View Full Version : Metrics?

sirius0

2007-Jan-24, 02:07 AM

What is a metric?

Take the oft mentioned Schwarzschild metric; is it just

Rs=2GM/(c^2) or is it the probably vast derivation behind it OR am I on the wrong track entirely?

publius

2007-Jan-24, 04:12 AM

What is a metric?

Take the oft mentioned Schwarzschild metric; is it just

Rs=2GM/(c^2) or is it the probably vast derivation behind it OR am I on the wrong track entirely?

Rs is the Schwarzschild radius. That's where the Schwarzschild metric blows up (in Schwarzschild spherical coordinates -- actually space-time itself is well-behaved there). Rs plays a big role in the actual metric, but it is not the metric itself.

Now, what is a metric. That is not so easy a question to answer from scratch.

What is the distance between two points (in regular Euclidean space)? Well Pythagorus gave us the formula for that a long time ago:

d^2 = (x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2

Where the x-y-z's are the coordinates, and 'd' is the distance between those two points. Sum of squares.

Now, that is for a straight line path between those two points (the shortest distance between two points is a straight line, as we all know, but that's only for Euclidean metrics, so we have to unlearn a lot of things in the long run.....). How about the distance for some arbitrary, curvy path? Well, you model that curve as a bunch of differential little straight lines and integrate them up over the path. The expression for the differential length is just the above in differentials:

ds^2 = dx^2 + dy^2 + dz^2.

Now that sucker is a metric. It is the 3D Euclidean metric in cartesian coordinates. A metric is the mathematical "rule" for finding the distance in a given space. Note the coefficients of the squares are all 1s. That is the expression in any cartesian coordinate system. But such systems could be rotated with respect with to each other or have their origins shifted. That will make the coordinates different, but the distance between the two same points is always the same. It is invariant, and coordinate independent.

Now, we got to space-time, a combination of the two that Mr. Einstein discovered had to be mixed up. The "distance" in space-time is this:

ds^2 = (ct)^2 - [ dx^2 + dy^2 + dz^2 ]

= (ct)^2 - dr^2

where in the second form, I've wrapped up the 3D spatial distance into one "dr" term. The distance (sometimes called a "norm") in space time is the difference between the squares of time and space. This is using a "positive time-like convention". Now, we've got some funny we don't see in Euclidean space. The "distance" squared can be zero or negative. Light happens to always follow such a zero "length" path. For a positive path, that "length" turns out to be the proper time experienced by a clock that travelled along that path.

Now, noting the coefficients, we've got +1, -1, -1, -1. We can write the above using a vector and matrix notation, and these coefficients appear along the diagonal of a 4x4 matrix, all other elements being 0. This is the Minkowski space metric.

Doing the same thing for Euclidean space, we get the identity matrix, all 1s down the diagonal of a 3x3 matrix.

So the Minkowski metric looks like similiar to Euclid, save for the difference in sign between the time and space parts. All 1s mean the metric is "flat", but the difference in sign means non-Euclidean. Space in Minkowski is Euclidean (the dr subpart) but the whole thing is a wee bit different.

It turns out, with a more complete and rigorous analysis of all this mess, that a diagonal metric matric means our basis vectors are orthogonal, and 1s mean the basis vectors are "normal", their magnitudes are 1.

In Euclidean space, we could choose some crazy non-orthonormal coordinate system where there were cross elements in the metric. Cross elements (ie some dy*dx terms in the ds^2 expression) mean our basis vectors aren't perpendicular.

Now, you smoke that over, and then we'll go on to "curvature". When we have curvature, the metric coefficients become functions of the coordinates themselves. GR is all about how mass-energy changes that Minkowski metric into "something more complicated". :)

-Richard

sirius0

2007-Jan-24, 05:09 AM

Ok I'm smoking. I understood perhaps 88%

Thank you I will let you know via this thread when this is digested.

Regards Chris

Wayne McCoy

2007-Jan-24, 06:12 AM

In topology, there is a more general description of a metric. Given a topological space (e.g., real Euclidean space) S, a metric is a function

d: SXS --> [Reals >=0]

satisfying for all x,y,z in S

d(x,y) = 0 if and only if x = y

d(x,y) = d(y,x)

d(x,y) + d(y,z) >= d(x,z) (triangle inequality)

This is general enough to be used in a space even where curvature is involved. A question of fundamental interest in topology is when can a metric be defined for a space which generates the topology of the space, when is it metrizable? Not every space can be metrized; however, a very broad class of metrizable and metric spaces is available.

Nereid

2007-Jan-24, 06:39 AM

The distance (sometimes called a "norm") in space timeWhy is it sometimes so called?

Does it relate to some hairy maths?

When we have curvature, the metric coefficients become functions of the coordinates themselves. GR is all about how mass-energy changes that Minkowski metric into "something more complicated".Is this one example of why GR is 'background dependent'?

More general question: how (where) does causality enter GR?

Nereid

2007-Jan-24, 06:40 AM

In topology, there is a more general description of a metric. Given a topological space (e.g., real Euclidean space) S, a metric is a function

d: SXS --> [Reals >=0]

satisfying for all x,y,z in S

d(x,y) = 0 if and only if x = y

d(x,y) = d(y,x)

d(x,y) + d(y,z) >= d(x,z) (triangle inequality)

This is general enough to be used in a space even where curvature is involved. A question of fundamental interest in topology is when can a metric be defined for a space which generates the topology of the space, when is it metrizable? Not every space can be metrized; however, a very broad class of metrizable and metric spaces is available.Are there - as far as you know - any spaces used in physics today which are known to be not metrizable?

publius

2007-Jan-24, 06:41 AM

Wayne,

Actually, in Minkowski space, the triangle inequality does not hold. Well, we can divide in two classes. For time-like norms, the triangle inequality is exactly *reversed*, but for space-like norms, it holds. The time-like distance between events A and C is always *greater than* the sum of the distances between A and B and B and C.

That is the most succinct way to explain the dreaded Twin "Paradox". The inertial path between two events always has more proper time elapse than any accelerated path. So space-time represents a type of "space" where the norm defintion has to be a little more relaxed than the above.

And finally, yes, there are very precise, rigorous ways to define a metric. And I will defer to those who are expert in it.

However, a problem I have is using those precise, rigorous ways to start teaching this. Start out with these highly mathematical defintions that have no obvious connection to anything "physical", and the student just sort of zones out and doesn't get it.

I like to start this stuff out with more physically grounded explanations, like starting out with the concept of "distance", then going on to show how that becomes generalized. Then, once the student understands that, you go to the more rigorous definitions to "polish it up" and tie up any loose ends.

