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Boundary Conditions for Extremal Black Holes from 2d Gravity
by Stephane Detournay, Thomas Smoes and Raphaela Wutte
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Submission summary
| Authors (as registered SciPost users): | Thomas Smoes · Raphaela Wutte |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202403_00003v1 (pdf) |
| Date submitted: | 2024-03-04 05:17 |
| Submitted by: | Wutte, Raphaela |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We devise new boundary conditions for the near-horizon geometries of extremal BTZ and Kerr black holes, as well as for the ultra-cold limit of the Kerr-de Sitter black hole. These boundary conditions are obtained as the higher-dimensional uplift of recently proposed boundary conditions in two-dimensional gravity. Their asymptotic symmetries consist in the semi-direct product of a Virasoro and a current algebra, of which we determine the central extensions.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 3) on 2024-4-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202403_00003v1, delivered 2024-04-09, doi: 10.21468/SciPost.Report.8854
Strengths
1. Connection between two and higher dimensional asymptotic symmetries.
2. Novel asymptotic symmetries for Kerr and Kerr-de Sitter spacetimes.
Weaknesses
1. Physical meaning of results not entirely clear.
2. Presentation is not very clear.
Report
The paper explores novel boundary conditions for black hole type spacetimes in three and four spacetime dimensions. The approach takes inspiration from a variety of analyses in two-dimensions leading to interesting asymptotic symmetry algebras. Of particular interest is the application to certain limits of Kerr-de Sitter black holes, which have not been analysed in the literature in much detail.
Requested changes
1. Provide a synthesis and summary of results, compared and contrasted to previous cases.
2. Make note of cases when central charges have non-standard reality properties, or are vanishing.
3. Provide some outlook on any uniqueness of lack thereof of the new boundary conditions.
Report #1 by Anonymous (Referee 4) on 2024-4-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202403_00003v1, delivered 2024-04-06, doi: 10.21468/SciPost.Report.8835
Strengths
New boundary conditions and asymptotic symmetry algebra are found for black holes in three and four dimensions including extremal Kerr black holes.
Weaknesses
lack of physical interpretations
Report
Boundary conditions are part of the definition of a quantum theory of gravity, and play an important role in the bottom-up approach of holography duality. The paper finds new boundary conditions for the near-horizon geometries of extremal black holes in three dimensions and four dimensions. The work is useful for understanding the holographic dual of extremal black holes including Kerr black holes. At the technical level, the paper is a higher dimensional uplift of boundary conditions in two dimensions. Expressions of the boundary conditions and asymptotic symmetry algebra have been written down explicitly and are useful for further study. The weak point of the paper is that there are not enough interpretations of the assumptions and results. I would like to recommend it for publication, provided that the authors consider the requested changes below.
Requested changes
1-In order to perform the mode expansion, the authors assume that $\tau$ in eq(34) is periodic. The authors should explain why such an identification is reasonable in a Lorentzian geometry, and what the implication in the putative dual theory. Relatedly, the central charges in eq.(25) depend on the Length $L$ which is arbitrarily chosen. The authors should explain why such central charges are physical.
2-The authors should provide more explanations on boundary conditions in different coordinates. Are these boundary conditions equivalent in the sense that they define the same quantum theory? If these boundary conditions are not equivalent, is there a preferred one? What is the interpretation of the relation eq.(39) that relates charges in different coordinates? How do we understand these from the perspective of the holographic dual?
3- What is the interpretation of imaginary level $\kappa$ such as in eq.(61)? Is the resulting dual theory non-unitary? If so, how do we make sense of it ?
Author: Raphaela Wutte on 2024-05-07 [id 4476]
(in reply to Report 1 on 2024-04-06)
We thank the referee for the careful reading of the manuscript and for their comments and questions. We have supplemented our paper with a conclusion to address their comments.
- We have addressed the assumption of Lorentzian time being periodic in the conclusion (paragraph 2), at the end of page 6 and on page 7 under equation (19). This is in addition to the existing explanation in the introduction (page 4 of the paper, paragraph 1). The periodicity of Lorentzian time in our paper should be understood as the Wick rotation of the periodicity in Euclidean time. We motivate switching angular and temporal cycle for the computation of the charges by assuming modular invariance of the putative dual quantum field theory. This introduces the length scale $L = I \beta$ in our charges, where $\beta$ is the temperature, hence it is not arbitrary. Our central charges are physical in the sense that these are central charges in general relativity that we compute with standard covariant phase space methods; however, the switching of the cycles requires the assumption that the putative dual theory is modular invariant, which is the case for warped conformal field theories or conformal field theories, but does not hold in general. In addition, by endowing the time coordinate with a complex period, we complexify our manifold.
- We have addressed this point in the conclusion (paragraph 2) and at the very end of section 2.4. In particular, relation (39) shows that the two algebras can be related by a mere redefinition of generators and are thus isomorphic. Hence, these boundary conditions are equivalent in the sense that they give rise to the same asymptotic symmetry algebra (up to isomorphisms). Relation (39), which, from a holographic perspective, corresponds to twisting the stress tensor in the boundary theory, can be understood to arise from the change of coordinates in the bulk from Schwarzschild-like to Eddington-Finkelstein-like coordinates.
- The central extension $\kappa$ in equation (61) is manifestly real, since $L$ is imaginary. In the case of the extremal BTZ and Kerr black holes, we can use the change of generators, eq. (39), to bring the algebra into a form where unitarity has been addressed in the literature. For the extremal Kerr and BTZ black hole, we find $c > 0$ and $k < 0$. For these values, unitary highest-weight representations do not exist. We comment more on the implications of this result for holography in the conclusion (second to last paragraph). In the case of the ultracold Kerr-dS black hole only the central extension $\kappa$ is nonvanishing and we cannot use the relation (39). For this case, we do not know whether unitary representations exist (see last paragraph of conclusion).
Author: Raphaela Wutte on 2024-05-07 [id 4475]
(in reply to Report 2 on 2024-04-09)We thank the referee for the careful reading of the manuscript and their comments. We have supplemented our paper with a conclusion to address their comments.