Black hole thermodynamics and topology

G.E. Volovik

REVIEW 4 major objections 4 minor 1 cited by

Black hole thermodynamics and topology

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that Reissner–Nordström black-hole entropy is $4\pi M^2$, independent of charge, because the inner and outer horizon topologies cancel the electrical contribution.

desk verdict The new topological step in this paper is unsound, and the charge-independence of RN entropy it claims is a restatement of prior work rather than a new result. read the letter →

arxiv 2505.20194 v3 pith:XJ4HPGDJ submitted 2025-05-26 gr-qc

G.E. Volovik This is my paper · ORCID

classification gr-qc

keywords blackholethermodynamicsReissner-NordströmhorizontemperatureEulercharacteristicChern-Gauss-Bonnettheoreminnerentropytopological

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the entropy of a Reissner–Nordström black hole is fixed by its mass alone, $S_{\rm RN}=4\pi M^2$, with no dependence on the electric charge. The argument runs by treating the outer and inner horizons as a correlated pair, each with its own near-horizon topology that fixes an inverse temperature through the Euler characteristic of the Euclidean spacetime. Adding the two inverse temperatures, $\beta_+ + \beta_- = 8\pi M$, cancels the charge dependence. If true, a charged black hole radiates with the same effective Hawking temperature $1/(8\pi M)$ as an uncharged black hole of the same mass, and the entropy equals the Schwarzschild value. The paper also uses the two-horizon entropy rule to fix the white-hole entropy as minus the black-hole entropy.

What carries the argument

The machinery is the Euler characteristic of the Euclidean spacetime region, computed by the Chern–Gauss–Bonnet invariant $\chi(\mathcal{M}) = \frac{1}{32\pi^2}\int (R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} - 4R_{\mu\nu}R^{\mu\nu})\sqrt{g}\,d^4x$, which the paper assumes equals $2$ for a near-horizon region of either horizon. For the Reissner–Nordström geometry the integrand reduces to a simple rational function of $r$, $M$, and $Q$, and integrating with $\chi=2$ yields the outer and inner inverse temperatures $\beta_\pm$. The new step is to demand the same topological rule for the inner horizon, whose inverse temperature comes out negative, and to combine the two temperatures additively in the co-tunneling radiation rate $w \propto e^{-(\beta_+ + \beta_-)E}$. That additive combination is what makes the charge cancel.

What would settle it

Compute the Gauss–Bonnet integral in equation (2) over only a small proper neighborhood of the inner horizon, $r\in(r_-, r_-+\varepsilon)$, and check whether its Euler characteristic equals 2; if it does not, $\beta_-$ is not $4\pi r_-^2/(r_- - r_+)$ and the sum $\beta_+ + \beta_- = 8\pi M$ is not the charge-free result.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a cancellation: applying the Chern–Gauss–Bonnet topological formula for horizon temperature separately to the outer horizon, $\beta_+ = 4\pi r_+^2/(r_+-r_-)$, and to the inner horizon, $\beta_- = 4\pi r_-^2/(r_- - r_+)$ (negative), and then adding them for the coherent tunneling process, gives $\beta = \beta_+ + \beta_- = 4\pi(r_+ + r_-) = 8\pi M$. Since $r_+ + r_- = 2M$ depends only on mass, the charge $Q$ disappears. The entropy assigned from this temperature, $S_{\rm RN}=4\pi M^2$, reproduces the Schwarzschild value and is supported by independent composition rules for the two-horizon entropies. The author presents this as evidence that the topological approach to horizon temperature works for multi-horizon systems and that the inner horizon cannot be ignored in black-hole thermodynamics.

Load-bearing premise

The load-bearing premise is that the inner horizon can be assigned the same topological weight as the outer horizon by integrating the Gauss–Bonnet expression from $r_-$ to infinity, even though that region is not the inner horizon's near-horizon neighborhood; if that assignment fails, the inner horizon's negative temperature no longer cancels the outer horizon's charge dependence and $S_{\rm RN}=4\pi M^2$ does not follow.

