Does the speed of moving object curve spacetime?

[tom.stoer]

Test particles have arbitrary "internal" qualities by definition, so I can't see how it helps.

The question is "Does the speed of moving object curve spacetime?". The approach is simple: we use the metric or geodesics to study spacetime curvature. Then we compare different mass distributions i.e. we study the effect of different mass distributions on geodesics.

One proposal I made a few days ago is to compare the metric and the geodesics of a stationary body and a (radially) collapsing body of the same mass. One finds that radially inward motion doies not affect spacetime curvature.

Another poroposal from sweet springs is to compare different mass distributions with identical total mass M and identical, static mass density ρ = const., but different internal d.o.f. In that case due to different energy-momentum density the internal motion (like temperature) may affect spacetime.

Another proposal was to look a the Kerr metric and interpret this as rotation. One finds that certain "force effects" extracted from the geodesics depend on the rotation i.e. the Kerr parameter, but that the "gravitational force term" itself doesn't.

A problem I mentioned was that the Kerr metric cannot be matched to a rotating fluid; therefore I proposed to use the Neugebauer–Meinel disk as a more realistic example for a rotating mass density instead. Unfortunately I couldn't figure out the "force effects".

So my conclusion is that in certain cases we can interpret the question "Does the speed of moving object curve spacetime?" using full solutions of GR with "internal d.o.f." and "internal motion". We do not need to study a two-body problem but rather a general solution of GR plus its effects on the vacuum metric and geodesics ouside. This is a valid and reasonable approach. In some cases one can even interpret the geodesics using "Newtonian force terms" and compare the effects for different "internal motion".
The answer is not so simple as can be seen in the Kerr case: of course the rotation of the Kerr solution does affect the spacetime metric but the "gravitational force" on a test body in a "Newtonian interpretaion" does not depend on it; the effects of frame dragging are something like a "Coriolis term".

Anyway - the answers are not so simple and by no means clear and unique: as the collasping star shows there is definately motion which does not affect spacetime curvature at all.

 

[yuiop]

The way I see it the OP question, strictly speaking, has no answer within GR because it implies a two body problem that GR has no way to answer in principle.

This is silly. If we had a two, three or million body problem it would be described by GR which is very very different from saying GR has no answer in principle. While the known solutions are based on a single massive body, this does not mean that GR cannot in principle have solutions for more than one massive body. It is just that finding solutions for multiple massive bodies are very difficult and possibly there are no simple analytical solutions, but we can use computers and numerical methods in some situations. To say that GR cannot provide answers for multi body problems even "in principle" is to suggest that GR does not describe the universe that we live in.

 

[yuiop]

One proposal I made a few days ago is to compare the metric and the geodesics of a stationary body and a (radially) collapsing body of the same mass. One finds that radially inward motion doies not affect spacetime curvature.

This is in direct contradiction to:

The alternative interpretation is to note that if we have perfect radial collapse of dust particles, so the temperature and other characteristics are unchanged until 'late' in the collapse, we see that a reduced size state with rapidly inward moving dust produces the same total curvature (ADM mass) as the expanded state with slow moving dust. However, compare this partially collapsed state to a state with identical number of dust particles beginning collapse from the reduced size (with only slow inward motion). This state will definitely produce less curvature and have lower ADM energy. One can argue that since the position, temperature of each dust particle and the number are identical, the only distinction is the rapidity of radial motion - which increases total curvature (as measured e.g. by ADM mass). Further, the amount of increase would be expected to be the increase in KE, which would have a primary proportionality constant of gamma.

Here is a sort of hybrid version. We have a solid massive sphere (not a black hole) surrounded by a collapsing sphere of dust. For a distant observer there is no change in the field or apparent mass of the combined solid and dust spheres as the dust collapses. When all the dust finally settles on the solid surface, the KE of the dust becomes heat but still there is no change in apparent mass. It is only when the heat is radiated away and the photons pass the distant observer that the gravitational mass appears to decrease. When the solid body and dust settles down to a temperature equal to its initial temperature (and the heat ernergy has radiated away) it will have less mass. This agrees with PAllen's analysis that collapsing dust sphere with particles that individually have high KE will have a greater ADM mass than a similar sized dust sphere that has just started collapsing with low KE particles.