-Richard

Nereid

2007-Jan-24, 06:46 AM

Wayne,

Actually, in Minkowski space, the triangle inequality does not hold. Well, we can divide in two classes. For time-like norms, the triangle inequality is exactly *reversed*, but for space-like norms, it holds. The time-like distance between events A and C is always *greater than* the sum of the distances between A and B and B and C.

[snip]Wouldn't a simple transformation make Minkowski space into one in which the triangle inequality did hold?

publius

2007-Jan-24, 07:04 AM

Why is it sometimes so called?

Does it relate to some hairy maths?

Yes. :) Basically, the idea of a norm is just "the length of a vector", a scalar term; it's magnitude. We can write any vector as that scalar length times a unit vector (vector whose norm, or length is 1). But, one can get much more general in what one means by vectors and their norms. So the norm becomes something more general that may or may not have anything to do with anything that could remotedly be considered a length.

For example, Fourier analysis. You can consider (certain classes) of functions as (usually infinite) sums of certain other functions. That sum can be seen as expressing a vector as a sum of its components. The "norm" (and the related inner product) becomes an integral of those functions over an interval.

This type of thing is the bread and butter of quantum mechanics. The "basis vectors" are eigenfunctions of certain differential operators, and any quantum state vector can be expressed as a "vector sum" of these basis functions.

Is this one example of why GR is 'background dependent'

More general question: how (where) does causality enter GR?

Those are questions best answered by, oh say a Kip Thorne, not a publius. :)

If we throw down a spherical mass in a Minkowski background space, we get Schwarszchild. But if we throw down another spherical mass in the field of the first mass, we get something different than the linear "sum of the two". That is the background depedence as my little brain understands it.

However, there is the Strong Equivalence Principle, which GR sastifies. If we plopped down a spherical mass in a uniform field (no tides), and transformed ourselves into a frame free-falling with that mass in the background field, it would look like Schwarzchild. This is why a local Cavendish experiment in the Earth's g-field gives the same local results as it would floating off in free space, far removed by any mass.

However, old publius wonders about the limits of that, especially when frame dragging is afoot big time.........

Now, about causality. Kip Thorne does indeed need to come here. I think, that means that no casual influence can get from point A to point B faster than light would. But how fast light gets from A to B is coordinate dependent.

-Richard

publius

2007-Jan-24, 07:19 AM

Wouldn't a simple transformation make Minkowski space into one in which the triangle inequality did hold?

No. You see, that's the thing. Minkowski is not a Euclidean space. It a crazy mixture of space and time. The space part of Minkowski is Euclidean space. Triangle inequalties hold there in that part as a subset.

The Minkowski norm, the "distance", the "interval" is a mixture of space and time. One of the features of a norm we want is that it be coordinate independent. All observers see the same "distance" between events, but when you break it down into time and space components for each, it is very much frame dependent.

So all observers will see the reverse triangle inequality for time-like intervals. They will see a regular triangle inequality for space-like intervals (negative norm).

Basically a time-like separation means there is some observer, some frame that can see those events as occuring at the same place, but at different times. In Minkowski, that is the just an observer travelling in a straight line at some velocity between those events.

A space-like separation means some frame sees those events as being simultaneous, occuring at the same time, but at different places.

-Richard

Ken G

2007-Jan-24, 08:34 AM

I think Richard's answers are, as usual, excellent, and I've learned a lot from them. Still, I think one gets more physical intuition by starting with the concept of proper time, and talking about metrics later on, rather than the other way around. My point is, I think a key lesson of relativity is that you don't have to worry about it when the laws of physics are expressed locally, i.e., in terms of an observer who is at the events in question. That is the only observer who can actually observe those events directly, all others have to use some degree of drawing inferences, and relativity is basically the rules for how to draw those inferences and translate them into the language of that nonlocal observer.

Now if you have two nearby (in space and time) events, then the simplest way to set up the physics is to have the same observer be at both of those events, to measure them. Then the key concept is proper time-- the time between those events as measured by that local observer. The laws of physics apply to that time, and there's no need for any relativity at this point. Relativity comes in when you want to conceptualize to what non-local observers would say is going on. They have to take the local observer's word for what happened (and after doing that for awhile they can learn to rely on light instead, but it's the same idea), but they must transform it into their own coordinate system to make sense of it. It turns out that when they make that transformation, the proper time gets broken up into a time interval and a length interval in the new coordinates, in just the way Richard said. And if the events are not nearby each other, you just have to take a series of small steps and "integrate" the time intervals and distance intervals, but always by transforming from the proper time of the local observer into these other coordinates. The invariant is the "norm", which you find using the "metric" (the metric is the rule for finding norms, i.e., proper times), and that guides this transformation. Although I'm not saying anything beyond what Richard said already, I find this approach more illuminating than talking about "distances" in spacetime.

You might wonder, what if you can't get the same observer to visit both events, as when they are spacelike separated? Then what does the "norm" mean? I'm not sure, but my feeling is, this is a totally different animal-- if there is no real observer that can be at both events, then there is no need to even have a "norm" connecting those points, there is no physical importance to negative norms. My guess is, all negative norms are created equal, can be arbitrary, and never affect the physics.

publius

2007-Jan-25, 03:28 AM

You might wonder, what if you can't get the same observer to visit both events, as when they are spacelike separated? Then what does the "norm" mean? I'm not sure, but my feeling is, this is a totally different animal-- if there is no real observer that can be at both events, then there is no need to even have a "norm" connecting those points, there is no physical importance to negative norms. My guess is, all negative norms are created equal, can be arbitrary, and never affect the physics.

Whether it means anything that is worth two cents or not, I don't know, but here's the way I think of the meaning of space-like "norms".

A time-like norm means there exists some (class of) frame in which the "distance" can be seen as "pure proper time". And that would be an observer moving between those events at whatever velocity is required to that it occurs at the same point at different times.

And for a space-like norm, that is turned around and means there exists some (class of) frames of whatever velocity in which those events occured at the same time, but were spatially separated. So in those frames, the "norm" is "pure proper distance".

So I think of time-line norm as "pure time", but space-like as "pure distance" equal to the square root of the absolute value. And never the twain can meet. If something is time-like, no frame can see it as all distance, and if space-like, not frame can see it as pure time.

Again, whether that is worth anything or not, I don't know.

-Richard

Ken G

2007-Jan-25, 04:29 AM

And for a space-like norm, that is turned around and means there exists some (class of) frames of whatever velocity in which those events occured at the same time, but were spatially separated. So in those frames, the "norm" is "pure proper distance".

