Black Hole Surface Gravity in Doubly Special Relativity Geometries
Round 1
Reviewer 1 Report
In this work, the authors study the notion of surface gravity in a momentum dependent Schwarzschild black hole geometry.
The manuscript is well written and technically sound. After some improvements I can recommend it for publication.
- Some variables are not defined carefully in the current version, please check all definitions.
- I suggest to update the list of references with the following items:
1. A Statistical Model of Information Evaporation of Perfectly Reflecting Black Holes, International Journal of Quantum Information, DOI: 10.1142/S0219749915600254, (2014).
2. Energy Transfer and Thermodynamics of Quantum Gravity Computation, Chaos, Solitons and Fractals, DOI: 10.1016/j.csfx.2020.100050, (2020).
3. Theory of Quantum Gravity Information Processing, Quantum Eng., DOI: 10.1002/que2.23, (2019).
4. Correlation Measure Equivalence in Dynamic Causal Structures of Quantum Gravity, Quantum Eng., DOI: 10.1002/QUE2.30, (2019).
5. A Survey on Quantum Channel Capacities, IEEE Communications Surveys and Tutorials 99, 1, DOI: 10.1109/COMST.2017.2786748
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper deals with the variant of quantum gravity based in momentum modified geometry which was previously suggested by the authors. It discusses the definition of the black hole horizons and an associated entropy, and clarifies several issues important for this approach. The paper is innovative and seems to be useful in the context of the search of non-contradictory quantum gravity. In my opinion it deserves publication in Universe
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
The paper aims to characterize the so called surface gravity in a novel approach to rainbow metrics, that presents some formal and conceptual advances in comparison to the original rainbow approach, since it is being formulated according to the geometry of the cotangent bundle. The inspiration for the metric actually comes from a mix between the curved momentum spaces paradigm and the use of spacetime curved tetrads in order to derive a cotangent bundle metric that depends on the base manifold coordinates and covectors (which are supposed to describe the momenta of particles that probe such spacetime, therefore in accordance with the rainbow geometries proposal).
The authors, then, characterize the Killing equation of such geometry in order to find an explicit expression of what would be the surface gravity in this geometry and also to compare different definitions that are known in the literature. Among the findings, the authors identify a special form of a rainbow metric that present equivalent expressions for the surface gravity (as a cotangent bundle metric that is conformal to its Riemannian version) and also a special basis in momentum space variables in which the dispersion relation preserves the massless shell. In this metric, the surface gravity of the Schwarzschild black hole would not depend on the momenta of the test particles and would coincide with the General Relativity (GR) result, thus opposing other approaches to the rainbow metrics idea.
However, an important hypothesis that is raised throughout the paper regards the momentum independence of the Killing vectors in such spacetime, which leads to a time-like Killing vector for the rainbow-Schwarzschild metric as equivalent to the GR case. In different approaches to this idea, for instance, the one followed in Ref.[arXiv:1407.8143], of one of the authors of this paper, in which the metric of the tangent bundle is considered through Finsler geometry (in which the bicrossproduct basis of k-Poincaré-inspired case is analyzed) the related Killing vectors are allowed to depend on vector coordinates, which indeed, lead to non-trivial boosts and apparently deformed translation generators even in the flat case (as can be verified in Eqs.(56) of the referred paper).
So, I consider that it would be important that the authors should comment on the generality (or lack of) of this ansatz, and if a fundamentally different result (regarding the momentum independence of the surface gravity in this ansatz) could be found had a different ansatz be considered (or if it could be assumed in any case).
Also, small typos were found, for instance
- Line 10: a non-unitary
- Line 71: Predicting what?
- “Remedy at this problem” or “Remedy this problem”?
- Line 96: to exist
- Line 110: in the context
- Line 147: curved
- Line after 173: Are the authors referring to Eq.(18) or (19)?
- Line 188: relationship between
- Also in line 188: It would be nice to express that one finds the same relationship of GR.
- After Eq.(35): Eddington-Finkelstein
So, I recommend that after the issue raised about the generality of the momentum-independence of the Killing vectors, a correction of the typos pointed out, and a new readout of the paper in order to find possible other typos by the authors, the paper should be accepted for publication, since it represents an interesting contribution to issue of furnishing a satisfactory description of rainbow metrics their relation with curved momentum spaces and the description of a physical quantity (surface gravity) that could be related to the thermodynamics of black holes from the perspective of quantum gravity phenomenology (although as pointed out by the authors, a DSR-QFT description would be needed in order to link this geometrical quantity to a thermodynamic one).
Author Response
Please see the attachment
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have adequately taken into account all the points made in the previous report. I can recommend the manuscript for publication.
Reviewer 3 Report
The authors have addressed the issues raised in the previous report satisfactorily. Therefore, I recommend the paper for publication.