Editorial extensions

If this is right

A charged Reissner–Nordström black hole radiates with effective inverse temperature $\beta = 8\pi M$, so its Hawking temperature is $1/(8\pi M)$, independent of $Q$.
The entropy of a charged black hole equals that of a Schwarzschild black hole of the same mass, so changing charge adiabatically at fixed mass changes no entropy.
The white-hole entropy of mass $M$ is $-S_{\rm BH}$, consistent with time-reversal antisymmetry and with the macroscopic tunneling rate $w\sim e^{-2S_{\rm BH}}$.
At extremality the two horizons merge, the topological invariant changes abruptly, and the entropy jumps from $4\pi M^2$ to zero.
The two-horizon topological treatment supports the topological explanation of the factor-2 difference between the cosmological-horizon temperature and the local de Sitter temperature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: The same two-horizon addition should apply to rotating black holes, whose inner Cauchy horizon would similarly cancel the angular-momentum dependence; the paper gestures at Kerr black holes through its references but does not work out this extension.
Inference: If the extremal entropy jump is real, near-extremal charged black holes should show a sharp discontinuity in radiation or tunneling rates as $Q$ approaches $M$, analogous to a topological phase transition; this is not tested in the paper.
Inference: A Euclidean path-integral computation of the full on-shell action for the Reissner–Nordström spacetime with both horizons included would independently confirm or refute $S=4\pi M^2$; the paper does not carry out that computation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This is a short paper by Volovik claiming that the entropy of the Reissner-Nordström black hole is S=4πM^2, independent of charge, and that this follows from applying the Chern-Gauss-Bonnet theorem to both horizons. The main result is not new—it is already in the author's Refs. 8–10. The genuinely new step is the topological derivation of a negative inner-horizon temperature β−, which then combines with β+ to cancel the charge dependence. That derivation is the soft spot, and it is fatal.

The paper is clearly written and honest that it starts from the assumption that the topological approach works. It also engages with the Hughes-Kusmartsev de Sitter work and with the prior multi-horizon literature. That is fair. But the new derivation does not survive inspection. Equation (2) writes a Gauss-Bonnet integral over r ∈ [r−, ∞) with χ=2 and the same Euclidean measure as the outer-horizon integral. In the RN metric, the Euclidean time circle has positive norm only for r>r+. Between the two horizons g_ττ<0, so the "Euclidean" section is not a Riemannian manifold there, and the domain includes the outer horizon as well. The Gauss-Bonnet theorem cannot be applied to that region. So β− is not derived; the cancellation β+ + β− = 8πM is unsupported. Without that sum, the paper's central result is just a restatement of the author's earlier, nonstandard claim.

The rest is also thin. The co-tunneling rate in Eq. (6) is asserted, not derived, and the entropy composition rule in Eq. (9) leans on Tsallis-Cirto statistics, which is not standard black-hole thermodynamics. The white-hole section is a byproduct, not independent evidence.

I agree with the reader's verdict. This is not a paper that overturns the area law; it is a speculative note that relies on a mistaken topological integral. If the author can justify Eq. (2) as some kind of analytic continuation, that would need to be spelled out carefully, but as written it is a gap.

Who is this for? Readers already sympathetic to Volovik's program—where the inner horizon contributes negative entropy and S=4πM^2—may find it a nice packaging, but it does not convince anyone who does not already accept the prior results. I would not cite it, and I would not bring it to my reading group. I would still send it to a referee, because the question of multi-horizon topology is real and the claim is prominent, and a referee should pin down exactly the failure I described. But my own recommendation would be reject.

Referee Report

4 major / 4 minor

Summary. The manuscript applies the Chern-Gauss-Bonnet topological invariant to the Reissner-Nordström black hole with two horizons. It assigns an Euler characteristic χ=2 to both the outer and inner horizons and obtains inverse temperatures β_+ and β_- from integrals of the curvature invariant over r∈[r_+,∞) and r∈[r_-,∞), respectively. Adding the two temperatures gives β=β_++β_-=8πM, from which the paper concludes that the black-hole entropy is S=4πM^2, independent of the electric charge. The same framework is extended to white holes with a sign flip in the temperatures. The paper explicitly states that this result agrees with the author's previous papers (Refs. 8--10).

