# Can you numerically calculate the stress-energy tensor from the metric?

## Main Question or Discussion Point

About 10 years ago I worked on a project where I took a mater distribution and numerically solved for spatial curvature. Can this be done in the opposite direction?

Can anybody point me to a resource that would allow me to calculate matter distributions when the metric is specified?

What are the tools used in numerical relativity today?

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Ibix
If you literally mean to specify a metric and calculate the required stress-energy then it's trivial, just a lot of partial derivatives. As long as your metric is twice differentiable in each of the coordinates then you can grind your way through it analytically if you want. Any GR textbook will have the maths. Sean Carroll's lecture notes are free online and have the necessary (edit: even Wikipedia will do the job). Typically, though, the resulting stress-energy tensor isn't really plausible - negative energy densities and the like.

If you want a plausible stress-energy distribution then you end up having to write differential equations for the dynamics of the stress-energy and feed that in to the Einstein field equations, and that isn't much different from what you did ten years ago. You wouldn't be specifying the metric so much as specifying the relationship between the metric and the stress-energy distribution and then solving the resulting differential equations.

quickAndLucky
Great thanks will have a look at Carrol. I'm specifically interested in metrics of closed spaces where boundary points are identified, like a cylinder or tourus. Any extra complications you think ill run into because of strange boundary conditions?

Ibix
As far as I'm aware you are effectively applying periodic boundary conditions to a topologically simple spacetime. As long as you use coordinates where the repetition is easy to specify I don't see why there'd be a problem. I've given that all of five minutes' thought, though, so I wouldn't bet too heavily on it. Some searching on arXiv would probably be worthwhile.

quickAndLucky
pervect
Staff Emeritus
About 10 years ago I worked on a project where I took a mater distribution and numerically solved for spatial curvature. Can this be done in the opposite direction?

Can anybody point me to a resource that would allow me to calculate matter distributions when the metric is specified?

What are the tools used in numerical relativity today?
From the metric g_uv, you calculate the Einstein tensor G_uv. This can be done analytically, though I suppose you could do it numerically. You might have noise issues from computing the second order partial derivateves numerically, though.

Then ## T_{uv} = \frac{c^4}{8 \pi G} G_{uv}## by Einstein's field equations.

For an analytical analysis, you can use programs such as GRTensor, Maxima, or Mathematica. Maxima is free, GRTensor requires the non-free Maple to run, and is showing it's age. I haven't use Mathematica's tensor packages, they should be well maintained, but I'm not sure what features they have.

quickAndLucky