Conservation of angular momentum in GR

[PAllen]

Can't argue the specifics, but agree 100% it's not proper to dismiss anyone who makes a serious argument without presenting their entire rationale first. While no doubt PAllen in #29 meant only the best, it would have been a bit kinder to A.Loinger to have quoted the remainder of the intro "..If we add non-gravitational forces, the conclusion remains the same, because the new trajectories do not possesses kinematical elements (velocity, acceleration,
time derivative of the acceleration, etc.) different from those of the geodesic motions."

I stopped quoting where I did for a very specific reason - the point at which a statement was made that differed with 70 years of virtually all other researcher's conclusions going back to Einstein and Infeld. Further, it is, in fact, pretty obvious that if all massive bodies follow geodesics exactly, there is no GW. So this is the fundamental starting point which must be disputed. And the way it is disputed is as I described: derive motion from the field equations rather than a-priori assuming the geodesic hypothesis.

Deriving motion from the field equations is a complex prodedure, but it has been done now dozens of different ways by many researcher's over decades, reaching common conclusions. There is no need to bring in non-gravitational forces to dispute the erroneous starting point.

 

[Q-reeus]

I stopped quoting where I did for a very specific reason - the point at which a statement was made that differed with 70 years of virtually all other researcher's conclusions going back to Einstein and Infeld. Further, it is, in fact, pretty obvious that if all massive bodies follow geodesics exactly, there is no GW. So this is the fundamental starting point which must be disputed. And the way it is disputed is as I described: derive motion from the field equations rather than a-priori assuming the geodesic hypothesis.

Deriving motion from the field equations is a complex prodedure, but it has been done now dozens of different ways by many researcher's over decades, reaching common conclusions.

Thanks for that clarification. As I say I can merely comment on the sidelines at that level. If I recall it right though Einstein himself was a doubter of GW's some twenty odd years after publishing the final version of GR. Lucky I believe to have his reputation saved by an anonymous journal referee who knocked back his 'rebuttal' of GW's as physically real. Subsequently someone persuaded him otherwise no doubt. And to his dying day refused to believe BH's were possible - maybe a sad case of pupil's supplanting teacher; don't know myself.

 

[PAllen]

Thanks for that clarification. As I say I can merely comment on the sidelines at that level. If I recall it right though Einstein himself was a doubter of GW's some twenty odd years after publishing the final version of GR. Lucky I believe to have his reputation saved by an anonymous journal referee who knocked back his 'rebuttal' of GW's as physically real. Subsequently someone persuaded him otherwise no doubt. And to his dying day refused to believe BH's were possible - maybe a sad case of pupil's supplanting teacher; don't know myself.

Yes, it is well known that Einstein never accepted that Black holes could form by realisitic physical processes. However, he didn't live to see the work of Chandresekhar, Penrose, and Hawking. I expect he would have been swayed.

 

[TrickyDicky]

I stopped quoting where I did for a very specific reason - the point at which a statement was made that differed with 70 years of virtually all other researcher's conclusions going back to Einstein and Infeld. Further, it is, in fact, pretty obvious that if all massive bodies follow geodesics exactly, there is no GW. So this is the fundamental starting point which must be disputed. And the way it is disputed is as I described: derive motion from the field equations rather than a-priori assuming the geodesic hypothesis.

So would you argue that the adimensional point that is the center of gravity of a massive body is not following a geodesic motion no matter how massive, is it not following an inertial path (in GR terms)?

 

[PAllen]

So would you argue that the adimensional point that is the center of gravity of a massive body is not following a geodesic motion no matter how massive, is it not following an inertial path (in GR terms)?

Yes, massive, non-singular bodies do not exactly follow geodesics. Out of curiosity, I researched a little when this was first convincingly shown, and it is even earlier than I thought. Vladimir Fock showed in 1939 that non-singular bodies that are massive enough to have self gravitation (again, no need for any non-gravitational forces to be considered) do not exactly follow geodesics.

