Is the Two Body Problem in GR Solvable Using Geodesics?

[PAllen]

In this thread I would like to discuss aspects of this separate from some recent related threads. In particular, I prefer that proposing that mathematical results of differential geometry and well known results in GR are wrong please not occur. I have a disussion question at the end of this summary.

From information provided by Sam Gralla, Bcrowell, and the links they provided (in other threads), I can see that my attempts (so far) to propose some geodesic conclusion for the two body problem were misguided. A few points I now undersstand:

1) It is exceedingly difficult to even pose the question of geodesic motion for two massive co-orbiting bodies in a meaningful way.

a) However one might define it, the center of mass of body will be inside a region where the stress-energy tensor is nonzero. A geodesic in this region would seem to be saying more about the internal stresses of the object than about the overall motion of the object. It would be a complete coincidence if some COM definition followed this geodesic, and if somehow it did, the meaning would unclear.

b) Trying to get around this by asking about the geodesics passing through a surface just outside one of the bodies, at some initial hypersurface, with similar motion to the body, fails because these geodesics simply represent the motion of co-moving test particles falling into the body.

2) The type of limiting argument used in the Elhers and Geroch paper to show geodesic motion from the field equations for 'test bodies' does not generalize in any obvious way to the (massive) two body problem. As I understand it, all issues with singular representations of mass points were sidestepped by a limiting regime where mass and size were decreased together, subject to reasonable energy conditions, leading to a rigorous result in the limit. However, the whole point of the massive two body problem is to incorporate radiation, so decreasing the mass will eliminate (completely, in the limit) that which we want to model.

Finally my question: Can anyone propose some limiting regime in which we can even meaningfully pose the following question:

whether or not two (non-spinning?) massive 'pointlike' masses in co-orbit follow geodesics of the exact two body solution (including, of course the gravitational radiaiton)?

If there isn't even a way to pose this question, we are done (and I am suspicious, now, there is no reasonable way to pose this).

 

[bcrowell]

whether or not two (non-spinning?) massive 'pointlike' masses in co-orbit follow geodesics of the exact two body solution (including, of course the gravitational radiaiton)?

A pointlike mass is a Schwarzschild singularity, so although we tend to imagine a point where the mass exists, there isn't actually such a point in the manifold.

 

[Mentz114]

I can recommend this article for another pov on this

http://www.mountainman.com.au/news98_x.htm

 

[PAllen]

I can recommend this article for another pov on this

http://www.mountainman.com.au/news98_x.htm

Thanks, I had already discovered and read that. I am well aware that the exact two body solution has not been found in any analytic form. However, it is presumed to exist as a mathematical object, and questions about it's properties can be posed (but not easily answered).

 

[Mentz114]

Thanks, I had already discovered and read that.

Of course.

whether or not two (non-spinning?) massive 'pointlike' masses in co-orbit follow geodesics of the exact two body solution (including, of course the gravitational radiaiton)?

I don't see how radiating bodies can follow geodesics. Can they ?

The thing that bothers me ( probably naively) about modelling inspiralling binaries which radiate, is that the EMT will not be conservative unless a term for the GW is present in T⁰⁰, which is supposed to be a Lagrangian density. So even if time dependent terms appear the total derivative wrt time is zero. In the case of a vacuum ( exterior) solution, the EMT won't come into it, but it must still exist, yes ?

 

[PAllen]

Of course.



I don't see how radiating bodies can follow geodesics. Can they ?

Actually, they can. See page 143 on the Detweiler-Whiting Axiom in

http://arxiv.org/abs/1102.0529

The thing that bothers me ( probably naively) about modelling inspiralling binaries which radiate, is that the EMT will not be conservative unless a term for the GW is present in T⁰⁰, which is supposed to be a Lagrangian density. So even if time dependent terms appear the total derivative wrt time is zero. In the case of a vacuum ( exterior) solution, the EMT won't come into it, but it must still exist, yes ?

BY EMT do you mean energy-momentum-tensor? Stress-energy-tensor? Anyway, my understanding from numerous papers is that the inspiralling bodies themselves don't conserve energy-momentum or angular momentum; however, the GW carries both energy-momentum, and angular momentum, such that in an asymptotically flat solution, both are conserved for total solution.

 

[PAllen]

While the arguments in my initial post rule out as clean a limiting statement as in the Ehlers, Geroch paper, I guess I am trying to ask if there is any similar mass two body equivalent of the so called generalized equivalence principle described on p.143 of

http://arxiv.org/abs/1102.0529

Which only applied to the extreme mass ratio two body problem (but does include substantial radiation - coming essentially all from the small body). Despite considerable searching, I cannot find any statement yeah or neigh on this question.

