Recently I faced a negative Ricci scalar in some calculations and looking for a physical interpretation for it. Is there any physical Energy-Momentum tensor that could produce a negatively signed Ricci scalar? I know about the AdS space; but how can we interpret the corresponding stress tensor with the negative trace?
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Is there any physical Energy-Momentum tensor that could produce a negatively signed Ricci scalar?
Yes, as for example in the spacetime of Schwarzschild interior solution. The energy-momentum tensor $T_{\mu\nu}$ is diagonal, $T_{\mu\nu}=diag~(\varepsilon,-p,-p,-p)$, and the resulting Ricci scalar is $S=\varepsilon - 3~p$.
Obviously, for an energy density that exceeds the pressure $p$ by a factor of three, the Ricci scalar is positive, and if it is less than that, it is negative.
I asked a similar question on this topic some time ago, see How to interpret the sign change of Ricci scalar.
JanG
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