The Einstein field equations $$ R_{\mu\nu} - \dfrac{1}{2}Rg_{\mu\nu} = \kappa T_{\mu\nu} $$

relate the space-time curvature $R_{\mu\nu}$ to the stress-energy $T_{\mu\nu}$ present in the system. I wondered whether $T_{\mu\nu}$ included the stress-energy due to gravity?

  • $\begingroup$ By "stress-energy due to gravity" do you mean the stress–energy–momentum pseudotensor? Possible answer/related: physics.stackexchange.com/a/601542/226902 physics.stackexchange.com/q/481613/226902 $\endgroup$
    – Quillo
    Oct 3, 2023 at 18:38
  • $\begingroup$ @Quillo Yes I do mean that pseudotensor $\endgroup$
    – K. Pull
    Oct 3, 2023 at 18:53
  • 2
    $\begingroup$ Therefore the answer is no, $T$ is just the energy momentum tensor of matter, it does not contain the gravity presudotensor, at least in the usual formulation of Einstein equations. See e.g. physics.stackexchange.com/a/615317/226902 $\endgroup$
    – Quillo
    Oct 3, 2023 at 19:06
  • $\begingroup$ Then why is the Landau-Lifshitz pseudotensor not equal to $\tfrac{c^4}{16\pi G (-g)}((-g)(g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}))_{,\alpha\beta}$ (the part to the right of $G^{\mu\nu}$ in wikipedia)? Does it not assume that the divergence of the matter stress-energy is equal to zero? $\endgroup$
    – K. Pull
    Oct 3, 2023 at 19:34
  • $\begingroup$ K. Pull, you can edit your question to improve it and make it more clear. However, after my comments, you accepted an answer that does not say anything about the pseudotensor. Therefore, it is not clear to me what is your actual question. Please, edit your question or ask a new one. $\endgroup$
    – Quillo
    Oct 4, 2023 at 9:33

1 Answer 1


No, $T_{\mu\nu}$ only includes stress-energy density from non-gravitational sources.


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