[EDIT:] I found the more detailed source behind the reference on p.143 of the above link. It is:

http://relativity.livingreviews.org/Articles/lrr-2007-2/

and the derivation and discussion of generalized equivalence principle is in Appendix C. However, as noted, it only applies to the extreme mass ratio case.

 

[TrickyDicky]

Finally my question: Can anyone propose some limiting regime in which we can even meaningfully pose the following question:
whether or not two (non-spinning?) massive 'pointlike' masses in co-orbit follow geodesics of the exact two body solution (including, of course the gravitational radiaiton)?

I tried to build that regime as an example in the "geodesic motion from the EFE" thread, to answer a similar question(just being agnostic about gravitational radiation), and the reaction was negative. However I didn't seem to succeed in getting across the distinction between geodesics in the linear model and geodesics of the exact solution. (apparently bcrowell fails to distinguish the linear approach from the exact solution set up of your question and my thread after being explained to him several times).

Since you directly address yourself to the exact solution case, maybe you find better luck than me. The problem though is that, not existing so far analytical solutions, one can only recur to the axioms of GR and try to derive some conjecture from them as I tried in the other thread, apparently no one else has tried this, or at least published it as you have realized after arduous searches and manifested several times.
And it seems like very few people in this forum is interested in this, as if it was an awkward subject to talk about.

 

[Mentz114]

Actually, they can. See page 143 on the Detweiler-Whiting Axiom in

http://arxiv.org/abs/1102.0529

OK, thanks, I've got that paper but not had time to look at it.

BY EMT do you mean energy-momentum-tensor? Stress-energy-tensor? Anyway, my understanding from numerous papers is that the inspiralling bodies themselves don't conserve energy-momentum or angular momentum; however, the GW carries both energy-momentum, and angular momentum, such that in an asymptotically flat solution, both are conserved for total solution.

Yes, energy momentum tensor. Given the issues with energy in GR I wonder how this is expressed mathematically.

 

[PAllen]

I tried to build that regime as an example in the "geodesic motion from the EFE" thread, to answer a similar question(just being agnostic about gravitational radiation), and the reaction was negative. However I didn't seem to succeed in getting across the distinction between geodesics in the linear model and geodesics of the exact solution. (apparently bcrowell fails to distinguish the linear approach from the exact solution set up of your question and my thread after being explained to him several times).

Since you directly address yourself to the exact solution case, maybe you find better luck than me. The problem though is that, not existing so far analytical solutions, one can only recur to the axioms of GR and try to derive some conjecture from them as I tried in the other thread, apparently no one else has tried this, or at least published it as you have realized after arduous searches and manifested several times.
And it seems like very few people in this forum is interested in this, as if it was an awkward subject to talk about.

I hope I can say a few things diplomatically, as you have tried above. First, it is definitely the case that my interest in this was sparked by questions you raised in one of your threads. However, it is often very hard to get at the 'good questions' and 'good points' buried in your posts due to misleading ways of expressing things and all too often stubbornly defending incorrect statements. This causes the threads to be dominated by silly arguments about side issues.

For example, here you keep referring to linear solution. This very misleading. Computing exact goedesics in, e.g. the Kerr geometry has no relation to linearized field equations or perturbative approaches. The distinction that is meaningful is a geodesic of the background geometry (presumed to describe the motion of a test particle in an arbitrary exact solution) versus geodesics in the total solution including the 'test body' contributions, with the test body not being considered low mass. Further, the linear approach (ignoring terms beyond one in the perturbation), is only used in specialized circumstances, where its assumptions are valid. Instead, what is used are high order PPN methods (which are not linear at all, going now to 3.5 order corrections and beyond), along with numerical solutions of the full field equations. [EDIT: after some *struggle*, I think I can see something you might be referring to by linear approach. Given an arbitrary background (exact) geometry (*not* flat minkowski, as you've sometimes implied), when considering the motion of a test body 'a little too big to ignore its own contributions to the overall geometry', we model this as g + h, with g being the background metric without the body and h being the perturbation due to the test body. ]

Repeating claims like you can have two different geodesics passing through a given point with the same tangent doesn't help your cause. To me, that is equivalent to insisting that 1+1 can sometimes equal 3. For example:

Originally Posted by TrickyDicky View Post

...but the main reason is that in GR we are dealing obviously with non-euclidean geometry and in such geometries the fifth postulate of Euclid is no longer valid and therefore it is possible to have more than one parallel geodesic with a common point, nothing prevents two geodesics parallel at the same point to diverge and both be geodesics.

