophy-- to inform the physicist what "makes sense" to do with the boundary conditions. Of course you can always judge them by their result, but then you lose the feel that you are doing "a priori" calculations-- a calculation that is only right because it gets the right answer is of a dubious nature when you have to know the right answer in advance to do the calculation.

DrRocket
2009-Apr-22, 10:59 PM
I believe you can think of the boundary conditions as whatever constraints you need to actually solve the Einstein field equations. So you always need them if you want to do G.R., they are not esoteric considerations. But what sets the boundary conditions is always the Big Unanswered Question thoughout physics, GR is by no means alone in that department. There is never a physics of boundary conditions-- they are whatever you care to make them, the physicist's fingerprints are always all over them. Sometimes it is more common sense than others, but it's always an entry point for philosophy-- to inform the physicist what "makes sense" to do with the boundary conditions. Of course you can always judge them by their result, but then you lose the feel that you are doing "a priori" calculations-- a calculation that is only right because it gets the right answer is of a dubious nature when you have to know the right answer in advance to do the calculation.

There is nothing magic about boundary conditions. Even we dumb mathematicians have run across them. The usual role of boundary conditions is not to permit solution of an ordinary differential equation (ODE) or partial differential equation (PDE) or system of equations. Typically the problem is that there are lot of solutions, although finding them explicitly may be out of reach. The boundary conditions then serve to select the specific solution that applies to the particular problem at hand. In some sense the boundary conditions are a minimal set of information that determines the solution uniquely, or at least restricts the class or admissable solutions. The utility is that the boundary conditions may sometimes be easily measureable or somewhat obvious from a physical perspective.

Here the idea is apparently that Mach's principle (vague though it may be) is realized by the concept of mass/inertia arising from boundary conditions; i.e. that the effect of the mass distribution of the universe can somehow be equivalent in a mathematical sense to some set of boundary conditions -- presumably some set of constraints on the boundary that is simpler than specifying the detailed mass distribution of the universe. That is appealing, since it gives some substance to Mach's idea, and also because it would presumably not change simply because the mass distribution changes -- planets and stars and galaxies go round and round changing the mass distribution but the concept of mass and inertial seems to stay the same.

Nice idea, but it still begs the question of to what "boundary" the conditions apply. This goes directly to your observation in the last sentence. To be a meaningful construct you need to know what the boundary is and what its significance is. I don't think this is a trivial question.

The problem here may simply be that none of us has seen, or at least remember, the details of the paper, so we are stumbling around in the dark. It is not a "standard topic", so it might take a specialist in that particular topic in relativity to really explain what is going on. If that is the case I don't think we can settle the question here, but even so I thank you and Publius for an enlightening discussion. I always learn things from you guys.

Ken G
2009-Apr-23, 03:15 AM
To be a meaningful construct you need to know what the boundary is and what its significance is. I don't think this is a trivial question.I'm sure it's not, but I would say that the "cosmological principle" is actually part of the boundary conditions. True the "principle" applies everywhere, not just at a "boundary", but boundary conditions can be generalized like that. As you say, all they really are are assumptions you need to make if you want to pick out one solution from many possible ones. I am thinking that the cosmological principle is itself quite Machian, so that right there is one of the ways Mach's principle shows up in the "boundary conditions"-- that is used all the time in cosmology, even though few cosmologists seem to think of themselves as "Machian" in philosophy. The latter might just relate to all the different things "Machian" could mean.

It is not a "standard topic", so it might take a specialist in that particular topic in relativity to really explain what is going on. If that is the case I don't think we can settle the question here, but even so I thank you and Publius for an enlightening discussion. I always learn things from you guys.Ditto. And this issue might actually be advanced enough that you need exactly one specialist (or "high priest")-- if you have two, you might not get the same answer twice!

DrRocket
2009-Apr-23, 04:24 AM
... but I would say that the "cosmological principle" is actually part of the boundary conditions. True the "principle" applies everywhere, not just at a "boundary", but boundary conditions can be generalized like that.

This may very well be a valid perspecive. But this part of the discussion started with the recollection that someone had published a paper (full of "tensor 'rthmetic" according to publius) that showed a method of invoking Mach's principle by means of boundary conditions. One might assume that method would be something more subtle than just declaring the principle to actually be a boundary condition.



And this issue might actually be advanced enough that you need exactly one specialist (or "high priest")-- if you have two, you might not get the same answer twice!

Maybe, if there are opinions involved. But if this is really mathematics, then it is a matter of what has been proved. That is normally quite objective. There are then only two answers:1) the answer is X and the proof goes like this ..... and 2) I dunno. Or maybe I dunno, but I think if your do something like this ... you could show that ....

But if there opinions involved, there is danger of getting two opinions from one guy. Both wrong. I have seen millions go down the drain over opinions.

Aside (and hopefully interesting story): The high priests in mathematics can be pretty impressive. When I was still in graduate school I was sitting in and AMS meeting in which Harry Furstenberg was giving a lecture on some topic in ergodic theory. Harry is himself a "high priest". I was having a hard time following him (I am being kind to myself with this description). But sitting in the student desk next to me (we were in a classroom) was this older oriental gentleman, who was asking lots of questions, good questions, and seemed to not only be following Furstenberg, but staying well ahead of him. Right after the lecture I caught my advisor at the door to the room and asked "Who is that guy ?". The response, after a quick glance into the room was "Oh! That's Kakutani!"

Kakutani might not ring a bell with you, but he was an absolute powerhouse in what I call "hard analysis". There are others, but he didn't take a back seat to anyone.

DrRocket
2009-Apr-23, 06:05 AM
I finally tracked down the fact that there is a theorem for embeddings of pseudo-Riemannian (or semi-Riemannian which means the same thing) manifolds that is quite similar to the Nash embedding theorem.

Here is a link to an Arxiv paper on the subject. But the fundamental papers are not available to me on line (this is not one of them). They are referenced in the paper at the link -- the references by C.J.S. Clarke and the monograph by R.E. Greene, references 7 and 9.

http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.4439v2.pdf

The 1970 monograph by Greene seems to be the one likely to be the most definitive, just because that is the nature of the AMS Monograph series.

The paper by Clarke also hits the mark, but only the abstract is available without some sort of a subscription.
http://rspa.royalsocietypublishing.org/content/314/1518/417.abstract

publius
2009-Apr-23, 07:00 AM
Here we go -- I think this was the paper:

http://arxiv.org/abs/hep-th/0612117

It's called "Mach's Holographic Principle". It was about what I dimly remembered, but didn't say anything because I wasn't sure. In EM, one can replace boundary conditions on the potentials with charge distributions.