Significance. If the central claim were correct, it would contradict the standard Bekenstein-Hawking area law S=A/4=πr_+^2 for charged black holes and would generalize the Hughes-Kusmartsev topological explanation to multi-horizon spacetimes. The paper makes a sharp, falsifiable prediction—charge independence of the entropy—and its algebra is transparent because r_++r_-=2M. However, the derivation is not sound as it stands: the inner-horizon integral in Eq. (2) is not a well-defined Euclidean invariant, the co-tunneling rate in Eq. (6) is asserted without derivation, and the final result is tuned to reproduce earlier work by the same author. The manuscript contains no independent consistency check against standard Euclidean path-integral thermodynamics.

major comments (4)

[Section II, Eq. (2)] The integral in Eq. (2) is not a well-defined Euclidean Gauss-Bonnet integral for the inner horizon. The Reissner-Nordström metric has Lorentzian signature between the two horizons (g_tt < 0 for r_- < r < r_+), so the Euclidean measure √g and the curvature integrand are evaluated on a region where the Euclidean continuation does not exist. Moreover, the integration domain r ∈ [r_-, ∞) includes the outer horizon r_+ and the entire inter-horizon band, not just a near-horizon neighborhood of r_-. Consequently, the value of β_- in Eq. (5) is not actually derived, and since Eq. (7) and the entropy result Eq. (8) depend linearly on β_-, the central claim loses its foundation.
[Section II, Eqs. (1) and (2)] The assignment χ(M)=2 for the inner horizon is assumed without independent justification. The paper does not show that the near-horizon topology of the inner horizon is D^2×S^2, nor does it explain why the same Euler characteristic applies to both horizons. Because the cancellation β_+ + β_- = 8πM in Eq. (7) is exactly what follows when both integrals are assigned χ=2, the result appears constructed rather than predicted. A derivation of χ=2 for the inner horizon from the geometry or from a boundary term is needed.
[Section III, Eq. (6)] The co-tunneling formula w ∝ e^{-β_+E} e^{-β_-E} is asserted without derivation. No microscopic model, amplitude calculation, or argument from coherent tunneling is given, and the use of a negative inverse temperature β_- inside a Boltzmann factor is non-standard. Since Eq. (6) is the only bridge from the horizon temperatures to the thermodynamic entropy, this missing derivation is a load-bearing gap. The paper should provide a concrete tunneling calculation or an alternative derivation of the combined rate.
[Section III, Eqs. (8) and (9)] The adiabatic argument and the composition rule in Eqs. (8) and (9) do not constitute independent evidence for charge independence of entropy. The adiabatic transformation by varying α at fixed M assumes that entropy does not change, which is precisely the point under dispute. The composition rule (√S(r_+) + √|S(r_-)|)^2 automatically equals π(r_+ + r_-)^2 = 4πM^2 because r_+ + r_- = 2M, so it is an identity that builds in the desired answer. The paper needs to confront the standard Euclidean-action result S = πr_+^2 and explain why it is inapplicable.

minor comments (4)

[Section II] The acronym "CGS" for the Chern-Gauss-Bonnet invariant is nonstandard and potentially confused with the centimeter-gram-second system; it should be "CGB".
[Section III] There is a duplicated word in the sentence following Eq. (6): "final state state" should be "final state".
[Section V] In the Conclusion, "topological quantum filed theories" is a typo for "topological quantum field theories".
[References] The reference list is heavily weighted toward the author's own papers, and the text states several times that the results agree with Refs. 8-10; the manuscript would benefit from a critical discussion of the standard area-law derivation and of whether Refs. 6-7 really support the inner-horizon integral as written.