 

[TrickyDicky]

This is silly. If we had a two, three or million body problem it would be described by GR which is very very different from saying GR has no answer in principle. While the known solutions are based on a single massive body, this does not mean that GR cannot in principle have solutions for more than one massive body. It is just that finding solutions for multiple massive bodies are very difficult and possibly there are no simple analytical solutions, but we can use computers and numerical methods in some situations. To say that GR cannot provide answers for multi body problems even "in principle" is to suggest that GR does not describe the universe that we live in.

Maybe you haven't noticed (you may want to take a look at the Beyond the Standard model subforum for instance), but there's close to consensus among scientists about considering GR an extremely good approximation to the universe we live in , but obviously not "the theory", as in the ultimate theory, there are clearly situations that are outside of the scope of GR and that don't describe the universe we live in. , regardless of your considering it silly or not.

As I said in a previous post , there are of course post-Newtonian computer numerical simulations but they depend on the Newtonian limit assumption. There is simply no practical exact solution of the EFE that deals with many-body problems. That has been known since little after the time Einstein came up with the EFE. There are no solutions in general relativity with test particle stress-energy.

 

[TrickyDicky]

The question is "Does the speed of moving object curve spacetime?". The approach is simple: we use the metric or geodesics to study spacetime curvature. Then we compare different mass distributions i.e. we study the effect of different mass distributions on geodesics.

The approach may be simple but as I said above if you are using a vacuum solution, the moving objects can only be test particles and there are no solutions in general relativity with test particle stress-energy.

 

[tom.stoer]

This is in direct contradiction to ...

It isn't; it's simply an approximation to study a collapse w/o any change in internal d.o.f., radiation etc. A collapsing sphere of dust w/o pressure is certainly only an approximation, therefore your hybrid version taking other effects into account is better.

All what I am saying is that during the collaps where the pressureless dust is a good approximation we have "kinetic energy" that does not affect spacetime (outside the initial sphere of dust).

Maybe you haven't noticed (you may want to take a look at the Beyond the Standard model subforum for instance), but there's close to consensus among scientists about considering GR an extremely good approximation to the universe we live in , but obviously not "the theory", as in the ultimate theory, there are clearly situations that are outside of the scope of GR and that don't describe the universe we live in. , regardless of your considering it silly or not.

We are talking about a problem which can be asked, discussed and solved perfectly within GR; there's no need for any physics beyond GR.

There is simply no practical exact solution of the EFE that deals with many-body problems. That has been known since little after the time Einstein came up with the EFE. There are no solutions in general relativity with test particle stress-energy.

We don't need this. I presented a couple of scenarios where motion is not due to "n-body solutions" but due to internal d.o.f., i.e. part of the stress-energy tensor. That's perfectly valid and we even have solutions for these scenarios.

The approach may be simple but as I said above if you are using a vacuum solution, the moving objects can only be test particles and there are no solutions in general relativity with test particle stress-energy.

I am not proposing a vacuum solution. A collapsing dust is not a vacuum solution; the Neugebauer–Meinel disk is not a vacuum solution. I am not discussing "test-particle-stress-energy" but I use geodesics of test particles to analyze these non-vacuum spacetimes; that's a big difference.

 

[TrickyDicky]

We don't need this. I presented a couple of scenarios where motion is not due to "n-body solutions" but due to internal d.o.f., i.e. part of the stress-energy tensor. That's perfectly valid and we even have solutions for these scenarios.

Your scenarios might be fine by themselves, I'm not debating that, I'm saying they don't address the OP at least the way it is written. The OP talks about a body not about a perfect fluid dust solution.

I am not proposing a vacuum solution. A collapsing dust is not a vacuum solution; the Neugebauer–Meinel disk is not a vacuum solution. I am not discussing "test-particle-stress-energy" but I use geodesics of test particles to analyze these non-vacuum spacetimes; that's a big difference.

