Does the speed of moving object curve spacetime?

[M8M]

Please consider the following scenario:

Assume we have an object (o) at rest with a mass = m. Hence, we can calculate by general relativity the gravitational force (g) on this mass. Now, assume we remain stationary, at the origin, and a velocity (v) is imparted to the mass along the x-axis. The velocity (v) is substantially near the speed of light (c).

This situation begs the question:

Is the gravitational force (g) on the fast moving mass (m) now proportionally higher by a magnitude γ? Is speed linked to gravitational effects? Please provide credible citations which address this issue.

Thanks.

 

[pervect]

What curves space-time is not mass, but the stress-energy tensor.

The gravitational force on a test-mass is a well defined concept in Newtonian mechanics, and in special circumstances in general relativity, but isn't well defined in general circumstances in general relativity. What is well defined in general in GR s the Riemann curvature tensor, which is more or less equivalent to the Newtonian idea of tidal force, i.e. force / unit length.

Because gravitational force isn't really well defined, I doubht you will find much in the way of references on the topic. There is a treatment of something that is well defined, the velocity induced by a flyby, however.

Olson, D.W.; Guarino, R. C. (1985). "Measuring the active gravitational mass of a moving object". American Journal of Physics 53 (7)

http://adsabs.harvard.edu/abs/1985AmJPh..53..661O

To quote from the abstract:

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel=γ(1+β^2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not γM but is approximately 2γM.

 

[Matterwave]

I think the easiest way to see if the velocity has an effect would be to use the Schwarzschild solution. One takes 2 observers, one which is stationary at Schwarzschild radius r (i.e. has 4-velocity u=(1,0,0,0)), and one observer which is moving but instantaneously at that same radius r (i.e. has 4-velocity u=(u0, u1, u2, u3)) and see which observer, if any, has a greater 3-acceleration.

Someone correct me if there's something wrong (something subtle I missed) with this scheme.

I have not carried out these calculations, but they shouldn't be too terribly difficult to do.

 

[M8M]

pervect:

Thanks for your reply. It is a bit of a concern that the Olson reference is listed as having a mere 2 citations. Clearly, it is more popular on the forums than in the scientific community. I wonder why? Further, if what you say is true this seems to be a glaring disconnect between Special Relativity and the General Theory.

 

[jtbell]

this seems to be a glaring disconnect between Special Relativity and the General Theory.

Do you mean the fact that gravitational effects do not in fact depend on the so-called "relativistic mass" $\gamma m_0$?

I would say rather that this is a glaring disconnect between popular and superficial introductory treatments of relativity that perpetuate the notion of "relativistic mass", and the professional physics community which mostly abandoned that notion decades ago.

 

[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. However, like said above, moving objects do generate different apparent forces onto test particles through the source term (stress-energy tensor) in the Einstein equation.

 

[A.T.]

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. However, like said above, moving objects do generate different apparent forces onto test particles through the source term (stress-energy tensor) in the Einstein equation.

Is it a correct approach to take the static field (say Schwarzshild metric) and apply length contraction to it, to obtain the field in a frame where the gravity source moves?

 

[M8M]

"The gravitational force on a test-mass is a well defined concept in Newtonian mechanics, and in special circumstances in general relativity, but isn't well defined in general circumstances in general relativity"

In studying relativity, I have noticed the following:

In Einstein's treatment of Special Theory we are always concerned only with objects with constant velocity. Then, supposedly acceleration is GR as being indistinguishable, in some respects, to a gravitational force.

But, in the General theory -- we can now talk about an object moving at any speed and any acceleration. If one conceptualizes space as behaving like a viscous fluid then it can be imagined that a stationary object would not produce waves in the medium; however, a rapidly moving object would generate a wave pattern in the fluid -- I'm guessing equations would be found in acoustics and fluid dynamics for this type of behavior.

If the following is true -- energy-momentum tells space-time how to curve and space-time tells energy-momentum how to move/distribute. Why? My understanding is that GR tries to kill the idea of a gravitational force originally, by equating with acceleration, then "injects" it somehow later in the theory. Is that true? Thanks.

 

[PAllen]

I think the easiest way to see if the velocity has an effect would be to use the Schwarzschild solution. One takes 2 observers, one which is stationary at Schwarzschild radius r (i.e. has 4-velocity u=(1,0,0,0)), and one observer which is moving but instantaneously at that same radius r (i.e. has 4-velocity u=(u0, u1, u2, u3)) and see which observer, if any, has a greater 3-acceleration.

Someone correct me if there's something wrong (something subtle I missed) with this scheme.

I have not carried out these calculations, but they shouldn't be too terribly difficult to do.

1) I would say it is more meaningful to calculate proper acceleration, not 3-acceleration (which corresponds to no measurable quantity, especially in SC coordinates for this purpose). Proper acceleration is what an accelerometer would measure.

2) While the u=(1,0,0,0) path has a well defined meaning as a static path (given the symmetry of the solution and the fact that it is static), you cannot come up with a preferred choice for the non-static path that aims to be 'straight as if the mass wasn't there', to compute proper acceleration for the path (I have tried this several times; there are plausible definitions - emphasis on the plural; each produces a different answer).