Circularity Check

4 steps flagged · score 8.0 of 10

The claimed charge-independent RN entropy is not an independent prediction: the inner-horizon χ=2 input is assumed, and the calculation is explicitly tuned to reproduce the author's prior Refs. 8–10, so S=4πM² is forced rather than derived.

self definitional [Section II, Eq. (2)]
"In the same manner the topological invariant for the inner horizon must provide the connection between its radius r− and the corresponding temperature β−:"

The inner-horizon inverse temperature β− is obtained by writing the same Chern–Gauss–Bonnet integral with χ(M)=2 and lower limit r−, exactly the value used for the outer horizon. No independent argument fixes χ=2 for the inner horizon over this domain, which includes the outer horizon and the inter-horizon region. Once this input is made, β− is algebraically determined, and β+ + β− = 8πM follows from r+ + r− = 2M. The central cancellation is therefore contained in the assumed input rather than derived from it.
fitted input called prediction [Section I and Section III, Eqs. (7)–(8)]
"Here we include into consideration both the inner and outer horizons and obtain the values of the temperature and entropy of RN black hole in agreement with Refs. 8–10."

The paper's stated goal is to reproduce the entropy value of Refs. 8–10, which are the author's own prior papers asserting S_RN=4πM². The calculation then adds the inner horizon in the precise way that makes Eq. (7) give β=8πM and Eq. (8) give S=4πM². The target result is thus the fitting condition, not an independently predicted outcome.

2 more flagged steps

self citation load bearing [Section III, Eqs. (7)–(8)]
"Thus the temperature, which determines the radiation rate from the RN black hole by co-tunneling process, does not coincide with the conventional Hawking temperature related to the outer horizon: 8–11 β = β+ + β− = 4π(r+ + r−) = 8πM . (7) Accordingly, the entropy of a black hole also does not depend on the charge: 8,9 SRN = 4πM 2 . (8)"

The co-tunneling rate formula (6) is asserted without derivation, and Eqs. (7)–(8) are attributed to Refs. 8–11, of which Refs. 8 and 9 are the author's own earlier statements of the same charge-independent entropy. The central claim is thus backed by a self-citation chain rather than by an independent calculation within this paper.
renaming known result [Section III, Eq. (9)]
"The modified Tsallis-Cirto δ = 2 statistics, which includes the negative entropy of the inner horizon, gives the following composition rule for entropy of the RN black hole in terms of the entropies of the two horizons: 16 SRN = (√S(r+) + √|S(r−)|)^2 = π(r+ + r−)^2 = 4πM 2 . (9)"

With S(r±) = ±πr±² and r+ + r− = 2M, the right-hand side is exactly 4πM² by algebra. Presenting this as a 'composition rule' from Ref. 16 (also by the author) renames the target entropy; the composition law is chosen to yield the desired r+ + r− sum, so it provides no independent confirmation of the charge-independent entropy.

full rationale

The central result S_RN = 4πM² is not a prediction from first principles in this paper. The pivotal step is Eq. (2), where the inner horizon is assigned the same χ=2 Euler characteristic as the outer horizon and the integral is taken from r− to infinity; no independent topological argument establishes χ=2 for that domain. The resulting β−, combined with β+ from Eq. (4), makes Eq. (7) reduce algebraically to 8πM, and Eq. (8) then gives 4πM². This is a fit to the author's previously published claim, as the introduction explicitly says the goal is to obtain 'agreement with Refs. 8–10.' The co-tunneling rate formula (6) is asserted rather than derived, and the cited support includes the author's own prior papers. The alternative derivation in Eq. (9) renames the same target via a composition rule that is algebraically identical to 4πM². Because the charge-independent entropy is forced by the assumed topological input and by the self-citation chain, the circularity score is 8. The independent content is minimal: the paper shows that if one accepts the author's prior entropy and the asserted co-tunneling sum, the topological language can be made to reproduce it, but it does not derive the result from an independent, externally falsifiable calculation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's conclusion rests on at least five unproven assumptions: the topological temperature formula, the validity of the Euclidean integral through a non-Euclidean region, the co-tunneling rate, the Tsallis-Cirto composition rule, and the adiabatic transformation. No new entities are introduced, but the central claim depends on ad hoc choices, especially the χ=2 assignment for the inner horizon.

