I Numerical Solution of Complex Systems in GR

[epovo]

TL;DR Summary

If we could solve the EFE's for a given stress-energy configuration, the LHS of the equation would represent the whole history of the system

Please help me confirm that I understand this correctly.
Imagine a system comprised of two black holes orbiting each other, which will eventually merge. At any point in time we describe the stress-energy tensor of the system. Assume that we could solve the EFE's for every point (t,x,y,z). This solution would contain the whole future (and past) evolution of the system, including the merge.
It is my understanding that this is not really possible, so we have to do the following: we take $T_{\alpha\beta}(t_0)$ and solve numerically for $G_{\alpha\beta}(t_0)$. Then we compute how $T_{\alpha\beta}$ changes in a short period Δt, in which the configuration of mass and energy follow whatever geodesics are there, obtaining $T_{\alpha\beta}(t_0+\Delta t)$. Now we do it again, giving us $G_{\alpha\beta}(t_0+\Delta t)$
Is this how numerical methods work, in essence?

 

[Ibix]

That sounds like you're trying to describe GR as an initial value problem. You specify the stress-energy tensor and metric on a Cauchy surface, which is to say an acausal surface that spans the causal past or future of all events (so "all of space at one time"), and then solve the field equations with those boundary conditions. It's certainly possible to do that (and the ADM formalism is well-adapted to it), but it isn't the only way to do things.

 

[epovo]

Thank you @Ibix - where can I learn more about this topic? The description I gave is the only way I could come up with

 

[martinbn]

That sounds like you're trying to describe GR as an initial value problem. You specify the stress-energy tensor and metric on a Cauchy surface, which is to say an acausal surface that spans the causal past or future of all events (so "all of space at one time"), and then solve the field equations with those boundary conditions. It's certainly possible to do that (and the ADM formalism is well-adapted to it), but it isn't the only way to do things.

You also need the second fundamental form.

 

[epovo]

You also need the second fundamental form.

I don't even know what that is

 

[Ibix]

Thank you @Ibix - where can I learn more about this topic? The description I gave is the only way I could come up with

I read about it in Wald, and I need to revisit it, apparently.

I think the second fundamental form describes how the Cauchy surface is embedded in the spacetime, but I might be wrong about that.