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George
2008-Apr-17, 02:15 AM
Ant analogy of the cosmos expansion.

[Revision: The analogy is contrary to what is held in modern cosmology regarding the expansion. The proper analogy would be one where the unfixed end of the rope is being pulled always at a constant velocity. So, as the ant walks toward the fixed end, it will be progressively advance into regions of the elastic rope that has less and less of a retractive velocity. Thus, the ant will always, eventually, reach the fixed end.

We have altered our dicussions to this revision, so it will be confusing if you join the fun at this point and don't realize we have changed to a constant pull rate (velocity) for the rubber rope.]




[Using the attachment.]

I can't see how the ant can reach its destination without a speed at least half that of the rope's speed where the ant starts.

Let the ant's velocity be 1 mps (meter per second), and the rope streching rate be set for 1 mps per meter. So, at P2, the rope velocity relative to P1 is 1 mps, and at P3 it is 2 mps.

Now, to simplify, let's not stretch the rope until after we give the ant the advantage of moving the first second from position P3. [Its goal is P1, of course.] The ant must stop after the first second and then give the next second to our rope stretchers (located off the page in some dark region. ;) ). What happens?

After the first second, our ant will first reach P2, but in the next second it will be pulled back to P3 since at P2 the rope stretch rate is 1 mps. Thus, the ant has accomplished nothing! This procedure could be done forever and we'll get the same results.

Note that the ant has not only the advantage of going first, which means it reaches a point that has less of a pull-back velocity, but also we are ignoring the fact that the pull-back rate will actually increase more than 1 mps as it gets further from P2 and gets closer to P3.

If we were to allow the ant more than 1.0 seconds for its travel time, then it will reach P1 eventually. If we, however, give the ant less time for its head start, then the ant looses ground.

I would expect the later case would be more of an accurate differential approach to the problem, so the ant would have to travel somewhat faster than 1/2 the pull-back rate relative to its destination.

I'm probably wrong, but why? :doh:

a1call
2008-Apr-17, 02:27 AM
Don't quite understand the question but have you considered that:

*- Pulling the right end of the rope at 1 m/s would mean that every other point on the rope will move decreasingly less towards the left reaching 0 m/s at the left attached-to-the-wall end?

George
2008-Apr-17, 03:01 AM
Don't quite understand the question... This analogy is a common example of how we can see light coming from regions that are deemed to be traveling away from us at a rate faster than light. I thought I would find a simple explanation and without much math. I was wrong, which is why I still don't get it.


*- Pulling the right end of the rope at 1 m/s would mean that every other point on the rope will move decreasingly less towards the left reaching 0 m/s at the left attached-to-the-wall end? The rope is made of rubber and the rate of pull on the end is not important except that it must produce a result of 1 meter per second for every 1 meter distance away from any other point. That is how the expansion is modeled for space. [Note the vectors (arrows) depicting the rope's movement rate.]

grant hutchison
2008-Apr-17, 07:46 PM
The free end of the rope moves one metre each second, but the other parts of the rope move at lesser speeds, proportional to their fractional distance along the length of the rope. So the ant moves steadily into regions that are moving more and more slowly relative to the fixed end of the rope.

Grant Hutchison

George
2008-Apr-17, 09:26 PM
The free end of the rope moves one metre each second, but the other parts of the rope move at lesser speeds, proportional to their fractional distance along the length of the rope. So the ant moves steadily into regions that are moving more and more slowly relative to the fixed end of the rope. Yet it appears to be negative progression. In the illustration, P2 is 1 mps, but P3 is 2mps. If the ant finds itself back a P3 after one second, or 2 sec. for example, then it will never reach P2. Our ant must travel faster to offset the stretching rate.

Ken G
2008-Apr-17, 09:33 PM
After the first second, our ant will first reach P2, but in the next second it will be pulled back to P3 since at P2 the rope stretch rate is 1 mps. Thus, the ant has accomplished nothing! This procedure could be done forever and we'll get the same results.
The problem is you are imagining that the "Hubble constant" is really constant-- but it's not in Grant's analogy, it is the inverse of the age of the stretching (as though a particular spot [/I] on the rope [/I]had a particular speed all the time, that's what Grant is describing and is relevant to a gravity-free cosmology). You are right that if the stetching was maintained at 1 m/s every meter, the ant would never get there, but that's an accelerating universe that can maintain a fixed Hubble "constant". Bad terminology catches another one. (And if there's acceleration due to dark energy, you are right-- the ant won't get there!)

grant hutchison
2008-Apr-17, 09:35 PM
Yet it appears to be negative progression. In the illustration, P2 is 1 mps, but P3 is 2mps. If the ant finds itself back a P3 after one second, or 2 sec. for example, then it will never reach P2. Our ant must travel faster to offset the stretching rate.Well, that's what's in your picture, now I look at it again. But that's not the caterpillar/rope problem as originally posed, and it's not the analogy with the expanding Universe.
In the original problem, the free end of the rope moves at one metre per second, increasing the length of the rope by that amount each second. A point halfway along the rope will move at half a metre per second. A point quarter way from the free end moves at three-quarters of a metre per second, and a point quarter way from the fixed end moves at quarter of a metre per second. The ant constantly reduces its fractional distance from the fixed end of the rope, and therefore constantly moves into regions with lower velocities. In fact, if your ant is moving at a metre per second along the rope, it will never be walking on a bit of rope that is moving away from the fixed end faster than the ant approaches the fixed end.

Grant Hutchison

George
2008-Apr-17, 09:41 PM
The problem is you are imagining that the "Hubble constant" is really constant-- but it's not, it is the inverse of the age of the stretching (as though a particular spot on the rope had a particular speed all the time). You are right that if the stetching was maintained at 1 m/s every meter, the ant would never get there, but that's an accelerating universe that can maintain a fixed Hubble "constant". Bad terminology catches another one... Hmmmm, I'm not out of the tunnel yet.

Isn't the Hubble constant kinda constant? Is there really that much variation in the rate both now and then?

Assuming it is significantly different, our ant could get closer but wherever the ant is "now" it will be subject to the "now" Hubble constant and begin to loose "ground", thus never reaching us. What am I not seeing?

grant hutchison
2008-Apr-17, 09:50 PM
Isn't the Hubble constant kinda constant? Is there really that much variation in the rate both now and then?It gives you the velocity difference for a certain separation between observers, at a certain moment in time ("now"). As time goes by, and the observers move farther apart at constant velocity, then the velocity difference per separation distance must go down: the relative velocity is the same, but the separation has increased. That is, the Hubble "constant" has to change with time, if all objects retain the same velocity.
For the constant to be constant, expansion would have to accelerate.

Grant Hutchison

George
2008-Apr-17, 09:53 PM
Well, that's what's in your picture, now I look at it again. But that's not the caterpillar/rope problem as originally posed, and it's not the analogy with the expanding Universe.
In the original problem, the free end of the rope moves at one metre per second, increasing the length of the rope by that amount each second. A point halfway along the rope will move at half a metre per second. A point quarter way from the free end moves at three-quarters of a metre per second, and a point quarter way from the fixed end moves at quarter of a metre per second. Yes, and I had hoped my illustration showed this proportionality, though I did not show a free end rate since it goes on and on. But, let's make the free end 4mps, then half-way (P3) will be 2 mps, and 1/4 of the way from P1 will be 1 mps, as illustrated.


The ant constantly reduces its fractional distance from the fixed end of the rope, and therefore constantly moves into regions with lower velocities. Not if the ant is being pulled-back faster than its forward motion. The ants speed of 1mps is only relative to the rope underneath it, but the rope underneat it at P3 is being pulled back at 2 mps.


It is In fact, if your ant is moving at a metre per second along the rope, it will never be walking on a bit of rope that is moving away from the fixed end faster than the ant approaches the fixed end. Why do you say that if the rope is moving away from the fixed end at twice the speed of the ant's speed upon it?

grant hutchison
2008-Apr-17, 10:05 PM
Not if the ant is being pulled-back faster than its forward motion. The ants speed of 1mps is only relative to the rope underneath it, but the rope underneat it at P3 is being pulled back at 2 mps.Even if the ant is being pulled back faster than its forward motion: it's still reducing its fractional distance along the rope, because the rope stretches both ahead and behind it. Every time it takes even a tiny step along the rope, it reduces its fractional distance from the other end.


Why do you say that if the rope is moving away from the fixed end at twice the speed of the ant's speed upon it?Because, as I pointed out at the start of that paragraph, the original problem stipulated that the rope increased in length by a metre per second, which means that the free end moves at a metre per second, and all other parts of the rope move at a lower speed. I don't really understand why you've added extra rope and extra velocities. But if you want the ant to walk on rope that's initially moving at two metres per second then, yes, it will initially move away from the fixed end, but will then move towards it again as it walks on to parts of the rope that are moving more slowly than one metre per second.

Grant Hutchison

Ken G
2008-Apr-17, 10:29 PM
Isn't the Hubble constant kinda constant?Note over times comparable to the age of the universe. The two heuristic possibilities to consider are if H was always 70 km/s/Mpc, which is if the expansion started very slowly and has been gathering speed ever since (and our own "speed" is increasing in proportion to our distance from a certain distant galaxy), or if it has always been 70 km/s/Mpc times 13.7 billion years / age, which would make our speed relative to some distant galaxy always the same. Note the "truth" is neither of those, but there is no distinction if "age" isn't changing much, so that's what people generally mean when they talk about "the Hubble law". It simply isn't consistent with caterpillars crossing the full length of the rope, because then "age" is changing a lot, and the analogy breaks down. What I mean is, you should probably understand the constant-speed expansion case, and the constant-H expansion case, as "benchmarks for comparison", but neither is right for the universe.

grant hutchison
2008-Apr-17, 10:57 PM
Let come at it with some sums specific to your scenario:
After the first second, our ant will first reach P2, but in the next second it will be pulled back to P3 since at P2 the rope stretch rate is 1 mps. Thus, the ant has accomplished nothing! This procedure could be done forever and we'll get the same results.The ant is certainly now two metres from the fixed end of the rope, but it is standing on a piece of rope moving at only 1m/s, the velocity of P2. It has moved to the distance of P3, but has not acquired the velocity of P3.
In the next second, it can walk a metre towards the fixed end (halving its distance) and then be pulled back by only half a metre (because half a metre is added behind and half a metre ahead, when the rope next stretches).
In the next second, it is a metre-and-a-half from home, and it can walk a metre: two-thirds of the remaining distance. This time it is pulled back by only a sixth of a metre.
In the next second, it can cover the remaining distance, which is less than a metre.

The game reduces to summing the series 1/2+1/3+1/4, and finding that the total is greater than one: so we know that the ant has to make only three one-metre journeys to get to the fixed end of the rope after it arrives at P2.

Grant Hutchison

George
2008-Apr-18, 12:14 AM
For the constant to be constant, expansion would have to accelerate. Yes, the observers will see greater and greater increases to separation velocities, thus, in effect, acceleration would be the simple result, even if the expansion rate were held constant for all time.

I think I am showing that in the use of my elastic rope. The ant seems to be accelerating away from the fixed point, P1.


Note over times comparable to the age of the universe. The two heuristic possibilities to consider are if H was always 70 km/s/Mpc, which is if the expansion started very slowly and has been gathering speed ever since (and our own "speed" is increasing in proportion to our distance from a certain distant galaxy), I may misunderstand you. Why would the “expansion” need to gather speed to produce increasing separation velocities? The expansion rate could simply stay the same and these separation velocities will increase with time, right? [Added: meaning they become further apart, so their velocity will be seen to be greater in accordance with the "constant".]


..or if it has always been 70 km/s/Mpc times 13.7 billion years / age, which would make our speed relative to some distant galaxy always the same. Why is that? If the expansion rate were fixed, then our velocity will be increasing by 70km/s whenever we become a Mpc further away than before (ignoring cluster gravity, of course).


Note the "truth" is neither of those, but there is no distinction if "age" isn't changing much, so that's what people generally mean when they talk about "the Hubble law". I had thought that the early deceleration period and today’s acceleration period did not vary the Hubble constant that much. [I recall the line in the supernova results as it bent slightly away from a constant value, thus demonstrating acceleration.]

Regardless, I assume it has little bearing on our oversimplified ant analogy. If I can’t get the ant to P1, these changes to the Hubble constant will just be another bridge too far for me, though I realize these changes will have an impact on the appropriate speed our ant must have to reach P1 for any given point along the rope.

George
2008-Apr-18, 12:28 AM
Because, as I pointed out at the start of that paragraph, the original problem stipulated that the rope increased in length by a metre per second, which means that the free end moves at a metre per second, and all other parts of the rope move at a lower speed. Ah, I see the hiccup. Actually, the rope expansion is a rate per unit length, like the Hubble Constant. The elastic rope is being pulled such that every meter of distance has an additional 1 mps value. This is shown for each P point. I did state it as such, but my wording is a little rough, sorry. [My rate of improvement with English is not out of line with this ant analogy. I'm still rooting for the ant. :)]


I don't really understand why you've added extra rope and extra velocities. But if you want the ant to walk on rope that's initially moving at two metres per second then, yes, it will initially move away from the fixed end, but will then move towards it again as it walks on to parts of the rope that are moving more slowly than one metre per second. Perhaps my problem must be in understanding how the expansion is seen to expand. Because, if the ant moves further away from the 2 mps position, it will find itself moving away even faster [away] as it is now further from P1. And, the further away the greater the velocity. So, the ant continues to loose ground.