For example, imagine a conductor immersed in the field of a given charge distribution. The boundary condition is the potential must be constant on that conducting surface. Or if "grounded" constant and equal to zero (well, reference at infinity) or held to some other constant constant and equal to a specified value. Well, that can be replaced with a charge distribution, exactly corresponding to the induced charge distribution by the field.

A simple way to see this is to imagine placing a battery between two arbitrary conducting surfaces. That's a boundary value problem on the potential equation. But it is equivalent to specifying a charge distribution on the surface of the conductors.

The method of image charges derives from this as well, being a variation on the same thing. One can place "imaginary" (image) point charges in various circumstances and get the same field (save for inside the conductor, or "outside the boundary"). This yields a much simpler way to solve for the potential and then the field.

Apparently one can do something similiar (but much more complicated) with GR, and replace boundary conditions with boundary stress-energy distributions. And that becomes the mass that Mach lives in.

And, for those interested, I do a search on "Mach's Principle" in the gr-qc section. You will find a lot of interesting papers.


-Richard

Ken G
2009-Apr-23, 02:37 PM
This may very well be a valid perspecive. But this part of the discussion started with the recollection that someone had published a paper (full of "tensor 'rthmetic" according to publius) that showed a method of invoking Mach's principle by means of boundary conditions. One might assume that method would be something more subtle than just declaring the principle to actually be a boundary condition.
But the principle is clearly a boundary condition, when boundary conditions are defined the way you did: whatever you need to assume to get a unique solution. As GR is a theory involving sources, you always need to specify what assumptions you are making about the distribution of those sources. You can express that constraint in terms of what it does at the boundary, but physically, it still has to show up somewhere in what you are assuming. One example of how that can get fouled up was the recent claims that using GR can relieve the need for dark matter in a galaxy. Since Newton's laws should get the same answer, it was always preposterous to imagine that GR could magically get something different, so I knew from the start the issue was a problematic boundary condition-- something was being assumed that was not recognized as such.

A classic example of how we embed our assumptions into boundary conditions is "periodic boundary conditions", which is a standard trick for doing simulations in a finite box that are intended to simulate what would happen in an unbounded homogeneous region. As is generally true, you decide what physics you are interested in, and you choose your boundary conditions appropriately-- in the case of periodic boundary conditions, you are not actually interested in a periodic system, you are interested in physics happening on a scale much smaller than your box, over a homogeneous region much larger than your box. It is then a matter of opinion on how best to accomodate that with boundary conditions, but periodic ones seem to have a good track record even though they are clearly unphysical if taken literally.

The situation is different in cosmology, there you are interested in the scale of everything, so you don't simulate a box at all, you simulate an ODE. But to turn it into an ODE, you make assumptions, that may certainly be called boundary conditions in the way we are using the term.

Maybe, if there are opinions involved. But if this is really mathematics, then it is a matter of what has been proved.

Would that it were so straightforward! That's true about a lot of the more clear-cut elements of mathematics, which is certainly the backbone of the endeavor, but opinion still plays an important role in mathematics, as in any human pursuit. Off the cuff, I can generate this list of ways opinion appears in mathematics:
1) in deciding if something has actually been proven (fortunately this is the easiest to establish, but for difficult proofs, it still requires opinion)
2) in deciding if the logic used to make the proof is adequate (some proofs might use higher-order logic, whereas the opinion of a mathematician might be that only a proof using first-order logic is acceptable for that theorem).
3) in deciding if the axioms used in the proof are the appropriate ones (this is the biggest issue in physics, look at string theory proofs versus quantum loop gravity proofs and so forth).
4) in deciding if what has been proven is of any importance or value (this is the opinion that goes into important things like job hiring, journal acceptances, invitations to give talks, etc.).

Ken G
2009-Apr-23, 02:51 PM
It's called "Mach's Holographic Principle". Yes, the holographic principle is a very interesting one, it gets used with black holes a lot because black holes act more like boundaries than like actual objects. As I understand it, the holographic principle is the idea that everything happening inside a bounded region leaves enough of a trace on the boundary that the complete information is encoded there. I don't know how it addresses the problem of finite and unbounded spaces though, since there it's hard to distinguish the "inside" of a boundary from the "outside"!


Apparently one can do something similiar (but much more complicated) with GR, and replace boundary conditions with boundary stress-energy distributions. And that becomes the mass that Mach lives in.I figured that would be the case, it just seemed reasonable. But I wouldn't know how to demonstrate it mathematically, so it sounds like this is what that paper did. Perhaps it turned that author into a Machian as well!

DrRocket
2009-Apr-23, 05:29 PM
Off the cuff, I can generate this list of ways opinion appears in mathematics:
1) in deciding if something has actually been proven (fortunately this is the easiest to establish, but for difficult proofs, it still requires opinion)
2) in deciding if the logic used to make the proof is adequate (some proofs might use higher-order logic, whereas the opinion of a mathematician might be that only a proof using first-order logic is acceptable for that theorem).
3) in deciding if the axioms used in the proof are the appropriate ones (this is the biggest issue in physics, look at string theory proofs versus quantum loop gravity proofs and so forth).
4) in deciding if what has been proven is of any importance or value (this is the opinion that goes into important things like job hiring, journal acceptances, invitations to give talks, etc.).

1) I don't buy this, for a couple of reasons. One is that many of the major theorems have actually been evaluated and reduced to symbolic logic are checkable and have been checked using some sophisticated computer algorithms to show the equivalence of the proof to a finite symbolic expression. Second is that the system of checks, for the major theorems of sufficient difficulty, involves seminars by experts at several locations over a period of time, and the resolution of concerns is NOT by weight of opinion, but rather by consensus in which those involved reach agreement. It is NOt a case in which there is any controversy.
It is the case that mathematicians who use results have in many cases verified all, or nearly all of the theorems used themselves. I did.

2) That is simply not how it works. High order logic is the rule and there is no controversy, at least outside of the work of logicians themselves. You have been delving into philosophy too much -- most mathematicians don't get involved in such issues. I can honestly say that I have never seen this issue come up, and I have dealt with more than a few mathematicians. I am one.

3) That may be how it works in physics but it is NOT how it works in mathematics. Virtually all mathematics is based on Zermelo-Frankel plus choice. Any deviation from that is clearly stated, and deviations like that are normally only found in work related directly to logic. In any case there is no question as to the "appropriate" axioms, the only issue is that the set of axioms invoked are clear. It is never the case that theorems using an unusual set of assumptions are then later applied to prove theorems in the more convention Zermelo-Frankel plus choice setting.

