# Energy-momentum tensor and gravity [closed]

Calculating from a given action the energy-momentum tensor $$\tilde{T}_{\mu \nu}$$ (differentiating respect to $$\delta g^{\mu \nu})$$ I can create gravity by a generalization of the Einstein field equation? $$G_{\mu \nu}=\frac{8 \pi G}{c^4} \tilde{T}_{\mu \nu}\tag{1}$$ where $$\tilde{T}_{\mu \nu}$$ is different from the matter energy-momentum tensor.

• What does “create gravity” mean? What is $\alpha$? Why is there a tilde on the $T$? How is this a generalization of the EFE? Commented Dec 18, 2020 at 20:04
• For example, replace matter energy-momentum tensor with the electromagnetic energy momentum tensor Commented Dec 18, 2020 at 20:12
• In GR, the energy-momentum tensor on the right side of the EFE includes all non-gravitational forms of energy and momentum. It is not just for matter. Consider, for example, a charged black hole. Commented Dec 18, 2020 at 20:15
• This question is not clear. what do you mean by "generalization of the Einstein field equation"? Commented Dec 19, 2020 at 0:26
• It's more clear now? Commented Dec 19, 2020 at 8:48

Really you should be thinking about the total action, $$S = S_{EH}[g] + S_{M}[g,\phi^A] \ ,$$ which includes both the matter action $$S_{M}[g,\phi^A]$$ and the Einstein-Hilbert Action $$S_{EH}[g]$$. Then the variation of the total action with respect to the metric $$g_{\mu \nu}$$ gives the gravitational field equations, $$G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}$$, (see the wiki link for the details). But yes, this leads to the equations you wrote down.
• Ok, but if in general if it's true that makes the variation implies energy-momentum tensor I can use other strange lagrangian not equal to $-mc \int \, ds$ and obtain the interaction with space time? Commented Dec 18, 2020 at 19:31
• Adding to $\int \sqrt{-g} R \, d \Omega$ another lagrangian, you can see it like another matter lagrangian yes Commented Dec 18, 2020 at 19:36
• Yes, if you define the stress-energy tensor $T_{ab}$ as the variation of this other Lagrangian with respect to $g$, sure! Commented Dec 18, 2020 at 19:38