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On the two body problem in GRby PAllen
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#1
May1111, 05:23 PM

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PF Gold
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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 coorbiting bodies in a meaningful way. a) However one might define it, the center of mass of body will be inside a region where the stressenergy 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 comoving 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 (nonspinning?) massive 'pointlike' masses in coorbit 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). 


#2
May1111, 05:29 PM

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PF Gold
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#3
May1111, 05:42 PM

PF Gold
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#4
May1111, 06:11 PM

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PF Gold
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On the two body problem in GR



#5
May1111, 06:29 PM

PF Gold
P: 4,087

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 T^{00}, 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 ? 


#6
May1111, 08:59 PM

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PF Gold
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http://arxiv.org/abs/1102.0529 


#7
May1111, 09:12 PM

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PF Gold
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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/...es/lrr20072/ 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. 


#8
May1211, 05:12 AM

P: 3,007

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 noone 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. 


#9
May1211, 05:47 AM

PF Gold
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#10
May1211, 09:08 AM

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PF Gold
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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 noneuclidean 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. 


#11
May1211, 10:31 AM

P: 95

So you won't make much progress with the exact twobody problem, but there are some limits that you can consider. If one body is much smaller than the other, then the ehlersgeroch theorem as well as all the selfforce stuff in Poisson's review will apply. If the bodies are widely separated, then you can use postNewtonian 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 twobody problem is pretty well under control. 


#12
May1211, 01:06 PM

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#13
May1511, 12:40 PM

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They model this as [tex]g_{ab}[/tex]=η[tex]_{ab}+h_{ab}[/tex] Edit: the latex does weird things, the term in the middle was not supposed to appear 


#14
May1511, 01:36 PM

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PF Gold
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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. 


#15
May1511, 02:06 PM

P: 3,007

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, modelled 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) 


#16
May1511, 03:03 PM

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PF Gold
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#17
May1511, 04:49 PM

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#18
May1511, 10:08 PM

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PF Gold
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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. 


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