On Black Hole Temperature in Horndeski Gravity

TL;DR

This paper proposes a modified black hole temperature in Horndeski gravity that accounts for different effective metrics experienced by gravitons and photons, ensuring the first law of thermodynamics holds.

Contribution

It introduces a new temperature definition based on an effective metric, resolving inconsistencies in black hole thermodynamics within Horndeski theories.

Findings

Modified temperature includes effects of the effective metric.

Corrects the first law of black hole thermodynamics.

Implications for Hawking radiation and species problem.

Abstract

It has been observed that for black holes in certain family of Horndeski gravity theories Wald's entropy formula does not lead to the correct first law for black hole thermodynamics. For this family of Horndeski theories speeds of propagation of gravitons and photons are in general different and gravitons move on an effective metric different than the one seen by photons. We show that the temperature of the black hole should be modified from surface gravity over $2\pi$ to include effects of this effective metric. The modified temperature, with the entropy unambiguously computed by the solution phase space method, yields the correct first law. Our results have far reaching implications for the Hawking radiation and species problem, going beyond the Horndeski theories.