4) The decision as to value is irrelevant to the question of the validity of a result. Yes it is important in career decisioins, and it is certainly the second most important issue (after correctness) in making decisioins for publication. But it has nothing to do with the question as to whether or not a theorem is valid. So this is an issue about the coduct of mathematicians but it is not an issue regarding mathematics itself.

DrRocket
2009-Apr-23, 05:48 PM
Here we go -- I think this was the paper:

http://arxiv.org/abs/hep-th/0612117


And, for those interested, I do a search on "Mach's Principle" in the gr-qc section. You will find a lot of interesting papers.


-Richard

Interesting. I did not see any indication that this paper had been submitted to a refereed journal. I am a little surprised.

I am still having trouble figuring out what the "boundary" is. Most of the paper appears to be working in a coordinate system, a coordinate chart in the mathematical terminology with the boundary applying to the boundary of the chart. But then there are some Penrose diagrams, which I don't understand, that sketch the boundary as well.

I don't understand how one can relate a boundary of a chart to the boundary of the entire manifold, unless the manifold is flat and there exists a global chart. And then the "boundary" is at infinity.

That is the problem that I have with the image charge analogy from electromagnetism. That technique does indeed work. I have used it myself, not for electromagnetism but for heat transfer. But in the case of say a plane of constant potential, you can set up image charges and establish that condition quite easily without having to specify boundary conditions. However, while replacing a plane of constant potential with a finite set of image chrges is convenient, it is used to model a phenomena in a half-plane, and what you give up is the ability to model all of "space-time". Basically you have changed the normal "space-time" to one that has symmetry with respect to a plane. Or another way to think about it is that you have this symmetrical space-time, and you model it in coordinate charts (one for reach half-plane) with boundary conditions on the charts. And this only works for a fictitious space-time with the necessary symmetry.

In the case of Mach's principle if the issue is to provide a model, one chart at a time, in isolation, without trying to patch those charts together to reconstruct the entire manifold, then I guess this might work. But I don't think that a local version of Mach's principle fits the usual version of the principle. And it Mach's principle is valid is must apply to the real universe, which is not particularly homogeneous at small scales.

Ken G
2009-Apr-24, 03:50 AM
Second is that the system of checks, for the major theorems of sufficient difficulty, involves seminars by experts at several locations over a period of time, and the resolution of concerns is NOT by weight of opinion, but rather by consensus in which those involved reach agreement. It is NOt a case in which there is any controversy.Opinion is not the same thing as controversy. It is not uncommon for opinions to reach consensus-- there is no "line in the sand" where an opinion is not an opinion any more just because everyone agrees. But I realize that if ten experts agree on something, it's not "just their opinion"-- the word "opinion" is a slippery one indeed, I see shades of gray where many see black and white.


It is the case that mathematicians who use results have in many cases verified all, or nearly all of the theorems used themselves. I did. There is no question that widely accepted proofs are very rarely found to be contradicted later on. Nevertheless, I suspect there must have been such cases. Even so, there is surely a "finite lifetime" that a wrong opinion can hold sway, I've no doubt that mathematics, like science, is highly self-correcting. And unlike science, mathematics only quite rarely undergoes "paradigm shifts", that's all true. Yet, it does, around issues like how applied should the goals be, and how serious a problem is not knowing if your endeavor is inconsistent or merely incomplete.


2) That is simply not how it works. High order logic is the rule and there is no controversy, at least outside of the work of logicians themselves. You have been delving into philosophy too much -- most mathematicians don't get involved in such issues. I can honestly say that I have never seen this issue come up, and I have dealt with more than a few mathematicians. I am one. No doubt high-order logic is the rule for the everyday activities of mathematicians, because those activities are interested in achieving results, so why hamstring themselves if there is no demonstrable cause (a reasonable opinion most working mathematicians share). But then, people use the arithmetic of real numbers all the time for the same reason, even though there is no proof that all real arithmetic is consistent (most simply trust that it is incomplete instead). That is a classic example of opinion in math: is it your opinion that real arithmetic is incomplete or inconsistent, or is it your opinion that it doesn't matter which because the problem hasn't had any effect day-to-day?


3) That may be how it works in physics but it is NOT how it works in mathematics. Virtually all mathematics is based on Zermelo-Frankel plus choice. Any deviation from that is clearly stated, and deviations like that are normally only found in work related directly to logic. In any case there is no question as to the "appropriate" axioms, the only issue is that the set of axioms invoked are clear. It is never the case that theorems using an unusual set of assumptions are then later applied to prove theorems in the more convention Zermelo-Frankel plus choice setting.I'm not sure what you mean when you claim that all math basically uses the same axioms. As you know, there are branches of mathematics like real analysis and complex analysis, and they use different axioms right from the outset. They don't step on each others toes, because the distinctions are clear, but nevertheless there is a wide array of theorems that apply to one but not the other. So if one wishes to know what is the appropriate theorem set for some application, one must decide if the application should be restricted to the reals, or if complex solutions will be regarded as meaningful. That does indeed come up often in physics, but since math is a tool for physics, the choices are being made on the mathematics side of things, and the tests of what was the right choice is made on the physics side. A classic example of this is people whose opinion is that there is something wrong about quantum mechanics because it relies on complex numbers. That opinion has nothing to do with experiment, it has to do with choices of axioms. Indeed, there is a whole class of people who call themselves mathematical physicists who basically do math all the time, and you can say they are not mathematicians they are physicists, but still they do math all the time.


4) The decision as to value is irrelevant to the question of the validity of a result. Yes it is important in career decisioins, and it is certainly the second most important issue (after correctness) in making decisioins for publication. But it has nothing to do with the question as to whether or not a theorem is valid. So this is an issue about the coduct of mathematicians but it is not an issue regarding mathematics itself.Again you attempt to place a firewall between the math and the mathematician. But math is what mathematicians do, and without mathematicians, there is simply no such thing as math. But we digress, we can certainly agree that math is based on proof far moreso than it is based on debate.