free parameters (1)

Inner-horizon Euler characteristic χ(M) = 2
Assumed equal to the outer horizon value; this choice produces the cancellation that yields S=4πM^2 and is not independently justified.

assumptions (5)

ad hoc to paper The Euler characteristic χ(M)=2 determines the inverse temperature of a horizon through Eqs. (1)-(2), with no boundary or conical-singularity corrections in the Chern-Gauss-Bonnet integral.
This is the core postulate borrowed from Ref. 4 and applied to both horizons; Eq. (1) and Eq. (2) treat the bulk curvature integral as exactly equal to 2.
domain assumption The Euclidean section remains valid when integrating the curvature invariant from r_- to infinity, including the interval where the Euclidean metric has negative signature.
Eq. (2) integrates over the entire radial range r > r_-, but for r_- < r < r_+ the Euclidean metric component f(r) is negative, so the integral is over a non-Euclidean region.
ad hoc to paper The total Hawking radiation rate from a two-horizon black hole is the product of the two independent Boltzmann factors, w ∝ e^{-β_+E} e^{-β_-E}.
Eq. (6) is asserted without derivation; it is not a standard result of quantum field theory in curved spacetime.
domain assumption The entropy composition rule S_RN = (√S_+ + √|S_-|)^2 from modified Tsallis-Cirto statistics holds for black hole horizons.
Eq. (9) imports a non-extensive entropy composition law that is not widely accepted in black hole thermodynamics.
ad hoc to paper Varying the fine structure constant α at fixed mass M transforms an RN black hole adiabatically into a Schwarzschild black hole without changing entropy.
Sec. III uses this imagined process to support S_RN = S_Schwarzschild; no dynamical model is given.

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Cite this review

Pith. "Pith review of Black hole thermodynamics and topology." pith.science (2026). https://pith.science/paper/XJ4HPGDJ

@misc{pith2026250520194,
  author       = {Pith},
  title        = {Pith review of: Black hole thermodynamics and topology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJ4HPGDJ}},
  note         = {Machine review of arXiv:2505.20194}
}

abstract

Recently the difference between the Gibbons-Hawking temperature $T_{\rm GH}$ attributed to the Hawking radiation from the de Sitter cosmological horizon and the twice as high local temperature of the de Sitter state, $T=H/\pi=2T_{\rm GH}$, has been discussed by Hughes and Kusmartsev from the topological point of view (see arXiv:2505.05814). According to their approach, this difference is determined by the Euler characteristic $\chi({\cal M})$ of the considered spacetime with Euclidean time. The invariant $\chi({\cal M})$ is different for the global spacetime ${\cal M}=S^4$ and for the manifold limited to a region near the horizon, ${\cal M}=D^2\times S^2$. Here we consider the application of the topological approach to Reissner-Nordstr\"om (RN) black holes with two horizons. Both the outer and inner horizons are characterized by their near-horizon topology, which determines the corresponding horizon temperatures. As a result of the correlation between the horizons, the entropy of the RN black hole is independent of its electric charge, being completely determined by the mass of the black hole. This demonstrates the applicability of the topological approach to the multi-horizon systems.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Thermodynamics of black and white holes in ensemble of Planckons
gr-qc 2025-06 conditional novelty 6.0 of 10

A toy model counting pairs of Planckons gives integer black hole entropy, negative white hole entropy, charge-independent Reissner-Nordstrom entropy, and a quantized cosmological constant.