George
2008-Apr-18, 12:40 AM
Let come at it with some sums specific to your scenario:The ant is certainly now two metres from the fixed end of the rope, but it is standing on a piece of rope moving at only 1m/s, the velocity of P2. It has moved to the distance of P3, but has not acquired the velocity of P3. I don't follow you. It starts at P3 where the rope velocity relative to the fixed end is 2 mps, though 1 mps relative to the ant.


In the next second, it can walk a metre towards the fixed end (halving its distance) and then be pulled back by only half a metre (because half a metre is added behind and half a metre ahead, when the rope next stretches). I should clarifiy that P2 and the other points are always positioned relative to the fixed end. In the 2nd second the ant is back to P3 (using only a fixed 1 mps P2 rate). Thus, the problem seems to remain.

grant hutchison
2008-Apr-18, 07:10 AM
I don't follow you. It starts at P3 where the rope velocity relative to the fixed end is 2 mps, though 1 mps relative to the ant.At the end of the first second, you tell us, the ant arrives at P2, which is then drawn back by a metre. It's now facing a two-metre rope, and standing on a point which is drawn back by a metre each second. But it immediately starts walking onto parts of the rope that will move at lower speeds.


I should clarifiy that P2 and the other points are always positioned relative to the fixed end. In the 2nd second the ant is back to P3 (using only a fixed 1 mps P2 rate). Thus, the problem seems to remain.I know this is how you're thinking about the problem, but that's not how the analogy works, as I've said. The way you're thinking about it requires each part of the rope to accelerate as time goes by, whereas the original problem requires a uniform expansion at constant velocity.
So you're modelling a Big Rip universe in which, indeed, light never makes forward progress against the increasing expansion.

Grant Hutchison

Jeff Root
2008-Apr-18, 08:35 AM
George,

This animation shows the expansion: expand3c.htm (http://www.freemars.org/jeff2/expand3c.htm)
Each dot moves across the screen at a different, constant speed. Each
dot moves away from its nearest neighbors at the same, constant speed.
Each gap between adjacent dots widens at the same, constant rate.

To fit the arrangement in your drawing, the ant could start at the dot
on the far right, and walk to the left until it reaches the center dot, but
any pair of dots will serve equally well as the start and end points.

-- Jeff, in Minneapolis

Acolyte
2008-Apr-18, 09:58 AM
I'm puzzled - to be a valid analogy, wouldn't the ant also be getting stretched? So it would be moving at a steadily increasing pace (as viewed from outside the frame) or at an equivalent pace (inside the frame) no matter what the acceleration of the rope was...

Or has doofus just put foot in widening facial gap?

Jeff Root
2008-Apr-18, 10:16 AM
The ant does not get stretched.

Little things like ants, people, planets, stars, solar systems, globular clusters,
galaxies, and galaxy clusters are held together by nuclear, electromagnetic,
and gravitational forces, so they do not expand. The expansion is seen only
between galaxies which are very far apart -- not in the same cluster. Not in
the same supercluster, for that matter. So two galaxies which are only a
couple of million light-years apart usually are not separating from each other
over the long term, while two galaxies which are a billion light-years apart
almost certainly are going their separate ways.

-- Jeff, in Minneapolis

Ken G
2008-Apr-18, 10:50 AM
I may misunderstand you. Why would the “expansion” need to gather speed to produce increasing separation velocities? Because that's the meaning of "increasing separation velocities". Pick two galaxies a billion LY apart at a time a few billion years after the Beginning. In a gravity-free universe (not a terribly bad approximation, as dark energy almost balances dark matter), those two galaxies will always be separating at the same rate, and will always perceive the same redshift. Yes? But their separations are increasing in proportion to age. So the Hubble "constant" they derive is inversely proportional to age. Indeed, that's pretty much how you get a 14 billion year old universe from a 70 km/s/Mpc value of H. In your picture, where H is always the same, we could not have any idea how old the universe was-- the expansion would have started very very slowly at first, and most of the age would have been spent in that early slow expansion. But in reality, the value of H was very large when the universe was very young, so the current value does accurately reflect the inverse age of the universe.


I had thought that the early deceleration period and today’s acceleration period did not vary the Hubble constant that much. That's true-- those aren't the main things that make it vary. It varies inversely to age even if there is no gravity at all, so no deceleration and no acceleration. Those just complicate things, and let H vary in other ways as well. In fact, the acceleration might make the Hubble constant "level off" at something like it's current value-- and indeed the photons will no longer get very far.

George
2008-Apr-18, 05:04 PM
I know this is how you're thinking about the problem, but that's not how the analogy works, as I've said. The way you're thinking about it requires each part of the rope to accelerate as time goes by, whereas the original problem requires a uniform expansion at constant velocity. Ah ha! That is indeed where I have it wrong. In my ant analogy, I do have the active end of the rope accelerating in order to maintain the 1 mps/meter rate both now and later. You, in accord with modern cosmology, see the end of the rope moving at a constant velocity, which as been illustrated before in your references and Jeff's recent one, too.

This means that points P2 and P3 will eventually loose their current retracting velocities from P1 as time progresses. So, any advancement the ant takes will eventually produce a net gain in velocity relative to the rope, which is what everyone has been trying to make clear.

It is confusing since this means that the magical force (i.e. Dark Energy) that is causal to the expansion is becoming less and less per unit area. I don't recall reading about that, though it may have been in bold, underlined letters and I just forgot.

This also means, if I now "have it" , that the expansion rate for any given point (not the initial end-of-rope pulling rate) had to have been greater in the past than now, even if we hold the Hubble Constant constant. Perhaps learning of the [actual cosmological] acceleration added to my confusion since acceleration would have made the ant analogy valid.


Pick two galaxies a billion LY apart at a time a few billion years after the Beginning. In a gravity-free universe (not a terribly bad approximation, as dark energy almost balances dark matter), those two galaxies will always be separating at the same rate, and will always perceive the same redshift. Yes? Yes, if Dark Energy per unit area is wimping out. If not, then "I don't get it", still.

It seems to be a question at to the rate at which subdimensional(?) space is infusing itself into 3D space. If the infusion rate is fixed for all time per unit area (volume), then our two galaxies here will accelerate away from each other, and not have a fixed rate. But if the total rate (not per unit volume) of new space infusion into our space is fixed, then the constant separation velocites should remain fixed. At first glance, this latter case seems less natural, but I am limited on free time to think about it. Is my logic right, regardless of actualality?

Jeff Root
2008-Apr-18, 06:28 PM
You, in accord with modern cosmology, see the end of the rope moving
at a constant velocity....

This means that points P2 and P3 will eventually lose their current
retracting velocities from P1 as time progresses.
In my animation, and I think in Grant and Ken's simplified scenarios,
the velocity of recession between any two points is constant, after
the initial instant.

My animation shows the situation without gravity and without dark
energy. Those cause relatively minor tweaks to the expansion, so
we can ignore them and still have an accurate description of the
expansion to first-order approximation. I believe that is what Grant
and Ken are describing.

Again, my animation shows constant speed for everything, at all
times, from the initial instant. Everything is in motion at the start,
and the motion never changes.



It is confusing since this means that the magical force (i.e. Dark Energy)
that is causal to the expansion is becoming less and less per unit area.
I don't recall reading about that, though it may have been in bold,
underlined letters and I just forgot.
That does not sound like the standard Big Bang model, which doesn't
specify a cause for the expansion.

There appear to be three regimes of expansion: The initial, overall
expansion, cause of which is unknown, and is essentially ballistic; the
extremely brief period of inflation hypothesized to have occurred during
the first second of the Big Bang, caused by some kind of phase change;
and the acceleration of the expansion, cause unknown, but given the
generic label "dark energy".

So my animation, and (I think) what Grant and Ken have said, have no
bearing on dark energy. I see no evidence that suggests anything at
all about what dark energy does.

However, interestingly to me, it is pretty close to my own ATM idea of
what is going on.



This also means, if I now "have it" , that the expansion rate for any
given point (not the initial end-of-rope pulling rate) had to have been
greater in the past than now, even if we hold the Hubble Constant
constant.
Again, I don't think you can get that from what we have said.
You can only get it by adding in gravitational slowing of the expansion.

-- Jeff, in Minneapolis

George
2008-Apr-18, 08:49 PM
In my animation, and I think in Grant and Ken's simplified scenarios,
the velocity of recession between any two points is constant, after
the initial instant.

My animation shows the situation without gravity and without dark
energy. Those cause relatively minor tweaks to the expansion, so
we can ignore them and still have an accurate description of the
expansion to first-order approximation.
Yes. It now appears to me as more of an explosion event, though that is also an inaccurate view, where the outer chunks are traveling proportionately faster than the inner chunks and simply continue on into empty space. [This analogy is for velocities only, not space adding to itself and carrying our chunks away.]

On a historical note, I have read some early BB theory descriptions where the use of "explosion" was often used, in lieu of the more modern view that sees an expansion, instead.


So my animation, and (I think) what Grant and Ken have said, have no
bearing on dark energy. I see no evidence that suggests anything at
all about what dark energy does. A good theory should offer a mechanism to explain the effects observed. Dark Energy is that mechanism, though not very well understood yet, apparently.


Again, I don't think you can get that from what we have said.
You can only get it by adding in gravitational slowing of the expansion. I am refering to fixed distances from our Earth. Billions of years ago, galaxies close to us must have had a much great velocity away from us than any galaxy would now have if it happens to be the same distance away from us as the first galaxy had a billion years ago.

Ken G
2008-Apr-18, 09:35 PM
It is confusing since this means that the magical force (i.e. Dark Energy) that is causal to the expansion is becoming less and less per unit area. I don't recall reading about that, though it may have been in bold, underlined letters and I just forgot. The normal expectation is that dark energy stays fixed per unit volume, because "space is space" if you will. So it will not be "wimping out", it will be messing up the original "Hubble Law with H changing like 1/age" and turning it into something much more like your "truly constant H" example. So in a sense, the model that has been suggested has more historical value than present value, and your picture is becoming more and more correct as the universe ages. Apparently.

It seems to be a question at to the rate at which subdimensional(?) space is infusing itself into 3D space. If the infusion rate is fixed for all time per unit area (volume), then our two galaxies here will accelerate away from each other, and not have a fixed rate. But if the total rate (not per unit volume) of new space infusion into our space is fixed, then the constant separation velocites should remain fixed. At first glance, this latter case seems less natural, but I am limited on free time to think about it. Is my logic right, regardless of actualality?Yes, and it is indeed the latter case that does not seem to be happening, though at one time it was thought that it was.

grant hutchison
2008-Apr-18, 09:47 PM
Just to give due credit, the "small creature on an elastic rope" puzzle is credited to Denys Wilquin of New Caledonia; it first appeared in the December 1972 edition of Science et Vie.
Martin Gardner reproduced it in his Mathematical Games column in Scientific American, and it is republished in his collection Time Travel and Other Mathematical Bewilderments. As Gardner renders it:
A worm is at one end of a rubber rope that can be stretched indefinitely. Initially the rope is one kilometer long. The worm crawls along the rope toward the other end at a constant rate of one centimeter per second. At the end of each second the rope is instantly stretched another kilometer. Thus, after the first second, the worm has traveled one centimeter and the length of the rope has become two kilometers. After the second second, the worm has crawled another centimeter and the rope has become three kilometers long, and so on. The stretching is uniform, like the stretching of a rubber band. Only the rope stretches. Units of length and time remain constant. We assume an ideal worm, of point size, that never dies, and an ideal rope that can stretch as long as needed.
Does the worm ever reach the end of the rope?Although Gardner writes "worm", the creature in the accompanying illustration is undoubtedly a caterpillar.
The number of seconds it takes the caterpillar to reach the end of the rope is given by the first value of n for which:

(1 + 1/2 + 1/3 + 1/4 + ... + 1/n) > 100,000

This gives:

n = e100000-γ

where γ is the Euler-Mascheroni constant = 0.57721...

That's a long time. :)

Grant Hutchison

George
2008-Apr-19, 08:08 PM
The normal expectation is that dark energy stays fixed per unit volume, because "space is space" if you will. So it will not be "wimping out", it will be messing up the original "Hubble Law with H changing like 1/age" and turning it into something much more like your "truly constant H" example. Yes, I now understand your inverse age statement, though I think you may be the first to express it meaningfully. I have read a book or two on Big Bang and, apparently, never picked-up on it, assuming any effort was given to explain it. Perhaps a Ho(t) might help soften the hard-sounding "constant" term. Of course, we all love oxymorons, and a variable constant is a dandy! ;) [Do we have a thread to serve as a repository for improved, or new, terms? :) (blue dwarfs, G2V white dwarfs, Theorinos, use of "precision", etc.]

I expect the inflationary period is a better example of where my thinking had been, where space behaved much more analogous to the elastic rope in the OP.

