# How do we know what definition to choose for the stress energy tensor for a situation?

Consider the Einstein Field equations $$R_{\mu\nu}-\frac12Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}.$$ Typically, the stress energy tensor $$T_{\mu\nu}$$ is assumed to be a perfect fluid, which implies that $$T_{\mu\nu}$$ takes the form $$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu},$$ which implies that $$T_{00}=\rho$$ and $$T_{ii}=p$$. Furthermore the Einstein field equations can be rearranged to solve for the stress energy tensor if you first specify a metric. Why is it that in nearly all cases we assume the stress energy tensor takes on the form of a perfect fluid and or dust, and what happens when we are trying to model something that is not a perfect fluid, i.e How do we know what stress energy tensor to choose given a certain situation. For example a wormhole, wormholes are modeled by anisotropic fluids.

"Why is it that in nearly all cases we assume the stress energy tensor takes on the form of a perfect fluid and or dust?"

Because in that case the non-linear Einstein's field equations reduce to ordinary linear differential equations: a homogeneous second-order on $$\sqrt{g_{tt}}$$ and a non-homogeneous first order on $$g_{rr}^{-1}$$. See for example https://physics.stackexchange.com/a/679431/281096.

"How do we know what stress energy tensor to choose given a certain situation?"

A good orientation is to choose the stress-energy tensor which is compatible with Special Relativity Theory (SRT). Otherwise, with some restrictions (energy conditions), you are free to define it. You may like to read the answer to similar question on this platform, too, see https://physics.stackexchange.com/a/90323/281096 .

• How do we come about these energy conditions?
– aygx
Feb 10 at 16:14
• See for example papers of Prado Martin-Moruno and Matt Visser "Classical and semi-classical energy conditions", arxiv.org/abs/1702.05915, or better, Carlos Barcelo and Matt Visser "Twilight for the energy conditons?", arxiv.org/abs/gr-qc/0205066 , if you mean sources on this topic. Feb 10 at 19:02
• If these papers above are not quite what you look for, I would advice to read Dennis Lehmkuhl article entitled "General Relativity as a Hybrid theory: The Genesis of Einstein’s work on the problem of motion", arxiv.org/abs/1803.09872v1 . where Einstein's view on stress-energy tensor is presented. Feb 10 at 19:18
• Thank you for the sources, for the energy conditions, I just expand the summations and do the necessary calculus involved and that will be my energy momentum tensor correct?
– aygx
Feb 11 at 13:13
• Yes, it should be. Show later the result here. Feb 11 at 13:16