[EDIT: And, continuing the historic trend of putting messy calculations on a more rigorous footing, here is a paper by a recent contributor to these forums verifying the main conclusions of Fock:

http://arxiv.org/abs/0806.3293
]

 

[TrickyDicky]

Yes, massive, non-singular bodies do not exactly follow geodesics.

This is IMO an important point worth to clarify. I would say that by the Equivalence principle, the adimensional center of mass of any massive body is considered "a local inertial frame" and must be following exactly a geodesic path. Is this statement not true? Or is the Equivalence principle not stating this? Or is the equivalence principle not valid here?

 

[PAllen]

This is IMO an important point worth to clarify. I would say that by the Equivalence principle, the adimensional center of mass of any massive body is considered "a local inertial frame" and must be following exactly a geodesic path. Is this statement not true? Or is the Equivalence principle not stating this? Or is the equivalence principle not valid here?

Why would the inside of massive body be expected to be (precisely) a local inertial frame? I would expect the opposite, in the sense the metric cannot be transformed to Minkowski form over the scale of interest (the massive body).

The principle of equivalence is problematic as a precise rule, rather than a useful guide, as Bcrowell and others here have helped me understand with pointers to various papers.

However, Clifford Will discusses his take on its usage for precisely this scenario of massive self gravitating bodies:

http://relativity.livingreviews.org/Articles/lrr-2006-3/

especially sections 3.1.2, 4.1.1 and 4.1.2

 

[TrickyDicky]

Why would the inside of massive body be expected to be (precisely) a local inertial frame? I would expect the opposite, in the sense the metric cannot be transformed to Minkowski form over the scale of interest (the massive body).

The center of mass is the single point on a structure which characterizes the motion of the object if the object shrinks to a point mass.
In a curved spacetime, obviously the center of mass is not following an SR inertial path, but a geodesic, or the straightes path in a curved space, but now you are saying that a massive enough body's trajectory,(wich can be characterized by that of its center of mass, as if it were a test particle)is not a geodesic, and I just don't understand what the reason is, all books I consult say in the absence of non-gravitational forces a body follows geodesic motion, can you explain with your own words why this is not the case?

The principle of equivalence is problematic as a precise rule, rather than a useful guide

But it's still valid, right? at the very least in its weak form.

 

[PAllen]

The center of mass is the single point on a structure which characterizes the motion of the object if the object shrinks to a point mass.
In a curved spacetime, obviously the center of mass is not following an SR inertial path, but a geodesic, or the straightes path in a curved space, but now you are saying that a massive enough body's trajectory,(wich can be characterized by that of its center of mass, as if it were a test particle)is not a geodesic, and I just don't understand what the reason is, all books I consult say in the absence of non-gravitational forces a body follows geodesic motion, can you explain with your own words why this is not the case?

But it's still valid, right? at the very least in its weak form.

I don't know of any simple justification. When a body must be treated as both a source of gravitation and responding to gravitation, The EFE are are complex, non-linear. I have read through a couple of such analyses of motion from the EFE, most carefully an old one by Synge (1960), and followed that different methods keep coming to the same conclusion. I was under the belief that center of mass is ill defined in GR, but have not really thought much about it. Birkhoff's theorem is only defined for non-rotating spherical bodies, which is what would lead me to think that the ability to treat any other body as equivalent to mass at a point is suspect.

As for the equivalence principle, in its most classic form (indistinguishability of an accelerating lab from a lab sitting on a planet) , it is true only locally, when all tidal effects can be ignored. Will describes 3 other formulations (all different from Einstein's original formulation). He argues that GR is the only known theory, consistent with experiment, that satisfies all of his 3 forms of equivalence principle. His strong one is specifically designed to be tested in the presence of massive self gravitating bodies that radiate and don't follow geodesics exactly.