Again, the OP was about the motion of a body. Not a fluid.

We are talking about a problem which can be asked, discussed and solved perfectly within GR; there's no need for any physics beyond GR.

What problem are you referring to? the OP's or your dust solutions? The latter are of course within GR.
And by the way by proposing fluid solutions you are implicitly admitting what I said about n-bodies and GR, in that case I don't know why you support yuiop on this.

 

[yuiop]

Again, the OP was about the motion of a body. Not a fluid.

As far as I know, people can and have done two body numerical computer simulations of colliding black holes, based on GR without introducing any new "beyond the standard model" theories.

 

[tom.stoer]

Your scenarios might be fine by themselves, I'm not debating that, I'm saying they don't address the OP at least the way it is written. The OP talks about a body not about a perfect fluid dust solution.

Again, the OP was about the motion of a body. Not a fluid.

What problem are you referring to? the OP's or your dust solutions? The latter are of course within GR.
And by the way by proposing fluid solutions you are implicitly admitting what I said about n-bodies and GR, in that case I don't know why you support yuiop on this.

TrickyDicky, what else but a localized fluid or dust should a "body" be?

 

[tom.stoer]

As far as I know, people can and have done two body numerical computer simulations of colliding black holes, based on GR without introducing any new "beyond the standard model" theories.

That's correct, but as I said above the black holes are in a sense not very good examples of "bodies". The are vacuum solutions. Nevertheless you are right, there are simulations of colliding black holes. One can show that the system emits gravitational waves i.e. propagating curvature.

 

[TrickyDicky]

As far as I know, people can and have done two body numerical computer simulations of colliding black holes, based on GR without introducing any new "beyond the standard model" theories.

That's correct, but as I said above the black holes are in a sense not very good examples of "bodies". The are vacuum solutions. Nevertheless you are right, there are simulations of colliding black holes. One can show that the system emits gravitational waves i.e. propagating curvature.

I've been careful in all my replies to make clear the caveat that there's linerization approaches that have more to do with approximmating a flat Newtonian solution than with the gravity as curvature of spacetime approach of the "full" nonlinear GR.

 

[tom.stoer]

Neither the BH collisions (afaik) nor the physical solutions like fluid or dust (here I am sure) rely on Newtonian approximation.

 

[TrickyDicky]

Neither the BH collisions (afaik) nor the physical solutions like fluid or dust (here I am sure) rely on Newtonian approximation.

BH collissions do , certainly fluid solutions don't, I never said they did, I said the way the OP was written I interpreted it as a vacuum solution scenario, I might have interpreted it wrong.

TrickyDicky, what else but a localized fluid or dust should a "body" be?

The way I understood the OP, he seemed to be talking about the motion of a discrete moving object, a point particle or test body, such objects are idealizations and by definition have no internal structure. This is the way this is usually discussed in the context of vacuum solutions.

Now, you claim the OP problem is not a vacuum problem and in that case of course we could talk about localized dust particles much in the same way galaxies are considered localized particles of a dust in FRW cosmologies, but in those cases the particles of the dust are infinite in number as points of a continuous pressureless fluidbut then again the OP said "we have an object...".

 

[tom.stoer]

I think we agree that interpreting it as a pointlike or test particle does not make sense.

So my idea was to discuss a context in which the question could make sense.

 

[TrickyDicky]

I think we agree that interpreting it as a pointlike or test particle does not make sense.

I'm sure we agree about many things
Honestly, I think we are in the semantic, interpretational camp right now (maybe the OP could clarify what he meant), if he really meant how the speed of an isolated body affects the curvature of spacetime I really think there is no answer within relativity.

So my idea was to discuss a context in which the question could make sense.

This is always a practical thing to do, as long as we are answering the problem asked, otherwise it is sometimes more honest to answer that the question may not have a clear answer in the context it was made.
We don't want to be like the drunk guy in that old joke that was looking for his keys at night near a streetlamp just because there was more light there not because he had lost them anywhere near the streetlamp.