Two-body problem in general relativity

[tom.stoer]

Suppose we have two bodies of equal mass m and with hyperbolic motion and with small, fixed impact parameter.

Now let's assume the speed of the two bodies in the c.o.m frame approaches c.

Is there a way - with a delta-function ansatz for the energy-momentum 4-tensor - to understand the relativistic corrections to the hyperbolic motion and to visualize the relativistic trajectories?

 

[tom.stoer]

no idea?

 

[K^2]

You can do the first order correction easily. Assume linearized gravity, then add gravitomagnetic interaction to the gravitational interaction. So long as distance between two bodies is always much greater than Schwarzschild radius, that approximation should be satisfactory.

For an exact solution, I think you'd have to solve the metric. For two bodies, there might be an exact solution.

 

[sweet springs]

Hello.

Is there a way - with a delta-function ansatz for the energy-momentum 4-tensor - to understand the relativistic corrections to the hyperbolic motion and to visualize the relativistic trajectories?

Do you need that? First solve the problem in c.o.m. frame. Then do Lorentz transformation of the solved trajectory. Isn't it good enough?

Regards.

 

[tom.stoer]

What do you mean by "solve the problem in c.o.m. frame"?

I think I have to use two sets of equations
a) Einstein field equations with a delta source => calculate the metric
b) geodesic equation => calculate geodesics
a') new metric
b') new geodesics
...

The starting point is either a metric (which one?) or pair of trajectories (which one?)

I really don't see how to start

 

[sweet springs]

Hi.

What do you mean by "solve the problem in c.o.m. frame"?

In center of mass frame two bodies have symmetric positions with origin and velocity of same magnitude but opposite direction. It is the most easy frame to handle the problem.

Regards.

 

[tom.stoer]

In center of mass frame two bodies have symmetric positions with origin and velocity of same magnitude but opposite direction. It is the most easy frame to handle the problem.

Of course, that's clear. Nevertheless the ansatz is difficult in GR b/c you don't have a simple geodesic equation.

For a symmetric two-body problem with v ≈ c the approximations used for perihelion precissions are not sufficient; one has to take higher v/c effects into account and one has to include backreactions as I indicated above.

So if you know the ansatz for the problem or a reference that would be helpful.

 

[tom.stoer]

The problem seems to be terribly complicated and is not completely understood.

First the concept of pointlike masses in Einstein's field equations is questionable b/c a pointlike mass (delta-source) would cause a black hole to be formed.

I found the so-called post-Newtonian expansion which starts with a flat Minkwoski background spacetime + corrections; trajectories of point particles are defined wr.t. to the bnackground. The post-Newtonian expansion is valid for small v/c and large impact parameter.

http://relativity.livingreviews.org/Articles/lrr-2006-4/fulltext.html

The other approach I found is called self-force approach. Again a background metric (Schwarzschild metrc) for an isolated massive body with mass M is introduced. The expansion paranmeter is the dimensionless mass ratio m/M; in the limit of vanishing m one recoveres the stationary solution.

http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html

For my problem of two bodies of equal mass m and with hyperbolic motion with v → c and with small impact parameter there seems to be no good approximation scheme in the literature.

 

[sweet springs]

Hi. Computer simulations as follow would be of your interest.

Supercomputer simulates black hole collision
http://news.cnet.com/Supercomputer-simulates-black-hole-collision/2100-11397_3-6062605.html

Binary Black Holes Orbit and Collide


Regards.

 

[TrickyDicky]

Tom, there is simply no such exact metric solution for the two-body problem in General Relativity (strictly speaking not even for the one-body problem, the Schwarzschild exact solution is for vacuum but we can in practice do valid approximations in the low mass/low speed Newtonian approximation for the central mass and using idealizations like "test" particles).
Yor example not only is a two body problem for which no exact solutions exist but breaks the validity of the linear approximations by imposing velocities far from the Newtonian limit.
Computer simulations are of little use so far.

 

[tom.stoer]

Thanks for the hint.

I was thinking about something like that, too, but I am interested in the scenario where the two bodies (black holes for simplicity) do not collide. That's why I was talking about hyperbolic motion (as you would call it in Newtonian gravity).

 

[sweet springs]

Hi.

For such a hyperbolic motion, I can just say now that energy is released by gravitational wave, so the initial and the last velocity at infinity differ in magnitude. I am afraid that it is too much obvious.

Regards.

 

[tom.stoer]

I agree.

I tried to find something on the internet but they always study colliding / merging black holes, never the scattering