Abstract

In 2015, Wall (arXiv:1504.08040) proposed an entropy formula for perturbatively dynamical black holes in higher curvature gravity. This entropy is defined to the first order of perturbations away from a stationary bifurcate black hole background, and it is proved to satisfy a first law and a linearised second law. It’s been a question whether such entropy definition is still valid if one includes matter fields, especially vector fields, as they will introduce seemingly problematic terms that may make Wall’s entropy prescription invalid. In our recent work, we considered the most general theories in which vector fields (including non-gauge fields) are non-minimally coupled to higher curvature gravity, and we proved that the “problematic” terms induced by vector fields vanish when integrated on compact black hole horizons at linear order in perturbation. This suggests Wall’s entropy proposal is valid in presence of vector fields, and a linearised second law also holds. We also calculated the entropy formula for f(Riemann, V, \nabla V) theories, as a general example.