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On the Euclidean Action of de Sitter Black Holes and Constrained Instantons
by Edward K. Morvan, Jan Pieter van der Schaar, Manus R. Visser
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Submission summary
Authors (as registered SciPost users):  Manus Visser · Jan Pieter van der Schaar 
Submission information  

Preprint Link:  https://arxiv.org/abs/2203.06155v2 (pdf) 
Date submitted:  20220414 14:36 
Submitted by:  van der Schaar, Jan Pieter 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We compute the onshell Euclidean action of Schwarzschildde Sitter black holes, and take their contributions in the gravitational path integral into account using the formalism of constrained instantons. Although Euclidean de Sitter black hole geometries have conical singularities for generic masses, their onshell action is finite and is shown to be equal to minus the sum of the black hole and cosmological horizon entropy. We apply this result to compute the probability for a nonrotating, neutral arbitrary mass black hole to nucleate spontaneously in empty de Sitter space, which separates into a constant and a "nonperturbative" contribution, the latter corresponding to the proper saddlepoint instanton in the Nariai limit. We also speculate on some further applications of our results, most notably the possible nonperturbative corrections to antipodal correlators in the de Sitter vacuum.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022727 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.06155v2, delivered 20220727, doi: 10.21468/SciPost.Report.5453
Strengths
1. Presents a new approximate linear relation between the difference of the entropies of Schwarschild de Sitter and de Sitter, and the mass of the black hole.
2. Derives the probability of BH pair creation in a novel manner, using the formalism of constrained instantons, and finds qualitatively similar results to previous literature.
Weaknesses
1. In the introduction, the derivation of equation (1.1) is presented as the main result of the paper, although this result was already reported in the reference [27]. I think this is a bit misleading.
2. The implementation of the constraints on the metric, through the constraint functional C[g], is not computed though it seems like a crucial part of the constrained instantons formalism.
Report
This submission presents some original results on the Schwarschildde Sitter geometry, and make use of the constrained instantons formalism, showing it leads to expected results at least qualitatively and hence bringing support to this method for computing gravitational path integral contributions. In my opinion, it suits the journal criteria and I recommend publication after the few requested changes points have been addressed.
Requested changes
1. First a typo: I think following its caption, the vertical axis on figure 6 should read $S_{SdS}/S_{dS}$ instead of the current legend.
2. A small comment: in the abstract and in section 2.2, the authors state that despite the conical singularities present in the SdS spacetime, the onshell action turns out to be finite. I'd like to point out the following papers: Barrow and Tipler, Nature 331 (1988) 31; Barrow 2019, arxiv: 1912.12926. There it was found that spacetimes with such a singularity have rather generically a finite onshell action; this was even proposed as a tradeoff situation, where one either gets an infinity in the onshell action, or a singularity in the spacetime. Additionally, I find that the denomination as “mild singularities” on page 9 is not precise, what do the authors mean technically? Is it that the singularities are simple poles and not essential singularities?
3. Just for the sake of completeness, I wonder if the authors also tried to fit other kinds of functions than the linear function for the difference of entropies between vacuum dS and the SdS solution, in order to compare how good the fits are, or if they have an argument for why this is not needed (except for the fact that the relative difference is of the order of the percent).
4. In Appendix B the authors show that their calculation of the negative mode for the SdS solution in the regime called "nearNariai", close to the solution with maximal mass, gives as in previous literature a negative mode in direction of decreasing black hole mass, but not the same quantitative coefficient for this negative mode. Do the authors have an idea what this discrepancy is due to?
5. Even if it is stated at the end of the introduction that [27] contains "similar results for the onshell action of Schwarschildde Sitter", I believe it should be cited before, when those results are stated (in (1.1) for example)
Author: Jan Pieter van der Schaar on 20220913 [id 2810]
(in reply to Report 2 on 20220727)
We thank the referee for the critical, positive, comments. We have prepared a revised version addressing the requested changes as best as we could, specifically:
 We fixed the typo in the caption of figure 6.
 We agree that the term 'mild singularities' on page 9 is vague and rephrased what we actually mean. Regarding the papers that were pointed out by the referee: although they are certainly interesting, we do not believe they are relevant to the specific Euclidean de Sitter problem we are studying.
 We added a footnote in the relevant section, explaining that the linear fit can be understood as a large d expansion and that we neglected higher order terms, de facto also pointing out that we did not try another fit.
 We checked once more what we wrote in appendix B, and although we were not explicit, out of politeness, we think it is pretty clear from the context that we believe Bousso and Hawking made a mistake. So we are inclined not to make any changes here.
 We agree, and we have moved the citation to the right place.
We hope to have responded appropriately to the referee's requests, and expect to upload a revised version for publication soon.
Report #1 by Anonymous (Referee 1) on 2022510 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.06155v2, delivered 20220510, doi: 10.21468/SciPost.Report.5058
Strengths
Schwarzschild de Sitter black holes are a wellstudied system, but the authors have been able to uncover two new and surprising features of the Euclidean solutions:
1. The vanishing of the term linear in beta in the Euclidean action.
2. The approximate linear dependence of the entropy difference on mass.
Weaknesses
I would appreciate a fuller discussion of the application of the constrained instanton method to these solutions. The authors point to the earlier paper [24] for more detail, but as this is an important part of their story, I think more explanation here would be welcome.
Report
I am happy to recommend publication the authors have found some interesting results, and this work has the potential to reinvigorate the exploration of black holes in de sitter as a probe of its microscopic description.
Requested changes
1. On the constrained instantons, I would appreciate in particular an explanation of the claim on page 16 that the solutions can be understood as smooth in this context.
Author: Jan Pieter van der Schaar on 20220715 [id 2663]
(in reply to Report 1 on 20220510)
First, let me apologise for the long delay in our reply to the referee report, which to some extent can be explained by ongoing followup work that actually pertains to the 'weakness' pointed out by the referee. We thank the referee for the otherwise mostly positive comments.
With regards to the identified weakness we agree that a fuller discussion of the application of constrained instantons would be beneficial and intend to update the manuscript accordingly, as soon as possible, in conjunction with our followup work. Specifically, with respect to the requested change, we will clarify or rephrase our supposed claim regarding the smooth nature of the solutions.
We expect to upload an updated version of the manuscript, addressing these points, after the Summer break.
Author: Jan Pieter van der Schaar on 20220920 [id 2835]
(in reply to Report 2 on 20220727)We thank the referee for the critical, positive, comments. We have prepared a revised version addressing the requested changes as best as we could, specifically:
We fixed the typo in the caption of figure 6.
We agree that the term 'mild singularities' on page 9 is vague and rephrased what we actually mean. Regarding the papers that were pointed out by the referee: although they are certainly interesting, we do not believe they are relevant to the specific Euclidean de Sitter problem we are studying.
We added a footnote in the relevant section, explaining that the linear fit can be understood as a large d expansion and that we neglected higher order terms, de facto also pointing out that we did not try another fit.
We checked once more what we wrote in appendix B, and although we were not explicit, out of politeness, we think it is pretty clear from the context that we believe Bousso and Hawking made a mistake. So we are inclined not to make any changes here.
We agree, and we have moved the citation to the right place.
We hope to have responded appropriately to the referee's requests, and expect to upload a revised version for publication soon.