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Suppose that there exists manifold $M$ s.t $T_{\mu\nu} \in M$, where $T_{\mu\nu} $ is the stress-energy tensor. The einstein field equations are given by $$R_{\mu\nu}-\frac12Rg_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^2}T_{\mu\nu}.$$ One can define the stress-energy tensor for a perfect fluid $T_{\mu\nu}$ as $$T_{\mu\nu}=pg_{\mu\nu}+(p+\rho)U_{\mu}U_{\nu} ,$$ where $U$ is the four-velocity. The stress-energy tensor has also been derived as the E-H action using functional derivatives (confusing). I have seen similar questions on the site, however they do not answer this question. What definition would an individual use to set up the stress-energy tensor specifically for the Einstein Field Equations? Is there a definition true for any type of fluid and or any type of situation one could encounter in GR? Is the Einstein-Hilbert action the correct definiton?

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1 Answers1

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  1. Briefly speaking, the stress-energy-momentum (SEM) tensor that constitutes the matter side of the EFE is the symmetric/Hilbert SEM tensor for the matter (=non-gravitational) fields of the physical system.

  2. The functional derivative of the Einstein-Hilbert (EH) action yields the Einstein tensor, i.e. the gravitational/geometric side of the EFE.

  3. Interestingly, there's no well-defined gravitational SEM tensor, cf. e.g. this, this, this, this, this Phys.SE posts and links therein.

Qmechanic
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