A local first law for black hole thermodynamics

@article{Frodden2011ALF,
  title={A local first law for black hole thermodynamics},
  author={Ernesto Frodden and Amit Ghosh and Alejandro Perez},
  journal={Physical Review D},
  year={2011},
  volume={87},
  pages={121503}
}
We first show that stationary black holes satisfy an extremely simple local form of the first law \delta E=\kappa(l) \delta A/(8 \pi) where the thermodynamical energy E=A/(8\pi l) and (local) surface gravity \kappa(l)=1/l, where A is the horizon area and l is a proper length characterizing the distance to the horizon of a preferred family of local observers suitable for thermodynamical considerations. Our construction is extended to the more general framework of isolated horizons. The local… 
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