Allow me to express my great appreciation for Ken and Grant (and Jeff,too) for the many times they've lent a helping hand. :clap: [This was one ditch I've needed out of for a long tome. ]

George
2008-Apr-19, 08:20 PM
Just to give due credit, the "small creature on an elastic rope" puzzle is credited to Denys Wilquin of New Caledonia; it first appeared in the December 1972 edition of Science et Vie. Interesting. I'm impressed with their intellectual approach. :)


The number of seconds it takes the caterpillar to reach the end of the rope is given by the first value of n for which:

(1 + 1/2 + 1/3 + 1/4 + ... + 1/n) > 100,000

This gives:

n = e100000-γ

where γ is the Euler-Mascheroni constant = 0.57721...

That's a long time. :)


It seems a little odd that it is quite this difficult mathematically. I did find a Wiki version (http://en.wikipedia.org/wiki/Ant_on_a_rubber_rope), too, but it looks cryptic.

I'll edit the OP to help others "eschew obfuscation".

Steve Limpus
2008-Apr-20, 06:43 AM
I'll edit the OP to help others "eschew obfuscation".

That it does George. :)

You might also enjoy this paper:

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf

George
2008-Apr-20, 06:49 PM
That it does George. :)

You might also enjoy this paper:

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf

Thanks, I skimmed only portions, but it's another love - hate paper.

I like this regarding redshift....


The key is to make it clear that cosmological redshift is not, as is often implied, a gradual process caused by the stretching of the space a photon is travelling through. Rather cosmological redshift is caused by the photon being observed in a different frame to that which it is emitted. In this way it is not as dissimilar to a Doppler shift as is often implied. The difference between frames relates to a changing background metric rather than a differing velocity.
I have never liked people putting poor little photons on the rack! "These folks are stretchin' the truth"... a proto-cosmologist and volunteer helochromologist. ;)

Regarding the expansion in their section on galactic cluster togetherness...


A clearer explanation is simply that on the scales of galaxies the cosmological principle does not hold, even approximately, and the FRW metric is not valid. The metric of spacetime in the region of a galaxy (if it could be calculated) would look much more Schwarzchildian than FRWlike, though the true metric would be some kind of chimera of both.
There is a strong taste of dust in my mouth right about here. :)

I was happier with my original rope view since now I have to think why such a fantasicly powerful force as the Dark Force does not accelerate galaxies. What kinda force generates no accelaration, or is space not friction free? :doh:

[I probably should start reading the Dark Energy and Expansion threads first, so no response is really necessary unless something y'all see something new here.]

Jeff Root
2008-Apr-20, 08:33 PM
George,

I agree with your opinions of the two quotes: The first is pretty clear; the
second pretty unclear, and the non-clarity is completely unnecessary.

It looks to me as though simple Doppler shift is the primary cause of the
cosmological redshift (gravitational redshift being a minor component),
but because it is so useful and so sensible to describe the geometric
relationships between far-flung parts of the Universe with co-moving
coordinates, the expansion gets attributed to the expansion of the
coordinate system itself, instead of the actual, physical cause(s).

Ignoring the acceleration of the expansion which was discovered ten
years ago now, it appears to me that the expansion is purely ballistic;
Matter is simply moving under its own inertia after having been given
an initial impulse about 14 billion years ago.

The acceleration could be caused by some kind of an expansion of
space itself, but there are other possible causes.

-- Jeff, in Minneapolis

George
2008-Apr-20, 08:52 PM
Ignoring the acceleration of the expansion which was discovered ten years ago now, it appears to me that the expansion is purely ballistic; Matter is simply moving under its own inertia after having been given an initial impulse about 14 billion years ago. That's what I now realize. However, this motion requires only impulse if we ignore gravity. Yet, Dark Energy is suppose to be 18x that of ordinary matter. Is that much force needed to maintain nearly fixed separation velocities?

grant hutchison
2008-Apr-20, 09:23 PM
It looks to me as though simple Doppler shift is the primary cause of the
cosmological redshift ...And yet Doppler red-shift and cosmological red-shift behave very differently. As Davis & Lineweaver (http://www.sciam.com/article.cfm?id=misconceptions-about-the-2005-03&page=4) point out:
According to the usual Doppler formula, objects whose velocity through space approaches light speed have redshifts that approach infinity. Their wavelengths become too long to observe. If that were true for galaxies, the most distant visible objects in the sky would be receding at velocities just shy of the speed of light. But the cosmological redshift formula leads to a different conclusion. In the current standard model of cosmology, galaxies with a redshift of about 1.5--that is, whose light has a wavelength 150 percent longer than the laboratory reference value--are receding at the speed of light. Astronomers have observed about 1,000 galaxies with redshifts larger than 1.5. That is, they have observed about 1,000 objects receding from us faster than the speed of light.
Grant Hutchison

George
2008-Apr-20, 10:20 PM
Ug, Grant. Don't tell me I'm wrong again. :(

Although it does seem that you are correct from what I have found (Wiki on SR Doppler (http://en.wikipedia.org/wiki/Relativistic_Doppler_effect)), I am not sure the way the expansion works will apply, as they seem to suggest, just as if the light were being emited from an object moving away from us in our own reference frame.

Consider our ant in its P3 position where it is loosing 1 mps. If the ant were in the P1 frame and not on an elastic rope, then it would never reach P1. The expansion of space, I think, should alter the SR Doppler formulation for a photon to incorporate an energy gain as it approaches us, just as our ant has a velocity gain. [I am shooting from the hip, admitedly. But, I can not sentence photons to the rack. :)]

grant hutchison
2008-Apr-20, 10:28 PM
Although it does seem that you are correct from what I have found (Wiki on SR Doppler (http://en.wikipedia.org/wiki/Relativistic_Doppler_effect)), I am not sure the way the expansion works will apply, as they seem to suggest, just as if the light were being emited from an object moving away from us in our own reference frame.Perhaps I misunderstand your intention, but what you've linked to is standard relativistic Doppler; cosmological red-shift behaves differently, as my Davis and Lineweaver quote describes.

Grant Hutchison

George
2008-Apr-20, 11:31 PM
Perhaps I misunderstand your intention, but what you've linked to is standard relativistic Doppler; cosmological red-shift behaves differently, as my Davis and Lineweaver quote describes. The results, at the quick glance I gave it, seem identical [to my Wiki reference]. Your reference says that the Doppler method requires that as v (velocity of light source) approaches c the wavelength goes to infinity. This I found to be true in the formula for relativistic Doppler. Are you saying this is not the case?

My hope is that the expansion itself, contrary to your link's view, makes a difference in how the Doppler shift is to be taken into account.

grant hutchison
2008-Apr-20, 11:40 PM
The results, at the quick glance I gave it, seem identical [to my Wiki reference]. Your reference says that the Doppler method requires that as v (velocity of light source) approaches c the wavelength goes to infinity. This I found to be true in the formula for relativistic Doppler. Are you saying this is not the case?My point was that relativistic Doppler red-shift goes to infinity as v approaches c, while cosmological red-shift does not.
When you wrote "it does seem that you are correct from what I have found", your link told only half that story. So I wasn't sure if you were missing my point, or making some other point.
I'm still not sure. :)

Grant Hutchison

publius
2008-Apr-21, 12:18 AM
George,

I'm gonna try to jump in here as this thread sort of caught my fancy. As I mentioned, I'm out in California now due to the sudden death of an aunt. (For those who didn't see my previous posts she fell in the street out here last week -- some blood vessels in her brain burst, causing massive brain damage. She died about 24 hours later). Anyway, I'm 3000 miles from home, the 3 hour time difference still has me sort of punch drunk, and I'm having to use an old Mac here for Internet access. This is the first time I've ever messed with a Mac. So I'm certainly not running on all eight cylinders, and don't be surprised if I get something wrong or otherwise say something plain stupid. Well, more stupid than my usual.

Anyway, what I wanted to ramble about is all this is a matter of coordinates. In inertial frames in flat space-time, the familiar SR rules about Doppler and everything else apply as normal. Go to a non-inertial frame even in flat space-time, say Rindler, and different rules apply. You still have a "relative velocity" contribution that you can pull out, but you have other stuff at work as well.

Now, go to curved space-time, and you get even more stuff. The "expanding universe" can be coordinatized in many different ways. And the language and mathematical formula for all this stuff depends on which coordinates you're using. But all will agree on what observers "see", that is, locally measure using their own ruler and clock. But those different coordinates will explain why those local observers see what they see in very different language.

You can take the standard comoving coordinates. There, your notion of "space" is metrically expanding with time, and that gives you a redshift according to that description. However, you can (for an idealized deSitter space-time at least) switch to a perfectly static set of coordinates. There things are different, and the redshift would be better described as "gravitational". It gives the exact same results for local observers.

The thing to appreciate is, with any space-time, there are invariants and then there are coordinates. The former is all that matters, and the latter just depends on your own choices. The rub is that things like space and time fall in the latter category. The fact that there is a redshift falls under the former, but "why", what is happenning to space and time, and relative motion between observers to make that light redshift the way it does comes under the latter category.

My point is, don't get to carried away with coordinate dependent descriptions of expanding universes like ants on rubber sheets. It works, but it's just one of many ways to describe it. It's not really "real", anymore than two observers in relative motion arguing over who is really moving and who is really stationary. All the ways of coordinatizing the expanding universe are no more or less real than that.


-Richard

RussT
2008-Apr-21, 01:06 AM
Hi Richard;

Very sorry to hear about your Aunt!

Might you be in Sacramento, or near there/here?

publius
2008-Apr-21, 02:25 AM
Hi Richard;

Very sorry to hear about your Aunt!

Might you be in Sacramento, or near there/here?

Thanks. No, I'm in the Oakland area (I think :lol: I was driving around, and it sort of hit me, "what the devil am I doing trying to drive around this far from home where I have no clue where I'm going or where I've been?").

My aunt was 86, and she truly lived a "full life". She was still getting around pretty good, still driving and doing all sorts of things. When she turned 80, she decided she wanted to go to London and took my mother with her for two weeks.

While a sudden bolt out of the blue death is hard on family and friends, thinking about it, it's a pretty good way to go. No extended illness and disability or lingering around.

Interestingly, the local authorities did order an autopsy, but that was more a formality as there is no indication of any foul play. There were no witnesses to her actual fall, which was on a public sidewalk (ironically, she was coming out of a doctor's office for a routine visit), they only found her lying there a little later and I think that's what triggered the autopsy.

There is one small question. According to the doctors, years ago, she had some uncontrolled high blood pressure (this would've been in her 50s, IIRC) for a while until they discovered it, and that certainly can weaken blood vessels. And advanced age does it too.

They say it could've gone either way. She could've misstepped or tripped and the trauma of the fall caused the blood vessels to rupture, or they could've ruptured spontaneously and that caused her to fall. I asked if this was a "big stroke" or an aneurism (sp?), and the doc said it was sort of "in between". It was a bigger event than a stroke, but not a full blown blow out of a large vessel, just smaller ones rupturing. Anyway, she went into a deep coma, with massive brain damage.

Luckily for my mother, the decision to pull the plug didn't have to be made. Had she pulled through, she would've been a vegetable. That decision would've come up shortly, but mercifully, she passed soon.

Doesn't matter what you call it, really, I don't guess.

-Richard

George
2008-Apr-21, 03:48 AM
My point was that relativistic Doppler red-shift goes to infinity as v approaches c, while cosmological red-shift does not.
When you wrote "it does seem that you are correct from what I have found", your link told only half that story. So I wasn't sure if you were missing my point, or making some other point.
I'm still not sure. :) My brevity quickly fails me. :) Your link surprised me with the statement that Doppler is not a fair model for large galactic redshifts. I found the wiki site which confirmed what I think he was saying, namely, that the SR Doppler equation shows that as a light source approaches c, then the wavelength approaches infinity.

Where we differ, I think, is in my hope that this SR Doppler is not applicable in the case of distant galaxies due to the expansion itself. This hope comes from the ant having the ability to reach its goal due to how the rope behaves. I see the ant gaining in relative momentum with the fixed end as time progresses. I have a hunch that a photon will gain momentum [actually, loose less momentum than the straight SR Doppler equation states], due to the behavior of the expansion. Is this view possibly correct? A simple yes would do nicely, however. :) [Chances are not great that I will understand any thorough answer.]

I'm glad to see your entry, Publius, though your loss is saddening.
T.S. Elliot once said, "What life have you if you have not life together". Life is about people most of all.

RussT
2008-Apr-21, 05:00 AM
Thanks. No, I'm in the Oakland area (I think :lol: I was driving around, and it sort of hit me, "what the devil am I doing trying to drive around this far from home where I have no clue where I'm going or where I've been?").

My aunt was 86, and she truly lived a "full life". She was still getting around pretty good, still driving and doing all sorts of things. When she turned 80, she decided she wanted to go to London and took my mother with her for two weeks.

While a sudden bolt out of the blue death is hard on family and friends, thinking about it, it's a pretty good way to go. No extended illness and disability or lingering around.

Interestingly, the local authorities did order an autopsy, but that was more a formality as there is no indication of any foul play. There were no witnesses to her actual fall, which was on a public sidewalk (ironically, she was coming out of a doctor's office for a routine visit), they only found her lying there a little later and I think that's what triggered the autopsy.

There is one small question. According to the doctors, years ago, she had some uncontrolled high blood pressure (this would've been in her 50s, IIRC) for a while until they discovered it, and that certainly can weaken blood vessels. And advanced age does it too.