But this is kind of my point. I think it's very confusing to have norms sometimes be proper time, and sometimes proper distance. I think the proper time is real, because it can be measured by the same observer, but proper distance is physically irrelevant-- if the two are not causally connected, who cares what their distance is? Indeed, the very nature of "simultaneity" of spacelike separation seems like an entirely arbitrary convention. So what I'm saying is, there really is no physical meaning to "Minkowski space" outside the light cone-- I don't think it ever shows up in any prediction of science. That frees you up to see the norm as purely the proper time-- what the observer's clock measures. You see, even the whole concept of distance is a bit weird-- you can't actually measure it locally, the observer is always at the distance origin. I think distance is a kind of mental extrapolation of the "real" quantity, which is proper time. Perhaps this is why space and time are so deeply connected.

satori

2007-Jan-28, 08:26 PM

i think you are right KenG,

the causality principle is deeper than propper GR , and Eigenzeit seems to reflect it in the GR scenario.

sirius0

2007-Jan-28, 08:45 PM

I think one point that had me stuck was the (ct) term as it looked to simple; and yet dimensionally it is distance isn't it? But surely the derivation of a time like dimension is more complex than dimensional analysis? or is it?

Argos

2007-Jan-29, 02:55 PM

Congrats Publius [and Ken]. I think your the explanation of 'metrics' has got as simple as it can be. :)

Ken G

2007-Jan-29, 03:21 PM

I think one point that had me stuck was the (ct) term as it looked to simple; and yet dimensionally it is distance isn't it? But surely the derivation of a time like dimension is more complex than dimensional analysis? or is it?

It has to be a bit more complex than that, because time is perceived differently by our brains, and because there's a different sign in the time part of the metric. I used to think time was a kind of add-on to the 3D space, but now I actually thinks it's the other way around-- time is the fundamental quantity (proper time, that is), and if you focus on one observer (the world according to Garp, as it were), then there is no real distance-- they are always at the same place. But stuff happens, and other causes percolate in that cojoin with the things you can cause, so we tack on a concept of "distance" to help organize these new causes that keep merging with what we're trying to do. You cannot affect me right now, but you can do something that will affect me soon, and I internalize that truth by conceptualizing a "distance" between us to explain why you can't affect me right now.

Distance has another important property also, which is that if another observer moves relative to me, they will synchronize their proper time a little different, and they will also conceptualize "distances" a bit different as well, and it turns out that the way I conceptualize my distance to their location is also the way to keep track of how their proper time is synchronized to mine. This is not a new property, but part of the necessary rules to make the cause and effect work out as before.

Thus I argue that when you are doing relativity, it is distance that is the organization principle-- we invent it, sort of like potential energy, to help keep track of "where the proper time went" when we conceptualize from one observer to other observers in a different frame who have their own proper time. There are apparently 3 degrees of freedom from whence to draw this reservoir of proper time, so we conceptualize a spatial universe that really doesn't exist in the way that time does, and it all centers on cause and effect. Just food for thought, of course, but I'm saying that time is the "real" entity, because cause and effect is about as close to reality as science can get, whereas space is the organizational principle, when doing relativity anyway (in daily life, it is much easier and more intuitive to treat both as real, and imagine that you can actually measure distances with a ruler!).

satori

2007-Jan-29, 03:54 PM

Sounds it could be deep, but might take some time to metabolize properly...

Sean Clayden

2007-Jan-29, 04:00 PM

Accuracy is the direct distance between point A and B. Weight is the measure of A against B.

Metrics are a measure of distance/weight relative to us.

What's your point ?

sirius0

2007-Jan-29, 08:36 PM

Congrats Publius [and Ken]. I think your the explanation of 'metrics' has got as simple as it can be. :)

They have certainly been helpful; thats for sure.

I am coming along Ok I think. Doing some background reading too.

Is it reasonable to say that while space seems different (distances etc) from different frames (at large cosmological distances apart) does it all balance out once you take the TIME to travel to the other frame? Of course both frames will have progressed but I mean that there is amaximum velocity .'. a minimum time to arrive; is this the timelike dimension in action? Actually just read Ken G's last post; Is what I am asking related/compatible? I think so. But what a spin around to put time as the centre peice. Then it is the dimension we measure most accurately (even locally)!

sirius0

2007-Jan-29, 08:45 PM

So is the accelerating recession also an organising principle? Ken. Is there a need in GR to keep those galaxies beyond the hubble sphere out of our frame? Imean the derivatives of displacement should also be organising principles yes? This allows our time never to to be compatible allowing them to receed superluminally?

Ken G

2007-Jan-29, 10:31 PM

But what a spin around to put time as the centre peice. Then it is the dimension we measure most accurately (even locally)!

Yes, I really think the local character of time is crucially intertwined with the local character of the laws of physics. Put differently, why does something have to first pass just "next to" us in order to then affect us? Because a cause temporally precedes an effect, and we can query "what happened" to an observer who was at the same place as both the cause and the effect. That is the fundamental truth, and it is most naturally classified under "time" not "distance". The things that are more naturally classified under distance are acausal and play no role in physics as we know it, so distance is a bit of a red herring in understanding reality, if you can believe that.

Ken G

2007-Jan-29, 10:36 PM

Imean the derivatives of displacement should also be organising principles yes? This allows our time never to to be compatible allowing them to receed superluminally?

I haven't gotten that far, frankly. I tend to think of time as the fundamental quantity, which is itself an organizing principle but has such a quintessentially real character that I would start there with an approach to reality. Then tack on distance as a kind of conceptualization to allow observers who were not present at the events in question to have some understanding of them, and then the derivatives come under the heading of how things behave-- i.e., those are the laws, moreso than the conceptualizations. Those are what reality hands us, given the conceptualizations already made. It sounds like you are asking if the laws of physics can be expressed in terms of their action on time, and to that I would say yes, indeed the very concept of "action" is an appropriate word here, in the classical mechanics sense. This is of course too murky to be of value in general, but it might lend insight into calculations that are already done but are not necessarily associated with physical insight.

publius

2007-Jan-30, 01:54 AM

My view, such as it as, is space-time is something more than just time (and space).

The thing is we're all moving through space-time, "compelled" to travel into our future by the ways the rules of existence are setup. That's the thing. Nothing is stationary in space-time, but moving.

Sitting here, we're all moving through space-time at the speed of light. 'ct' is distance we move along our world line in what we perceive as 't' units of time.

Draw an x-t Minkowski space-time diagram, place your spatial location at x = 0. Your world line is just x(t) = 0, a flat line. But you're moving along that line at a "speed" of c. Now, draw a line x = vt, which represents an observer moving at some velocity v relative to you who passed right by you at t = 0.

He moves along that line at a speed of 'c' as well. But you'll see, if he's moving at 'c', and you're moving at 'c', the projection of his position along your t axis is moving *less* than 'c' (and vice versa for yours along his -- but be careful with these diagrams, but this is Minkowski, not Euclid and the various projections and components of vectors DO NOT WORK like they do in Euclid -- the diagram is just a helpful device, but don't get apply Euclidean relations like you would for two rotated x-y coordinate systems).

