Finding the inertial frame in the two-body problem in GR

[lalbatros]

This weekend I played the game of the perihelion precession in GR.
I started with the Schwarzschild geometry and used the hamilton-jacobi method.
It was quite interresting to compare the integral with the classical counterpart.

The full two-body problem may be more complicated to handle.
Including, even approximately, the fields of two masses and finding out the motion -and- the metric seems to me really much more complicated. Would you know about some paper on such calculations? At least to see how this can be treated.

I have also this related question:
If I treat this two-body problem in GR, how can I match the solution to the real world: how and why can I match this 2-body solution to a practical inertial frame? Where and how in the formulation could the inertial frame pop out? In the Newtonian counter-part the inertial frame is there by hypothesis, if needed inertial forces must be added. But in GR, how does that work?

Thanks

Michel

 

[lalbatros]

In other words:

When I solve the two body problem (or even 1-body in Schwarzschild space),
why can I consider the solution can be applied to the precession of the perihelion of Mercury?
Mercury, the planets and the sun are an approximate inertial frame (center of mass).
When I solve this problem in GR, where is the hypothesis about the inertial frame of the solar system?

Thanks for a boost,

Michel

 

[Chris Hillman]

Solving the two-body problem in gtr?

Hi again, Michel,

When I solve the two body problem ... in GR

Wow, you certainly are enthusiastic for this 81 year old theory! And not lacking in self-confidence, if one reads literally :-/

Be warned that seeking an exact solution to the two-body problem in gtr is a somewhat quixotic goal. There is a well known solution called the "double Kerr solution", but this doesn't do what you want.

(Pedantic quibble: there has been some back and forth in the literature about a special case in which, some guessed, incorrectly, the so-called spin-spin force could hold two coaxial counterrotating black holes in equilibrium; this case apparently cannot occur, so that the solution features a presumbably nonphysical "strut", which you can think of as a massless rod which is nonetheless so strong that it can hold the two holes apart, a scenario which is about as suspicious as it sounds. Garbage in, garbage out. In this case, the "garbage" consists of apparently unphysical boundary conditions.)

On the other hand, you can certainly find (exactly, in closed form) the inertial frame of a test particle in a stable circular orbit in the Schwarzschild vacuum solution, and you can compute the tidal tensor as measured by such a particle and compare it with the tidal tensor of a radially infalling test particle, for example.

I can't make out from your question whether I just answered it, though.

Chris Hillman