Reference graph

Works this paper leans on

29 extracted references · 14 canonical work pages · cited by 1 Pith paper

[1]

Gibbons and S

G. Gibbons and S. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 , 2738 (1977)

work page 1977
[2]

Volovik, First law of de Sitter thermodynamics, JETP Letters 121 , issue 10 (2025), arXiv:2504.05763

G.E. Volovik, First law of de Sitter thermodynamics, JETP Letters 121 , issue 10 (2025), arXiv:2504.05763

arXiv 2025
[3]

Maxfield and Z

H. Maxfield and Z. Zahraee, Holographic solar systems and hydrogen atoms: non-relativistic physics in AdS and its CFT dual, JHEP 11 (2022) 093

work page 2022
[4]

Hughes and F.V

J.C.M. Hughes and F.V. Kusmartsev, Topological Origin of Horizon Temperature via the Chern-Gauss-Bonnet Theorem. arXiv:2505.05814

arXiv
[5]

Stefano Liberati and Giuseppe Pollifrone, Entropy and topology for gravitational instantons, Phys. Rev. D 56 , 6458--6466 (1997), arXiv:hep-th/9708014

arXiv 1997
[6]

Emel Altas and Bayram Tekin, Hawking temperature as the total Gauss–Bonnet invariant of the region outside a black hole, Eur. Phys. J. C 83 , 414 (2023)

work page 2023
[7]

From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited

C. Fairoos and Chiranjeeb Singha, From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited, arXiv:2505.16309

work page Pith review arXiv
[8]

G.E. Volovik, Effect of the inner horizon on the black hole thermodynamics: Reissner-Nordstr\"om black hole and Kerr black hole, Modern Physics Letters A 36 , 2150177 (2021), arXiv:2107.11193

arXiv 2021

Show all 29 references

[9]

Volovik, Macroscopic quantum tunneling: from quantum vortices to black holes and Universe, ZhETF 162 , 449--454 (2022), JETP 135 , 388--408 (2022), arXiv:2108.00419

G.E. Volovik, Macroscopic quantum tunneling: from quantum vortices to black holes and Universe, ZhETF 162 , 449--454 (2022), JETP 135 , 388--408 (2022), arXiv:2108.00419

2022 arXiv
[10]

Chiranjeeb Singha, Pritam Nanda and Pabitra Tripathy, Hawking temperature of black holes with multiple horizons, Gen. Rel. Grav. 55 , 106 (2023)

2023
[11]

Space Sci

Zhongxu Zhai and Wenbiao Liu, Nernst theorem and Hawking radiation from a Reissner-Nordstr\"om black hole, Astrophys. Space Sci. 325 , 63--67 (2010)

2010
[12]

Tsallis and L.J.L

C. Tsallis and L.J.L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73 , 2487 (2013)

2013
[13]

Tsallis, Black Hole Entropy: A Closer Look, Entropy 22 , 17 (2020)

C. Tsallis, Black Hole Entropy: A Closer Look, Entropy 22 , 17 (2020)

2020
[14]

Volovik, Tsallis-Cirto entropy of black hole and black hole atom, Pis’ma v ZhETF 121 , 260--262 (2025), JETP Letters 121 , 243--248 (2025), arXiv:2409.15362

G.E. Volovik, Tsallis-Cirto entropy of black hole and black hole atom, Pis’ma v ZhETF 121 , 260--262 (2025), JETP Letters 121 , 243--248 (2025), arXiv:2409.15362

2025 arXiv
[15]

Manoharan and N

Manosh T. Manoharan and N. Shaji, Reconciling fractional entropy and black hole entropy compositions, Eur. Phys. J. C 85 , 373 (2025), arXiv:2501.18151

2025 arXiv
[16]

Volovik, Extended Tsallis-Cirto entropy for black and white holes, arXiv:2505.05178

G.E. Volovik, Extended Tsallis-Cirto entropy for black and white holes, arXiv:2505.05178

arXiv
[17]

Carroll, Matthew C

Sean M. Carroll, Matthew C. Johnson and Lisa Randal, Extremal limits and black hole entropy, JHEP11(2009)109

2009
[18]

Joscha Henheik, Bipul Poudyal and Roderich Tumulka, How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem, arXiv:2409.00677
[19]

Joshi and S

P.S. Joshi and S. Bhattacharyya, Primordial naked singularities, JCAP01(2025)034

2025
[20]

