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Does the speed of moving object curve spacetime?

I only wanted to introduce two simple spacetimes a) w/o and b) w/ a "kinetic energy" of a mass distribution.

I can't do this using test bodies (as discussed) and I can't do this using one single object (a star) in motion, b/c this is fake. A single object in an otherwise empty spacetime can't move w.r.t. this spacetime. It defines the entire spacetime (typically Schwarzsschild).

 Recognitions: Gold Member M8M queried about gravitational forces resulting from different translational velocities between two objects. Suppose we set the one object rotating instead of moving translationally. Would there be a way [or is it even useful] in GR to describe such spacetime curvature effects differently between these cases since the 'invariant SET based spacetime curvature' changes [from the stationary] with rotational energy and momentum but not with translational?? I'm wondering here about our descriptions of curvature, not the magnitude of the effects.

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 Quote by Naty1 M8M queried about gravitational forces resulting from different translational velocities between two objects. Suppose we set the one object rotating instead of moving translationally.
In M8M's OP, it looks to me like the "object with mass m" is supposed to be a test body; i.e., its effect on the overall curvature of the spacetime is too small to be significant. If that's the case, I'm not sure how to add internal angular momentum (or "spin") to the test body and have it have any effect. When I've seen angular momentum is ascribed to test bodies in GR problems, it's always orbital angular momentum, due to the trajectory of the body around the central mass.

 Quote by Naty1 Would there be a way [or is it even useful] in GR to describe such spacetime curvature effects differently between these cases since the 'invariant SET based spacetime curvature' changes [from the stationary] with rotational energy and momentum but not with translational?? I'm wondering here about our descriptions of curvature, not the magnitude of the effects.
If the "object with mass m" in the OP is *not* supposed to be a test body, if it is supposed to have significant effects on its own on the overall curvature of the spacetime, then yes, obviously those effects are different if the object has angular momentum as well as mass. If gravity throughout the spacetime were weak enough, one could approximate the solution as one object (the one that gives rise to the "gravitational force" in the OP) with some mass M, described by a Schwarzschild solution centered on that object, and another object (the "object with mass m" in the OP) with mass m and angular momentum L, described by a Kerr solution centered on *that* object. This would not be an exact solution, but it could be a starting point for an approximation scheme.

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 Quote by tom.stoer It's not a typical scenario, but you can separate the effects cleary. You have a non-stationary mass distribution with stationary spacetime. So my conclusion is that the idea of an 'object moving through spacetime and distorting spacetime' (due to its velocity, not only due to its presence!) is misleading in GR.
This interesting example shows the difficulty of interpreting the thread titular question. I can propose an interpretation where it supports it, in contrast to Tom's interpretation of the same scenario (about which there is not doubt about the actual physics).

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.

 Recognitions: Science Advisor Another example to study this effect could be Kerr black holes with identical M but different J. In that case the orbits of test particles in equatorial plane feel J via frame dragging. I think it's hard to identify something like a "force-term" which could be used to "measure the gravitational pull" on these test particles. My conclusion that the idea of an 'object moving through spacetime and distorting spacetime' due to its velocity is misleading in GR still holds. There may be some limiting cases where this idea could be useful, but in general I think it's neither unique nor consistent. @PAllen: I think you interpretation goes into the same direction.
 Recognitions: Science Advisor Here's a paper studying "Newtonian forces in the Kerr geometry". http://adsabs.harvard.edu/abs/1990JApA...11...29C Eq. (7-9) show the relevant expressions for "gravitational force", "centrifugfal force" and "Coriolis force". A naive inspection shows that the "gravitational force" is Kerr-parameter independent and that all other terms are highly supressed for large radius. Therefore my interpretation is that the "gravitational force" of the Kerr BH is due to its mass, not due to its angular momentum! Therefore the "rotational energy" of the Kerr BH does not generate an additional "relativistic mass" which contributes to the "gravitational force". Caveat: a) In the Kerr BH there is no "rotating mass distribution". b) It is unknown whether the Kerr metric may serve as exterior solution for spacetime around a rotating massive object like star. We know that it works for the J=0 Schwarzschild case: here it's possible to match the Schwarzschild exterior to a fluid interior solution. A similar matching for a rotating fluid interior is not known So my example may be somehow unrealistic. One would nee to study the rotating fluid case which I do not know. Perhaps the so-called Neugebauer–Meinel disk could serve as a more realistic example, but I was not able to figure out something like geodesic equations, their asymptotics for large r and a possible interpretation as "force terms"
 This is an interesting question and several reasonable answers and comments have been offered, but I don't think the crux of this problem has been mentioned, or else everyone is answering purposedly in a naive way that pretends to ignore the limitations of GR (except for passionflower lucid comment). 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. Just in case someone wonders why this is a two body problem in the context of relativity:to determine properly the speed of an object one needs a second object, otherwise one cannot determine whether it is moving or not. All the examples presented here that use test bodies motion wrt to a central mass (actually all based in the vacuum solutions) cannot in principle address the OP.

