Generalized second law for non-minimally coupled matter theories

We establish the generalized second law (GSL) within the framework of higher curvature gravity theories, considering non-minimal couplings in the matter sector. Our proof pertains to the regime of linearized fluctuations around equilibrium black holes, aligning with previous works by Wall and Sarkar. Notably, while prior proofs addressed various gravity theories such as Lovelock theory and higher curvature gravity, they uniformly assumed minimally coupled matter sectors. In this work, we extend the proof of the linearized semi-classical GSL to encompass scenarios involving non-minimal couplings in the matter sector. Our approach involves a proposal for evaluation of the matter path integral in the expectation value of the stress tensor, adopting an effective field theory treatment for the higher derivative couplings. We leverage the recently established outcome regarding the linearized second law in such theories to substantiate our argument.