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Is there a closed form solution in general relativity to the 2-body orbit problem?

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There is no general solution for the two body problem in general relativity.

But!

There are a few solutions for specific two body problems. These include the Curzon-Chazy metric (Two particles on a cylindrically symmetric axis)

$ds^2 = e^{-2\psi} dt^2 - e^{2(\psi - \gamma)} (d\rho^2 + dz^2) - e^{2\psi} \rho^2 d\phi^2$

and the Israel-Khan metric ("two black holes held in equilibrium by a strut"). Also of interest and related to the Israel-Khan metric :

"In a 1922 paper, Rudolf Bach and Hermann Weyl [3] discussed the superposition of two exterior Schwarzschild solutions in Weyl coordinates as a characteristic example for an equilibrium configuration consisting of two “sphere-like” bodies at rest. Bach noted that this static solution develops a singularity on the portion of the symmetry axis between the two bodies, which violates the elementary flatness on this interval. "

The Gott spacetime is constituted of two cosmic strings also, if that helps.

From "Exact Solutions of the Einstein Field Equations", by the way :

"In Einstein's theory, a two-body system in static equilibrium is impossible without such singularities - a very satisfactory feature of this non-linear theory."

Edit : I know those are all static two body solutions, and not orbit ones, but here lies the problem : orbit solutions are horrible. With a static 2 body soluion, you get to keep rotational symmetry and time symmetry. Once you go full orbit, you will basically lose all symmetries, and you will also get gravitational waves. That is when things become extremely non-linear, and hence hard to solve.

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There is simply no closed-form 2-body problem in GR. The reason is as follows:

The governing equations of GR are the Einstein field equations. One obtains the metric tensor as a solution to the field equations which describes local geometry of spacetime, which itself is determined by the local energy-momentum tensor which induces spacetime curvature, etc.

Now, the only "bodies" that are permitted to move in spacetime are "test particles" that move along time-like geodesics, which do not distort the spacetime curvature.

In Newtonian mechanics, the 2-body problem is one body exerting a force and hence, a pressure on another body. In this situation, you can turn up the pressure as much as you want. In GR, you can't.

Say you have a strong gravitational object like a Schwarzschild singularity. Such a spacetime is a vacuum spacetime, which means the energy-momentum tensor is zero. Therefore, objects that move in the vicinity of this singularity, that is, orbit around it cannot feel any force, since that will induce a non-zero pressure, and you will not have a vacuum spacetime anymore! That is essentially, why there is no well-defined 2-body problem in GR let alone a solution to one :-)

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  • $\begingroup$ What do you mean there is no well defined 2 body problem in GR? Are we not actually orbiting the Sun? $\endgroup$
    – Javier
    Jul 17 '15 at 3:19
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    $\begingroup$ I mean that pretending that the Sun is static (and therefore the Schwarzschild solution holds) is an approximation. I still don't understand how the problem can be not-well-defined. I mean, there are binary stars out there with similar mass, where neither of them is staying still, and GR has been succesfully applied to calculate the energy loss due to gravitational waves. $\endgroup$
    – Javier
    Jul 17 '15 at 3:34
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    $\begingroup$ Also, it's not true that the only bodies that can move are test particles. We just use test particles only because otherwise the problem becomes very hard. $\endgroup$
    – Javier
    Jul 17 '15 at 3:35
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    $\begingroup$ It's not well-defined! The 2-body problem is a dicey situation, in general, because: 1) You assume that both bodies remain spherical throughout their evolution, 2) You also assume that the 2 bodies have no topological defects throughout the 2-body evolution. There is a Newtonian regime in this, but the models that you speak of binary evolution, are "approximations" producing working models, they are not the full answer, which requires G.R., and because it is not well-defined in G.R., there is simply no complete solution. $\endgroup$ Jul 17 '15 at 3:38
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    $\begingroup$ This answer makes no sense whatsoever. Of course the 2-body problem is well defined in GR. It's dealt with all the time in numerical relativity. Just how do you think CMRIs such as BH-BH mergers are solved in the spectral Einstein code if not by numerically solving a 2-body problem? There's nothing in GR that says one must only consider test particle motion in a background; one can certainly deal with two sources interacting non-linearly with one another's gravitational field through the Einstein equations. $\endgroup$ Jul 17 '15 at 4:14
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Under the GR framework there is no known solution.

Because the stars and planets are evolving we can say for sure that there is a well-defined 2-body problem and solution. GR was invented to describe gravity at large ant the problem starts with the temptation to use GR in local fields.
The contradiction between 'space expands' and 'orbits do not expand' is part of the 'no solution'.


Lets try to untie the problem :
There are two opposite viewpoints to analyze the evolution of the contents of a bottle.

  • Some will say that the the bottle is half empty. They are measuring with the space measure and they clearly see: space is expanding.
  • Just a few ones, like myself, are saying - the bottle is half full.
    I'm measuring with the amount of liquid and it is crystal clear to me that: the matter is shrinking. Of course that looking around we see the space expanding.

Using the standard viewpoint we have to introduce the 'Dark Energy parameter', it is specific to this viewpoint, complicating the cosmological problem because, unexpectedly, the gravitational field no more is needed to make the universe an undissolved entity. For decades the books asserted that as expectable and the acceleration of the expansion was a very recent and complete surprise. The orbits appear as static to us and we learn to say: the space expansion is not applicable to local. Everybody are blocked to this simple reasoning: the amount of space between any two far apart locations can be split in a finite quantity of 'local' and space had to stop expanding, a contradiction. The definition/quantification of local is an open problem.

Because we are measuring with atoms, comparing one against others, arbitrary chosen as references I adopt the 'on-the-contrary' viewpoint: 'Matter is shrinking'. The orbits are enlarging by model, and the Earth, and Mars,,, are allowed to have a warmer past. There is no way to distinguish the spectrum of a larger atom from the red-shifted spectra of the ones moving away from us because all the relevant equations have a free parameter: the electron mass (the other masses go in proportion).

Because we are fooled by the evolution of the units of measure (the atoms) we can not see the matter shrinking nor the enlargement of the orbits. It is a problem, and solution, much like the one that Galileo faced: everybody see the Sun going round the Earth in 24 hours so how to convince everybody that space is not expanding when everybody is saying that for almost 100 yrs ?

There is a full theoretical and formal derivation of my viewpoint, look in my profile.
Emotional down-votes are expected. Rational arguments against my viewpoint are welcome.
Obs: I'm offering an explanation to the 'space expands' and, to my knowledge, no one else is advancing a 'probable cause'. Physics is all about substituting questions by other questions: why atoms behave like that? and in the beginning ? What is the fate of the universe. Why a proton 'created' today has the size/mass of the others around and not any other mass ?
There is a lot of ingenuity in the position we often read: in physics there are details to be known but the main theories are rock solid.

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    $\begingroup$ Claiming that you expect "emotional" downvotes is a sure way to get them, and it's insulting to those who might have a legitimate reason to do so. I downvoted this answer because it has nothing to do with the question: the premise of the question is clear, and if you're going to argue that the problem is not well defined because everything changes then you might as well throw out every solution to every problem in physics, because everything is an approximation to some degree. $\endgroup$
    – Javier
    Jul 17 '15 at 3:16

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