They say it could've gone either way. She could've misstepped or tripped and the trauma of the fall caused the blood vessels to rupture, or they could've ruptured spontaneously and that caused her to fall. I asked if this was a "big stroke" or an aneurism (sp?), and the doc said it was sort of "in between". It was a bigger event than a stroke, but not a full blown blow out of a large vessel, just smaller ones rupturing. Anyway, she went into a deep coma, with massive brain damage.

Luckily for my mother, the decision to pull the plug didn't have to be made. Had she pulled through, she would've been a vegetable. That decision would've come up shortly, but mercifully, she passed soon.

Doesn't matter what you call it, really, I don't guess.

-Richard

Thanks for the response Richard, and again, sorry for your loss!
Oakland is a little over an hour from Sac, AND, Not a place you want to just drive around randomly!!! Just like many large cities in the US, it has beautiful and well developed (Oakland Hills, where the fires were many years ago) and historical places to visit, But there are certainly neighborhoods that are dangerous to be 'just driving around in'...Take care.

I defintely think that you are right about the going quick part though, as one of my great fears is a long prolonged, dragged out debilitating kind of process, which I could be facing, just like my Dad did with Emphasema (sp). I am not on Oxygen yet, but soon.

Anyway, take care and have a safe trip home.

Neverfly
2008-Apr-21, 05:27 AM
Thanks for the response Richard, and again, sorry for your loss!
Oakland is a little over an hour from Sac, AND, Not a place you want to just drive around randomly!!! Just like many large cities in the US, it has beautiful and well developed (Oakland Hills, where the fires were many years ago) and historical places to visit, But there are certainly neighborhoods that are dangerous to be 'just driving around in'...Take care.

I defintely think that you are right about the going quick part though, as one of my great fears is a long prolonged, dragged out debilitating kind of process, which I could be facing, just like my Dad did with Emphasema (sp). I am not on Oxygen yet, but soon.

Anyway, take care and have a safe trip home.

When I first moved to Austin, I ended up over on the East side of 12th street. (I'm a white guy) And when I was approached by a particular crowd, a bit of conversation revealed that I was from Oakland- East Side. No one messed with me after that bit of knowledge came to light...

publius
2008-Apr-21, 06:01 AM
Thanks and don't worry. My "spidey sense" usually keeps me out of places where I have no business. Basically, I just tooled around Alameda a bit, which is a beautiful little town. The architecture is striking and so far I've been well out of trouble. Everyone so far is just nice as they can be, and I do have local chaperones when we venture too far. :)

It's been a year or two since I last flew, and I tell you, it just ain't much fun anymore. Taking your shoes off and all that crap is just ridiculous. And it is just me or are the planes getting smaller and more cramped? :lol: I'm no aviation buff, so I can't tell you what plane it was, but the first leg out to Houston was on the this dinky little puddle jumper that was just about a sardine can.

Well, the only real complaint I had was a bad case of "airplane ears", which was brought on in that little puddle jumper. I've always been sort of chronically "stuffed up" -- one of my nostrils always seems to be clogged, and it varies from side to side -- and my ears always pop with pressure changes. But this time it was pretty bad, with some ear pain.

It was funny, when we landed in Houston, my ears were plugged up and hurting and stayed that way the whole layover. The leg to Oakland was on a 737, and the funny thing was my ears cleared right up once we got up to altitude and I had no problems since.

What I think was going on was it was a *repressurization* problem. That is, the pressure in my middle ear managed to equalize at low pressure, but stopped up quickly due to a sudden increase in pressure on landing (my Eustachian tubes may be blocked in some crazy check valve way :) ). When we got back up, the lower external pressure cleared it up, and then coming back down, the pressure change was slower.

I fear I'll have the same problem coming back and will be home with stopped up ears.

-Richard

Jeff Root
2008-Apr-21, 08:48 AM
It looks to me as though simple Doppler shift is the primary
cause of the cosmological redshift ...
And yet Doppler red-shift and cosmological red-shift behave
very differently. As Davis & Lineweaver point out:


According to the usual Doppler formula, objects whose velocity
through space approaches light speed have redshifts that approach
infinity. Their wavelengths become too long to observe. If that
were true for galaxies, the most distant visible objects in the
sky would be receding at velocities just shy of the speed of light.
But the cosmological redshift formula leads to a different conclusion.
In the current standard model of cosmology, galaxies with a redshift
of about 1.5--that is, whose light has a wavelength 150 percent longer
than the laboratory reference value--are receding at the speed of
light. Astronomers have observed about 1,000 galaxies with redshifts
larger than 1.5. That is, they have observed about 1,000 objects
receding from us faster than the speed of light.
Okay, I will simply deny that, and assert that all of those
galaxies, and the matter which emitted the cosmic background
radiation, were moving away from our current position at less
than the speed of light at the time the light was emitted.
As you suggested, "just shy of the speed of light."

Why should I use an ad hoc cosmological redshift formula instead
of the special relativistic Doppler formula?

-- Jeff, in Minneapolis

grant hutchison
2008-Apr-21, 08:51 AM
Okay, I will simply deny that ...Feel free.

Grant Hutchison

Steve Limpus
2008-Apr-21, 08:53 AM
Sorry for your loss, Richard.


...but soon.

The Maori have a saying when facing a mortal adversary:

'Kia Kaha'
Forever Strong

Go well friend.

Ken G
2008-Apr-21, 04:45 PM
Why should I use an ad hoc cosmological redshift formula instead
of the special relativistic Doppler formula?
Because presumably you will want a consistent picture, which means along with your Doppler formula, you will need concepts like time and distance. There are ways to supply such concepts such that you can use the Doppler formula and interpret v as the derivative of distance with respect to time, but problems will immediately crop up. Your distance will not agree with any distance indicators if you want your time to agree with any single clock, for example. If that doesn't sound like a problem to you, then you are fine, you can just use a different clock for every galaxy that you have a redshift for, and interpret the redshift as a Doppler shift in the frame of that one particular clock. That's the difference between a "reference frame" (one clock) and a "coordinate system" (an array of clocks). But as was pointed out by publius above, what is interesting in physics is not the language you can support by choosing a coordinate system, it is what invariant results you obtain by choosing a clock.

Jeff Root
2008-Apr-21, 11:33 PM
There are ways to supply such concepts such that you can use the
Doppler formula and interpret v as the derivative of distance with
respect to time, but problems will immediately crop up. Your distance
will not agree with any distance indicators if you want your time to
agree with any single clock, for example. If that doesn't sound like a
problem to you, then you are fine, you can just use a different clock
for every galaxy that you have a redshift for, and interpret the redshift
as a Doppler shift in the frame of that one particular clock.
That actually sounds pretty good, at first blush.

What is the alternative to using different clocks in different places?
It isn't obvious that such an alternative is possible.

-- Jeff, in Minneapolis

George
2008-Apr-22, 01:45 AM
There are ways to supply such concepts such that you can use the Doppler formula and interpret v as the derivative of distance with respect to time, but problems will immediately crop up. Your distance will not agree with any distance indicators if you want your time to agree with any single clock, for example. Grant gave a reference claiming SR Doppler will not work, but I suspect it does work with galaxies having the same time in expanding space. The SR Doppler failure for redshift would only be if we impose the Earth "reference frame" upon these calculations. Is this view remotely correct?

Ken G
2008-Apr-22, 03:20 AM
That actually sounds pretty good, at first blush.

What is the alternative to using different clocks in different places?Not different clocks in different places, different clocks for every different redshift that you are trying to interpret as a Doppler shift. That's the problem, you'll need a coordinate system that monkeys with time in a way that no single observer can observe. That's the problem with interpreting it as "really" a Doppler shift for all the galaxies at once.

Jeff Root
2008-Apr-22, 03:47 AM
Not different clocks in different places, different clocks for every
different redshift that you are trying to interpret as a Doppler shift.
Oh. I don't understand, but I agree that sounds no good.

Can you describe the problem, if I measured redshifts of galaxies at all
distances, and the cosmic background radiation, and tried to calculate
the speeds and distances using the relativistic Doppler formula? Where
would I run into trouble? What would the results be?

-- Jeff, in Minneapolis

George
2008-Apr-22, 07:55 PM
Restating, is the use of SR Doppler a failure regardless of the use of reference frames, or does some non-obtuse model allow its use? Either Grant's reference on SR Doppler is right or it assumes the use of a simple reference frame only, and understates the broader story.

[I am only interested in a yes or no taste, so I'm not asking for the whole enchilada. :)]

Ken G
2008-Apr-23, 09:57 PM
Can you describe the problem, if I measured redshifts of galaxies at all
distances, and the cosmic background radiation, and tried to calculate
the speeds and distances using the relativistic Doppler formula? Where
would I run into trouble?You could measure all the redshifts, and pick distance indicators, and place the whole thing in a grid of distances and speeds, certainly. But the problem is if you did that, you would lack a consistent dynamical description that unified those numbers, so they'd be scientifically meaningless. You would be looking at a universe with no discernable laws, they'd be so scrambled by your choice of coordinates, and no way to predict how an Earth observer would see things a billion years from now (that's not a particularly useful prediction, but let's just take it as an example) that would not be tantamount to converting to a sensible reference system first. So it's not that it would be wrong, it's that it would be a jumble of garbled information.

For example, if you got a redshift corresponding to 1,000 km/s at some distance, and tracked that redshift and distance every year for a billion years, you would generally not find that the speed you infer corresponded to the rate of change of your distance. You could select a reference frame (say get in a rocket) so that it might work for a little while for one galaxy, but it would not extend universally to the other galaxies you see, simply because it is not part of a unified and self-consistent description of universal dynamics. The only theory that we know of that succeeds at that is general relativity, and that theory admits no global inertial frames that you could use to specify coordinates where the observed redshifts are interpretable entirely as velocities. Instead, all the interesting dynamics would have to be subsumed into your coordinate system, and knowing how to do that would require solving general relativity, post-engineered to look like an arbitrary coordinate system with all kinds of awkward and specialized characteristics simply to allow you to interpret redshifts as speeds. You'd just get out what you put in, with zero predictive or unifying characteristics outside of what emerges from the general relativity you have worked so hard to obscure.

George
2008-Apr-23, 10:52 PM
There is something humbling about space itself being so empty yet so full of it that it almost laughs at me for my lack of incomprehensibility. My advice to astronomers is to set your Dark Matter pattern recognition software to "grin shapes". :)

If we attempted to use our own inertial reference frame, it seems to me we would get the Hubble Constant and it would vary inversely with age (as you have shown before). The problem I see is that it fails the further out we go. Grant's reference stated SR Doppler fails and the Wiki refernence states the equations that reveal why: as v --> c, then the redshift --> infinity. But, I assume that this is only true if we erroneously attempt to set everything within our own fixed inertial frame, as you show this is the wrong view.

About now, the ant seems to fall off the rope in this analogy. [Like all analogies, they loose their efficacy the more you stretch them ;). This becomes especially true regarding the ant's momentum, which stays fixed to the rope and, oddly enough, blueshifts after its intial redshift relative to the fixed end. Ug. Appropriately enough, that happens as the ant reaches the end of its rope. :D]

In addition, I was hoping GR allowed us to pick any reference frame and get similar results. No doubt it does, but they are all equally and invariably wrong (ignoring some minor anisotropy); we now have an infinite number of wrong answers. [Why do I suddenly fell like I'm taking a major chemistry exam? :)]

Does the best model lie in co-moving frames? If so, is there a pedagogical approach that a certain analogical ant can handle?

publius
2008-Apr-23, 11:56 PM
George,

Depends on what you mean by "best". :lol:

And I'm not exactly sure what you mean by all wrong answers. GR can model the observed behavior of our universe very well, adding in that cosmological constant/dark energy term. Plus dark matter, but you can count that as regular mass-energy as far as GR's source stress-energy is concerned. It's something different from the matter we're familiar with, of course, but it's just mass-energy as far as GR cares. And then there's the all important boundary/initial conditions. But any rate, what it observed agrees very well with valid GR solutions. Uncle Al done pretty good.

GR works, but it works the way it does, and not the way we might want it to work. The way we sometimes want things to work depends on notions of space and time and stuff embedded it in that we hold on to from our Newtonian/Galilean intuition that just don't apply. And that's what's going on here with you, I think. :) You've got a picture (of ants and ropes) in your head that doesn't work the way you're imagining.

I wish I could get this ant and rope stuff out of your head. It's entirely a coordinate dependent thing. :lol: Me, I don't sweat it. The co-moving coordinates are using a certain global clock, which amounts to a certain defintion of "now", and then defines space as "slices" (hyperslices) of that space-time at constant global clock time. Distances are thus defined along those slices.

And that is arbitrary. There are other ways to coordinatize. Other ways to slice it where the notion of space doesn't expand, it's just world lines of test particles fly apart due to global gravity.

The notion of distance and recessional velocity and all that good stuff would be very different there, but it would predict the exact same observations because it is describing the same space-time. It's just using different coordinates. It's like polar vs cartesians descriptions of a sphere, for instance.

Cosmologists like the comoving, expanding space description for a variety of reasons. And more power to them. However, the trouble there is these notions of distance, and expansion, and recessional velocity and all that get engrained and take on a reality beyond what they deserve. GR doesn't care about that. The invariants, the local "physical observerables", don't either.

However, our minds want to explain those locals in global terms, and there you go with the ants and ropes. There space is expanding between "spatial points" with time, and light emitted from one point to another gets stretched.

