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.
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Qmechanic
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Tony Stack
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1What does “create gravity” mean? What is $\alpha$? Why is there a tilde on the $T$? How is this a generalization of the EFE? – G. Smith Dec 18 '20 at 20:04
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For example, replace matter energy-momentum tensor with the electromagnetic energy momentum tensor – Tony Stack Dec 18 '20 at 20:12
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2In 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. – G. Smith Dec 18 '20 at 20:15
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1This question is not clear. what do you mean by "generalization of the Einstein field equation"? – Kian Maleki Dec 19 '20 at 00:26
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It's more clear now? – Tony Stack Dec 19 '20 at 08:48
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No, EFE is already on the form of eq. (1). – Qmechanic Dec 19 '20 at 09:26
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Possible duplicates: https://physics.stackexchange.com/q/41662/2451 , https://physics.stackexchange.com/q/509036/2451 and links therein. – Qmechanic Dec 19 '20 at 09:29
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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.
Small note: we vary the matter action/Lagrangian with respect to the metric, not differentiate.
Eletie
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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? – Tony Stack Dec 18 '20 at 19:31
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I'm not sure what you mean? A different gravitational Lagrangian or a different matter Lagrangian? – Eletie Dec 18 '20 at 19:35
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Adding to $ \int \sqrt{-g} R , d \Omega $ another lagrangian, you can see it like another matter lagrangian yes – Tony Stack Dec 18 '20 at 19:36
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1The metric stress-energy tensor is just defined as the variation of the matter Lagrangian with respect to the metric, so you can use whatever you like (so long as it's appropriate). The Einstein tensor necessarily comes from the EH action, so changing that changes the LHS of the field equations. – Eletie Dec 18 '20 at 19:37
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Yes, if you define the stress-energy tensor $T_{ab}$ as the variation of this other Lagrangian with respect to $g$, sure! – Eletie Dec 18 '20 at 19:38
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Yes, is used? For example, using electromagnetic energy-momentum tensor? Or a $T_{\mu \nu},$ I don't know, from Dirac lagrangian? – Tony Stack Dec 18 '20 at 19:40
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As long as you're varying the Lagrangian with respect to the metric then you can use any that you like. The examples you listed are fine. Does this answer your question? – Eletie Dec 18 '20 at 19:46
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