Ken G
2009-Apr-24, 04:01 AM
I don't understand how one can relate a boundary of a chart to the boundary of the entire manifold, unless the manifold is flat and there exists a global chart. And then the "boundary" is at infinity.I don't know what they are doing either, but my guess is that they are merely trying to be more careful about what is being assumed about the boundary than similar GR calculations normally are. In other words, they are not doing anything different from a standard GR cosmology, they are doing something more than that. They are tracing the full ramifications of the kinds of calculations that are normally done simply out of necessity to find a reasonable seeming solution.

publius
2009-Apr-24, 04:53 AM
I wish I could be more help, but I don't understand it either beyond the vague things I've said above. We'd need a real expert to weigh in and translate (if the expert could :) ).


-Richard

DrRocket
2009-Apr-24, 05:21 AM
I'm not sure what you mean when you claim that all math basically uses the same axioms. As you know, there are branches of mathematics like real analysis and complex analysis, and they use different axioms right from the outset. They don't step on each others toes, because the distinctions are clear, but nevertheless there is a wide array of theorems that apply to one but not the other.

This is not true. Real and complex analysis are based on precisely the same set of axioms. Given Zermelo-Frankel plus choice (and I'm not sure at what stage you even need choice, but transfinite induction is used without much comment) you can develop everything. ZF gives you the natural numbers, and the rest is nothing but construction.

As a matter of practice, the subject usually starts with the Peano Postulates, but strictly speaking the foundation is ZF plus choice. If you would would like to see the construction of everything -- integers, rationals, reals, complex numbers -- just from the Peano Postulates (which basically say that the natural numbers exist and behave as you normally think that they do) the little book Foundations of Analysis by Landau presents the development in Landau's signature telegraphic style. You can also see it done in Naive Set Theory by Halmos. It is an interesting exercise, particularly seeing the completeness of the reals emerge from the process via the use of Dedekind cuts.

There is one school of mathematics, and "school" is really a poor choice of words, called the constructivists, led by Erett Bishop (http://en.wikipedia.org/wiki/Errett_Bishop) who managed to develop quite a bit of modern mathematics without the use of the axiom of choice. This is not really so much a controversy as an attempt to see what could be done without the axiom of choice. He is not a controversial figure within mathematics. He is in fact a respected mathematician. It is just that most mathematics is done under the assumption that the axiom of choice is true. It is logically independent of the rest of the axioms. Anything that Bishop has proved using constructivist methods is valid to all other mathematicians, but the converse is not true. His work simply shows how far a clever guy can go without the axiom of choice, and also serves to point out the importance of the axiom to the remainder of mathematics.

The only issue with choice (this axiom simply says that given a family of non-empty sets there is a function defined on that family for which the value on each set is a member of the set -- you can "choose" a member from each non-empty set) is that it has some decidedly non-intuitive implications, the most famous being the Banach-Tarski theorem. The Banach-Tarski theorem, sometimes called the Banach-Tarski paradox, says that given a sphere of diameter X, and an arbitrary number e>0 it is possible to decompose X into seven disjoint sets, and by rigid motion reassemble them inside a sphere of diameter e. This is sometimes states as saying that you can cut up the sun and put it in your pocket. The trick here is that proof merely shows the existence of the decomposition, not how to do it explicitly and the sets involved are so weird (not measurable in the sense of Lebesque) that the idea of volume is not even defined for them. Here is a Wiki article on the theorem, which states it in a slightly different, but equivalent form. http://en.wikipedia.org/wiki/Banach-Tarski_Theorem



So if one wishes to know what is the appropriate theorem set for some application, one must decide if the application should be restricted to the reals, or if complex solutions will be regarded as meaningful. That does indeed come up often in physics, but since math is a tool for physics, the choices are being made on the mathematics side of things, and the tests of what was the right choice is made on the physics side.

Not at all. This is a purely physical decision. The mathematics of real analysis and complex analysis are completely consistent with one another. It is true that some theorems assume the complex numbers in the hypothesis and are not true if "real" is replaced by "complex", but that merely reflects the fact that the real and complex number systems have different properties. That is a consequence of definitions, and constructions based on those definitions, but has nothing whatever to do with the axioms.

There are actually very few axioms in mathematics. Zermelo-Frankel plus choice suffices for all of the mathematics used by virtually all physicists and engineers. I can't think of any exceptions off the top of my head. Frankly, outside of work of logicians, there are very very few times that any other set of axioms is even discussed by most mathematicians. Once in a long while someone finds an application for the continuum hypothesis, and what happens then is the theorem is stated as explicitly assuming the continuum hypothesis (or perhaps assuming that it is false), but such theorems are basically curiosities and they are not used further without the assumption being re-stated.

Virtually any mathematics text you are likely to have seen is based on ZF plus choice and nothing more. Algebra, Real Analysis, Complex Analysis, Functional Analysis (and the sub-disciplines involving ordinary and partial differential equations, Banach and Hilbert spaces, all topological vector spaces, Fourier analysis, theory of distributions), measure theory and probability, point-set topology, algebraic topology, differential geometry, algebraic geometry ..... all are based on ZF plus choice. Accept any one and you logically accept them all.


A classic example of this is people whose opinion is that there is something wrong about quantum mechanics because it relies on complex numbers. That opinion has nothing to do with experiment, it has to do with choices of axioms.

They may think it has to do with choices of axioms, but it has nothing to do with the axioms of mathematics. All of the mathematics used in conjunction with quantum mechanics is based on ZF plus choice. Now, there may be choices in the assumptions on which quantum mechanics is based, but those relate to the physics and not the mathematics.

That is really a bit amusing. If physicists have trouble with assumptions that are not related to experiment, and are most certainly not related to the mathematics, then it seems to me that they are on rather shaky ground. They have no mathematical leg on which to stand. And by rejecting experiment, they are certainly outside the mainstream of physics. If you don't believe simple mathematics, and you reject the empiricism of physics are you either a mathematician or a physicist ?





Indeed, there is a whole class of people who call themselves mathematical physicists who basically do math all the time, and you can say they are not mathematicians they are physicists, but still they do math all the time.

There is actually a difference between physicists, even mathematical physicists, and mathematicians. Sometimes that difference is quite small, as in the case of Witten (a physicist) and Penrose (a mathematician). Sometimes they cross the lines rather smoothly. But in fact, most of the time, when mathematical physicists are working they are doing physics, and they do not follow the rigor required in mathematics (this is not a bad thing for physics in most cases, and it does permit one to forge ahead even when there are some important details that one does not know how to handle).

