Entropy increases at linear order in scalar-hairy Lovelock gravity

Jie Jiang,Ming Zhang

doi:10.1007/jhep04(2020)148

Abstract

In this paper, we investigate the second law of the black holes in Lovelock gravity sourced by a conformally coupled scalar field under the first-order approximation when the perturbation matter fields satisfy the null energy condition. First of all, we show that the Wald entropy of this theory does not obey the linearized second law for the scalar-hairy Lovelock gravity which contains the higher curvature terms even if we replace the gravitational part of Wald entropy with Jacobson-Myers (JM) entropy. This implies that we cannot naively add the scalar field term of the Wald entropy to the JM entropy of the purely Lovelock gravity to get a valid linearized second law. By rescaling the metric, the action of the scalar field can be written as a purely Lovelock action with another metric. Using this property, by analogy with the JM entropy of the purely Lovelock gravity, we introduce a new formula of the entropy in the scalar-hairy Lovelock gravity. Then, we show that this new JM entropy increases along the event horizon for Vaidya-like black hole solutions and therefore it obeys a linearized second law. Moreover, we show that different from the entropy in F (Riemann) gravity, the difference between the JM entropy and Wald entropy also contains some additional corrections from the scalar field.

Highlights

Scalar-hairy Lovelock gravityWe consider a gravitational theory containing a real scalar field φ conformally coupled to Lovelock gravity
The most important thing is to see whether the higher curvature corrections spoil the second law of black hole thermodynamics and the entropy satisfies the second law of black hole thermodynamics
In this paper, we investigate the second law of the black holes in Lovelock gravity sourced by a conformally coupled scalar field under the first-order approximation when the perturbation matter fields satisfy the null energy condition

Summary

Scalar-hairy Lovelock gravity

We consider a gravitational theory containing a real scalar field φ conformally coupled to Lovelock gravity. The action of this theory in D-dimensional spacetime is given by [31]. It is not difficult to verify that this theory is invariant under the conformal transformation: gab → Ω2gab and φ → Ω−1φ It can be regarded as a natural generalization of Lovelock gravity with a non-minimal coupling scalar field. This theory admits a scalar-hairy black hole solution where the scalar field is nonvanishing and regular outside of the singularity [33,34,35,36].

Linearized second law of the scalar-hairy Lovelock gravity

Wald entropy

Jocobson-Myers entropy

Conclusion