## SciPost Submission Page

# Gravitational Thermodynamics of Causal Diamonds in (A)dS

### by Ted Jacobson, Manus R. Visser

### Submission summary

As Contributors: | Theodore Jacobson · Manus Visser |

Arxiv Link: | https://arxiv.org/abs/1812.01596v2 |

Date submitted: | 2019-07-20 |

Submitted by: | Jacobson, Theodore |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Gravitation, Cosmology and Astroparticle Physics |

### Abstract

A maximally symmetric causal diamond is a solution to Einstein's equation with a cosmological constant. We establish a Smarr formula for such diamonds and a "first law" for variations to nearby solutions. The latter relates the variations of the bounding area, spatial volume of the maximal slice, cosmological constant, and matter Hamiltonian. The total Hamiltonian is the generator of evolution along the conformal Killing vector that preserves the diamond. To interpret the first law as a thermodynamic relation, it appears necessary to attribute a negative temperature to the diamond, as has been previously suggested for the special case of the static patch of de Sitter spacetime. With quantum corrections included, for small diamonds we recover the "entanglement equilibrium" result that the generalized entropy is stationary at the maximally symmetric vacuum at fixed volume, and we reformulate this as the stationarity of free conformal energy with the volume not fixed.