[Edit: If you're willing to tackle it, I really got the impression the Sam Gralla paper I linked to is much more careful than any prior presentation I've read. Since you seem particularly sensitive to 'ignored details' (not a bad thing at all), this paper might be more satisfying than other treatments. Repeating the link:
http://arxiv.org/abs/0806.3293 ]

 

[bcrowell]

The center of mass is the single point on a structure which characterizes the motion of the object if the object shrinks to a point mass.
In a curved spacetime, obviously the center of mass is not following an SR inertial path, but a geodesic, or the straightes path in a curved space, but now you are saying that a massive enough body's trajectory,(wich can be characterized by that of its center of mass, as if it were a test particle)is not a geodesic, and I just don't understand what the reason is, all books I consult say in the absence of non-gravitational forces a body follows geodesic motion, can you explain with your own words why this is not the case?

Amplifying on PAllen's response to this question: --

If the books you are referring to are carefully written, they should not say that "in the absence of non-gravitational forces a body follows geodesic motion." They should say that "in the absence of non-gravitational forces a body with negligible mass follows geodesic motion."

The basic reason is the one I gave in #34.

-Ben

 

[bcrowell]

[Edit: If you're willing to tackle it, I really got the impression the Sam Gralla paper I linked to is much more careful than any prior presentation I've read. Since you seem particularly sensitive to 'ignored details' (not a bad thing at all), this paper might be more satisfying than other treatments. Repeating the link:
http://arxiv.org/abs/0806.3293 ]

Another rigorous paper is this one: Ehlers and Geroch, http://arxiv.org/abs/gr-qc/0309074v1

It is nontrivial to even formulate what is meant by geodesic motion of a test body, and a lot of the Ehlers and Geroch paper is taken up with that.

Another thing to note is that if energy conditions are violated, you can't necessarily prove geodesic motion of test bodies.

I made an attempt here http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3) to explain the Ehlers and Geroch argument in a way that would be accessible to people who are not big-time GR technicians. It's hard to say whether I succeeded, since I'm not a big-time GR technician myself :-)

 

[TrickyDicky]

I don't know of any simple justification. When a body must be treated as both a source of gravitation and responding to gravitation, The EFE are are complex, non-linear. I have read through a couple of such analyses of motion from the EFE, most carefully an old one by Synge (1960), and followed that different methods keep coming to the same conclusion. I was under the belief that center of mass is ill defined in GR, but have not really thought much about it. Birkhoff's theorem is only defined for non-rotating spherical bodies, which is what would lead me to think that the ability to treat any other body as equivalent to mass at a point is suspect.

It is true that the center of mass is ill defined in GR, according to Krasinski "Introduction to general relativity": " In relativity, so far there is not even a generally accepted definition of the centre of mass, although work on this problem is being done. Thus, when we consider the orbits of point bodies in relativity, we are in fact extending the theory into the domain in which it has not been worked out yet. Nevertheless, these results agree with observational tests."
In practice orbits are considered geodesic paths and I haven't found any reason to think this shouldn't apply to the binary pulsar orbit. A reference where this is stated explicitly would help.

As for the equivalence principle, in its most classic form (indistinguishability of an accelerating lab from a lab sitting on a planet) , it is true only locally, when all tidal effects can be ignored. Will describes 3 other formulations (all different from Einstein's original formulation). He argues that GR is the only known theory, consistent with experiment, that satisfies all of his 3 forms of equivalence principle. His strong one is specifically designed to be tested in the presence of massive self gravitating bodies that radiate and don't follow geodesics exactly.

That is a valuable opinion but it's just C. Will's, not necessarily the consensus in GR.

[Edit: If you're willing to tackle it, I really got the impression the Sam Gralla paper I linked to is much more careful than any prior presentation I've read. Since you seem particularly sensitive to 'ignored details' (not a bad thing at all), this paper might be more satisfying than other treatments. Repeating the link:
http://arxiv.org/abs/0806.3293 ]

The notion of self-gravitation is not without its own conceptual problems , since the gravitational field can be made to vanish for a free-falling body following a geodesic path or just by choosing the appropriate coordinates.

