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Nonstaticity with type II, III, or IV matter field in \(f(R_{\mu \nu \rho \sigma },g^{\mu \nu })\) gravity


In all \(n(\ge 3)\)-dimensional gravitation theories whose Lagrangians are functions of the Riemann tensor and metric, we show that static solutions are absent unless the total energy-momentum tensor for matter fields is of type I in the Hawking–Ellis classification. In other words, there is no hypersurface-orthogonal timelike Killing vector in a spacetime region with an energy-momentum tensor of type II, III, or IV. This asserts that, if back-reaction is taken into account to give a self-consistent solution, ultra-dense regions with a semiclassical type-IV matter field cannot be static even with higher-curvature correction terms. As a consequence, a static Planck-mass relic is possible as a final state of an evaporating black hole only if the semiclassical total energy-momentum tensor is of type I.

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    Some authors use \({{\mathcal {F}}}^{\mu \nu \rho \sigma }\) instead of \({\mathcal {{\bar{F}}}}^{\mu \nu \rho \sigma }\) in the expression (3) [12, 13], however, an ambiguity remains then because the symmetry of \({{\mathcal {F}}}^{\mu \nu \rho \sigma }\) is nontrivial. Indeed, even in the Einstein-Hilbert case (\(f=R\)), two equivalent expressions \(f=R_{\mu \nu \rho \sigma }g^{\mu \rho }g^{\nu \sigma }=R_{\mu \nu \rho \sigma }(g^{\mu \rho }g^{\nu \sigma }-g^{\mu \sigma }g^{\nu \rho })/2\) give different \({{\mathcal {F}}}^{\mu \nu \rho \sigma }\).


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The author thanks Yuuiti Sendouda for helpful communications.

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Correspondence to Hideki Maeda.

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Maeda, H. Nonstaticity with type II, III, or IV matter field in \(f(R_{\mu \nu \rho \sigma },g^{\mu \nu })\) gravity. Gen Relativ Gravit 53, 90 (2021).

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