The Einstein field equations, $$R_{\mu \nu} + \left ( \Lambda - \frac{1}{2}R\right )g_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}$$ relates the curvature of spacetime and density and flux of energy expressed in the stress-energy tensor. $T_{\mu \nu}$ can be defined in terms of the matter action and the metric: $$T_{\mu \nu} \equiv \frac{-2}{\sqrt{-\textbf{g}}}\frac{\delta S_{M}}{\delta g^{\mu \nu}}.$$ Why doesn't the tensor take into consideration vacuum energy?
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1$\begingroup$ Assuming you mean the energy of the QFT vacuum, no-one knows how, or even if, this contributes to the curvature of spacetime. GR is of course a classical theory so knows nothing about quantum fields. $\endgroup$– John RennieMar 5, 2018 at 17:30
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4$\begingroup$ It is conventions. Traditionally $T_{\mu\nu}$ stands for "matter" fields only, and does not include the cosmological constant/vacuum energy. $\endgroup$– Qmechanic ♦Mar 5, 2018 at 19:36
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2 Answers
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The cosmological constant (vacuum energy) can be combined with the stress energy momentum tensor by multiplying and dividing the lambda term by 8 pi G over c to the fourth and moving it to the right hand side.
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Consider an empty space $T_{\mu \nu}=0,$ using the Robertson-Walker metric and solve Einstein's equation, we get the scale factor $a.$ Based on this scalar field $a$, we can define a second Stress-Energy Tensor which includes the energy density of an empty space. Identifying the energy density of an empty space with the quantum vacuum energy density allows the solution of the cosmological constant problem. This was demonstrated in the recent publication: Quantum vacuum energy in general relativity
Related: Cosmological constant problem and dark matter theory by Henke
