# Generalised second law (GSL) The generalised second law generalises the [[0005 Black hole second law|BH second law]] by requiring the *total* entropy of the black hole and matter fields outside the black hole to increase. The matter field outside is originally considered to be classical, but it generalises to quantum fields. The quantum version implies a quantum [[0225 Singularity theorems|singularity theorem]]. ## Refs - classical - [[Bekenstein1972]] - [[Bekenstein1973]] - [[Bekenstein1974]] - [[1975#Hawking]] - explicitly showing that the Bekenstein bound holds: [[SchifferBekenstein1989]] - [[1986#Bombelli, Koul, Lee, Sorkin]] - one of the very earliest proposals for splitting the entropy to be BH + quantum matter field outside ## Proofs - review: - [[2009#Wall (Jan)]] - main proof - [[2011#Wall]]: proves GSL for semiclassical quantum fields falling across a causal horizon, minimally coupled to general relativity; generalises automatically to interacting fields, after it was established that their modular Hamiltonian takes the same form on such null hypersurfaces - [[KoellerLeicheNauerLevineShahbazi-Moghaddam2017]][](https://arxiv.org/abs/1702.00412) - [[CasiniTesteTorroba2017]][](https://arxiv.org/abs/1702.00412) - [[Wall201007]][](http://arxiv.org/abs/1007.1493) - [[2024#Faulkner, Speranza]]: proof using gravitational algebra - [[2024#Ali, Suneeta]]: GSL for product state using modular theory - [[2024#Kirklin]]: beyond the semiclassical regime ## Related - [[0004 Black hole entropy|black hole entropy]] - [[0212 Quantum extremal surface|QES]] - [[0418 Bekenstein bound|Bekenstein bound]]