Non-linear equation of motion for higher curvature semiclassical gravity

TL;DR

This paper derives a fully non-linear semiclassical equation of motion for higher curvature gravity theories using thermodynamic properties of causal horizons, advancing beyond previous linearized results.

Contribution

It introduces a non-linear formulation of semiclassical gravity for higher curvature theories, employing two approaches based on horizon thermodynamics and quantum focusing.

Findings

Derived non-linear equations of motion for higher curvature gravity

Established equilibrium conditions via quantum focusing

Extended previous linearized results to a non-linear regime

Abstract

We derive the non-linear semiclassical equation of motion for a general diffeomorphism-invariant theory of gravity by leveraging the thermodynamic properties of closed causal horizons. Our work employs two complementary approaches. The first approach utilizes perturbative quantum gravity applied to a Rindler horizon. The result is then mapped to a stretched light cone, which can be understood as a union of Rindler planes. Here, we adopt the semiclassical physical process formulation, encapsulated by $\langle Q\rangle = T \delta S_{gen}$ where the heat-flux $\langle Q\rangle$ is related to the expectation value of stress-energy tensor $T_{ab}$ and $S_{gen}$ is the generalized entropy. The second approach introduces a "higher curvature" Raychaudhuri equation, where the vanishing of the quantum expansion \(\Theta\) pointwise as required by restricted quantum focusing establishes an equilibrium condition, \(\delta S_{\text{gen}} = 0\), at the null boundary of a causal diamond. While previous studies have only derived the linearized semiclassical equation of motion for higher curvature gravity, our work resolves this limitation by providing a fully non-linear formulation without invoking holography.