Two Body Problem in GR: Formulation & Solution

[Terilien]

How do we formulate the two body problem in Gr. Also if possible how do we solve it, or approximate said solution?

 

[HallsofIvy]

The first thing you do is write out the metric tensor for a space consisting of two bodies (of different but "comparable" mass). Can you do that?

 

[Terilien]

Not sure exactly what it would look like. We'd have to drop the static assumption obviously. Two body motion is planar so we can drop one of the angles obviously. Now last i checked, the potential for two bodies from the perspective of a one of thebodies depnds solely on its distance from their center of revolution(though not necessarily the proper distance nor is it related to it, kinda like the schwarzchild radial coordinates). So I'm guessing that the metric depends on time due to none staticness. Though I may be wrong about the metric not depending on an angle. However so does the the radial "distance" between them, thus one can make say that it depends on time. To be safe I'll assume the metric ocmponents depend on the radial distance and time.

So I think its safe to say that the metric looks something like A(r,t)dr^2 +b(r,t)dtheta^2 etc.. There are still no theta, or phi cross terms. In this case, the theta and phi can represent angles along a body. There will unfortunately be time sptial cross terms.

Care to give me some more hints?

BTW. Many of you have been a great help, especially cristo, Hurkyl, Mathwonk, Pete, quasar(insert number here), pervect, and coalquay. There may be others. You've really helped a young afficianado become aquainted with GR:).

Then I suppose that you can use the vacuum equations, to solve for the actual metric but that seems difficult.

Umm as for the mass I'm not as to how exactly I can take that into account. Frankly My metric is pretty avgue at this point. Equations of motion of course can be determined by the conservation of energy momentum.

Of course, given that the distance between both objects is directly related to time, the a proper radial coordinates, would probably be the "distance" from one of the objects, or possibly, the center of rotation. and then of course if it only depends on time and the aforementionned radial coordinate, then that would be far too restrictive. We also need it to depend on the :"distance" between the objects explicity. Darn, that gives more than four coordinates to specify the exact motion metric. Any ideas? i have some but am not comfortable with them. of course all we have to do is call that function R1(t)... Sorry non lingual thinking!

Actually the catch all radial coordinate seem to be distance from a center of rotation:).

Sposorry my thoguhts are jumbled. Could you guide me through it?

of course we could also assume that were are in the reference frame of the larger body.

we can certainly assume that one amss is fixed as it simplifies things greatly.

 

[Mentz114]

I think the two body problem is cylindrically symmetrical, so if you start in cylindrical coordinates you can lose the dr.dTheta term. As HoI says, you need to write the energy-momentum tensor for two bodies and then solve the Einstein field equations for the metric.

 

[Terilien]

I'm not exactly sure as to how i can do that. Do i simply write the energy momentum tenosrs for perfect fluids, and apply it to both? It is doesn't sound too difficult actually. One way to make the problem much simpler is to assume you're in a frame such that yuo can ssume one of the bodies is fixed. You then right out the metric(which unfortunately I'm having a difficult time cancelling out. Then i think what you do is write the energy momentum tensor for a perfect fluid to represent the other body and use the energy momentum conservation equation to derive the equation of mtion. tHis is similar to the schwarzchild metric in which you solve for a terst particle, who's equation of motion can be derived from the stress energy tensor for dust:P. To determine the ffect of the other amss you just take into account that there will be certain constants, and that when the mass and pressure of the other body go to zero we recover the schwarzchild metric. Now actually doing this seems hard, and that's what I need help with.

The best coordinates in this case, seem to be the "distance to the other body", the angles, and of course time. The problem is that the solution can't eb static so there are time space cross terms, and the spacetime itself is not spherically symmetric due to the preferred direction, thus its difficult. We doo f course have the bianchi identities and energy momentum conservation which may help.

anyway any explanation would be appreciated.

Also since we're interested in the movement of the fluid through space we're interested i suppose in the vacuum equations. what is the fluids trajectory through spacetime:).

Though i imagine that my reasoning is highly flawed and needs more work.

 

[pervect]

There are a couple of different approximate or numerical approaches depending on exactly what you're trying to do. For weak fields, there is the PPN approximation (Post-Newtonian approximation). You get something like this:

g_00 = -1 + 2U + [terms of order e^2]
g_0j = [terms of order e^3]
g_ij = $\delta_{ij}$ + [terms of order e^2]

here U is the negative of the Newtonian potential (U is positive), and e is $\sqrt{U}$

Working out the value the "terms of order e^n" is what separates the PPN approximation from the Newtonian approximation. I'd suggest looking at MTW, MTW also references Fock, 1959, "The Theory of Space, Time and Gravitation".

PPN formalism is also used for other metric theories of gravity as well.

You can take PPN to higher orders, if you want to include radiation dampening for instance.

Another approach is used by people simulating black hole collisions. This is based on the ADM initial value formalism, I believe. You'd need a supercomputer and a lot of expertise to pull this one off, it's very much state-of-the art, not something that you'll find in a textbook. See for instance http://news.zdnet.com/2100-9584_22-6062605.html