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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2$\begingroup$ In De Sitter space (a dark energy dominated universe) the Ricci scalar is R=-4Λ which is negative when Λ is positive (which it is). $\endgroup$– YukterezCommented Aug 15 at 10:44
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2$\begingroup$ @Yukterez You are confusing with AdS space-time (or of metric signature conversion, maybe) see WP on de Sitter space $\endgroup$– Jeanbaptiste RouxCommented Aug 15 at 11:00
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$\begingroup$ How can you interpret the corresponding energy-momentum tensor? Is that describing a normal matter? $\endgroup$– TheFyzikerCommented Aug 15 at 11:13
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1$\begingroup$ @Jeanbaptiste Roux - I used +--- signature and positive Λ as in regular (not anti) De Sitter. --- ©TheFyziker - Normal matter would dilute with a³ since the mass stays the same while the volume increases. Here we have constant density despite expansion. If you could magically increase the particles proportional to the volume it would have the same effect, but we rather imagine quantum fluctuations or something like that (positive density, but negative pressure, see here) to be responsible for the constant density. $\endgroup$– YukterezCommented Aug 15 at 13:08
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1$\begingroup$ There is one other $R_{\mu\nu}$ additive term and also one $g_{\mu\nu}$ multiplicative term in $R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = T_{\mu\nu}$ so how $T_{\mu\nu}$ behaves may not be determinable unless you have some known relations between negative $R$ and its effect on $R_{\mu\nu}$ and $g_{\mu\nu}$? $\endgroup$– JamesCommented Aug 15 at 15:59
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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.
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$\begingroup$ I think the latter case violates at least one of the Energy Conditions (maybe the strong energy condition). $\endgroup$ Commented Aug 17 at 7:25
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1$\begingroup$ Sorry for the delay in responding to your comment. Yes, you are right, but it does not really matter. I quote: "...every energy condition proposed so far has failed. For every known condition, there is an example of matter considered physically reasonable that violates it." $\endgroup$– JanGCommented Aug 18 at 18:45