The bottom-line is much of this seems to be not fully worked out yet, and no generally accepted conclusion has been drawn, in this context, leaning on recent papers not yet fully validated to reject completely a perhaps valid premise is not appropriate IMO.
It just seems unfair to anybody's work, no matter how flawed it seems to dismiss it just by saying that his premise is clearly wrong because it is obvious that GW exist when his purpose is precisely to prove that they don't exist.

 

[TrickyDicky]

I made an attempt here http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3) to explain the Ehlers and Geroch argument in a way that would be accessible to people who are not big-time GR technicians. It's hard to say whether I succeeded, since I'm not a big-time GR technician myself :-)

There seems to be some conflating between geodesic in curved spacetime and inertial paths in flat spacetime here. For instance when you assert: "The world-line of a such a body therefore depends on its mass, and this shows that its world-line cannot be an exact geodesic, since the initially tangent world-lines of two different masses diverge from one another, and these two world-lines can't both be geodesics."
First, in flat space parallel inertial paths remain paralle but in curved space all initially parallel geodesics diverge.
Second, I thought all bodies, no matter their mass are affected in the same way by gravity (curvature), isn't that the principle of equivalence between inertial mass and gravitational mass? or the reason a hammer and a feather fall at the same time on the moon's floor.
Also, why can't both be geodesics?, in a sphere longitude lines departing from the poles diverge and they are all geodesics. The divergence of the masses is derived from the fact that in a real setting gravitational fields are not uniform, orbiting bodies have a near spherical shape so when acting as sources of curvature they produce different geodesics for different giving rise to tidal forces, in fact that is the reason we feel the curvature, and what geodesic deviation measures.

Also it is obvious that not all points in a massive body follow a geodesic path,(maybe this is what you and Pallen mean) the further from the center of mass the more so, at the surface of the body, that is going to be rotating, a test particle at rest is not allowed to follow geodesic motion, that is felt as an acceleration that is commonly described "as gravity" but that in fact is due to non-gravitational forces, like the EM forces that keep matter united or the Pauli exclusion principle affecting fermions.

The cited Geroch paper seems to be in agreement with what I'm saying.On the other hand if all massive bodies, (which is the same that saying all bodies) fail to follow geodesic paths that would mean the only type of motion is non-geodesic motion and this doesn't seem to be right in GR.Perhaps it would be interesting to discuss this in a new thread since it is not directly related to the OP.

 

[bcrowell]

There seems to be some conflating between geodesic in curved spacetime and inertial paths in flat spacetime here. For instance when you assert: "The world-line of a such a body therefore depends on its mass, and this shows that its world-line cannot be an exact geodesic, since the initially tangent world-lines of two different masses diverge from one another, and these two world-lines can't both be geodesics."
First, in flat space parallel inertial paths remain paralle but in curved space all initially parallel geodesics diverge.

Initially parallel geodesics at a distance from one another diverge. Initially parallel geodesics at the same point are the same geodesic. (This follows from the fact that the geodesic equation is a second-order differential equation, and there are uniqueness theorems for the solutions of such equations.)

Second, I thought all bodies, no matter their mass are affected in the same way by gravity (curvature), isn't that the principle of equivalence between inertial mass and gravitational mass? or the reason a hammer and a feather fall at the same time on the moon's floor.

This is the same issue we keep coming back to. That is only an approximation for small masses.

Also, why can't both be geodesics?, in a sphere longitude lines departing from the poles diverge and they are all geodesics.

They aren't parallel at the pole.

The cited Geroch paper seems to be in agreement with what I'm saying.

No, you've misunderstood the paper.

On the other hand if all massive bodies, (which is the same that saying all bodies) fail to follow geodesic paths that would mean the only type of motion is non-geodesic motion and this doesn't seem to be right in GR.

Geodesic motion is the limiting case where the body's mass is small. This is standard stuff that you can find in any GR textbook. If you tell us what GR textbook(s) you own, we can point you to the relevant material in them.