However, using deSitter space-time as a toy model, we can go to a perfectly static set of coordinates where nothing is changing with time. Our notion of "space" there is not expanding. However, there is a global gravitational field that causes a gravitational redshift. And that global field is radially "ripping", pulling everything apart equally in all directions. With time, stuff accelerates away and gets to moving. If something emits light, there is gravitational plus a doppler component that gives a redshift.

And that redshift agrees exactly with the expanding space coordinates for what a local observer would measure using his own local coordinates.

So don't worry so much about these ants and ropes and expanding space. It's all, well, relative.

-Richard

George
2008-Apr-24, 01:52 AM
The way we sometimes want things to work depends on notions of space and time and stuff embedded it in that we hold on to from our Newtonian/Galilean intuition that just don't apply. And that's what's going on here with you, I think. :) That's pretty close, but don't assume there is a whole lot "going on" with me. :) I am surprised that SR Doppler seem to fail, perhaps I am discounting the gravitational factors far too much, too.


You've got a picture (of ants and ropes) in your head that doesn't work the way you're imagining. I wish I could get this ant and rope stuff out of your head. No problem, ants and I don't really get along very well. [I hope you lack the experience of a fire ant attack.] :)


The notion of distance and recessional velocity and all that good stuff would be very different there, but it would predict the exact same observations because it is describing the same space-time. It's just using different coordinates. It's like polar vs cartesians descriptions of a sphere, for instance. That sounds like a good analogy, except I can mentally adjust cartesian to polar and back again. I have no handle yet on space, though receding galaxies with matching clocks (co-moving flying carpets carrying galaxies as space between the carpets oozes-up between Planck cracks in itself) vs. receeding galaxies in our own unique frame where galactic times are different. The latter seems wrong based on the clocks alone, though I'm kinda partial to the reference frame. :)


However, our minds want to explain those locals in global terms, and there you go with the ants and ropes. There space is expanding between "spatial points" with time, and light emitted from one point to another gets stretched. Oh mercy, the lament of the photons. Everyone makes light of them. ;) There just has to be a better way than stretching a photon, but not a galaxy (hardly), and, apparently, sucking the blood (energy) out of it in a mysterious way. [Where does its energy go? [/rhetorical]]


However, using deSitter space-time as a toy model, we can go to a perfectly static set of coordinates where nothing is changing with time. Our notion of "space" there is not expanding. However, there is a global gravitational field that causes a gravitational redshift. And that global field is radially "ripping", pulling everything apart equally in all directions. With time, stuff accelerates away and gets to moving. If something emits light, there is gravitational plus a doppler component that gives a redshift. This makes some sense, though I feel compelled to see a concentrated gravitational field else I don't see the redshift. Thus, I now have a unique spot in space.... but, I don't see any ants, so that should be an improvement, right?


So don't worry so much about these ants and ropes and expanding space. It's all, well, relative. This encourages me, but I've been a bit too corny already. :)

Here's another try for me. It would seem there is a decrease in the gravitational field with respect to time. When the Bang banged, the gravitational field strength of the universe was significantly higher than today. So, photons, in a sense, are continually climbing out of the well and are loosing their potential energy. If so, and if the expansion suddenly stopped, then SR Doppler would work, perhaps maybe. Since I don't recall anyone suggesting this, my confidence is high that I should have no confidence.

publius
2008-Apr-24, 07:58 AM
That's pretty close, but don't assume there is a whole lot "going on" with me. :) I am surprised that SR Doppler seem to fail, perhaps I am discounting the gravitational factors far too much, too.


SR only works in inertial frames in flat space-time. SR Doppler would fail for say two observers in a simple Schwarzschild field as well. SR Doppler would also fail for accelerating, non-inertial observers as well. (One can work it out from a global inertial frame and calculate what the accelerating observer sees in an instantaneous inertial frame, which is actually easier, but if you're doing things all from the non-inertial frame, you've got more than SR going on, you've got some GR machinery required there).

IOW, don't be so surprised that SR doesn't work outside its domain. :) If space-time is flat, well, stick with inertial frames and don't mess with "curved" coordinates of accelerating frames and avoid the additional complexity. However, when space-time is curved, not flat, then you've got to buck your ears and go with the full GR machinery. When there is real curvature afoot, SR fails you globally (locally, which means a region small enough that the curvature isn't significant, you can use SR).




No problem, ants and I don't really get along very well. [I hope you lack the experience of a fire ant attack.] :)


Oh, I know all about fire ants. I manage to get popped a couple of times per year now. Luckily I've never been swarmed, just let one or two of the little buggers get on me.

They're getting worse and worse. Last year, I asked a chemical salesman what was the best stuff he had for fireants, and he recommened a product named "Esteem". It's a growth regulator in the standard "grits" bait form. What the little molecule does is interfere with the growth cycle from larva to adult, sorting of stunting their growth, which messes up their operations. It worked pretty darn good last year. One treatment kept activity in the yard down to nearly nothing. They're ramping up again this year, and another treatment will be one of the first jobs I need to do when I get back home. :)

Fire ants are very clever, plus there's zillions of 'em in a colony, and it's very difficult to get a poison in there to get to the queen. IIRC, it takes weeks for new source to propagate to the queen, and if there's the slightest suspicion during the meantime, they kick out all the ants that touched the food source. And their sense of "smell", the ability to detect strange molecules is to a dog as a dog is to us! Makes it darn difficult to get them to take in and spread poisons. :) The growth regulator is a different approach that seems to work well. But they'll soon adapt to that in a few years and they'll have to come up with something else.

What were we talking about........:lol: I'm getting too tired and will pick up this tomorrow. Or sometime.

-Richard

George
2008-Apr-24, 12:59 PM
...if you're doing things all from the non-inertial frame, you've got more than SR going on, you've got some GR machinery required there.
That's what I thought.


IOW, don't be so surprised that SR doesn't work outside its domain. :) If space-time is flat, well, stick with inertial frames and don't mess with "curved" coordinates of accelerating frames and avoid the additional complexity. However, when space-time is curved, not flat, then you've got to buck your ears and go with the full GR machinery. When there is real curvature afoot, SR fails you globally (locally, which means a region small enough that the curvature isn't significant, you can use SR). I thought of what you said yesterday about the radial "ripping" and realized you may have been telling me what I thought no one had [probably had but it went over my head]. So I'm wondering if your "full GR machinery" is needed because of the gravitational effect on our photons. That seems likely, but is it ok for certain Newtonian-minded engineers to think of the redshift in photons as if they are exchanging a little color for potential energy, like a rock gains PE when it moves into a lesser gravitational field when thrown up?


They're getting worse and worse. Last year, I asked a chemical salesman what was the best stuff he had for fireants, and he recommened a product named "Esteem". It's a growth regulator in the standard "grits" bait form. I may try that. Some people use simple cornmeal, but there are some dedicated products we prefer.

The fire ants we have are the little bitty ones. 40 years ago, they use to be quiet large and I found one mound over one meter high. But, the little guys have taken over as they marched up from Mexico. They even beat the killer bees in getting here.


What were we talking about........:lol: You're right, we should talk now about ropes. ;)

tommac
2008-Apr-24, 03:26 PM
This feels similar to paying off a mortgage.

tommac
2008-Apr-24, 03:38 PM
My understanding of this problem is that regardless of the expansion there are still finite distances that are to be walked. The piece that is may be deceiving for some is that the expansion also pulls the ant along also so that the ant is moving much faster than its initial speed relative to its starting point. The further the ant moves the greater its relative velocity from the starting point. At one point the ant will be moving faster than expansion of the rope ( relative to the midpoint ) Eventually it will make it.

George
2008-Apr-24, 07:40 PM
At one point the ant will be moving faster than expansion of the rope ( relative to the midpoint ). This is a bit confusing since the ant's velocity relative to any point on the rope is always the same, 1 mps.

astromark
2008-Apr-24, 08:32 PM
If I might try to clarify this in my own tripping over my foot style of bumbling logic.This idea of ant walking along a ever stretching rope. Can I call it a Bungee cord?, for that allows the expansion I perceive as fact. The left hand end of our fictitious rope is stationary. It being the beginning point, start, We might call it the Big Bang. While the rope stretches away to the right, ([ :) not a political trend.]). The right hand end of our rope is unseen today as the stretch has taken it some 13.7 billion light years distant and may well be racing away at greater than C velocity. Into infinity. If. At some point along our rope the ant walked right he will never reach that end. Its accelerating away faster than the ant can go. The end does not exist. Back to reality... norr does the rope., and the ants can be dealt to... :(

grant hutchison
2008-Apr-24, 09:32 PM
Astromark, the whole point of this thread is the mathematical demonstration that the ant does reach the other end of the cord.

Grant Hutchison

George
2008-Apr-24, 10:02 PM
If. At some point along our rope the ant walked right he will never reach that end. Its accelerating away faster than the ant can go. The end does not exist. Back to reality... norr does the rope., and the ants can be dealt to... :( This is a new direction, literally. :)

At this point, we have changed the original scenario such that the hand pulling on the rope is pulling at a steady rate. So, keep this in mind to eschew obfuscation. :)

But, if we go back to my erroneous analogy of the expansion, then I would agree with the idea that the ant will not make it for the same reasons it would not have made it to the other end. Relativity explains it since we could change the reference frame fixing the pulling end and watching the "fixed" end accelerate away from us. I think I'm right.

astromark
2008-Apr-24, 11:06 PM
Astromark, the whole point of this thread is the mathematical demonstration that the ant does reach the other end of the cord.

Grant Hutchison

Yes and I can see that yes the ant would be on a moving ever faster platform and that given time enough he could theoretically reach the right hand end. 'No, he could not.'
It would seem to me that he will be moving ever faster as he moves right. So is the rope. He therefor could never reach that end that is ever accelerating away. Just as we would never catch him up. I can see him now, red shifted to a horrible crimson red. Where's the ant past.:)
Now I except that my logic may be wrong and the ant may not be out of reach yet... but we would need to hurry.... because we can not reach C. we would need to cover that rope very swiftly... but then if the rope we are transversing is already exceeding C then maybe we will catch the ant. Unfortunately tho, the end of the rope is gone and its end unobtainable. Can you agree with this?

and yes this is just a thought experiment. What might be proved to be possible in mathematics may not actually be so simple.... and yes you could use mathematics to prove this. but my head is beginning to ache at just the thought.:)-:)

grant hutchison
2008-Apr-24, 11:17 PM
Now I except that my logic may be wrong and the ant may not be out of reach yet... but we would need to hurry.... because we can not reach C. we would need to cover that rope very swiftly... but then if the rope we are transversing is already exceeding C then maybe we will catch the ant. Unfortunately tho, the end of the rope is gone and its end unobtainable. Can you agree with this?I'm not sure when the problem turned into our catching the ant, rather than the ant reaching the end of the rope.

The ant reaches the end of the rope, assuming the rope expands at constant velocity, and the ant walks at constant velocity relative to the part of the rope it is traversing. That was the original problem, and the derivation of the solution has already been given.

Grant Hutchison

astromark
2008-Apr-24, 11:18 PM
Oops! ... just spotted you changed the rules...:) Yes. the ant 'can'.

but no he cant. thanks for this interesting puzzle... I may never sleep.

tommac
2008-Apr-25, 12:59 AM
This is a bit confusing since the ant's velocity relative to any point on the rope is always the same, 1 mps.

wrong. The ant moves away from parts of the rope that is in its past at its speed plus the speed of the expansion of the part of the rope that is behind it.

Neverfly
2008-Apr-25, 03:02 AM
wrong. The ant moves away from parts of the rope that is in its past at its speed plus the speed of the expansion of the part of the rope that is behind it.
Bold Mine:

Originally Posted by George
This is a bit confusing since the ant's velocity relative to any point on the rope is always the same, 1 mps.

George
2008-Apr-25, 03:14 AM
wrong. The ant moves away from parts of the rope that is in its past at its speed plus the speed of the expansion of the part of the rope that is behind it. My phrasing was a bit muddy, sorry. Neverfly saw it as I meant it, but it would have been better to say that the ant will always see itself traveling 1 mps as it looks down upon the rope.

publius
2008-Apr-25, 06:40 AM
That's what I thought.

I thought of what you said yesterday about the radial "ripping" and realized you may have been telling me what I thought no one had [probably had but it went over my head]. So I'm wondering if your "full GR machinery" is needed because of the gravitational effect on our photons. That seems likely, but is it ok for certain Newtonian-minded engineers to think of the redshift in photons as if they are exchanging a little color for potential energy, like a rock gains PE when it moves into a lesser gravitational field when thrown up?




The best way to say it is we need the full GR machinery because our global space-time is curved, not flat. And there can be some confusion there. There is a difference between curvature of space-time, and the curvature of "space", which depends on how we slice it with coordinates. In the comoving coordinates, it appears our space is "flat". Space-time is the invariant construct, not how you split it into space and time. That can be very confusing sometimes.

Take deSitter space-time. You can write in the FLRW comoving form, and have a perfectly flat spatial part (albeit expanding with time). But switch the static coordinates, and your notion of space there is very much curved. But the curvature of the space-time is the same, because it's just different ways of mapping the same space-time.

