# Generalised focusing theorem and dynamical horizon entropy in diffeomorphism-invariant theories

## TL;DR

This paper reviews recent advances in light-ray focusing and horizon thermodynamics within diffeomorphism-invariant gravity theories, establishing a generalized focusing theorem and horizon entropy that extend classical results to more complex fields.

## Contribution

It introduces a generalized focusing theorem and horizon entropy applicable to a broad class of gravity theories, including those with higher-spin fields, under specific conditions.

## Key findings

- Null-null gravitational equations generalize Raychaudhuri equation.
- Horizon entropy (Wall entropy) is non-decreasing under null energy condition.
- Higher-spin fields require a new focusing condition for the theorem to hold.

## Abstract

I summarise recent progress on light-ray focusing and horizon thermodynamics in general diffeomorphism-invariant theories of gravity coupled to bosonic matter. In pure gravity and with scalar or vector fields, the null-null gravitational equation of motion on a linearly perturbed Killing horizon generalises the Raychaudhuri equation, defining a generalised expansion that never increases under the null energy condition. This proves a generalised focusing theorem and defines an increasing horizon entropy (Wall entropy). When higher-spin fields are present, the generalised focusing theorem persists subject to a "higher-spin focusing condition", which I propose as a physical consistency constraint on higher-spin theories.