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Gravitational Thermodynamics of Causal Diamonds in (A)dS

by Ted Jacobson, Manus R. Visser

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Submission summary

Authors (as registered SciPost users): Theodore Jacobson · Manus Visser
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-10-28
Date submitted: 2019-07-20 02:00
Submitted by: Jacobson, Theodore
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
Approach: Theoretical


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.

Published as SciPost Phys. 7, 079 (2019)

Submission & Refereeing History

Reports on this Submission

Anonymous Report 1 on 2019-10-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1812.01596v2, delivered 2019-10-18, doi: 10.21468/SciPost.Report.1240


1) interesting, with many potential ramifications
2) thorough and complete
3) clearly written, except in a few odd spots
4) candid - when the authors don't know something, they say it


1) nothing, except possibly that it's lengthy
-1) but this length was probably unavoidable


This paper tells everything anyone will need to know about the thermodynamics of causal diamonds in symmetric spaces. There is nothing I could possibly suggest to improve about this manuscript because it is interesting, comprehensive, clear and honest. I suggest publication essentially without changes.

Requested changes

The only place that I found confusing was the two paragraphs below equation (4.8). The explanation why matter can appear both as an energy or an entropy variation, and which terms are explicit and/or implicit, is rather disorienting.

I would suggest clarifying this fragment. This is a request and not a demand; it's entirely optional.

  • validity: top
  • significance: top
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Author:  Theodore Jacobson  on 2019-12-09  [id 669]

(in reply to Report 1 on 2019-10-18)

We thank the referee for suggesting we clarify why matter can appear explicitly in the first law either via its energy variation or via its entropy variation, but not both. This is indeed an important point. We have revised the presentation, attempting to explain it more clearly.

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