My point is although it can drive you crazy to get your head wrapped around it, "space" is just an arbitrary, coordinate dependent construct. (which is why I don't worry too much about the various descriptions like ants and ropes and such :) ).

And finally, yes, you can think of a photon loosing energy as it climbs out of a gravitational well. However, you don't have to. In static (actually, just stationary) space-times, things work our very neatly. Consider a simple Schwarzschild field with two stationary observers, one deep down and the other high up.

They exchange light signals each says is fixed at some frequency according to his own local clock. The guy high up sees a redshift in the light coming up, while the guy deep down sees a blueshift.

In your Newtonian language, you can say the photons lost or gained potential energy. However, in the coordinate picture the photons didn't loose or gain anything. Their energy remained the same according to all observers.

The guy high up says the low down guy's clock is running slow and so he overestimates the frequency and energy of his transmitter. Because of the way space-time works, the path of a null geodesic, the wavelength stretches a bit as it climbs out of the well. But the energy remains the same.

And vice versa for the guy deep down. He says the high guy's clock is running way too fast, and so he underestimates the frequency and energy of his transmitter. The wavelength compresses as it comes down, but the energy remains the same.

Same thing with falling particles. The energy doesn't change. What happens is, at it falls, it's clock slows down so it's coordinate rest energy decreases. But it started out with higher energy, and the difference is "converted" to kinetic energy. :)

In stationary/static space-times, there is no such thing as gravitational energy. When you lift a weight, the work you're doing goes to speeding up the weight's clock (you're increasing it's "kinetic energy of motion through time" :lol: which is one way to think of rest energy). When you drop a weight and it does work, that energy comes from slowing down the clock as it falls.

Anyway, that's what Newtonian gravitational potential energy really is in the GR view, the difference in "coordinate rest energy". Maybe I shouldn't have mentioned that, because it can be confusing and blow your mind at first. But once you pick up the pieces and get in GR's warped frame of mind, it's beautiful. Slick and clever.

Now, deSitter space-time is completely static (even though you can use comoving style coordinates where it looks like space is expanding, and thus the metric is changing with time), and so the above tricks will work there.

It would be a bit more complicated and beyond me to show it, but we could do the same thing and say nothing gains or looses energy, even the redshifted photons.

However, our real LCDM universe is not deSitter, there's real mass-energy there in addition to lambda, and it turns out the space-time is dynamic, not stationary. It's close, and we (well the high priests who can actually do the math) could coordinatize in something very similiar to static deSitter. However, there would be a little time variation in there. It would be something different from the comoving expanding space variation, but still something would be changing with time in some fashion.

And so that means that the energy does appear to change -- we can do this "coordinate rest energy" being converted back and forth, nor keep photons at the same coordinate energy even as their wavelength stretches or compresses.

We've got to have a notion of gravitational energy there. (and that is a humdinger of a thing to do, actually. Turns out there is no invariant notion of gravitational field energy that can be defined. One observer can define something, but it doesn't agree with other observers in any way that be made globally consistent).

-Richard

Ken G
2008-Apr-25, 01:45 PM
Same thing with falling particles. The energy doesn't change. What happens is, at it falls, it's clock slows down so it's coordinate rest energy decreases. But it started out with higher energy, and the difference is "converted" to kinetic energy. :) Indeed, there is a "cute" way to imagine why the object falls in the first place, which is to say that falling experiences constructive interference in its wave function. The constructive interference stems from an extremum in the "action", which is like proper time here.

The reason falling represents an extremum of proper time is as follows. Imagine you are constrained to pass between two points in a certain time in the lab frame. Normally you maximize your proper time by moving at a fixed speed. But if there is gravity and the final point has time "running slower", you get more proper time elapsed by spending a higher fraction of the laboratory time in the higher-up regions where time is faster-- i.e., by starting slow and speeding up. But if you start too slow and speed up too much, you won't suitably "time dilate" the lab clocks relative to yours, so you won't maximize your proper time. Somehow it all works out!

tommac
2008-Apr-25, 02:32 PM
My phrasing was a bit muddy, sorry. Neverfly saw it as I meant it, but it would have been better to say that the ant will always see itself traveling 1 mps as it looks down upon the rope.

Hmmm ... but that point moves, so that complicates your statement( what is the distances we mark on the rope ) . Relative to the ant itself the ant is moving at 1 mps. Relative to any part of the rope it moves faster.

tommac
2008-Apr-25, 02:38 PM
As one falls into a gravity well and space-time curves, doesn't he BECOME further away from a point in a vacuum ( relative to the one in the gravity well )?

Conversely doesn't the light produced from a gravity well need more energy ( as there is a time dilation from the curvature of space-time) so that when the light exits the gravity well it blue shifts?



The best way to say it is we need the full GR machinery because our global space-time is curved, not flat. And there can be some confusion there. There is a difference between curvature of space-time, and the curvature of "space", which depends on how we slice it with coordinates. In the comoving coordinates, it appears our space is "flat". Space-time is the invariant construct, not how you split it into space and time. That can be very confusing sometimes.

Take deSitter space-time. You can write in the FLRW comoving form, and have a perfectly flat spatial part (albeit expanding with time). But switch the static coordinates, and your notion of space there is very much curved. But the curvature of the space-time is the same, because it's just different ways of mapping the same space-time.

My point is although it can drive you crazy to get your head wrapped around it, "space" is just an arbitrary, coordinate dependent construct. (which is why I don't worry too much about the various descriptions like ants and ropes and such :) ).

And finally, yes, you can think of a photon loosing energy as it climbs out of a gravitational well. However, you don't have to. In static (actually, just stationary) space-times, things work our very neatly. Consider a simple Schwarzschild field with two stationary observers, one deep down and the other high up.

They exchange light signals each says is fixed at some frequency according to his own local clock. The guy high up sees a redshift in the light coming up, while the guy deep down sees a blueshift.

In your Newtonian language, you can say the photons lost or gained potential energy. However, in the coordinate picture the photons didn't loose or gain anything. Their energy remained the same according to all observers.

The guy high up says the low down guy's clock is running slow and so he overestimates the frequency and energy of his transmitter. Because of the way space-time works, the path of a null geodesic, the wavelength stretches a bit as it climbs out of the well. But the energy remains the same.

And vice versa for the guy deep down. He says the high guy's clock is running way too fast, and so he underestimates the frequency and energy of his transmitter. The wavelength compresses as it comes down, but the energy remains the same.

Same thing with falling particles. The energy doesn't change. What happens is, at it falls, it's clock slows down so it's coordinate rest energy decreases. But it started out with higher energy, and the difference is "converted" to kinetic energy. :)

In stationary/static space-times, there is no such thing as gravitational energy. When you lift a weight, the work you're doing goes to speeding up the weight's clock (you're increasing it's "kinetic energy of motion through time" :lol: which is one way to think of rest energy). When you drop a weight and it does work, that energy comes from slowing down the clock as it falls.

Anyway, that's what Newtonian gravitational potential energy really is in the GR view, the difference in "coordinate rest energy". Maybe I shouldn't have mentioned that, because it can be confusing and blow your mind at first. But once you pick up the pieces and get in GR's warped frame of mind, it's beautiful. Slick and clever.

Now, deSitter space-time is completely static (even though you can use comoving style coordinates where it looks like space is expanding, and thus the metric is changing with time), and so the above tricks will work there.

It would be a bit more complicated and beyond me to show it, but we could do the same thing and say nothing gains or looses energy, even the redshifted photons.

However, our real LCDM universe is not deSitter, there's real mass-energy there in addition to lambda, and it turns out the space-time is dynamic, not stationary. It's close, and we (well the high priests who can actually do the math) could coordinatize in something very similiar to static deSitter. However, there would be a little time variation in there. It would be something different from the comoving expanding space variation, but still something would be changing with time in some fashion.

And so that means that the energy does appear to change -- we can do this "coordinate rest energy" being converted back and forth, nor keep photons at the same coordinate energy even as their wavelength stretches or compresses.

We've got to have a notion of gravitational energy there. (and that is a humdinger of a thing to do, actually. Turns out there is no invariant notion of gravitational field energy that can be defined. One observer can define something, but it doesn't agree with other observers in any way that be made globally consistent).

-Richard

tommac
2008-Apr-25, 02:40 PM
It must be in my wording because this is the effect that I have been trying to discuss when people tell me that space-time does not compress but rather only space disappears as it gets smaller. I agree with both of these posts.


Indeed, there is a "cute" way to imagine why the object falls in the first place, which is to say that falling experiences constructive interference in its wave function. The constructive interference stems from an extremum in the "action", which is like proper time here.

The reason falling represents an extremum of proper time is as follows. Imagine you are constrained to pass between two points in a certain time in the lab frame. Normally you maximize your proper time by moving at a fixed speed. But if there is gravity and the final point has time "running slower", you get more proper time elapsed by spending a higher fraction of the laboratory time in the higher-up regions where time is faster-- i.e., by starting slow and speeding up. But if you start too slow and speed up too much, you won't suitably "time dilate" the lab clocks relative to yours, so you won't maximize your proper time. Somehow it all works out!

George
2008-Apr-25, 06:01 PM
And finally, yes, you can think of a photon loosing energy as it climbs out of a gravitational well. However, you don't have to. In static (actually, just stationary) space-times, things work our very neatly. Consider a simple Schwarzschild field with two stationary observers, one deep down and the other high up.

They exchange light signals each says is fixed at some frequency according to his own local clock. The guy high up sees a redshift in the light coming up, while the guy deep down sees a blueshift.

In your Newtonian language, you can say the photons lost or gained potential energy. This seems to be an understanble analog for the cosmological shift; it can be seen as a shift due to a gravitational field gradient. Potential energy itself is pretty weird, but it is used so much and it is a simple construct that we all have comfortable familiarity with it.


However, in the coordinate picture the photons didn't loose or gain anything. Their energy remained the same according to all observers.

The guy high up says the low down guy's clock is running slow and so he overestimates the frequency and energy of his transmitter. Because of the way space-time works, the path of a null geodesic, the wavelength stretches a bit as it climbs out of the well. But the energy remains the same. Nuts, I'm back in the ditch with lots of ants. If we throw a rock upward it too doesn't loose energy, ignoring air friction, because it gains PE equal to its loss of KE. Can we look at the photon's wavelength, as well as, its place within the gravitational field and see the same thing? When photons were actually measured to blueshift as they were shot downward from a tall building, was it the difference in their measuring oscillator times (due to the gravitational field gradient) the only reason they compared wavelenths between the two observers and found a difference? Is this what you are saying?


Same thing with falling particles. The energy doesn't change. What happens is, at it falls, it's clock slows down so it's coordinate rest energy decreases. But it started out with higher energy, and the difference is "converted" to kinetic energy. :) Suddenly, there appears a short climb spot out of the ditch. :) I think this is close to what I am saying, right? Except I don't see much of a clock factor for particles.


In stationary/static space-times, there is no such thing as gravitational energy. When you lift a weight, the work you're doing goes to speeding up the weight's clock (you're increasing it's "kinetic energy of motion through time" :lol: which is one way to think of rest energy). When you drop a weight and it does work, that energy comes from slowing down the clock as it falls. That is hard to imagine. If I may be a little more georgepormorphic, is the pain difference between an anvil sitting on my foot and one dropped on my foot due to the difference in clock rates within the gravitational field? That hardly seems likely, but GR is nothing I know much about.


Anyway, that's what Newtonian gravitational potential energy really is in the GR view, the difference in "coordinate rest energy". Maybe I shouldn't have mentioned that, because it can be confusing and blow your mind at first. But once you pick up the pieces and get in GR's warped frame of mind, it's beautiful. Slick and clever. I can't quite tell if my mind is blown, but I am detecting quite a draft.

George
2008-Apr-25, 06:07 PM
Hmmm ... but that point moves, so that complicates your statement( what is the distances we mark on the rope ) . Imagine the ant with a wheel along side of it that has a speedometer attached to the wheel. The ant will see the speedometer always read the same speed, 1 mps.


Relative to the ant itself the ant is moving at 1 mps. How can our ant be moving away from itself?

grant hutchison
2008-Apr-25, 07:09 PM
I can't quite tell if my mind is blown, but I am detecting quite a draft.:clap:
I'm going to steal that. :)

Grant Hutchison

tommac
2008-Apr-25, 07:41 PM
Imagine the ant with a wheel along side of it that has a speedometer attached to the wheel. The ant will see the speedometer always read the same speed, 1 mps.

How can our ant be moving away from itself?

Yes ...
I meant in its own frame of reference but to nothing else.

tommac
2008-Apr-25, 07:48 PM
Yes ...
I meant in its own frame of reference but to nothing else.

In its perception of the world it is moving constantly at that rate.

publius
2008-Apr-25, 08:25 PM
Is this what you are saying?

............

That is hard to imagine. If I may be a little more georgepormorphic, is the pain difference between an anvil sitting on my foot and one dropped on my foot due to the difference in clock rates within the gravitational field? That hardly seems likely, but GR is nothing I know much about.

I can't quite tell if my mind is blown, but I am detecting quite a draft.


:) Yes, that's pretty much what I'm saying, although there are some more little details and complications there (mainly about different relative radial ruler lengths that come in as well as clock rates) we'd need to dot all the 'i's and cross all the 't's.

