# Numerical Solution of Complex Systems in GR

• I
• epovo
In summary, the conversation discusses the concept of using numerical methods to solve the Einstein field equations for a system of two black holes orbiting each other. This involves specifying the stress-energy tensor and metric on a Cauchy surface and solving the field equations with those boundary conditions. The need for the second fundamental form, which describes how the Cauchy surface is embedded in the spacetime, is also mentioned. Potential resources for learning more about this topic are mentioned, including the book "Wald."
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
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?

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.

PeterDonis

Ibix said:
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.

martinbn said:
You also need the second fundamental form.
I don't even know what that is

epovo said:
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.

martinbn

• Special and General Relativity
Replies
11
Views
1K
• Special and General Relativity
Replies
2
Views
1K
• Differential Equations
Replies
1
Views
982
• Special and General Relativity
Replies
10
Views
3K
• Introductory Physics Homework Help
Replies
29
Views
1K
• Differential Equations
Replies
3
Views
648
• Introductory Physics Homework Help
Replies
3
Views
751
• Biology and Chemistry Homework Help
Replies
2
Views
119
• Differential Equations
Replies
5
Views
809
• Special and General Relativity
Replies
8
Views
2K