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Entropy of causal diamond ensembles

by Ted Jacobson, Manus R. Visser

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Authors (as registered SciPost users): Manus Visser
Submission information
Preprint Link: scipost_202303_00042v1  (pdf)
Date submitted: 2023-03-31 21:28
Submitted by: Visser, Manus
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
  • Quantum Physics
Approach: Theoretical


We define a canonical ensemble for a gravitational causal diamond by introducing an artificial York boundary inside the diamond with a fixed induced metric and temperature, and evaluate the partition function using a saddle point approximation. For Einstein gravity with zero cosmological constant there is no exact saddle with a horizon, however the portion of the Euclidean diamond enclosed by the boundary arises as an approximate saddle in the high-temperature regime, in which the saddle horizon approaches the boundary. This high-temperature partition function provides a statistical interpretation of the recent calculation of Banks, Draper and Farkas, in which the entropy of causal diamonds is recovered from a boundary term in the on-shell Euclidean action. In contrast, with a positive cosmological constant, as well as in Jackiw-Teitelboim gravity with or without a cosmological constant, an exact saddle exists with a finite boundary temperature, but in these cases the causal diamond is determined by the saddle rather than being selected a priori.

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Reports on this Submission

Anonymous Report 1 on 2023-4-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202303_00042v1, delivered 2023-04-08, doi: 10.21468/SciPost.Report.6997


The paper attempts to formulate the thermodynamic properties of the static patch more clearly by introducing an auxiliary boundary, of the type considered in the early work of York. The methodology of Euclidean gravity is employed, and various spacetime dimensions as well as vanishing and non-vanishing cosmological constants are considered.

Although the problem is considered at the level of the semiclassical saddle, which only exists for $\Lambda>0$, it seems that Dirichlet boundary conditions are chosen at the boundary to formulate the problem, and it would be good to state that more explicitly to contrast other boundary conditions considered for such auxiliary boundaries (e.g. conformal boundary conditions discussed in 0704.3373, 1805.11559, 2103.15673, 1110.3792 among other places). I encourage the authors to expand on this but leave it as an option.

The paper nicely expresses salient differences between $\Lambda=0$ and $\Lambda>0$ and contributes to the general question of horizon thermodynamics. I support the publication.

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Author:  Manus Visser  on 2023-04-23  [id 3609]

(in reply to Report 1 on 2023-04-08)

We thank the referee for the suggestion that we expand on the issue of boundary conditions. The issue of boundary conditions is an important one. For the present paper we feel it is sufficient to emphasize that the Dirichlet choice has been made, that this choice might not be dynamically viable, that we expect the results will likely not depend on that choice, and that it would be interesting to investigate that question. To this end, we have replaced the opening of section 2.1 with text that addresses these points.

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