In the potential energy picture, you think of energy as being stored and released from the "field". Lift a weight up and do work against the field. Drop it and that energy returns as kinetic or other work depending on the setup. In the static GR view, however, that energy difference doesn't go to a "field", it just "stays with the weight".

Take a heavy anvil -- mc^2 there is a huge amount of energy. As you raise and lower it, the clock rate changes by a miniscule factor. But that small factor times mc^2 is a pretty large amount, and that's what breaks your foot when you drop it.

And I'll stress this just a coordinate picture, just doing book keeping in certain coordinates extended globally. IOW, I'm doing something similiar to the ant and rope thing. :) Globally, we have to do what Ken like to call "cockamamie physics". That is oscillators speed up or slow down when we move them, physical systems behave differently depending on their location and all that. Locally, everything is normal, but as we extend things out in "space", it goes cockamamie. But we get to conserve energy and momentum -- it all stays with the particles, or gets exchanged with ineractions between them. Ie the anvil falls and hits your foot. Some of its rest energy was converted to kinetic energy which was then transferred to your foot. Now, different observers, with their different rulers and clocks will disagree on how much energy and momentum things have to begin with, but they can all keep it with the particles in this fashion.

In the PE view, the "field" transferred energy to the anvil as it fell. In the static GR view, there was no transfer, just energy that was there all along, but got "warped" around due to the curvature. :)

Again, this is one way to do things. There are others. One is just "defer to local observers who can measure things" and declare energies and momenta are what they measure. Then, the photon looses or gains energy, falling particles gain energy, etc from the field in a PE-like fashion.


-Richard

George
2008-Apr-25, 08:52 PM
:clap:
I'm going to steal that. :) Good to see it. I try to hold a lot of cards, but this is my only strong suit. :)

George
2008-Apr-25, 09:08 PM
:) Yes, that's pretty much what I'm saying... You were suppose to say no, and draw a simple PE-KE analogy so I could move along my semi-Newtonian way.


In the potential energy picture, you think of energy as being stored and released from the "field". Lift a weight up and do work against the field. Drop it and that energy returns as kinetic or other work depending on the setup. In the static GR view, however, that energy difference doesn't go to a "field", it just "stays with the weight". I must admit this sounds better than the mysterious PE alternative. I've never noticed the "field" change much no matter how high I've jumped.


Take a heavy anvil -- mc^2 there is a huge amount of energy. As you raise and lower it, the clock rate changes by a miniscule factor. But that small factor times mc^2 is a pretty large amount, and that's what breaks your foot when you drop it. This looks pretty meaty, but the only possible variable is the mass. If you are treating the mass as being relativistic due to the change in clock rates as the height of the anvil changes, which makes some sense, does this difference match what my foot feels? Is it really that equivalent in energy difference compared to the PE-KE analogy?


In the PE view, the "field" transferred energy to the anvil as it fell. In the static GR view, there was no transfer, just energy that was there all along, but got "warped" around due to the curvature. :) I like the clock time differential better, but you are probably saying the same thing, right?

[I'll be out for a day. Thanks for all the help to all of y'all.]

publius
2008-Apr-25, 11:12 PM
This looks pretty meaty, but the only possible variable is the mass. If you are treating the mass as being relativistic due to the change in clock rates as the height of the anvil changes, which makes some sense, does this difference match what my foot feels? Is it really that equivalent in energy difference compared to the PE-KE analogy?

I like the clock time differential better, but you are probably saying the same thing, right?




Well, the variable *is* the clock rate, well, I should say the variable is the metric. In Schwarzschild, this is the expression for the coordinate rest energy:

E = mc^2 *sqrt(1 - R/r) = mc^2 * sqrt(g_00)

Here, R = 2GM/c^2, the Schwarzschild radius, and r is the radial distance coordinate of a far away observer. The spatial coordinates are spherical centered around the mass. Both that and the time coordinate are technically those of an observer at "spatial infinity" where space-time becomes asymptotically flat. He has the fastest clock rate, and he is our reference.

g_00 is the time-time part of our metric, which can be represented as a 4x4 matrix. g_00 is then the upper left corner of the matrix and governs stationary clock rates. That is, dT/dt = sqrt(g_00), where T is tau, the local proper time. I'm not certain this energy formula holds for the most general possible cases where the metric is doing everything it could be doing, but that's at least what it reduces to for simple static space-times.

That is the coordinate rest energy. If we're in flat space-time, the domain of SR, g_00 is simply one, and we have the familiar E = mc^2. But here, were not in Minkowski's Kansas anymore, and we've got a non-zero g_00.

As r --> infinity, g_00 goes to one, but for r finite, g_00 is less than one. If we have a test mass at some 'r', you have a difference from there to infinity of mc^2*(1 - sqrt(1 - R/r), and that's where the apparent potential energy comes from.

Now, the way things work, the transform to coordinates of local stationary observers cancel the metric fators out so each observers thinks his own clock rate is unity (and his own ruler is unity as well) locally. If we have some observer sitting at r1, then he says the coordinate rest energy is this:

mc^2*sqrt[ (1 - R/r) / (1 - R/r1) (For simplicity we're still using our own radial coordinate 'r', the transform to his own notion of radial coordinate would make it more complex. :) )

And you can see that is simply scaling things to his own local clock rate. When r = r1 and the test particle is sitting with him locally, he says the rest energy is mc^2. At infinity, he says it is higher than that by a factor of sqrt[1/(1 - R/r1)].

Note that all these expressions blow up at r = R, the event horizon of a Schwarzschild black hole. There's no real problem there, it's just our notion of stationary fails. Nothing can be stationary there, and so we can't think of observers sitting still at r <= R at any time on our external clocks.

But for an observer hovering asymptotically close to the horizon, and feeling a force asymptotically approaching infinity, he would say the object falling in from infinity was moving asymptotically at light speed and had infinite energy. That's that factor of 1 - R/r1 in the denominator going to zero. However, we the distant observer would not say that energy is infinite, but that the rest energy part went to zero (stationary clocks stop at the horizon), and all the energy was converted to kinetic energy.

That may be a lot to digest, and don't worry about it too much, I just wanted to show how this works out, how the this little coordinate energy game is played.

-Richard

astromark
2008-Apr-26, 01:33 AM
It would be a very boring world if we were all thinking the same...
But do we need to be sooo... different.
I see questions and answers. some of the questions have been answered in hundreds of ways, and yet...This Ant on the stretching rope has been explained and explained again.
Remembering that it was only introduced as a tool to help those understand who do not. My advise to those would be... just close your eyes and imagine what the universe is doing. Expanding. How that looks is entirely related to your point of observation.
From a point on that rope near that ant. Here he comes, or goes depending on your position. Relative to him. (IT). From off the rope he could flash past you at the speed of light. I do not see any thing new here, is there?
How many ways can we say the same thing?

My brother found it amusing to shout at me...' If you do not agree with me. Its because you are not listening.'
Having my own opinion was obviously forbidden.. many years have past. little has changed.
I see it happening here.

Neverfly
2008-Apr-26, 01:53 AM
It would be a very borring world if we were all thinking the same...
But do we need to be sooo differant.
I see questions and antswers. some of the questions have been antswered in hundreds of ways, ant yet...This Ant on the stretching rope has been explained and explained again.
Remembering that it was only antroduced as a tool to help those understant who do not. My advise to those would be... just close your eyes and amagine what the universe is doing. Expanttng. How that looks is antirely related to your point of obsevation.
From a point on that rope near that ant. Here he comes, or goes depanting on your position. Relative to him. (IT). From off the rope he could flash past you at the speed of light. I do not see antything new here, is there?
How many ways can we say the same thing?

My brother found it amusing to shout at me...' If you do not agree with me. Its becouse you are not listening.'
Having my own apinion was obviously forbiden.. many years have past. little has changed.
I see it happening here.
Hmmm:think:
Hey:lol:
It still works....!

astromark
2008-Apr-26, 02:06 AM
:)-:) Oops Lol...and I fixed it...sort of.
Yes it works...

Michael Noonan
2008-Apr-26, 02:20 AM
New research (http://www.nytimes.com/2008/04/25/science/25math.html?_r=2&ref=science&oref=slogin&oref=slogin) indicates mathematics works better without the assistance of reality.

Neverfly
2008-Apr-26, 02:26 AM
New research (http://www.nytimes.com/2008/04/25/science/25math.html?_r=2&ref=science&oref=slogin&oref=slogin) indicates mathematics works better without the assistance of reality.

I think that article has the wrong premise- The idea behind the train is not to teach kids math.
It is to teach kids how to apply their math to real world situations.

Neverfly
2008-Apr-26, 02:28 AM
:)-:) Oops Lol...and I fixed it...sort of.
Yes it works...

Sorry.. I couldn't resist:D

I hope you didn't think I was being pedantic.

Anyway...

Sorry for the antics. Let's get back on topic.

Michael Noonan
2008-Apr-26, 03:01 AM
Sorry.. I couldn't resist:D

I hope you didn't think I was being pedantic.

Anyway...

Sorry for the antics. Let's get back on topic.

Well I should have anticipated that.

Still back on track ...

:lol:

Neverfly
2008-Apr-26, 03:04 AM
George is gonna kill me...

Bullet

proof

vest

goes

here
.

tommac
2008-Apr-26, 03:14 AM
Bold Mine:

wrong

Neverfly
2008-Apr-26, 03:22 AM
Originally Posted by Neverfly
Bold Mine:

Originally Posted by George
This is a bit confusing since the ant's velocity relative to any point on the rope is always the same, 1 mps.
wrong
I put it in bold to show you were you were mistaken.

George already clarified this...

Instead of just blurting out "Wrong." Why don't you take like fifty seconds to explain why you think so?

You are wrong. Read back over the posts- and you might see that.
The ant- perceives 1mps relative to him.
That is what the ants speedometer would tell him.

Now, if the ant knows the rate of expansion- etc- he can do some calculations- but this does not change the ants perception.

ETA: I adjusted the quoting to reflect properly- If you could please do that when you quote. Your original post above mine just showed a quote from me saying, "Bold Mine" and nothing else. It took me a minute of staring at it to figure out what on Earth you were talking about...

tommac
2008-Apr-26, 03:32 AM
i disagree


I put it in bold to show you were you were mistaken.

George already clarified this...

Instead of just blurting out "Wrong." Why don't you take like fifty seconds to explain why you think so?

You are wrong. Read back over the posts- and you might see that.
The ant- perceives 1mps relative to him.
That is what the ants speedometer would tell him.

Now, if the ant knows the rate of expansion- etc- he can do some calculations- but this does not change the ants perception.

ETA: I adjusted the quoting to reflect properly- If you could please do that when you quote. Your original post above mine just showed a quote from me saying, "Bold Mine" and nothing else. It took me a minute of staring at it to figure out what on Earth you were talking about...

Neverfly
2008-Apr-26, 03:43 AM
i disagree



Instead of just blurting out "Wrong." Why don't you take like fifty seconds to explain why you think so?


Or in this follow-up post- Spell out why you disagree.

publius
2008-Apr-26, 06:48 AM
Indeed, there is a "cute" way to imagine why the object falls in the first place, which is to say that falling experiences constructive interference in its wave function. The constructive interference stems from an extremum in the "action", which is like proper time here.

..........

Somehow it all works out!

Ken,

That's actually exactly how the geodesic equation of motions are determined. By extremizing the proper time (which is equivalent to the action -- the equivalence principle cancels out the 'm' cleanly so that they are equivalent).

You have your metric ds^2 expressed in your coordinates, time plus your space coordinates. A world line, a single dimensional path, through the space-time then can be parameterized by some parameter, 'tau' call it, and thus your world line is thus a series of unkown equations x(tau), y(tau), z(tau), and t(tau).

You plug that into your ds expression from the metric and extremize that path integral ds(tau) by variational methods (although it's been a while and I'm rusty, I found the calculus of variations to be some of the most beautiful math I'd come across, but it has been a while). (And it turns out that "tau" is just the proper time along the world line, and s = c*tau of course). The resulting differential equations are the geodesic equations. That even works for null geodesics, where s/tau is always 0, there's just some trick you do to switch to some other parameter somehow. And the spatial geodesics (hold t constant in your coordinates) work in a similiar fashion, although the extreme flips. On the time-like side it's the *maximum* 's', proper time, but on the space-like side, it's the "shortest distance" between the events at constant time.

So the geodesics are just the same principle as the "shortest distance" between two points, just with the complication (because your "norm" can be positive or negative) that it's "longest distance" on the time-like side, the triangle inequality being flipped right around. Sort of weird, but really the same thing just generalized more.

And, of course, it's just the "Least Action" at work as well as you noted, which is a very deep principle that seems to underlie everything. It's amazing how it's there lurking behind everything, with vastly different mathematical constructs, such as GR vs quantum theory!

IIRC, Feynman loved to show how just about anything could be derived from Least Action (or Extreme Action, or better Stationary Action, I should say).


Indeed the EFE itself comes from this. Space-time itself curves due to the presence of mass-energy so that the action (Einstein-Hilbert action it's called) of that space-time is stationary. But when you do that, you're running with the Big Boys, doing some high powered, high priest 'rithmetic shore nuff. But it's still conceptually varying a path integral, but a humdinger of path integral involving big messes of tensor expressions.