He's "motion through time" *looks slower than yours* because he's travelling at different angle through space-time than you are.

One's own "local proper time" is just this universal compulsion to "move at c" through space-time.

c is more than the speed of light -- that's just a little trick of stuff whose lack of rest mass makes that motion through time look like pure motion through distance to everyone. c is somehow the "speed" everything is compelled to move along world lines.

-Richard

Ken G

2007-Jan-30, 02:23 AM

I've always liked this "everything moves at c" approach to spacetime, but here's what I'm now advocating. Instead of talking about covering a distance ct, just say you are moving through time t, not distance ct. The coefficient is then 1, so everything moves at speed 1 through proper time (that's the "speed" along the wordline that publius mentioned, but it's not a speed, it's a rate of time). Then distances come in like d/c to put them in the same units. It's the flow of time that's the "real" thing here, and should not have any arbitrary constants (like c) in it. The c should be viewed as coming in downstairs in d/c, since because d is just a conceptualization anyway, it's not surprising we chose the wrong units for it and now need a universal conversion factor. With that minor (and purely pedagogical) change, you can still keep everything publius said, what I think is the new insight is that all physics is most naturally local, so does not involve distance at all, and you then bring in distance as a pure conceptualization that is only needed because things happen in places where you are not (but some observer could connect you to those things, for all the things that can affect you, and for that observer, it's not distance, it's proper time!). Indeed, when you begin to see the fundamental importance of time (I claim), you also see why everything travels at the same rate through spacetime-- you are just seeing the flow of time in other guises.

publius

2007-Jan-30, 03:50 AM

Yes, as is now, we write the space-time interval/displacement/"distance" in terms of distance units. That s is in distance units, so we have to multiply the time part by c. It would work just as well, maybe better conceptually as Ken is thinking, to take s/c as the norm, and let it be in time units. We'd then have factors of 'x/c' in the spatial part.

It's really all a matter of *units*, anyway. Let c = 1, and the unit distinction between distance and time goes away. That's what you're doing when you speak of distances in light-time units, anyway, but it may not have been so obvious because we're "hung up" on c being a speed that light travels.

However, there is the difference in sign, the "signature" of Minkowski, and any time-like displacement means the "distance/time" must be positive (or negative if you use the space-like convention, which was another sort of conceptual error that was made as the formulation was being worked out, but that's all water under the bridge).

Anyway, the so-called "geometric units" that most of the "high priest" relativists use in their calculations are doing all the above anyway by setting c = 1 (and just about every constant, such as G, as well that they can).

When you do that, whether your base unit is space or time is entirely up to you.

And doing that, you'll notice some interesting things, a velocity is dimensionless, just a ratio, which can be time per time or distance per distance, however you like. :) But note acceleration has dimensions of 1/time or 1/distance............ That means force has dimensions of mass per distance or mass per time.

If g = GM/r^2 = 1/distance. Then GM has units of distance (or time). If you say G is 1, the mass has units of distance. So a force is dimensionless itself in geometric units, a ratio of "distance/time" itself.

-Richard

sirius0

2007-Jan-30, 04:18 AM

Then Work or energy has units of mass; which makes sense.

But didn't your earlier arguaments turn time into a vector (different angles etc). Torque and Joules have the 'same' units but one is a dot product the other a cross product. I mean is this sort of dimensional analysis allowed?

publius

2007-Jan-30, 05:19 AM

Then Work or energy has units of mass; which makes sense.

But didn't your earlier arguaments turn time into a vector (different angles etc). Torque and Joules have the 'same' units but one is a dot product the other a cross product. I mean is this sort of dimensional analysis allowed?

Yes, you can make a (4-)vector out of anyone else's proper time, and express its components in your coordiantes (your space and time directions). Note that you can express any vector in this form:

A = a*N, where N is a *unit vector* in the direction of A, and 'a' is the scalar magnitude. Now, where do you put the dimension of A? You can put it on 'a', the scalar magnitude, or 'N' the unit vector.

Traditionally, and it makes better sense when you work with these kinds of expression, the dimension, the units go with the scalar, so these unit vectors are dimensionless.

In Minkowski space, we'd express 'tau' units of proper time as some

tau*T, where T is the unit vector of his "time direction" expressed in our coordinates.

tau would carry the time unit, not T. To get that in terms of space distance vector we'd just put c in front of it S= c*tau*T = sT. You can also express the "four velocity" in the form V = cT, expressly showing the "magnitude" of one's "motion through time" is always c. Now, proper acceleration, is dV/dtau and you can see, since c, the "speed" is always constant, an acceleration is merely a change in the "direction of one's proper time vector".

Now, about torque. One way to look at that is torque is actuall work per angular displacement. Since angular displacement is dimensionless, the torque part carries all the units.

-Richard

Ken G

2007-Jan-30, 05:20 AM

It's really all a matter of *units*, anyway. Let c = 1, and the unit distinction between distance and time goes away. It is certainly true that you can choose such a unit system. What I'm saying is that it doesn't really matter where you put the c, it's only there in the first place because we conceptualized distance in such a way that had an arbitrary proportionality constant in it. It is still the rate of flow of proper time that is fundamental, and that is 1 in all unit systems for distance and time.

The clues that proper time is fundamental is that time "flows", not distance, and things tend toward greater entropy with time, not with distance. Also, there is 1 dimension of time, eliminating the ambiguity you have with distance. All this is likely related to the fact that time is the parameter that governs cause and effect, not distance. I see cause and effect as the central organizing principle of physical phenomena, and it is what we use to achieve a time ordering. Put differently, all of physics happens over timelike separations, and what happens over spacelike separations is more in the realm of initial conditions, no laws of physics govern it.

And doing that, you'll notice some interesting things, a velocity is dimensionless, just a ratio, which can be time per time or distance per distance, however you like. I think the above fundamental differences remain, which is why I think it's a shame that metrics are generally described as distance measures rather than time measures, under the assumption that there really is no important conceptual differences between them.

If g = GM/r^2 = 1/distance. Then GM has units of distance (or time). If you say G is 1, the mass has units of distance. So a force is dimensionless itself in geometric units, a ratio of "distance/time" itself.

That's interesting, I'm not sure what to make of unitless forces. Acceleration seems natural to be a rate, but inertia being a time, and forces being unitless, inspires no insights at the moment, it's curious.

BigDon

2007-Jan-30, 05:49 AM

I wish this thread came with a wiki so I could get definitions without bothering people.