There is nothing subtle about the issue - there is no way at all to make your proposal correspond in any unique way to a measurable quantity.

 

[Mentz114]

In studying relativity, I have noticed the following:

In Einstein's treatment of Special Theory we are always concerned only with objects with constant velocity.

In SR, it is possible to deal with objects experiencing acceleration ( those that have curved worldlines). Non-accelerating (inertial) frames are distiguished as being equivalent
in that Newton's laws apply in all of them.

Then, supposedly acceleration is GR as being indistinguishable, in some respects, to a gravitational force.

Yes, that is the principle of equivalence in one form. But replace the final 'force' with 'acceleration'.

But, in the General theory -- we can now talk about an object moving at any speed and any acceleration.

There are two kinds of acceleration in the presence of gravity. 'Coordinate acceleration' is the kind you can't feel when you are in free-fall but accelerating, the other is 'proper acceleration' caused by an applied force. This you can measure with an accelerometer and f=ma applies.

If one conceptualizes space as behaving like a viscous fluid then it can be imagined that a stationary object would not produce waves in the medium; however, a rapidly moving object would generate a wave pattern in the fluid -- I'm guessing equations would be found in acoustics and fluid dynamics for this type of behavior.

Treating spacetime as a medium is not very productive and probably unnecessary. But gravitational waves can be deduced from GR.

My understanding is that GR tries to kill the idea of a gravitational force originally, by equating with acceleration,

Gravitational forces (except tidal effects) don't exist in GR because forces cannot be 'transformed away' by going into free fall. This was Einstein's great insight.The huge difference between GR and Newtonian dynamics is that GR is a kinematic theory. The spacetime metric already contains all possible paths, so solving the equations of motion is different entirely.

I suggest you look at an introductory GR text, because going into GR with a Newtonian ideas to the fore is likely to cause serious confusion.

 

[Naty1]

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. However, like said above, moving objects do generate different apparent forces onto test particles through the source term (stress-energy tensor) in the Einstein equation.

This is one appropriate explanation. What seems confounding, however, is that two objects of equal rest mass energy but traveling at different velocities follow different worldlines. So how do we explain that apparently different 'curvature':

From my notes of other discussions in these forums:

PeterDonis: “..if one is inferring from either the relativistic mass or the stress-energy tensor that an object's behavior as a source of gravity depends on its state of motion relative to you, one is inferring incorrectly.

The key is that gravitational curvature IS observer INdependent and is reflected as curvature of the spacetime manifold ("graph paper" as described below). Frame dependent curvature (observer dependency) is an observer variable overlay on top of this fixed background curvature. [Mass, energy, momentum, and pressure, properly defined.…in the rest frame of the system….. DO affect spacetime curvature.

[my comment: The above description is a view point that does not seem to match any particular component of ‘curvature’ within the EFE...because no single component fully describes the spacetime curvature... ]

The stress-energy tensor takes into account both the object's rest energy and the object's motion (and pressure and internal stresses in the object) in such a way that the [space time] curvature caused by the object is frame-invariant. The sign of the metric is opposite for the timelike and spacelike terms, so you generally expect the timelike and spacelike components to have somewhat opposite effects. But this doesn’t mean that a hot object doesn’t have different trajectories in a gravitational field than an identical cold object. In addition, that
Light follows [null] geodesics is an approximation… different light colors have
slightly different trajectories resulting from their different energies.

So if you write down the EFE in a frame in which the object is at rest, the stress-energy tensor in that frame (for an object like an electron that has no internal pressure or stresses) does contain *only* the object's rest energy density.

Lots more here:
https://www.physicsforums.com/showthread.php?t=548148
DrGreg explained this way:

When we introduce gravitation, the graph paper itself becomes curved. (I am talking now of the sort of curvature that cannot be "flattened" without distortion. The curvature of a cylinder or cone doesn't count as "curvature" in this sense.) Now we find that it is impossible to draw a square grid to cover the whole of the curved surface.
If we switch to a non-inertial frame [an accelerated observer] but still in the absence of gravitation), we are now drawing a curved grid, but still on the same flat sheet of paper. Thus, relative to a non-inertial observer, an inertial object seems to follow a curved trajectory through spacetime, but this is due to the curvature of the grid lines, not the curvature of the paper which is still flat.So, to summarize, "spacetime curvature" refers to the curvature of the graph paper, regardless of observer, whereas visible [apparent] curvature in space is related to the distorted, non-square grid lines drawn on the graph paper, and depends on the choice of observer. In the absence of gravity, spacetime [the graph paper itself] is always "flat" whether you are an inertial observer or not; non-inertial observers will draw a curved grid on flat graph paper.

When we talk of curvature in spacetime (either curvature of a worldline, or curvature of spacetime itself) we don't mean the kind of curves that result from using a non-inertial coordinate system, i.e, non-square grid lines in my analogy.