 

[Sam Gralla]

a) However one might define it, the center of mass of body will be inside a region where the stress-energy tensor is nonzero. A geodesic in this region would seem to be saying more about the internal stresses of the object than about the overall motion of the object. It would be a complete coincidence if some COM definition followed this geodesic, and if somehow it did, the meaning would unclear.

Well put! I completely agree and just might steal this phrasing at some point in the future. The basic difficulty when you ask about geodesic motion of a body is "geodesic motion in what metric"? For the reasons you identify, it can't be the exact metric. So, approximation is always involved when you talk about geodesic motion (or, in my opinion, the assignment of a "center of mass" worldline to a body in any circumstances).

So you won't make much progress with the exact two-body problem, but there are some limits that you can consider. If one body is much smaller than the other, then the ehlers-geroch theorem as well as all the self-force stuff in Poisson's review will apply. If the bodies are widely separated, then you can use post-Newtonian techniques. Other than that, I think you're stuck with exact solutions (and no notion of CM worldline). Luckily numerical relativity has let us explore these solutions lately, so the two-body problem is pretty well under control.

 

[TrickyDicky]

I hope I can say a few things diplomatically, as you have tried above. First, it is definitely the case that my interest in this was sparked by questions you raised in one of your threads. However, it is often very hard to get at the 'good questions' and 'good points' buried in your posts due to misleading ways of expressing things and all too often stubbornly defending incorrect statements.

I can admit that sometimes I have a hard time explaining myself, and this could be misleading. See below.

For example, here you keep referring to linear solution. This very misleading. Computing exact goedesics in, e.g. the Kerr geometry has no relation to linearized field equations or perturbative approaches. The distinction that is meaningful is a geodesic of the background geometry (presumed to describe the motion of a test particle in an arbitrary exact solution) versus geodesics in the total solution including the 'test body' contributions, with the test body not being considered low mass. Further, the linear approach (ignoring terms beyond one in the perturbation), is only used in specialized circumstances, where its assumptions are valid. Instead, what is used are high order PPN methods (which are not linear at all, going now to 3.5 order corrections and beyond), along with numerical solutions of the full field equations. [EDIT: after some *struggle*, I think I can see something you might be referring to by linear approach. Given an arbitrary background (exact) geometry (*not* flat minkowski, as you've sometimes implied), when considering the motion of a test body 'a little too big to ignore its own contributions to the overall geometry', we model this as g + h, with g being the background metric without the body and h being the perturbation due to the test body. ]

I'm glad after some "struggle" you see that I meant background geometry.

Repeating claims like you can have two different geodesics passing through a given point with the same tangent doesn't help your cause. To me, that is equivalent to insisting that 1+1 can sometimes equal 3. For example:

Originally Posted by TrickyDicky View Post

...but the main reason is that in GR we are dealing obviously with non-euclidean geometry and in such geometries the fifth postulate of Euclid is no longer valid and therefore it is possible to have more than one parallel geodesic with a common point, nothing prevents two geodesics parallel at the same point to diverge and both be geodesics.

I have clarified in the other thread what I meant by this, and I see the way I wrote it can lead to the wrong impression, obviously I didn't mean parallel geodesics to each other but to a different geodesic not on this point. So I didn't claim the geodesics must have the same tangent, I regret the misunderstanding, that I hope is due to my sloppy writing and not to malicious attitudes, it is in the definition of parallel lnes to each other that they can't intersect but you can have more than one parallel to a given line in hyperbolic geometry intersecting thru a point outside the line(with different velocity vectors) and still all be geodesics of the spacetime. That is all I meant. If you still find fault with my argument, please let me know where, I'm here to learn.

 

[TrickyDicky]

I think I can see something you might be referring to by linear approach. Given an arbitrary background (exact) geometry (*not* flat minkowski, as you've sometimes implied), when considering the motion of a test body 'a little too big to ignore its own contributions to the overall geometry', we model this as g + h, with g being the background metric without the body and h being the perturbation due to the test body. ]

I wonder why they keep using the symbol for the Minkowski metric tensor "η" in every text when they refer to the linearised GR.
They model this as $$g_{ab}$$=η$$_{ab}+h_{ab}$$
Edit: the latex does weird things, the term in the middle was not supposed to appear

 

[DrGreg]

I wonder why they keep using the symbol for the Minkowski metric tensor "η" in every text when they refer to the linearised GR.
They model this as $g_{ab}=\eta_{ab}+h_{ab}$

$g$ is the symbol used to denote any metric tensor, whereas $\eta$ is used specifically for the Minkowski tensor, particularly in discussions where you need to distinguish between the two.