Ken G

2007-Jan-30, 05:59 AM

It's no bother-- just ask.

sirius0

2007-Jan-30, 06:01 AM

Thanks Richard nice explanation. Another question. (anyone)

Else where you (Richard) stated that the time dimension ((ct) i assume) swaps with a spatial dimension when space inside a black hole's event horizon is considered. I was going to ask you in that thread but it seems relevant here. Is light actually in this state as a special (as you said massless) case within our space? And therefore light behaves from our point of view as mass when in a black hole. Is this another explanation for the blackness? Or am I starting to become a tripper?

sirius0

2007-Jan-30, 06:04 AM

I wish this thread came with a wiki so I could get definitions without bothering people.

I started this thread to get definitions. No one minds. Wikipedia didn't have anything on metrics. Go for it!

sirius0

2007-Jan-30, 06:16 AM

Wayne,

That is the most succinct way to explain the dreaded Twin "Paradox". The inertial path between two events always has more proper time elapse than any accelerated path. So space-time represents a type of "space" where the norm defintion has to be a little more relaxed than the above.

And finally, yes, there are very precise, rigorous ways to define a metric. And I will defer to those who are expert in it.

However, a problem I have is using those precise, rigorous ways to start teaching this. Start out with these highly mathematical defintions that have no obvious connection to anything "physical", and the student just sort of zones out and doesn't get it.

I like to start this stuff out with more physically grounded explanations, like starting out with the concept of "distance", then going on to show how that becomes generalized. Then, once the student understands that, you go to the more rigorous definitions to "polish it up" and tie up any loose ends.

-Richard

I agree Richard but I also know enough to know that Wayne's definition should have some value to me could either Wayne or another give an inbetween step or explanation.

My brain is a little lazy I have only been doing engineering for the last few years so I need to brush up on vectors matrices cross products tensors ..... Actually don't worry I will just re-read some stuff.

publius

2007-Jan-30, 06:18 AM

I started this thread to get definitions. No one minds. Wikipedia didn't have anything on metrics. Go for it!

Wiki does have tons of stuff on relativity, both special and general, you've just got to find the magic search phrases, and pay attention to the "see also" links at the bottom of each page you do find. Things aren't as cross referenced and TOC'd as well as they could be. Search on "General Relativity" and you should get sort of a main article with links to all the others.

From what I seen, they have some fairly expert GR contributors who write the GR articles, and they will "spew" tons of terse tensor math out all the time. Wiki has a problem with, uh, how shall I tactifully put it, "obvious non experts" on other subjects who sometimes mess up the articles. But fortunately GR is so advanced, even "arcane", that nobody but the real experts even try to write the stuff. :)

So, you get a lot of advanced GR math done as though its child's play that the reader should see as obvious without explanation, but thankfully, they do know what they're talking about. Some of the *interpretations* of that math might certainly be open to debate, of course. (Mostly Ken's favorite word, pedagogy).

Anyway, if you spend some time following the "see also" link trees at the bottom of GR articles you do find, you can find a lot of stuff.

-Richard

sirius0

2007-Jan-30, 06:36 AM

Oh yes, I am. But I started this thread because not knowing what a metric was I didn't know how to be subtle in my search. The Schwazschild radius was all I could find that seemed to come close to words I was seeing on the forum so that is why my first post opened with this.

publius

2007-Jan-30, 07:19 AM

Thanks Richard nice explanation. Another question. (anyone)

Else where you (Richard) stated that the time dimension ((ct) i assume) swaps with a spatial dimension when space inside a black hole's event horizon is considered. I was going to ask you in that thread but it seems relevant here. Is light actually in this state as a special (as you said massless) case within our space? And therefore light behaves from our point of view as mass when in a black hole. Is this another explanation for the blackness? Or am I starting to become a tripper?

Well, basically, everyone can be a "tripper" or sound like one when they try to put into words what the *math of GR* "means". :) Light, travels along a "null path" in space-time, all space-times, whether "flat or curved" ie ds = 0 for light. The path light follows in your coordinates is whatever it is it to make the "length" always 0. Interpret that however you wish. :)

A good text to get you prepared for GR, although the connection won't be obvious until you, well, get into GR, is "Vector Calculus", by Marsden and Tromba. I never had a course that used that text, but some course was using it one year, and I noticed it the university book store and picked it up after looking through it. It is one of the better texts out there. My real passion has always been EM, and knowing vector calculus goes hand in hand.

But that text has some dang good stuff, really Ken's "pedagological interpretations" that are darn good for learning GR.

Think of a curve in 2D space, say a parabola. At every point, that curve has a tangent line (tangent *vector* in a particular coordinate system). Curves have properties that are independent of the coordiante system. For example, take y = x^2. That's one coordinate system where we have a nice parabola that "starts" at the x axis (a double root at x = 0) and does it's thing. But, if we rotate our coordinate system 90 degrees, that becomes

y = sqrt(x), a multi-valued function (taking y as the indepedent variable and x as the independent, after rotating).

But it's the same darn curve. The only difference there is how our x-y plane, our *basis vectors*, our coordinate system is aligned/defined.

In GR, "space-time" can be thought of as such a curve. It turns out our local coordinates, the natural rulers and clocks of some observer at a given point on that curve (some location in space-time) have to do with *the tangents to that curve* at any point.

Consider that parabola. It has a tangent at every point. Let that tangent be your new x-axis at a point, and a normal to it your y-axis. Express the parabola in terms of the coordinates along the tangent. That's what GR is all about! Consider something moving down that parabola at consant *speed* (arclength per time). Now, consider the projection of that motion along the tangents at two different points on the curve. Very different motions.

Any local observer's rulers and clocks are the *tangent vectors* of a curve. Everything moves through space-time at 'c' along some curve. Different observers see different projections of that motion along their different tangent vectors (spaces).

That is the basic "game" of GR, just done in more abstract multidimensional, non-Euclidean ways. When you get the simple 1D curve in the 2D Euclidean space ideas of tangent vectors and projections down pat, you know the basic conceptual machinary of GR.

So finally, after all that, what is meant (and it might very well be "tripping") by "time becomes a space coordinate), is that in the Schwarzchild metric, which is desribing a curved 3D-1T "surface", the time tangent vector beyond the horizon, is "parallel" with the radial spatial vectors outside the horizon. And one of those spatial direction vectors inside is aligned with our time direction outside.

-Richard

satori

2007-Jan-30, 10:03 AM

The clues that proper time is fundamental is that time "flows", not distance, and things tend toward greater entropy with time, not with distance. Also, there is 1 dimension of time, eliminating the ambiguity you have with distance. All this is likely related to the fact that time is the parameter that governs cause and effect, not distance. I see cause and effect as the central organizing principle of physical phenomena, and it is what we use to achieve a time ordering. Put differently, all of physics happens over timelike separations, and what happens over spacelike separations is more in the realm of initial conditions, no laws of physics govern it.

Ken G,

a little bird with a little brain has been sitting on your garden tree and was well pleased by your tune...

Argos

2007-Jan-30, 01:31 PM

As Publius says, Wiki has some pretty good stuff.

