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Entropy of causal diamond ensembles
by Ted Jacobson, Manus R. Visser
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Authors (as registered SciPost users):  Theodore Jacobson · Manus Visser 
Submission information  

Preprint Link:  https://arxiv.org/abs/2212.10608v3 (pdf) 
Date submitted:  20230214 10:51 
Submitted by:  Visser, Manus 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
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 hightemperature limit, in which the saddle horizon approaches the boundary. This hightemperature 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 onshell Euclidean action. In contrast, with a positive cosmological constant, as well as in JackiwTeitelboim 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
Report #1 by Anonymous (Referee 1) on 2023322 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2212.10608v3, delivered 20230322, doi: 10.21468/SciPost.Report.6941
Report
This paper is an attempt to put the work of the authors' reference [30] (BDF) within a proper thermodynamic context  by using the argument of York (reference [32]) for introducing an artificial boundary term that is held at a given temperature to define the canonical ensemble. While I'm sympathetic to the motivation there are several issues in the authors' work that I do not quite follow.
1. It appears to me that the heart of the paper (or at least that which is most relevant to BDF) is section 3.1. There they consider an "approximate saddle" with metric in eqn. (3.1). Here $\theta$ is supposed to be Euclidean time in which case I would have thought that the inverse temperature is to be identified by the requirement that there is no conical singularity at $\rho=0$. But this would have given $\beta = 2\pi$ but the authors have an additional factor $\epsilon$ which is the radius of the disc. This needs some explanation I think. It seems to me that $\epsilon$ is a red/blue shift factor but it's not clear how it comes about from eqn. (3.1)
2. If this is indeed a blue shifted temperature, isn't that the same effect as in the usual calculation of say the black hole temperature? But then what is the thermodynamics that one is investigating. In the black hole case one sets up the system as observed at infinity and the relevant temperature is that measured there  which gives the Hawking temperature. Clearly the temperature at the horizon which is infinite makes no sense. In fact I would consider that to be in the regime where the effective field theory breaks down. Why isn't that the case here?
3. The calculation of BDF appears to be based on a exact Euclidean saddle point. They use the ADM formalism and set the bulk contribution to the Hamiltonian to zero as is the case for a solution and the entire contribution comes from the boundary term. Why is the authors' calculation only an "approximate" saddle? It seems to me that in the limit $\epsilon$>0 one recovers the saddle point action of BDF.
I would like to see some clarification of these issues before proceeding further with this review.
Author: Manus Visser on 20230326 [id 3515]
(in reply to Report 1 on 20230322)We thank the referee for the useful comments. Here is our reply: 1. We choose this temperature so that the boundary of the saddle geometry will lie at $\rho = \epsilon$. We have modified the text so as to hopefully clarify the logic. 2. As explained in section 2, we are following here York’s method of introducing an artificial boundary at which the ensemble parameters are specified, which need not lie at infinity. We agree with the referee that if $\epsilon$ were taken arbitrarily small the boundary would eventually lie in a regime where the effective field theory breaks down. Our presentation was misleading as to our intention. We stated something about the limit $\epsilon/r_0 \to 0$, but we meant only that we are working in the regime where $\epsilon/r_0\ll1$. We have modified the text to make this clear. 3. We agree. This is discussed in the introductory paragraph of section 3, in the final paragraph of section 3.1, and in the final paragraph of the paper. The reason the saddle is approximate is, as explained in the first paragraph of section 3.1, the metric (3.1) is not the same as the Euclidean causal diamond of BDF (which is flat Euclidean space). As explained in the penultimate paragraph of section 2.1, we cannot make use of the BDF metric since it is not the saddle of any well defined canonical (or microcanonical) ensemble. Since it is not, its statistical interpretation remains, to us at least, unclear. The goal of our paper was to see to what extent one can retain the essence of the BDF calculation and result while situating it as the answer to the entropy calculation in a welldefined statistical ensemble. Upon rereading our section 2, we still feel that this reasoning and perspective is explicitly and clearly explained there.