 

[bcrowell]

The bottom-line is much of this seems to be not fully worked out yet, and no generally accepted conclusion has been drawn, in this context, leaning on recent papers not yet fully validated to reject completely a perhaps valid premise is not appropriate IMO.

No, it has been fully worked out, there is a generally accepted conclusion, and it's been a generally accepted conclusion since roughly 1940 (some history here http://en.wikipedia.org/wiki/Sticky_bead_argument#Einstein.27s_double_reversal ).

It just seems unfair to anybody's work, no matter how flawed it seems to dismiss it just by saying that his premise is clearly wrong because it is obvious that GW exist when his purpose is precisely to prove that they don't exist.

You've misread what PAllen wrote in #29. He didn't assume gravitational waves. He described calculations that assume only the validity of the Einstein field equations.

 

[TrickyDicky]

This is the same issue we keep coming back to. That is only an approximation for small masses.

The Weak Equivalence principle is an approximation for small masses?

They aren't parallel at the pole.

Ok change diverge by converge, are they not parallel at the equator?

No, you've misunderstood the paper.

Maybe so, but is it not trying to prove that small bodies follow geodesic motion?, how does that disagree with my trying to shed light about why shouldn't any size body's trajectory, which is a unidimensionalline regardless its size, be considered a geodesic motion?.

Geodesic motion is the limiting case where the body's mass is small.

I know the standard reason given to assert this is that massive bodies have a self-gravitation that interferes with the background curvature, but this seem counterintuitive when it is also asserted that freefalling bodies gravitational field vanishes at the appropriate coordinates, and that their inertial mass is equivalent to their gravitational mass.

 

[TrickyDicky]

No, it has been fully worked out, there is a generally accepted conclusion, and it's been a generally accepted conclusion since roughly 1940 (some history here http://en.wikipedia.org/wiki/Sticky_bead_argument#Einstein.27s_double_reversal ).

I was referring to the center of mass in GR, what does that have to do with the GW and Einstein story?

You've misread what PAllen wrote in #29. He didn't assume gravitational waves. He described calculations that assume only the validity of the Einstein field equations.

In fact I was thinking of your post #34 when I wrote that. I misquoted Pallen.

 

[PAllen]

The Weak Equivalence principle is an approximation for small masses?

Here is a reasonably precise statement, with appropriate caveats, from Clifford Will:
"An alternative statement of WEP is that the trajectory of a freely falling “test” body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition "

Ok change diverge by converge, are they not parallel at the equator?

Bcrowell was clear that he was talking about parallel at the same point. An easy way to see that this must be true (though Bcrowell already gave you a perfectly clear one) is to use the parallel transport definition of geodesic: a curve that parallel transports its tangent vector. There is one curve for each tangent vector at a given point.

I know the standard reason given to assert this is that massive bodies have a self-gravitation that interferes with the background curvature, but this seem counterintuitive when it is also asserted that freefalling bodies gravitational field vanishes at the appropriate coordinates, and that their inertial mass is equivalent to their gravitational mass.

Are you saying that the earth, which is free falling around the sun, may be treated as if it has no gravitational field? I've never heard such a claim. Can you justify it or give a reference for what this could mean.

 

[TrickyDicky]

Here is a reasonably precise statement, with appropriate caveats, from Clifford Will:
"An alternative statement of WEP is that the trajectory of a freely falling “test” body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition "

So once again, are you saying that the WEP is not valid for massive bodies? only for idealized test particles? just asking...

Bcrowell was clear that he was talking about parallel at the same point. An easy way to see that this must be true (though Bcrowell already gave you a perfectly clear one) is to use the parallel transport definition of geodesic: a curve that parallel transports its tangent vector. There is one curve for each tangent vector at a given point.

He also said if they are at the same point they are the same geodesic.

Are you saying that the earth, which is free falling around the sun, may be treated as if it has no gravitational field? I've never heard such a claim. Can you justify it or give a reference for what this could mean.

