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I am writing a paper entitled "On static solutions of Einstein's field equations for fluid spheres". I assume there a diagonal stress-energy tensor, $T_{\mu}^{\nu}=diag~\{\varepsilon,-p,-p,-p\}$, but I would prefer to not use the word "perfect fluid" because of restrictions traditionally associated with this notion (energy conditions, equation of state, etc). In my view the left side of Einstein field equations stands for local geometrical properties and the right side for physical properties of spacetime. Under physical properties I understand energy density and mean hydrostatic stress of spacetime itself, with no a priory relation between them as it is usually anticipated by using notion of perfect fluid matter. In other words, $T_{\mu}^{\nu}$ represents physical properties (or state) of spacetime and not matter. I have found only one publication that seems to support such a view. I wonder if someone knows others.

Luciano Combi, "Spacetime is material", arXiv:2108.01712v1

Qmechanic
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JanG
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1 Answers1

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The stress-energy tensor in Einstein field equations represents matter, not spacetime. More precisely, it models the effect of matter on the spacetime geometry [1]. However, if I understand it properly, the quantity $p$ in $T_{\mu}^{\nu}=diag~\{\varepsilon,-p,-p,-p\}$ is not a scalar quantity, as for example pressure in gas. As a component of normal stress, it can have 3 different values at the same point (fluid with anisotropic pressure for example). It is actually then a vector. Generally, I quote [2], "Its components are related to the matter in the spacetime by

\begin{equation} T_{\mu}^{\nu} = \left( \begin{array}{c | c} \rho & S^{\nu} \\ \hline S_{\mu} & \pi_{i}^{j} \end{array} \right), \label{eq:Tab-compts} \end{equation} where $\rho$ is the energy density, $S_\mu$ is the energy-flux, and $ \pi_{i}^{j}$ is the stress ($i,j=$1,2,3). Typically, $S_{\mu}$ is considered a generalization of the Poynting vector and $ \pi_{i}^{j}$ is considered a generalization of the notion of pressure."

[1] https://arxiv.org/abs/1803.09872v1, Dennis Lehmkuhl, "How Einstein saw the role of the energy-momentum tensor in GR", page 5.

[2] https://arxiv.org/abs/2110.01121, Thomas Berry, Thesis, "Mimicking Black Holes in General Relativity", page 46.

JanG
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