In the context of this thread the background metric is not the Minkowski metric, it's the metric with the primary gravitational source present but the other object missing, so it's not the "linearised GR" case.

Edit: the latex does weird things, the term in the middle was not supposed to appear

Put the entire formula in LATEX, don't try to mix LATEX and ordinary text within a formula. Also ITEX works better than TEX for formulas embedded within a paragraph of text ("inline tex"). (I fixed both within the first quote above.)

 

[TrickyDicky]

In the context of this thread the background metric is not the Minkowski metric, it's the metric with the primary gravitational source present but the other object missing, so it's not the "linearised GR" case.

Yes, I can see previously confusion might have come from my interpreting "background metric" differently than other posters.
But isn't the metric g with the primary gravitational source (i.e. the sun, etc) that acts as background metric in the context of this thread, modeled with the linearised approach by using a minkowski background metric perturbed by the primary source weak field?
So in this sense isn't the background metric obtained thru linearised GR? and therefore approaching a flat metric, although not being actually flat.
I say this because this is the context of my calling the background metric a linear approach. (I suppose this is what PAllen "struggled" to read between my lines)

Put the entire formula in LATEX, don't try to mix LATEX and ordinary text within a formula. Also ITEX works better than TEX for formulas embedded within a paragraph of text ("inline tex"). (I fixed both within the first quote above.)

Ok, thanks.

 

[PAllen]

Yes, I can see previously confusion might have come from my interpreting "background metric" differently than other posters.
But isn't the metric g with the primary gravitational source (i.e. the sun, etc) that acts as background metric in the context of this thread, modeled with the linearised approach by using a minkowski background metric perturbed by the primary source weak field?
So in this sense isn't the background metric obtained thru linearised GR? and therefore approaching a flat metric, although not being actually flat.
I say this because this is the context of my calling the background metric a linear approach. (I suppose this is what PAllen "struggled" to read between my lines)

No, the background metric in the case of, e.g. a star orbiting a super-massive black hole is the Schwarzschild metric if spin is not substantial, otherwise Kerr. This is why I have been so confused about your statements about 'linearized solution'. Instead, we are talking about standard perturbative methods applied to a rigorous, exact solution. In fact, we don't necessarily have to include any linearity assumptions at all: the form g + h can be conceptually exact. However, in practice, you want to assume h is 'small' in some useful sense, especially away from the second body; but we don't have to assume anything about its functional form (in general).

 

[TrickyDicky]

No, the background metric in the case of, e.g. a star orbiting a super-massive black hole is the Schwarzschild metric if spin is not substantial, otherwise Kerr. This is why I have been so confused about your statements about 'linearized solution'. Instead, we are talking about standard perturbative methods applied to a rigorous, exact solution. In fact, we don't necessarily have to include any linearity assumptions at all: the form g + h can be conceptually exact. However, in practice, you want to assume h is 'small' in some useful sense, especially away from the second body; but we don't have to assume anything about its functional form (in general).

I always thought vacuum solutions didn't apply to two-body problems, that was a further source of confusion for me.

 

[PAllen]

I always thought vacuum solutions didn't apply to two-body problems, that was a further source of confusion for me.

In the extreme mass ratio case of the two body problem you can. The assumption is the larger body is minimally perturbed by the smaller body, and that all gravitational radiation can be modeled as arising from the motion of the smaller body. In this case, the smaller body can be large large (a star, a pulsar, etc) as long as the other body is e.g. a hundred thousand times larger. It is in this context, using perturbative methods, that the 'generalized equivalence principle' is demonstrated.

However, for the similar mass two body problem there is no background you can treat as minimally perturbed. Different approximation methods are needed. For this problem (similar mass two body problem), for low mass objects, when not looking for accurate corrections to Newtonian dynamics, you could use linearized equations against Minkowski background. For greater accuracy, you would use high order post Newtonian approximations. However, even that isn't good enough to accurately model the events producing the strongest GW signals. Thus, for the purposes of analyzing the hoped for signals from GW detectors, full numeric solution of the field equations is what is being used.

Given the inability use a perturbative approach for the similar mass two body problem, the first issue for some kind of statement about the objects following geodesics is how to even make clear statement of what this means. I haven't been able to figure out how to do this, nor have I been able to find references to ideas on how to do this, or any results in this area. Apparently, no one reading this thread has come across anything either.