Non-linear equation of motion for higher curvature semiclassical gravity

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 〈 Q 〉 = T δ S g e n where the heat-flux 〈 Q 〉 is related to the expectation value of stress-energy tensor T a b and S g e n is the generalized entropy. The second approach introduces a “higher curvature” Raychaudhuri equation, where the vanishing of the quantum expansion Θ pointwise as required by restricted quantum focusing establishes an equilibrium condition, δ S 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.