Physicists are sometimes rather loose with mathematics. For instance, I have seen statements in physics books about the convergence of Fourier series, stating for instance that they converge for continuous (periodic)functions, that are simply false. Badly false. There are, for instance, continuous functions for which the Fourier series diverges at every rational number. I also have a bit of a problem with the statement that one sees in books on general relativity or cosmology that the openness or closedness of the spatial slices of space-time are cleanly implied by curvature alone. There are manifolds of negative curvature that are closed, but they are apparently discarded for unstated "physical" reasons -- Wald at least acknowledges their existence but rejects them on the basis that they are realized by the usual topological technique of "cutting and pasting", or in mathematical terms as quotient spaces -- which is how you get things like the Moebius strip and the Klein bottle, but also a perfectly normal way to get a torus.

But the physicists gain an advantage sometimes by being a bit loose. They are able to press ahead and discover new physics where a mathematician might be held back cleaning up details. The physicist is thus able to penetrate deeply down a path, and thereby determine if a "clean up" is justified on the basis of the prize at the end of the path. Heaviside, for instance, employed methods in vector analysis that provided great benefit to physics, and that were made precise from a mathematical perspective only much later.



Again you attempt to place a firewall between the math and the mathematician. But math is what mathematicians do, and without mathematicians, there is simply no such thing as math. But we digress, we can certainly agree that math is based on proof far more so than it is based on debate.

But such an interesting digression.

I do place a firewall between math and the mathematician. I also place a firewall between physics and the physicist. Math is not just what is done by anyone who calls himself a mathematician. There are some nut cases who may call themselves mathematicians, but are not and who most certainly are not doing mathematics. Witten is not a mathematician. He is a physicist. With a Fields Medal. Nash is a mathematician. With a Nobel Prize. In economics. Economics is not mathematics. (Nash should have received a Fields Medal, but for his work on embeddings and not for his work in game theory that resulted in the Nobel Prize.) There are also some nut cases who call themselves physicists and most certainly are not doing physics.

There is even work by real physicists, that outside of their expertise, is pretty questionable. I have read quite a bit of stuff by Hannes Alfven. His work in plasma physics, particularly the early work, is stellar -- he received a Nobel Prize for it. His work toward the end of his life, on cosmology, is pretty shaky, and is the inspiration for the EU cult (though they have badly misinterpreted much of what he said).

It is similar for mathematicians, except that the nature of mathematical proof does not really permit the publication of incorrect fantasy in the literature. One of the most powerful mathematician ever was Alexander Grothendieck. His mathematical work is unbelievably deep. But his writings outside of mathematics, and his retreat from humanity (he is reportedly now a hermit somewhere in France) are a bit bizarre.

DrRocket
2009-Apr-24, 05:39 AM
I wish I could be more help, but I don't understand it either beyond the vague things I've said above. We'd need a real expert to weigh in and translate (if the expert could :) ).


-Richard

Out of curiousity, is there a list of people that you think might have this specialized expertise ?

The only name that leaps to my mind, who is alive and might have thought about this sort of thing, is Roger Penrose.

There are some pretty high-powered geometers (real, first-class, stratospheric, high priests) who could probably handle this easily, but they have not written anything that I know of about GR per se -- Richard Hamiltonl, Grigori Perleman, Pierre Deligne.

Hawking, maybe, but he seems to be almost gone.

None of these guys are likely to show up on a bulletin board.

Edit: I forgot one guy. Shlomo Sternberg. And in fooling around a bit, I found a book available at his homepage at Harvard that talks about general relativity an geometry. I doubt that it even addresses the question at hand. but I thought folds reading this thread might be interested in this resource. http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf

Ken G
2009-Apr-24, 05:38 PM
The only issue with choice (this axiom simply says that given a family of non-empty sets there is a function defined on that family for which the value on each set is a member of the set -- you can "choose" a member from each non-empty set) is that it has some decidedly non-intuitive implications, the most famous being the Banach-Tarski theorem. The Banach-Tarski theorem, sometimes called the Banach-Tarski paradox, says that given a sphere of diameter X, and an arbitrary number e>0 it is possible to decompose X into seven disjoint sets, and by rigid motion reassemble them inside a sphere of diameter e. This is sometimes states as saying that you can cut up the sun and put it in your pocket.That's quite interesting, I may have heard of that but had forgotten.

The mathematics of real analysis and complex analysis are completely consistent with one another. I never said they weren't, that's what I meant by not "stepping on each other's toes".

It is true that some theorems assume the complex numbers in the hypothesis and are not true if "real" is replaced by "complex", but that merely reflects the fact that the real and complex number systems have different properties. That is just what I am saying.

That is a consequence of definitions, and constructions based on those definitions, but has nothing whatever to do with the axioms.I am not distinguishing a definition from an axiom, they have the same place as the starting points from which the logical ramifications issue. A definition is an axiom, it is that which we accept as true without proof, and lives entirely within the context of whatever endeavor is consistent with those axioms. Perhaps you are using the word in a more formal way, so simply take all my points from above and append your definition of "definition" to your definition of "axiom", and that is what I am saying about opinion as to what is going to produce useful results. If you will allow that said opinion applies to the definitions that are made, I am happy to reserve "axiom" to the meaning you take-- my point is still just the same.

Frankly, outside of work of logicians, there are very very few times that any other set of axioms is even discussed by most mathematicians. That is certainly an important and interesting point for many types of discussions about mathematics, and I benefit from having it clearly stated, but it is still not relevant to my points about the role of opinion in formal mathematics.


Virtually any mathematics text you are likely to have seen is based on ZF plus choice and nothing more.Well, it is certainly also based on the definitions that get used, and what is viewed as an allowed animal to consider! That's the difference between real algebra and complex algebra, they are just different things (even though they don't contradict each other). It would then be purely a matter of opinion as to which was worth one's while to consider, in various contexts or even in complete generality. Until one starts doing physics, that is, at which point it starts to become more clear what is going to be important and beneficial-- though sometimes it can take a while to realize it.


That is really a bit amusing. If physicists have trouble with assumptions that are not related to experiment, and are most certainly not related to the mathematics, then it seems to me that they are on rather shaky ground. You were not aware of this?

They have no mathematical leg on which to stand.Neither did Einstein when he said that "God doesn't roll dice", yet that didn't stop you from pointing out the quote as a kind of definitive statement of how science can be done in some cases. Again, it is the crucial difference between what is actually science, and what is the motivation used by a human scientist (ergo, a philosopher) along the way. So we are both arguing that the endeavor has a kind of "in principle" character, an ideal if you will, and in science that is sheer instrumentalism and in mathematics it is sheer logic. I'm also pointing out that both science and math also have an "in practice" flavor, which is where human opinion and foibles and philosophies (and whether or not God rolls dice) come into play. So the common ground is the recognition of value in distinguishing the ideal from what is actually accessible to us.