Joshi, Low mass naked singularities from dark core collapse, JCAP07(2024)053

Chandrachur Chakraborty, Sudip Bhattacharyya and Pankaj S. Joshi, Low mass naked singularities from dark core collapse, JCAP07(2024)053

2024
[21]

Virbhadra, Distortions of images of Schwarzschild lensing Phys

K.S. Virbhadra, Distortions of images of Schwarzschild lensing Phys. Rev. D 106 , 064038 (2022) arxiv:2204.01879 [gr-qc]

2022 arXiv
[22]

Yen Chin Ong, The Case For Black Hole Remnants: A Review, arXiv:2412.00322

arXiv
[23]

Mihalis Dafermos, The Stability Problem for Extremal Black Holes, In: Black Holes Inside and Out 2024, Visions for the future of black hole physics, arXiv:2410.14414

2024 arXiv
[24]

Quantum Analogues: From Phase Transitions to Black Holes and Cosmology

G.E. Volovik, Quantum phase transitions from topology in momentum space, in: "Quantum Analogues: From Phase Transitions to Black Holes and Cosmology", eds. William G. Unruh and Ralf Sch\"utzhold, Springer Lecture Notes in Physics 718 , 31--37 (2007), cond-mat/0601372

2007 arXiv
[25]

Volovik, Exotic Lifshitz transitions in topological materials, PHYSICS-USPEKHI 61 , 89--98 (2018), arXiv:1701.06435

G.E. Volovik, Exotic Lifshitz transitions in topological materials, PHYSICS-USPEKHI 61 , 89--98 (2018), arXiv:1701.06435

2018 arXiv
[26]

Gerard ’t Hooft, Quantum clones inside black holes, arXiv:2206.04608

arXiv
[27]

Gerard ’t Hooft, Alternative theory for the quantum black hole and the temperature of its quantum radiation, arXiv:2410.16891

arXiv
[28]

Douglas Stanford and Zhenbin Yang, Firewalls from wormholes, arXiv:2208.01625

arXiv
[29]

Ecaterina Howard, Geometric Aspects of Extremal Kerr Black Hole Entropy, Journal of Modern Physics 4 , 357-363 (2013)

2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Gibbons and S

G. Gibbons and S. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 , 2738 (1977)

work page 1977

[2] [2]

Volovik, First law of de Sitter thermodynamics, JETP Letters 121 , issue 10 (2025), arXiv:2504.05763

G.E. Volovik, First law of de Sitter thermodynamics, JETP Letters 121 , issue 10 (2025), arXiv:2504.05763

arXiv 2025

[3] [3]

Maxfield and Z

H. Maxfield and Z. Zahraee, Holographic solar systems and hydrogen atoms: non-relativistic physics in AdS and its CFT dual, JHEP 11 (2022) 093

work page 2022

[4] [4]

Hughes and F.V

J.C.M. Hughes and F.V. Kusmartsev, Topological Origin of Horizon Temperature via the Chern-Gauss-Bonnet Theorem. arXiv:2505.05814

arXiv

[5] [5]

Stefano Liberati and Giuseppe Pollifrone, Entropy and topology for gravitational instantons, Phys. Rev. D 56 , 6458--6466 (1997), arXiv:hep-th/9708014

arXiv 1997

[6] [6]

Emel Altas and Bayram Tekin, Hawking temperature as the total Gauss–Bonnet invariant of the region outside a black hole, Eur. Phys. J. C 83 , 414 (2023)

work page 2023

[7] [7]

From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited

C. Fairoos and Chiranjeeb Singha, From Nonextremal to Extremal: Entropy of Reissner-Nordstr\"om and Kerr black holes Revisited, arXiv:2505.16309

work page Pith review arXiv

[8] [8]

G.E. Volovik, Effect of the inner horizon on the black hole thermodynamics: Reissner-Nordstr\"om black hole and Kerr black hole, Modern Physics Letters A 36 , 2150177 (2021), arXiv:2107.11193

arXiv 2021