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Register a new userGeneralized proof of the linearized second law in general quadric corrected Einstein-Maxwell gravity
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Although the entropy of black holes in any diffeomorphism invariant theory of gravity can be expressed as the Wald entropy, the issue of whether the entropy always obeys the second law of black hole thermodynamics remains open. Since the nonminimal coupling interaction between gravity and the electromagnetic field in the general quadric corrected Einstein-Maxwell gravity can sufficiently influence the expression of the Wald entropy, we check whether the Wald entropy of black holes in the quadric corrected gravity still satisfies the second law. A quasistationary accreting process of black holes is first considered, which describes that black holes are perturbed by matter fields and eventually settle down to a stationary state. Two assumptions that the matter fields should obey the null energy condition and that a regular bifurcation surface exists on the background spacetime are further proposed. According to the two assumptions and the Raychaudhuri equation, we demonstrate that the Wald entropy monotonically increases along the future event horizon under the linear order approximation of the perturbation. This result indicates that the Wald entropy of black holes in the quadric corrected gravity strictly obeys the linearized second law of thermodynamics.
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Since the entropy of stationary black holes in Horndeski gravity will be modified by the non-minimally coupling scalar field, a significant issue of whether the Wald entropy still obeys the linearized second law of black hole thermodynamics can be proposed. To investigate this issue, a physical process that the black hole in Horndeski gravity is perturbed by the accreting matter fields and finally settles down to a stationary state is considered. According to the two assumptions that there is a regular bifurcation surface in the background spacetime and that the matter fields always satisfy the null energy condition, one can show that the Wald entropy monotonically increases along the future event horizon under the linear order approximation without any specific expression of the metric. It illustrates that the Wald entropy of black holes in Horndeski gravitational theory still obeys the requirement of the linearized second law. Our work strengthens the physical explanation of Wald entropy in Horndeski gravity and takes a step towards studying the area increase theorem in the gravitational theories with non-minimal coupled matter fields.
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We consider a static self-gravitating perfect fluid system in Lovelock gravity theory. For a spacial region on the hypersurface orthogonal to static Killing vector, by the Tolmans law of temperature, the assumption of a fixed total particle number inside the spacial region, and all of the variations (of relevant fields) in which the induced metric and its first derivatives are fixed on the boundary of the spacial region, then with the help of the gravitational equations of the theory, we can prove a theorem says that the total entropy of the fluid in this region takes an extremum value. A converse theorem can also be obtained following the reverse process of our proof. We also propose the definition of isolation quasi-locally for the system and explain the physical meaning of the boundary conditions in the proof of the theorems.
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