And by rejecting experiment, they are certainly outside the mainstream of physics. If you don't believe simple mathematics, and you reject the empiricism of physics are you either a mathematician or a physicist ? Those who reject quantum mechanics (like Einstein) did not "reject" any experiments, or any mathematics. They simply rejected the idea that quantum mechanics was a good (or the best possible) description of reality, because of its reliance on non-real definitions.

There is actually a difference between physicists, even mathematical physicists, and mathematicians. Sometimes that difference is quite small, as in the case of Witten (a physicist) and Penrose (a mathematician). Sometimes they cross the lines rather smoothly. But in fact, most of the time, when mathematical physicists are working they are doing physics, and they do not follow the rigor required in mathematics (this is not a bad thing for physics in most cases, and it does permit one to forge ahead even when there are some important details that one does not know how to handle).
I agree that rigor is not a bad way to distinguish mathematics from mathematical physics. The physicist has the result in mind, the mathematician cares more about the sanctity of the process.

Heaviside, for instance, employed methods in vector analysis that provided great benefit to physics, and that were made precise from a mathematical perspective only much later. Yes, that is certainly a valid point.


Math is not just what is done by anyone who calls himself a mathematician.That is so, but it is also true that math is always done by a mathematician, or a computer programmer telling a computer what a mathematician does. Math has rules, but they have to be rules that humans can follow or they have no meaning at all. Ergo, one can never separate the math from the mathematician. Now there's mathematical logic for you.
I have read quite a bit of stuff by Hannes Alfven. His work in plasma physics, particularly the early work, is stellar -- he received a Nobel Prize for it. His work toward the end of his life, on cosmology, is pretty shaky, and is the inspiration for the EU cult (though they have badly misinterpreted much of what he said). But that's part of the problem right there. In history, there have been many cases of "nut cases" who turned out to be right. Not nearly as often as they turned out to be wrong, but the point is, science is inseparable from human foibles, even though it tries very hard to be (and one way we make that separation is recognizing the role of instrumentalism, which is my prevailing point here). Math tries even harder, and succeeds even more, but never completely.

DrRocket
2009-Apr-24, 10:13 PM
That's quite interesting, I may have heard of that but had forgotten.

Proabably not. It is not a theorem that I would expect you to have seen. It created quite a stir when it was first proved, but now even professional mathematicians typically hear of it "around the water cooler" and not in a formal class.


I am not distinguishing a definition from an axiom, they have the same place as the starting points from which the logical ramifications issue. A definition is an axiom, it is that which we accept as true without proof, and lives entirely within the context of whatever endeavor is consistent with those axioms.

But it is most important that you do distinguish between definitions and axioms. It is, in fact crucial.

Axioms are accepted as "true", as postulates, on faith, because of intuitive appeal or for whatever reason blows your skirt up. They are the foundation on which everything else is built, and if an axiom is later rejected for any reason, everything built on it comes tumbling down. There are 8 axioms in the Zermelo Frankel system and 9 when the Axiom of choice is added http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html ALL of the mathematics used in physics is based on those 9 axioms (ZF+C for short) and nothing more.

Definitions are simply that, definitions. They are neither true nor false, because they assert nothing. This is important. There is no such thing as a false definition. The worst that can be done with a definition is to formulate one that is self contradictory, so that you wind up with a complicated definition of the empty set. But such a definiton is not false, it is merely trivial.

I can define the natural numbers, the rationals, the reals and the complex numbers using nothing more than ZF+C as a basis. There is no article of faith involved in those number systems -- if you accept ZF+C then logically, you must accept all of those number systems as valid mathematical entities. All of those and a lot more -- everything I noted in the earlier post for starts.

Here is a quick example. Suppose we have progressed to the point at which you have accepted the real numbers (remember I can construct them from nothing more than ZF+C) , and understand the defintion of an ordered pair (which can also be stated in terms of set theory and requires nothing more than ZF+C). Consider the set of all ordered pairs of real numbers (a,b). Not DEFINE operations of addition and multiplication of ordered pairs as follows:

(a.b) + (c,d) = (a+c, b+d) (Edited to correct typo pointed out by Ken G)

(a,b)*(c,d) =(ac-bd,ad+bc)

where the operations inside the parentheses are just addition, subtraction and multiplication of real numbers, which we have assumed that you know about.

Then I have just constructed, using these definitions, the complex numbers. All the other results of complex analysis follow from this defintion (and it is a definition, not an assumption and not an axiom) and ZF+C.

Whether or not you believe that the complex numbers accurately model reality, or some bit of physics, is an entirely different question. But in terms of the mathematics this is all that is needed to develop all of complex analysis. Note that I have assumed nothing as "true" other than ZF+C.




Perhaps you are using the word in a more formal way, so simply take all my points from above and append your definition of "definition" to your definition of "axiom", and that is what I am saying about opinion as to what is going to produce useful results. If you will allow that said opinion applies to the definitions that are made, I am happy to reserve "axiom" to the meaning you take-- my point is still just the same.

As noted above there is a tremendous difference between an axiom and a definition.

What I can agree with is, not that axioms and definitions are similar in any way, but that it is certainly a matter of opinion as to whether a concept that has been defined is a useful concept.

Not only is that judgement merely a matter of opinion, so is the classification of a result as "useful" or not. In many cases it is in fact a matter of taste.

Let's take a subject that we have discussed in this thread, Riemannian geometry. Riemannian geometry is, like the rest of mathematics, based on ZF+C. In terms of axioms that is all that is needed. Beyond that you have the definition of a manifold, with constructions to show that they exist in the mathematica sense, the definition of a differential structure, with more examples, the definition of the algebra of smooth functions, the definition of a vector field as a derivation on tha algebra, the definition of the Lie bracket, the definition of a tensor field, the definition of an alternating tensor, the definiton of a differential form, the definition of a metric, the definition of a connection ..... The result is a mathematical theory that may or may not have an application to physics. When first developed by Riemann the applications were not so evident. Einstein, with a little help from his friends, managed to cast his theory of relativity in this language and voila' you have general relativity, so apparently it did turn out to be useful. But originally it was Riemann trying to distill the essence of Gauss's theory of curves on surfaces to higher dimensions.