 Quote by clamtrox A moving object does not curve space-time any differently from a stationary object. This is because the curvature is a geometric quantity, and therefore invariant under coordinate transformations.
This would seem to make sense geometrically, except it contradicts the fact that geodesics with different velocities follow different paths and thus correspond to different spacetime curvatures.
Then again this would be a many-body problem outside of the scope of GR. Remember also in relativity we cannot distinguish if an isolated object is moving or stationary.

 Of course since we are approaching this from the "in principle" GR side I'm not even mentioning the calculational post-newtonian approximations that "GR simulations" are based on when dealing with two or more bodies. Those simulations are much more "newtonian" than anything we could discuss based on curved spacetime geometries intuition. They all rely on the assumption- that is considered valid by mainstream in practice and that Einstein himself used as guiding principle (almost as much as the equivalence principle) when seeking the EFE- that the newtonian limit is valid for GR in general (certainly it seems to be in the solar system context). I'm not really sure if this assumption is part of the theory, I'm not even sure it is axiomatized in a way that it can be ascertained, I always thought GR is basically just the EFE in a certain manifold.
 Hi. Say here are three balls of same mass and thus same gravity source. You cut out these balls. The first one is packed with solid matter. The second one contains very hot gas inside. Mass and kinetic energy of gas generates gravity. The third one is cavity mirrored inside. Photons are packed inside. Non mass energy of photon generates gravity. Regards.

 Quote by sweet springs Hi. Say here are three balls of same mass and thus same gravity source. You cut out these balls. The first one is packed with solid matter. The second one contains very hot gas inside. Mass and kinetic energy of gas generates gravity. The third one is cavity mirrored inside. Photons are packed inside. Non mass energy of photon generates gravity. Regards.
What do you mean by the same gravity source?

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 Quote by sweet springs The first one is packed with solid matter. The second one contains very hot gas inside. Mass and kinetic energy of gas generates gravity. ...
Good point.

We don't necessarily need to study a two-body problem; we simply need to compare mass distributions with identical "stationary density" ρ=const. but different "internal" motion. Like the three balls, like the Neugebauer–Meinel disk, like the collapsing dust sphere etc.

I think this is a way to interpret the question and to find answers for some special scenarios.

 Quote by tom.stoer Good point. We don't necessarily need to study a two-body problem; we simply need to compare mass distributions with identical "stationary density" ρ=const. but different "internal" motion. Like the three balls, like the Neugebauer–Meinel disk, like the collapsing dust sphere etc. I think this is a way to interpret the question and to find answers for some special scenarios.
Test particles have arbitrary "internal" qualities by definition, so I can't see how it helps.

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 Quote by TrickyDicky 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 definitly motion which does not affect spacetime curvature at all.

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 Quote by TrickyDicky 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.

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 Quote by tom.stoer 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:
 Quote by PAllen 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.

 Quote by yuiop 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.

 Tags dilation, gravitational, mass, relativistic, velocity