The linearized second law for any higher curvature gravity with the scalar and the electromagnetic fields

Abstract: The first law of black hole thermodynamics is suitable for any diffeomorphism invariant gravity, and the entropy in the first law is the Wald entropy which is highly dependent on the non-minimal coupling interactions in the theory of gravity. However, whether the Wald entropy still satisfies the second law needs to be investigated. The entropy of black holes obeying the linearized second law in arbitrary high-order curvature gravity is given, which can be written as the Wald entropy with correction terms. It indicates that the Wald entropy is not commonly obeying the linearized second law for any high-order curvature gravity. When the interactions of gravity with matter fields are included in the theory of gravity, the entropy of black holes obeying the linearized second law has not been obtained in this case. Considering any high-order curvature gravity with the scalar and the electromagnetic fields, from the Raychaudhuri equation, the entropy obeying the linearized second law is generally obtained, which can be expressed as the Wald entropy with correction terms as well. The entropy does not include the contribution from the electromagnetic fields, and the correction terms contain the contribution from the minimal coupling interaction between gravity and the scalar fields. Since the entropy satisfying the linearized second law depends only on the non-minimal coupling interaction of gravity in previous research, this result upends our understanding of the entropy of black holes obeying the linearized second law in any gravitational theory with matter fields.