We are talking about its path, and I'm saying the Earth's path is a geodesic. The Earth is of course a source of curvature too.

 

[Q-reeus]

So in the context of full GR treatment, when talking of 'small body' vs 'large body' re any departure from geodesic motion of co-orbiting masses, is this basically referring to mass as the only important determinant, or spatial extent? If the latter, wouldn't this imply it all hinging on 2nd and higher order extended body correction terms, making the rationale for GW generation quite different from the linearized theory where point masses are assumed?

 

[PAllen]

He also said if they are at the same point they are the same geodesic.

No, this is what Bcrowell said, that I was justifying a different way:

"Initially parallel geodesics at a distance from one another diverge. Initially parallel geodesics at the same point are the same geodesic. (This follows from the fact that the geodesic equation is a second-order differential equation, and there are uniqueness theorems for the solutions of such equations.)"

I was saying the parallel transport definition of a geodesic makes it obvious that this must be so.

 

[Mentz114]

So in the context of full GR treatment, when talking of 'small body' vs 'large body' re any departure from geodesic motion of co-orbiting masses, is this basically referring to mass as the only important determinant, or spatial extent? If the latter, wouldn't this imply it all hinging on 2nd and higher order extended body correction terms, making the rationale for GW generation quite different from the linearized theory where point masses are assumed?

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

 

[cosmik debris]

Are you saying that the earth, which is free falling around the sun, may be treated as if it has no gravitational field? I've never heard such a claim. Can you justify it or give a reference for what this could mean.

I'm guessing he means that the connection can be made to vanish for a body in geodesic motion.

 

[Q-reeus]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum...

OK thanks, but it's all been centering here about small departures from geodesic motion tying in or not with GW emission, and if spatial extent was a prerequisite, seemed very unlikely there would be even close to a match to the linearized GR quadrupole formula. Guessing the Hulse-Taylor system was basically treated as two orbiting point masses re orbital period decline at least - maybe extended bodies approach for more refined calcs. :zzz:

 

[bcrowell]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

I don't think this is true. The body radiates gravitational waves at a rate that is proportional to the square of its mass. Therefore its trajectory depends on its mass, and can't be a geodesic.

 

[TrickyDicky]

So in the context of full GR treatment, when talking of 'small body' vs 'large body' re any departure from geodesic motion of co-orbiting masses, is this basically referring to mass as the only important determinant, or spatial extent? If the latter, wouldn't this imply it all hinging on 2nd and higher order extended body correction terms, making the rationale for GW generation quite different from the linearized theory where point masses are assumed?

Yes, it seems contradictory that GW are derived from linearized , Newtonian limit theory where point masses are assumed and the center of mass is completely accepted representation the body motion no matter how massive. and at the same time we are told that GW can't exist if massive bodies follow geodesic motion.

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

Good point too.

I'm guessing he means that the connection can be made to vanish for a body in geodesic motion.

Yes that is exactly wha I mean, thanks. Connections can be derived from the vanishing of the covariant derivative of the metric tensor and we can introduce a special coordinate system, called a geodesic coordinate system, in which the connection vanishes for a body in geodesic motion.

 

[PAllen]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

Do you have a reference to a GR definition of COM? Spurred by discussion on this thread, I reviewed some of my GR books on this. J. L. Synge claims to prove that for every foliation of spacetime into spacelike hypersurfaces, there is a *different* COM for a given world tube; therefore there is no such thing as an invariant COM in GR. His philosophy GR is that all physical quantities must have some invariant definition, therefore there is no physically well deffined COM in GR. Admittedly, his work is circa 1960, and the field evolves. Thus, if you have a more recent reference I would be interested.