Distance (http://en.wikipedia.org/wiki/Distance)

Metrics (http://en.wikipedia.org/wiki/Metric_%28mathematics%29)

Vector Space (http://en.wikipedia.org/wiki/Vector_space)

Vectors in Physics (http://en.wikipedia.org/wiki/Vector_%28spatial%29)

Metric Tensor (http://en.wikipedia.org/wiki/Metric_tensor)

Minkowski Space (http://en.wikipedia.org/wiki/Minkowski_space)

Frame of Reference (http://en.wikipedia.org/wiki/Frame_of_reference)

Inertial Frame of Reference (http://en.wikipedia.org/wiki/Inertial_frame_of_reference)

Tranformation (http://en.wikipedia.org/wiki/Transformation_(mathematics))

Lorentz Transformation (http://en.wikipedia.org/wiki/Lorentz_transformation)

If you follow all related links, you have material for a whole year [or more] :)

Ken G

2007-Jan-30, 04:41 PM

To sum up, thinking more about metrics has led me to see that there are actually three pedagogically separate (but physically equivalent) ways to answer the question: What is a metric? (in special relativity-- let's leave off general relativity for now, these insights are enough to digest in the Minkowski metric!)

1) (the standard answer): a metric is a prescription for identifying the "invariant" quantity that separates two events as viewed from different inertial reference frames. It's nice to think in terms of invariants, but you have to link space and time in a not very intuitive way, and the geometry you get is non-Euclidean, so it all takes a lot of getting used to.

2) (the Euclidean answer): a metric is a way to transform from the proper time of the observer who was actually at both events (call this the proper time between the events themselves), to the proper time of observers who were not at both events (call this a nonlocal proper time). The transformation only requires that you know the distance between the nonlocal observer and the events at the times in question, and the geometry is entirely Euclidean if you think of proper times as lengths of line segments, and distances as perpendicular displacements from the proper time between the events. This approach is tantamount to taking the dx^2 part of the metric equation and pulling over to the side of the ds^2. You no longer treat distance and proper time on equal footing, and the proper time of the events themselves is viewed as a special frame, but you only have Euclidean geometry to deal with.

3) (the proper time-centered approach): a metric is a way to define a concept of spatial distance such that physics works out the same for all observers in their own proper time. In this view, all physics is local, and only depends on proper time, but you need to conceptualize to nonlocal events to figure out how they may affect you in the future (that's how spatial distance enters into the laws of physics), so you invent a concept of distance that is guided by the metric formula. In this approach, you pull the dx^2 part of the metric over to the side of the ds^2, but then you also pull the ds^2 over to the side of the dt^2! You then divide by the coefficient of dx^2 so it's a formula for dx^2. This creates a new algebra for comparing proper times, and treats the proper times as the fundamental observable-- the metric just gives you your working concept of spatial distance. Note here there is no longer any need to view time and space as part of a unified spacetime-- they are always separated by opposite sides of the equals sign, like any law. Again, this whole approach views space as a mental construct that is not actually measurable-- only proper time is.

Of course approach (1) is easier to talk to others with, because it's the standard, and approach (2) used Euclidean geometry which more people are used to, but I much prefer approach (3) because I feel it is by far the most physical. Note that none of this is ATM, because all the mathematics works the same, we're just playing with physical interpretations.

Squashed

2007-Jan-30, 05:18 PM

...

Thus I argue that when you are doing relativity, it is distance that is the organization principle-- we invent it, sort of like potential energy, to help keep track of "where the proper time went" when we conceptualize from one observer to other observers in a different frame who have their own proper time. There are apparently 3 degrees of freedom from whence to draw this reservoir of proper time, so we conceptualize a spatial universe that really doesn't exist in the way that time does, and it all centers on cause and effect. Just food for thought, of course, but I'm saying that time is the "real" entity, because cause and effect is about as close to reality as science can get, whereas space is the organizational principle, when doing relativity anyway (in daily life, it is much easier and more intuitive to treat both as real, and imagine that you can actually measure distances with a ruler!).

But isn't it also feasible to think of it the other way around: distance is "real" and time is an artifact of distances?

Squashed

2007-Jan-30, 05:30 PM

...

And doing that, you'll notice some interesting things, a velocity is dimensionless, just a ratio, which can be time per time or distance per distance, however you like. :) But note acceleration has dimensions of 1/time or 1/distance............ That means force has dimensions of mass per distance or mass per time.

If g = GM/r^2 = 1/distance. Then GM has units of distance (or time). If you say G is 1, the mass has units of distance. So a force is dimensionless itself in geometric units, a ratio of "distance/time" itself.

-Richard

I was thinking along those lines when I created my "Squashed Universe" thread but it just seemed so foreign that I never thought it through. I figured people thought I was crazy enough so I refrained from proposing it and confirming their suspicions.

Ken G

2007-Jan-30, 05:37 PM

But isn't it also feasible to think of it the other way around: distance is "real" and time is an artifact of distances?

I puzzle with that question, it's true, but I don't think they are conceptually equivalent. The key point is, there is just one dimension of time, and it has an "arrow". There are three dimensions (at least) of space, and they do not have an arrow. All this is connected to how we define the concepts of time and space, and the importance of the concept of cause and effect. I realize that the mathematics does not need to see any of these differences, but I am thinking physically, and I see them as fundamentally different, and time as the more physical, while space is more of an invention. I'm sure an even deeper cut will unearth the ways that time is also an invention, and many have thought about that already, but we choose how deep we want to go, and I think there's a physically intuitive place where time is "real" and space is "imaginary", like the quaternions.

publius

2007-Jan-30, 07:29 PM

Ken,

I would say the "arrow of time" has to do with everything being compelled to move along its world line. Measurements of time and space are relative, and mixed up in the Minkowski manner. So for that reason, I don't worry with if we measure space in time units, or vice versa, or leave them mixed up.

Forces can change the (relative) direction of the arrow of time, but they do not change its magnitude.

The larger "space" we're in has a measure, 's'. We have to move through that space according to our local arrow of time.

-Richard

sirius0

2007-Jan-30, 08:30 PM

As Publius says, Wiki has some pretty good stuff.

Distance (http://en.wikipedia.org/wiki/Distance)

Metrics (http://en.wikipedia.org/wiki/Metric_%28mathematics%29)

Vector Space (http://en.wikipedia.org/wiki/Vector_space)

Vectors in Physics (http://en.wikipedia.org/wiki/Vector_%28spatial%29)

Metric Tensor (http://en.wikipedia.org/wiki/Metric_tensor)

Minkowski Space (http://en.wikipedia.org/wiki/Minkowski_space)

Frame of Reference (http://en.wikipedia.org/wiki/Frame_of_reference)

Inertial Frame of Reference (http://en.wikipedia.org/wiki/Inertial_frame_of_reference)

Tranformation (http://en.wikipedia.org/wiki/Transformation_(mathematics))

Lorentz Transformation (http://en.wikipedia.org/wiki/Lorentz_transformation)

If you follow all related links, you have material for a whole year [or more] :)

My apologies to Wikipedia; obviously I had used a dodgy strategy when searching for metric! Thank you Argos for those references I will go through them all!