Like I said above, it's all very beautiful and slick and profound seeming, but as I get older I realize (and we've discussed this) that we tend to read too much into reality and get carried away with our mathematical constructs. It may be neat, but maybe not "Mind of God" neat. :)

-Richard

George
2008-Apr-26, 07:14 PM
Well, the variable *is* the clock rate, well, I should say the variable is the metric. In Schwarzschild, this is the expression for the coordinate rest energy:

E = mc^2 *sqrt(1 - R/r) = mc^2 * sqrt(g_00)
Would this equate to our PE value? Say the anvil is 116 kg (a 2.1.3 anvil, 255 lbs.). At 1 meter we have 126 J of extra energy compared to the rest anvil on my, say, left foot. For my right foot to withstand the fall of the dropped anvil will require no small feat. [That's for your Neverfly, ;); puntificate on that one.]

You don't have to crunch it for me since I ask only to see how I can tie the two models together. The results experienced by my feet must be the same using either model, of course.


As r --> infinity, g_00 goes to one, but for r finite, g_00 is less than one. If we have a test mass at some 'r', you have a difference from there to infinity of mc^2*(1 - sqrt(1 - R/r), and that's where the apparent potential energy comes from. Is the difference between the anvils a delta r, and best taken from our observer at infinity? I can't connect this to the 126J PE value.



And you can see that is simply scaling things to his own local clock rate. When r = r1 and the test particle is sitting with him locally, he says the rest energy is mc^2. At infinity, he says it is higher than that by a factor of sqrt[1/(1 - R/r1)]. I agree but I am not seeing the equivalence connection to Newton's results. [I had a hunch an anvil would turn into an appropriate metaphor. :wall: I do appreciate your efforts, nevertheless.]

publius
2008-Apr-26, 07:52 PM
George,

It's fairly simple to see:

The difference in rest energy from above is:

mc^2[1 - sqrt(1 - R/r)]

That is, that's the difference in the rest energy at infinity and the rest energy sitting at rest at some 'r' down in the well. And I claim that is just the Newtonian potential energy, or more precisely that it reduces to that in the weak field (R/r very small) limit.

We'll apply the binomial theorem -- sqrt(1 + a) = (1 + a)^(1/2) expands to 1 + 1/2 a + ....... For small 'a', we can ignore the higher order terms (the .....). Here our 'a' = (-R/r) so sqrt(1 - R/r) ~ 1 - R/2r for small R/r. So plugging this approximation in the above, our difference in rest energy is thus:

mc^2[ 1 - (1 - R/2r) ] = mc^2 * R/2r.

Well R, the Schwarzschild radius is 2GM/c^2. The 2 and the c^2 cancel and we're left with simply:

m * GM/r

And that is just the Newtonian potential energy.

Slick, ain't it? :lol: Actually I should've shown this in the previous post, but I know this stuff works out and so I don't worry about it.


-Richard

Ken G
2008-Apr-26, 08:03 PM
Again, this is one way to do things. There are others. One is just "defer to local observers who can measure things" and declare energies and momenta are what they measure. Then, the photon looses or gains energy, falling particles gain energy, etc from the field in a PE-like fashion.Yes, as you know I really like to "defer to the local observer", I think it is part of the structure of science to let local observers decide their reality, and treat everything else as a coordinate transformation into the language (i.e., not the reality, which is local) of other observers. What we call "physics" is then a kind of union of what is real to us and what is useful language about what is real to someone else (possibly someone hypothetical).

Indeed, in this spirit, not only should we seek an observer at the same place as the "anvil", but even one that is instantaneously moving with the anvil, i.e., we should let the anvil "speak for itself". That tells us what is the reality of an anvil (or as I like to say, the projection of what is real about an anvil onto our objective capability to model an anvil). Then, as necessary, we may transform to local observers who are moving, and decide that the anvil has more energy for those observers, and then we transform to nonlocal observers at other gravitational potentials, and again we shift the energy. In that light, all these energy changes are just coordinate transformations, and there is no mystery why energy is not conserved, because coordinate changes don't conserve energy. The only energy conservation for a falling anvil is that it's still the same anvil.

In this sense, there is nothing "magic" about energy conservation, all the magic is in the way we do the coordinate transformations-- and in a sense we put that in ourselves as part of how we conceptualize the whole business. We are the magicians, and that's probably the proper place to look for magic (i.e., sleight of hand). So when we state the "universal principle" that "energy is conserved", we are really just saying "an anvil is an anvil", and then looking in the mirror and marveling at all the beautiful baggage we added to that as we conceptualized the whole situation.

a1call
2008-Apr-26, 09:39 PM
I can't quite tell if my mind is blown, but I am detecting quite a draft.



:clap:
I'm going to steal that. :)

Grant Hutchison

Sorry Grant,
From what we have seen from you, you won't get many occasions where you can use it.

George
2008-Apr-26, 11:11 PM
We'll apply the binomial theorem -- sqrt(1 + a) = (1 + a)^(1/2) expands to 1 + 1/2 a + ....... For small 'a', we can ignore the higher order terms (the .....). Here our 'a' = (-R/r) so sqrt(1 - R/r) ~ 1 - R/2r for small R/r. So plugging this approximation in the above, our difference in rest energy is thus:

mc^2[ 1 - (1 - R/2r) ] = mc^2 * R/2r.

Well R, the Schwarzschild radius is 2GM/c^2. The 2 and the c^2 cancel and we're left with simply:

m * GM/r

And that is just the Newtonian potential energy.
:clap: Impressive. Thanks. There's hope.

George
2008-Apr-26, 11:34 PM
Indeed, in this spirit, not only should we seek an observer at the same place as the "anvil", but even one that is instantaneously moving with the anvil, i.e., we should let the anvil "speak for itself". That tells us what is the reality of an anvil (or as I like to say, the projection of what is real about an anvil onto our objective capability to model an anvil). Then, as necessary, we may transform to local observers who are moving, and decide that the anvil has more energy for those observers, and then we transform to nonlocal observers at other gravitational potentials, and again we shift the energy. In that light, all these energy changes are just coordinate transformations, and there is no mystery why energy is not conserved, because coordinate changes don't conserve energy. The only energy conservation for a falling anvil is that it's still the same anvil. Yes, this makes some sense. My thermo is very rusty, so I could be wrong, but I think your view is elegant if for no other reason than the fact that the anvil does not get hotter when work is performed to elevate it. [I remember long ago wondering where the meat was in potential energy. It seemed contrived, but effective.]

Still, the causal issue and the beauty of forces seem to present a more palatable model, though at some point relativistic terms are required.

[I think I'm going to go drink some milk, now. ;)]

tommac
2008-Apr-27, 01:48 AM
Or in this follow-up post- Spell out why you disagree.

I believe that I disagree with everything you have to say.

publius
2008-Apr-27, 06:26 PM
Yes, this makes some sense. My thermo is very rusty, so I could be wrong, but I think your view is elegant if for no other reason than the fact that the anvil does not get hotter when work is performed to elevate it. [I remember long ago wondering where the meat was in potential energy. It seemed contrived, but effective.]

Still, the causal issue and the beauty of forces seem to present a more palatable model, though at some point relativistic terms are required.

[I think I'm going to go drink some milk, now. ;)]

George,

We wouldn't say the anvil gets hotter or colder as its coordinate rest energy changes anymore than we would say a moving anvil is hotter than an anvil at rest (the rest energy is sort of kinetic energy of motion through time if you like, but that's sort of just word games that aren't really rigorous, of course). When something is above absolute zero, all the particles have a bit more thermal energy over their rest energy.

When I think of doing thermodynamics in curved space-time, well, my mind begins to feel that big draft. It's a big enough mess even for SR (what it the "coordinate temperature" of something moving near c, when it's clock appears very slow and it's contracted along the line of motion). Go to curved space-time, and it gets even more cockamamie.

So, if I need to know the anvil's temperature, I am definitely going to defer to the local observer in the rest frame of the anvil. Whatever temperature he measures is good enough for me. :)

And you may wonder about the thermal energy of the anvil. Well, just think of the total energy as being subsumed in some notion of 'm', and as we raise and lower it, the difference in clock rates gives us that 'm'*GM/r potential looking energy.

-Richard

George
2008-Apr-27, 08:15 PM
We wouldn't say the anvil gets hotter or colder as its coordinate rest energy changes anymore than we would say a moving anvil is hotter than an anvil at rest (the rest energy is sort of kinetic energy of motion through time if you like, but that's sort of just word games that aren't really rigorous, of course). When something is above absolute zero, all the particles have a bit more thermal energy over their rest energy. Yes, I was not saying otherwise, however. Often when work is done on an object, we see heat generated. Now arises another draft maker, at least for me, known as entropy. Let's just say friction is the cause and limit our experiental travel to a walk. :) But, I knew the anvil would not get hotter regardless of the pedagogical approach, though the "why" gets poorly explained.

Engineers, even today I would bet, use PE and KE ideas regarding anvil, or any mass, lifts and falls. The math cruncing and chewing can be accomplished with a minimal number of bites compared to a gravitational field strength variation approach, and, especially, a GR approach. It is a little difficult to teach a freshman that raising something upward gives it more potential energy, though no microscope nor thermometer will see any energy change. [I speak as student, not teacher, admittedly.] Tell me something now has more energy and I want to measure it. Embracing what you show in field differnce is the superior view unless simplicity is prefered for pedagogical purposes within anvilolgy. :)


And you may wonder about the thermal energy of the anvil. Well, just think of the total energy as being subsumed in some notion of 'm', and as we raise and lower it, the difference in clock rates gives us that 'm'*GM/r potential looking energy. Nuts, I was half-way out the door and you bring this up. :) You may be sorry, but I am going to tell you what I think I'm getting from a few of your statements. I'll make it mercifully brief.

Since, per Einstein, space-time is meshed together, the g_00 time variation can tell us the difference in distance variation due to time's known link to space. Further, the time dilation between two regions within a gravitational field reveals the energy difference, too.

Jeff Root
2008-Apr-28, 04:58 AM
I don't recall having any difficulty understanding the concept of potential
energy. But then, I don't recall whether I learned about it in 11th grade
physics, or earlier, outside of school.

I can easily tell the difference in potential energy between two things by
converting that potential energy to kinetic energy. I can easily see, hear,
and feel differences between the effects of a 10 cm fall and the effects
of a 20 cm fall. But I'm pretty sure that even the most precise hydrogen
maser clocks, some of the most (maybe THE most) precise measuring
instruments in existence, could not even detect the gravity-caused time
difference between two points near Earth's surface, vertically separated
by only 10 cm.

-- Jeff, in Minneapolis

tommac
2008-May-05, 07:15 PM
I was using the ant and the rope idea to demonstrate how light could eventually make it of a black hole but Grant threw a wrench into a perfectly beautiful piece of ATM and made me realize that the ant and the rope does not work in this case.

The premise was that the black hole was stretching back space-time in a similar way to the expansion of the universe so if light can get to us from a galaxy that is receding super-luminaly then it should be able, given enough time get out of a black hole.

However there is a difference and it is not ant and the rope. It seems that it is more like a treadmill where light can keep running but not get anywhere ...

Is that because there is no fixed point to get to? And no boost from the expansion of the point that I left from?

George
2008-May-05, 09:29 PM
However there is a difference and it is not ant and the rope. It seems that it is more like a treadmill where light can keep running but not get anywhere ...

Is that because there is no fixed point to get to? And no boost from the expansion of the point that I left from?
The only reason the ant reaches its destination is due to the fact that with every step it takes it is stepping into a portion of the rope that is receding at a lesser speed away from the fixed end of the rope. This assumes that the other end of the rope is being pulled at a fixed, non-accelerating, rate, which is how the expansion is viewed (ignoring the slight variation due to the recent discovery of accelearation from supernova data).

tommac
2008-May-06, 12:04 AM
The only reason the ant reaches its destination is due to the fact that with every step it takes it is stepping into a portion of the rope that is receding at a lesser speed away from the fixed end of the rope. This assumes that the other end of the rope is being pulled at a fixed, non-accelerating, rate, which is how the expansion is viewed (ignoring the slight variation due to the recent discovery of accelearation from supernova data).

So the key is that there is no fixed end. It is like rope is just being fed in at greater than the speed of light.

George
2008-May-06, 01:51 AM
So the key is that there is no fixed end. It is like rope is just being fed in at greater than the speed of light. Fixed just means a set reference point that you elect to say is not moving, and you imagine standing there and looking out in every direction to see what is moving relative to this arbitrary position. If you imagine a distant point that is traveling away at the speed of light and the ant is marching toward you, then the ant will reach a place on the rope that is moving less than the speed of light since every step that ant takes puts the ant in a spot that is moving away from the fixed point at a speed less than before.

tommac
2008-May-06, 02:08 AM
Fixed just means a set reference point that you elect to say is not moving, and you imagine standing there and looking out in every direction to see what is moving relative to this arbitrary position. If you imagine a distant point that is traveling away at the speed of light and the ant is marching toward you, then the ant will reach a place on the rope that is moving less than the speed of light since every step that ant takes puts the ant in a spot that is moving away from the fixed point at a speed less than before.

I get the initial thing ... but why doesnt this also work for a black hole? I think the deal is that there is no fixed point ... and that there is just a new block of space that pops in between you and the event horizon. It travels at least as fast as you. You can not progress on the rope so you can not use the expansion to help speed you along.