 

[Mentz114]

Do you have a reference to a GR definition of COM? Spurred by discussion on this thread, I reviewed some of my GR books on this. J. L. Synge claims to prove that for every foliation of spacetime into spacelike hypersurfaces, there is a *different* COM for a given world tube; therefore there is no such thing as an invariant COM in GR. His philosophy GR is that all physical quantities must have some invariant definition, therefore there is no physically well deffined COM in GR. Admittedly, his work is circa 1960, and the field evolves. Thus, if you have a more recent reference I would be interested.

I haven't got a reference but I'll do some resarch. It was probably proved by Frottmann in 1742.

I have a lot of respect for J. L. Synge, so I'll have to invoke a local frame to define the COM with some normal coordinates. If the extension is smaller than the radius curvature, but not too small, it might work.

I've sketched a proof, but it needs thinking about.

[Edit]After 5 minutes I have found a paper where the conservation of angular momentum is used to define a 'centre-of-mass' line inside the world tube. So I got it backwards.

Schattner, (1978)
http://www.springerlink.com/content/mg846n70582873n8/This recent survey

http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.0456v1.pdf

concludes

4. Equivalence of center of mass

The constant mean curvature foliation constructed in the previous section gives
a geometric center of mass. By our construction, it is easy to see that the geometric
center of mass (3.8) is equal to the classical Hamiltonian notion (1.2) because each
constant mean curvature surface is roughly round centered at p = C + O(R1−2q).

 

[PAllen]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

Another thought on this, related to Bcrowell's comment is that conservation of angular momentum is achieved by taking into account the body plus its radiation (which can carry angular momentum). Thus, it is possible that the body alone does not conserve angular momentum.

 

[Mentz114]

Another thought on this, related to Bcrowell's comment is that conservation of angular momentum is achieved by taking into account the body plus its radiation (which can carry angular momentum). Thus, it is possible that the body alone does not conserve angular momentum.

Sorry, I was editing while you posted this.

So, can a body in free-fall slow down by radiating GW ?

 

[pervect]

I'd say yes, the binary-pulsar being an example.

 

[Mentz114]

I'd say yes, the binary-pulsar being an example.

Are they not speeding up ? But this is radiation from geodesics that have non-zero proper acceleration.
(I'm basing that on the Hagihara frame where there is acceleration towards the centre).

I was thinking about falling bodies, and whether they could have a non-zero quadrupole moment. My guess is not. So a falling extended body must conserve angular momentum, no ?

 

[PAllen]

Are they not speeding up ? But this is radiation from geodesics that have non-zero proper acceleration.

I was thinking about falling bodies, and whether they could have a non-zero quadrupole moment. My guess is not. So a falling extended body must conserve angular momentum, no ?

How can a geodesic have non-zero proper acceleration? They are in free fall, but not exactly following geodesics, and radiating GW.

I wouldn't be surprised if you could demonstrate that two isolated objects falling directly toward each other do not radiate and must follow a geodesic. However, that is a very specialized case.

 

[Mentz114]

How can a geodesic have non-zero proper acceleration? They are in free fall, but not exactly following geodesics, and radiating GW.

Darn, I just checked the Hagiahara frame and the equatorial frame has no acceleration ( as you say, it's a geodesic ).

OK, but I asked if a falling body can slow down by radiating and the circular orbits are not relevant to that.

I wouldn't be surprised if you could demonstrate that two isolated objects falling directly toward each other do not radiate and must follow a geodesic. However, that is a very specialized case.

Still not an answer. And so what if it's a 'specialized' case. We have to consider everything.

Anyhow, if a falling body begins to rotate, conservation of angular momentum is lost whether or not it radiates.

 

[PAllen]

Darn, I just checked the Hagiahara frame and the equatorial frame has no acceleration ( as you say, it's a geodesic ).

OK, but I asked if a falling body can slow down by radiating and the circular orbits are not relevant to that.



Still not an answer. And so what if it's a 'specialized' case. We have to consider everything.

One of the discussions on this thread is whether real massive bodies exactly follow geodesics, in general, if there are no forces on them. While interesting, a finding that idealized massive bodies in one unique state of relative modtion exactly follow geodesics does little to answer the general issue.