Ken G

2007-Jan-30, 09:01 PM

I would say the "arrow of time" has to do with everything being compelled to move along its world line.I agree, and that's all the physics one needs, the rest is semantics. But semantics are important, because they relate to how we think about things. We can define time and distance to be anything we want for the purposes of doing relativity, I'm saying, let's keep to the definitions that mean something to us already. When we do that, the fact that objects must move along their world line is very clearly connected to the concept of time, because we all perceive ourselves as moving through time (which we see as a linear dimension) even as we sit on our duffs, but we certainly do not always see ourselves as moving through space (which we perceive as 3-dimensional). That's why I argue the motion along the world line is "really" a motion in proper time, regardless of whatever purely mathematical definitions we choose to apply to Minkowski calculations. It always gives me insight to think physically instead of mathematically, and here that means thinking in terms of observers. Observers are always in the same place, they can only conceptualize about other places, but they can measure time.

Forces can change the (relative) direction of the arrow of time, but they do not change its magnitude. Indeed, when we demote distance to something else, not a part of "spacetime" but rather just a way to organize rules about the geometry of the various proper times, then you are led to say that forces don't change the direction of proper time, because proper time is now one-dimensional, they change the relative rate of flow of proper time. That's all velocity is, when there's no distance-- it's just a comparison of the rate of flow of proper time for two different observers. The only thing physical is time dilation, the rest is just an illusion of the coordinates. Physics is then a set of rules about how time dilation works, and it all happens "here", though we also have to conceptualize distance to organize how events can affect us that are not happening "here". But that's just kind of make believe-- if it isn't here, where is it? Somewhere arbitrary, that we choose a nice way to keep track of. That's why time is different from distance-- time is here.

The larger "space" we're in has a measure, 's'. We have to move through that space according to our local arrow of time.

That's the concept of space as applied to spacetime, so you are saying lets generalize space to include time, and that's the common approach. I'm saying that I am realizing it is more physical to do the opposite-- get rid of space altogether, except as a way of conceptualizing the behavior of proper times that are not here, for obsevers not at the same place, like the Wizard of Oz behind the curtain. This view is also going to be consistent with general relativity, because there is no action at a distance.

I think another way to say all this that might help it click is that if we first choose an observer, then we can have events that occur at different places, and we need to have a concept of spatial distance to accomodate that fact. But that's only if we need to use that observer-- any two events that are causally connected, so the laws of physics can be applied to them, will have some (hypothetical) observer who was at both places, i.e.., for whom the two events happened at the same place. That's why we never need action at a distance, in a nutshell. So if you want to know the physics of those events, just ask that observer, and use the rules of relativity to transform to the original observer who was asking the question. Then the physics is all in the proper time of that second observer, and everything else is just a transformation. The first observer will only measure their own proper time, so that part of the transformation is all that is real-- the distance part is just part of the algebra of the transformation. If you think of distance as real, then you say the distance explains why the proper times were different. If you don't, you just assert that the proper times were different because you measured them to be different, and distance is just a placekeeper for that difference. In short, the difference in proper times caused you to need some concept of distance to distinguish them, not the other way around.

I'm saying that I see distance as being like potential energy-- you see at the top of a hill a rock that has no kinetic energy, and other rock rolling down that does, and you ask, where did the energy come from? You can either just say "it appeared because forces do work", or you can say "it came from the release of potential energy". In the latter case, you'd say the potential energy was there first, and in the former, you'd say there's no need for any such thing except as a bookkeeping device. We normally think of distance as like the former, but I'm saying we can also think of it like the latter, and it may be insightful to do so.

(edited to add some more explanatory stuff at the end)

sirius0

2007-Jan-30, 11:41 PM

My apologies to Wikipedia; obviously I had used a dodgy strategy when searching for metric! Thank you Argos for those references I will go through them all!

And I thought the distance link would be a breeze :)

Whew! Still it's great to be learning!

I remember when I did this maths (unfortunately I DID the maths which can be some way away from learning it) it was never in my memory called a metric.

Also the axies can be warped to get curves straight. I recall this was considered a form of mapping. The most common use was logarithmic axies so that the slope gave the expodential constant and I think the intercept gave the multiplying constant. A*e^(slope).

Does GR allow for the possibillity of axies that actually do warp? Or is it a system of pedantically straight orthoganal dimensions that lets space go to hades over large distances? (Of course this is ok as hades is not in our frame of refference; I understand this) I was just wondering if any one has proposed a non-static/dynamic set of axies. I am also wondering (and this is connected in my mind) if instead of often explaining gravities effect on space as a rubber table with depressions if it might have some pedagogical benefit to use a more optical scenario; say like water drops on a sheet of glass. It is an idea forming as I write.

publius

2007-Jan-31, 12:16 AM

GR basically lets you use whatever coordinates your little heart desires. For example, because of the spherical symmetry, Schwarzschild is derived in spherical spatial coordinates. That is not "straight". So you decribe warped space-time with warped coordinates. :)

There is mathematical machinery that deals with "real curvature" of the space-time itself vs "apparent curvature" due to coordinates. We can easily visualize "curvy space coordinates", like spherical coordinates, and see how even though we're using those, the space it is describing is flat.

However, throw time in the mix, and it gets complicated to visualize and think about. Rindler is all about a type of coordinates where the time coordinate "isn't straight".

On definition (via hard, observer centric Equivalence Principle guided thinking) is there is "gravity" whenever geodesics (the paths objects with no forces acting on them follows) are not "straight" in your coordinates.

And taking that to heart, in spherical coordinates, there is "gravity", because something moving in a straight line does NOT move in linear functions of the coordinates. But no one would really call that gravity -- it would just be silly.

Well, it turns out the Rindler (or Born, or other accelerating transforms) are the same darn thing with time thrown in the mix. Yet, because of the way we think, we're more likely to call that "psuedo gravity" at least. But really, mathematically, when we get comfortable with throwing time in and mixing it all up, Rindler's gravity is as "silly" as curved spatial coordiantes.

Now, *real gravity* means the space-time is "really curved", and not just a coordinate trick. There is no transform that will get back to you Minkowski globally.

Roughly, in Newtonian terms, this corresponds to gradients in the gravitational field, not the field itself, which is sort of well, thought provoking. With EM, derivatives of the potential give us a field, and that sucker is real. Well, with gravity, it's gradients of the gradients of the potential (metric) that is the "real field", so to speak.

