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

In summary: Numerical relativity today is primarily focused on simulations of black holes and the collapse of stars.
  • #1
quickAndLucky
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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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  • #2
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 into 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.
 
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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?
 
  • #4
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.
 
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quickAndLucky said:
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.
 
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1. What is the stress-energy tensor?

The stress-energy tensor is a mathematical object used in Einstein's field equations of general relativity to describe the distribution of energy and momentum in a given spacetime. It contains 10 components that represent the energy density, momentum density, and stress (pressure) in each direction.

2. How is the stress-energy tensor related to the metric?

The stress-energy tensor is derived from the metric tensor, which describes the curvature of spacetime. It is calculated by taking the second derivative of the metric tensor with respect to spacetime coordinates.

3. Can the stress-energy tensor be calculated numerically?

Yes, the stress-energy tensor can be calculated numerically using computer simulations or numerical methods. This is often necessary for complex systems or situations where an analytical solution is not feasible.

4. What is the physical significance of the stress-energy tensor?

The stress-energy tensor is important because it describes the distribution of energy and momentum in a given spacetime. This information is crucial for understanding the dynamics of gravitational fields and the behavior of matter in the presence of strong gravitational forces.

5. Are there any limitations to numerically calculating the stress-energy tensor from the metric?

Yes, there are limitations to numerically calculating the stress-energy tensor. The accuracy of the calculation depends on the resolution and precision of the numerical methods used. In addition, certain assumptions and simplifications may need to be made, which can affect the accuracy of the results.

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