Well, it is certainly also based on the definitions that get used, and what is viewed as an allowed animal to consider! That's the difference between real algebra and complex algebra, they are just different things (even though they don't contradict each other). It would then be purely a matter of opinion as to which was worth one's while to consider, in various contexts or even in complete generality. Until one starts doing physics, that is, at which point it starts to become more clear what is going to be important and beneficial-- though sometimes it can take a while to realize it.

You are confusing the opinion as what is appropriate as a model with what is valid mathematics. Those are two completely different questions. Physics relies heavily on mathematics to function. Mathematics does not rely on physics at all, it merely finds in physics some interesting problems. Not all mathematical problems come from physics, but some of the most interesting ones do. But mathematics could in principle be conducted without any interaction with physics at all. It would be poorer for that, but there is no logical or functional dependency.


Neither did Einstein when he said that "God doesn't roll dice", yet that didn't stop you from pointing out the quote as a kind of definitive statement of how science can be done in some cases. Again, it is the crucial difference between what is actually science, and what is the motivation used by a human scientist (ergo, a philosopher) along the way. So we are both arguing that the endeavor has a kind of "in principle" character, an ideal if you will, and in science that is sheer instrumentalism and in mathematics it is sheer logic. I'm also pointing out that both science and math also have an "in practice" flavor, which is where human opinion and foibles and philosophies (and whether or not God rolls dice) come into play. So the common ground is the recognition of value in distinguishing the ideal from what is actually accessible to us.

Those who reject quantum mechanics (like Einstein) did not "reject" any experiments, or any mathematics. They simply rejected the idea that quantum mechanics was a good (or the best possible) description of reality, because of its reliance on non-real definitions.
I agree that rigor is not a bad way to distinguish mathematics from mathematical physics. The physicist has the result in mind, the mathematician cares more about the sanctity of the process.
Yes, that is certainly a valid point.

Bold added. We agree with the statement in bold. That assertion is critical.

There is a big difference between what is accepted physics, and the conjectures and notions that go through the mind of a researcher trying to change what is accepted. Einstein was a researcher (duh!). His statement needs to be understood in that light.

His rejection of quantum mechanics, was not, as you say a rejection of the experimental evidence. It was a statement of the direction in which he thought research ought to be pursued -- in this case to develop a deterministic theory that would supplant the stochastic theory that we know as quantum mechanics. That was a perfectly reasonable statement in that context -- of course, it seems that it was wrong given what we know today. A good reseacher has lots of ideas, and most of them will turn out to be wrogn.


Not nearly as often as they turned out to be wrong, but the point is, science is inseparable from human foibles, even though it tries very hard to be (and one way we make that separation is recognizing the role of instrumentalism, which is my prevailing point here). Math tries even harder, and succeeds even more, but never completely.

I would agree that in practice human foibles intrude on science and mathematics. But usually those foibles are overcome. They become problematics when arguments occur over opinion as opposed to over interpretation of real data, and when people are unwilling to say "I don't know". There seems to be a resurgence in publications of science in popular press presenting opinions as facts when the right answer is "I don't know."

Ken G
2009-Apr-25, 03:16 AM
Axioms are accepted as "true", as postulates, on faith, because of intuitive appeal or for whatever reason blows your skirt up.Exactly, they are accepted as true for any reason at all, like "it is my opinion that this should be an axiom"! Just like definitions. There is a distinction, but I'm talking about that fact that math simply cannot even get off the ground without opinion entering at the very start.

They are the foundation on which everything else is built, and if an axiom is later rejected for any reason, everything built on it comes tumbling down. There are differences between axioms and definitions, I accept that, my point is that what I was saying does not need to make the distinction. So any distinction you draw between them, no matter how valid, is irrelevant to the argument I am making about how mathematics does rely on opinion, even at its very core (where the definitions are, if not what you are distinguishing as axioms on the grounds that they are "clearly right", regardless of opinion, even though that is itself a bit of a stretch.)


ALL of the mathematics used in physics is based on those 9 axioms (ZF+C for short) and nothing more.Except the definitions, without with physics would be completely sterile and lame, as would math for that matter. Those axioms are a fairly obvious set of things you are allowed to do with sets, and nothing more. It is nice to lay out all the things that we are allowed to do with sets, but they are a long way from leading to anything interesting or useful in either math or physics. For that, you need to make choices about what you think will be useful definitions.


Definitions are simply that, definitions. They are neither true nor false, because they assert nothing.They do not have a truth value, but that is irrelevant to this entire discussion. They are the bricks, if the logical axioms are the mortar.


This is important. There is no such thing as a false definition. No, that isn't important, it is irrelevant. Definitions do not derive their importance from being true, they derive their importance from being the stuff off which a theory is made, and they are all pure choice, pure opinion. Not opinion that they are true, because that notion has no meaning for definitions, the opinion is that they are useful or have meaning-- a far more important property than being essentially obviously true like the axioms.
There is no article of faith involved in those number systems -- if you accept ZF+C then logically, you must accept all of those number systems as valid mathematical entities. Nor did I ever claim otherwise, that is not the nature of opinion in mathematics.


(a.b) + (c,d) = (a+b, c+d)You meant = (a+c, b+d). And yes, that is the complex numbers. But it all just begs the real question here-- does this algebra correspond to anything fundamentally important or real? Certainly it is highly mathematically convenient in many areas, but all those conveniences might somehow be replaceable, possibly, by some other system of algebra, applied very differently, that seems more realistic. That's the objection with quantum mechanics. Personally I don't see any purpose in that kind of position, but that's just my opinion. This is exactly the way I used opinion in regard to mathematics, about what is really the most productive avenue to make definitions, following some kind of intuition or goal... or opinion.

Whether or not you believe that the complex numbers accurately model reality, or some bit of physics, is an entirely different question.I realize that, and it is precisely the question that is relevant to the issue of opinion in mathematics. That is just what I am saying.


What I can agree with is, not that axioms and definitions are similar in any way, but that it is certainly a matter of opinion as to whether a concept that has been defined is a useful concept.Then we are actually in complete agreement, because that is just what I have been saying.

You are confusing the opinion as what is appropriate as a model with what is valid mathematics. Those are two completely different questions.Actually, it is you who are making that confusion. I've been completely clear on that distinction throughout the above.


I would agree that in practice human foibles intrude on science and mathematics. But usually those foibles are overcome.Yes, they are in time. Both physics and math are highly self-correcting, and math has a more careful way of progressing so it requires that capability far less often than physics.
There seems to be a resurgence in publications of science in popular press presenting opinions as facts when the right answer is "I don't know."Indeed there is, and I'm not sure it was ever not the case.