And the gradients are sometimes called the tidal forces or tidal field. So the language "real gravity is tidal gravity" is used.

The main equation of GR, the big bad Einstein Field Equation is about finding the metric from the source mass-energy-momentum sources (gravity depends not just on the mass-energy, but the mass-energy current AND the momentum and "momentum currents", but the latter are very weak and higher order effects).

Now, something called the Riemann tensor tells you if you have real, tidal curvature of the metric. The Riemann tensor is quite a thing to behold. Since it involves second derivatives of the metric, it is a **RANK 4 tensor**, a freaking 4-dimensional matrix/array with 256 components. Fortunately, the number of independent degrees of freedom is much less than that (think about the relation of partial derivatives, ie d^z/dxdy = d^2z/dydx), and there are various meaningful decompositions of that sucker that can have meaning.

Anyway, real gravity is that mess, and you can use whatever coordinates suit your fancy.

-Richard

publius

2007-Jan-31, 12:39 AM

And while I'm rambling about this mess, here's something you can do with Riemann's tensor, the thing that tells you about the "real gravity" you've got.

First, a little Maxwellian side-bar. When we write Maxwell in the modern vector form, due to the Heaviside, the dube staring at you spookily (itchy trigger finger, he looks like....) from my avatar, we've got these two vector field things, E and B, which are functions of space and time, and the equations tells you what various spatial and temporal derivatives of those fields are doing in terms of the sources.

But, you can go to a 4-vector form (and you have to make E and B part of *one thing*, a field tensor, because E and B alone are not proper 4-vectors), and merge the spatial and temporal derivatives together and get a nice 4-tensor equation with one source term, the charge and current together.

One thing you learn early on is the notion of potentials, and how equations for the potentials are generally easier to solve than the equations for the field directly.

With Maxwell, in the 3-vector/time form, we get two equations for two potentials, one a vector A, and the other a scalar, V. In the merged 4-vector form, we get one equation for one 4-vector potential. The EM field tensor is a derivative operator on the 4-potential.

With gravity, the "potential" is actually the metric, and this potential is a rank-2 4-tensor, rather than a simple rank 1 4-vector. That was one of the big things about gravity -- it can be shown that gravity must be at least a rank-2 tensor field, no simple vector potential field can describe it.

The EFE is written in the nice terse, 4-tensor form for the metric/potential, the equivalent of writting EM in than terse 4-potential form, with space and time mixed together.

You might wonder if you could go the other way and maybe write gravity in some Maxwell-like form with the spatial and temporal parts separated.

You can.....................

-Richard

publius

2007-Jan-31, 01:04 AM

I apologize for all this rambling. A light bulb goes off in my head, "aha, I'll tell him about this neat thing", but then I realize it will take 100 paragraphs of setup to explain what the thing is and why its neat. *sigh*.

Yet another side-bar. GEM, for gravito-electromagnetism. THe EFE (Einstein field equation) can be linearized in the weak field limit, by ignoring small terms and making a linear set of diffy Qs out of it. When you do that, you get something that looks almost exactly Maxwellian.

We get 'g', which plays the role of the electric field, and a 'B_g', which is a magnetic-like aspect of gravity, the gravitomagnetic field. g is then called the "gravitoelectric" field. By ignoring the small terms and linearizing, we've effectively turned the metric, the rank-2 potential into a rank-1 4-potential using the first row of the metric tensor. g_00 is the "scalar potential", and the three g_0i (i = 1 to 3) look like a gravitomagnetic vector potential A. The sources are the mass and the mass currents (we ignore the momentum/pressure contributions), just like the sources of EM are charge and charge currents.

Now, that's the Maxwell looking linearization. There are some rather neat things about that, but the gravitomagnetic field is so small, that non-Newtonian aspects of the regular 'g' part will kick in well before B_g is strong enough to really matter. GEM is worthless in a practical sense, but very valuable to see how gravity has a Maxwell like aspect, and how gravitomagnetism works to a linear approximation.

----------

After all that setup, I can now make my "neat point" about Riemann. Taking the full non-linear equations, and writing out a 3space-1time Maxwell inspired set of equations, you can pull the following "fields" out:

E_g = "electrogravitic tensor"

B_g = "magnetogravitic tensor"

T_g = "topogravitic tensor".

It turns out T_g is just the transpose or something of E_g, but so you can have two "field" E_g and B_g that you write some equations for. The terminology is close to GEM, but note the "gravito" part goes after the electric and magnetic terms here.

These fields are rank-2 3x3 tensors, as opposed to 3D vectors. The E_g (in a free-falling frame) is the tidal tensor. B_g is a type of gravitomagnetic tidal tensor that governs the gravitomagnetic effects. T_g governs "other stuff that there because of spatial curvature".

Inspired by GEM we want to write a Lorentz force like expression for gravity:

a = g + v x B_g, and that's what GEM gives us. However, in reality we have this:

a = [something_like g] + [something like v x B_g] + [other crap]

E_g gives us the "something like g" part, B_g gives us the "something like v x B_g] and T_g gives us the "other crap".

Now, you see why Einstein's hair looked like it did..........:D

-Richard

Argos

2007-Jan-31, 12:37 PM

And I thought the distance link would be a breeze :)

Whew! Still it's great to be learning!

Yes it is. :) You might like to check this one (http://math.ucr.edu/home/baez/einstein/) too.

And btw, I think this is one of the top threads in the board currently. Nice job by Publius et al.

sirius0

2007-Feb-25, 11:57 PM

I think I am ready for curved space and how a metric relates to this.

I am wondering if any one has devised a metric that allows the axes themselves to change, roll, precess as a function of distance?

publius

2007-Feb-26, 07:03 AM

Sirius,

When you say "curved space", there is some ambiguity there about what you mean. Taken literally, curved space means the *spatial part* of the space-time is curved. Now, what is space and what is time is coordinate dependent, so really, curved space (or curved time) is in the eye of the observer. But curved space-time is something coordinate independent.

In the other thread, you mentioned the "bowl" picture of the "fabric of space" between curved, and you imagine an orbiting test particle "banking" around that bowl. That can be misleading. Well, it's trying to paint a helpful picture, but if you take it too literally, you'll go astray.

That bowl picture is actually what is called an embedding diagram, and the standard picture is actually a diagram of the *spatial curvature*, leaving time out completely. What that is doing is taking constant coordinate time (for some observer), holding it constant, and looking at what space is doing. If the metric is static, that has meaning, but it is still coordinate dependent. Much, if not most of the path you see a test particle follow depends on the time part of the metric -- how time is curving as well as space, so these embedding diagrams just miss all of that.

I just want to make this clear about these "fabric of space" things. Don't get too carried away with them.

More later.............

-Richard

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