 

[tom.stoer]

Let's make a simple example.

Consider a spherical, stationary mass-distribution, e.g. a massive star, a study its spacetime metric, curvature and geodesics of test bodies.

Now consider a spherically symmetric collapse of this star. e.g. to a neutron star or a black hole. During this (spherically symmetric) collapse the spacetime metric and curvature do not change; the test bodies feel nothing else but the stationary spacetime as before.

Therefore at least in this situation the change in the energy momentum tensor due to the collapse and the huge additional kinetic energy of the collapsing matter does not act gravitationally.

 

[Passionflower]

Let's make a simple example.

Consider a spherical, stationary mass-distribution, e.g. a massive star, a study its spacetime metric, curvature and geodesics of test bodies.

Now consider a spherically symmetric collapse of this star. e.g. to a neutron star or a black hole. During this (spherically symmetric) collapse the spacetime metric and curvature do not change; the test bodies feel nothing else but the stationary spacetime as before.

Therefore at least in this situation the change in the energy momentum tensor due to the collapse and the huge additional kinetic energy of the collapsing matter does not act gravitationally.

It is true that it is a simple example but to suggest that this situation is typical I find frankly misleading as an explanation whether a moving object curves spacetime.

 

[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.

 

[PeterDonis]

My contribution to this for what it's worth: I think it's helpful to carefully distinguish different possible meanings for the term "moving object", since different meanings lead to different answers to the question of what is affected by "motion":

(1) Consider a spacetime containing a single gravitating body, such as a star. Observer A is at rest relative to the body; observer B is moving relative to it. Both observers are "test bodies", which do not contribute to the curvature of the spacetime. In this case, spacetime curvature is the same for both observers, even though one perceives the gravitating body to be "at rest" but the other perceives it to be "moving". The spacetime itself, the geometric object, is the same for both observers. So in this sense, the "motion" of the gravitating body does not change anything.

(2) However, observers A and B in the above scenario *can* feel different "forces" due to the gravitating body. For example, suppose that A is "hovering" motionless at a given radius, while B is passing by on a free-fall trajectory at a speed close to the speed of light (relative to the gravitating body and A) with an impact parameter equal to the radius at which A is hovering. The effective "acceleration" of B towards the gravitating body will be *larger* than the "acceleration due to gravity" that is measured for test objects dropped radially by A--in the limit as B's speed goes to the speed of light, the "acceleration" for B will be twice as large. ("Acceleration" is in quotes because it's coordinate acceleration, not proper acceleration--both B and the test objects dropped by A are in free fall.) So in this sense, the "motion" of a *test object* does change things.

(3) Now consider a different scenario, where we have a gravitating body that is undergoing proper acceleration--say a star which is ejecting mass in a definite direction, so that it behaves as if it had a rocket engine attached to it. (Suppose a very advanced alien civilization has caused the star to do this for purposes unknown to us.) The spacetime around this star will be *different* than the spacetime in #1 above, and this could be observed by noting the different "forces" observed by A' and B', observers analogous to A and B in #1 and #2 above in their motion relative to the star (A' is at rest relative to the star, B' is passing by at near the speed of light). So again in this sense, "motion" does change things--but now it is "motion" of the gravitating body in an invariant sense, i.e., "motion" due to actual proper acceleration, as opposed to just a change in reference frame.

 

[Passionflower]

(1) Consider a spacetime containing a single gravitating body, such as a star. Observer A is at rest relative to the body; observer B is moving relative to it. Both observers are "test bodies", which do not contribute to the curvature of the spacetime. In this case, spacetime curvature is the same for both observers, even though one perceives the gravitating body to be "at rest" but the other perceives it to be "moving". The spacetime itself, the geometric object, is the same for both observers. So in this sense, the "motion" of the gravitating body does not change anything.

However when A and B are not test bodies then certainly the spacetime is no longer stationary. Then relatively moving bodies certainly influence spacetime.

As in a previous writing of yours you cannot use test bodies to prove they have no gravitational influence because test bodies by definition have no gravitational influence.

It is like saying I want to prove that jumping in a swimming pool causes no waves because when a test swimmer jumps in we see no waves.

 

[PeterDonis]

However when A and B are not test bodies then certainly the spacetime is no longer stationary. Then relatively moving bodies certainly influence spacetime.

In the sense that a spacetime with multiple gravitating bodies in it is a different spacetime, a different geometric object, than a spacetime with only one, yes, certainly this is the case. I didn't mean to imply that I had covered all the possible spacetimes that might bear on the question; I was only trying to cover the simplest ones.

 

[tom.stoer]

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).

 

[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.

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.

 

[PeterDonis]

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.

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.

 

[PAllen]

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.

 

[tom.stoer]

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.

 

[tom.stoer]

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"

 

[TrickyDicky]

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.

 

[TrickyDicky]

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.

 

[TrickyDicky]

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.

 

[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.

 

[Passionflower]

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?

 

[tom.stoer]

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.

 

[TrickyDicky]

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.

 

[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.