Sam5
2009-Apr-25, 04:42 AM
Moved from thread about speed of Earth rotation...

I might be stealing this thread but I don't want to start a new one, but another quick question; Isn't the universe expanding due to the force of the Big Bang?


I’ve done some research on the history of this topic, and understanding that history can help clear up some confusion, since many of the “expanding universe” ideas have become modified over the years, especially due to recent astronomical observations.

The basic “Big Bang” concept dates as far back as Newton, who hypothesized (in letters to friends and associates) that since all the gravity in the universe should be pulling all the stars and nebula inward toward each other, there might have been an initial “projectile force” that started everything moving outward and apart in an expanding universe that started at the first moment of creation, i.e. at the initial moment of what we now call the “Big Bang”.

But not much was written in detail about such a “projectile” event concept until the 20th Century, after the redshifts of the galaxies were discovered by Vesto Slipher in 1915 and later confirmed by Hubble in the late 1920s. However, Newton’s basic BB idea was noted in other science and literary works of the 19th and 18th Centuries.

Here are the comments of Jean Jacques Rousseau (1712–1778), re: Newton’s “projectile force”, which was also sometimes called a “projectile impulse”:
http://www.bartleby.com/34/4/1.html
“Newton discovered the law of attraction; but attraction alone would soon have reduced the universe into one solid mass: to this law, therefore, he found it necessary to add a projectile force, in order to account for the revolution of the heavenly bodies.”

This invention of the “projectile force” idea was a precursor to Einstein’s invention of his “cosmological constant” idea of 1917, which was designed to keep the universe static and neither expanding nor contracting (he added the constant before he knew of the galaxy redshifts).

In William Paley’s 1803 astronomy book, Paley mentioned Newton’s expansion and “projectile impulse” ideas. This is a copy of a page from my 1803 copy of his book:
http://i28.tinypic.com/2cp26om.jpg
“But many of the heavenly bodies, as the sun and fixed stars are stationary. Their rest must be the effect of an absence or of an equilibrium of attractions. It proves also that a projectile impulse was originally given to some of the heavenly bodies, and not to others. But further; if attraction act at all distances, there can be only one quiescent center of gravity in the universe: and all bodies whatever must be approaching this center, or revolving around it. According to the first of these suppositions, if the duration of the world had been long enough to allow it, all its parts, all the great bodies of which it is composed, must have been gathered together in a heap round this point.”

This is what Ralph Waldo Emerson said about it in 1844:
”What shall we say of this omnipresent appearance of that first projectile impulse,”
http://www.bartleby.com/5/114.html

Alexander von Humboldt’s “Cosmos” series (4 or 5 volumes), published early in the 19th Century and re-printed many times during that Century, discussed the possibility of an expansion cause by some kind of initial “projectile” force.

Edgar Allen Poe read Humboldt’s books and was fascinated with the idea of an expanding and a possible later contraction of the universe. In 1848 he wrote a highly philosophical and speculative essay about it titled “Eureka”:
http://xroads.virginia.edu/~HYPER/POE/eureka.html

When Einstein came along he thought the stars were basically “static” in space, which was a common view back then. But he later accepted Hubble’s redshift studies and accepted the “expansion”.

For many years the expansion was thought to have been caused by an initial “projectile event”, which became known as the “Big Bang” in the 1960s. By the ‘80s and ‘90s, when faint distant galaxies were discovered that indicated a radial speed that might be faster than light speed, the concept of the galaxies “moving through space” was changed to the concept of “expanding space” that was “carrying the galaxies along with the expansion.”

This latter idea had already been mentioned in books and magazine articles written since the 1920s, such as this one from 1932. The “projectile force” “fireworks” version is on page 1, and the “expansion of space” version is on page 2:

http://blog.modernmechanix.com/2007/01/11/blast-of-giant-atom-created-our-universe/?Qwd=./PopularScience/12-1932/big_bang&Qif=big_bang_0.jpg&Qiv=thumbs&Qis=XL#qdig

http://blog.modernmechanix.com/2007/01/11/blast-of-giant-atom-created-our-universe/?Qwd=./PopularScience/12-1932/big_bang&Qif=big_bang_1.jpg&Qiv=thumbs&Qis=XL#qdig

http://blog.modernmechanix.com/2007/01/11/blast-of-giant-atom-created-our-universe/?Qwd=./PopularScience/12-1932/big_bang&Qif=big_bang_2.jpg&Qiv=thumbs&Qis=XL#qdig

The latest discoveries suggest that the expansion might be speeding up somewhat, and no one is sure why. This increase in the rate of expansion tends to de-emphasize the initial “projectile force” concept. Others here are more up to date on the latest developments in the newer expansion theories.

speedfreek
2009-Apr-27, 12:44 AM
Here is a list of papers in chronological order reflecting the history of the recent controversy over the use of the term "expanding space".

The paper "Eppur si espande" addresses the original question Jeff posted, but there have been subsequent objections to the conclusions of that paper.

Inflation and the Cosmic Microwave Background (http://arxiv.org/abs/astro-ph/0305179)

Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe (http://arxiv.org/abs/astro-ph/0310808)

Is space really expanding? A counterexample (http://arxiv.org/abs/astro-ph/0601171)

A direct consequence of the expansion of space? (http://arxiv.org/abs/astro-ph/0610590)

Eppur si espande (http://arxiv.org/abs/astro-ph/0612155)

Expanding Space: the Root of all Evil? (http://arxiv.org/abs/0707.0380)

Coordinate Confusion in Conformal Cosmology (http://arxiv.org/abs/0707.2106)

Cosmological Radar Ranging in an Expanding Universe (http://arxiv.org/abs/0805.2197)

A short answer to critics of our article "Eppur si espande" (http://arxiv.org/abs/0812.3266)

Eppur si muove (http://arxiv.org/abs/0812.3972)

Ken G
2009-Apr-27, 12:51 AM
And I must say that in my view, every one of those papers is "barking up the wrong tree". They each shoot down the argument of the last by replacing it with an arbitrarily different argument that can equally be shot down and on the same grounds (that it takes its conclusions too literally), instead of simply recognizing that the question "is space actually expanding" is scientifically ill-posed. But in terms of this thread, those papers are correct in pointing out that special relativity cannot explain cosmological redshifts, expressly because SR does not include the inputs of the mass distribution, and from basic GR we find that this is quite the central issue in our universe. Some other universe might not have that problem, but that hypothetical issue is only of general pedagogical concern but is not relevant to the question in this thread (which is about our universe).