Gravitational focusing and horizon entropy for higher-spin fields

Previously, the Raychaudhuri equation and the focusing theorem in General Relativity were generalised to diffeomorphism-invariant theories of gravity coupled to scalar and vector fields on linearly perturbed Killing horizons. The Wall entropy can be extracted from the generalised focusing equation and it satisfies the first and the second laws of thermodynamics. In this paper, we further extend the discussion of gravitational focusing on the horizon to include arbitrary bosonic fields with spin $s \geq 2$. These higher-spin fields introduce indefinite terms into the generalised focusing equation, obstructing the proof of the focusing theorem and the existence of an increasing horizon entropy. To resolve this issue, we propose a higher-spin focusing condition that eliminates these indefinite terms, thereby restoring the focusing theorem and the associated thermodynamic laws. We speculate that the focusing condition could be a necessary condition for the physical consistency of higher-spin theories.