Quantization of Black Hole Entropy for Black Holes in Subtracted Geometry

Malcolm J. Perry; Mirjam Cvetic

REVIEW 3 major objections 5 minor 43 references

Quantization of Black Hole Entropy for Black Holes in Subtracted Geometry

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For black holes in subtracted geometry, the horizon area is quantized in units of $8\pi l_{\mathrm{Pl}}^2$, so one absorbed quantum changes the outer-horizon area by an integer multiple of that unit.

desk verdict The paper's 8π area quantization is attainable, but only after fixing the dE_L = dM vs E_L = M/2 inconsistency in the printed derivation. read the letter →

arxiv 2504.14247 v1 pith:VL2RVCDQ submitted 2025-04-19 hep-th gr-qc

Mirjam Cvetic , Malcolm J. Perry This is my paper · ORCID

classification hep-thgr-qc MSC 83C5781T4083C45 PACS 04.70.Dy04.60.-m11.25.Hf

keywords blackholeentropyareaquantizationquasinormalmodessubtractedgeometryKerr/CFTcorrespondencesupergravityholesholographicduality

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 claims that the horizon area of black holes in subtracted geometry—a family of four-dimensional supergravity black holes that share the horizon areas and surface gravities of Kerr-type asymptotically flat black holes—takes only discrete values, spaced by $8\pi l_{\mathrm{Pl}}^2$. The claim is reached by solving the quasinormal-mode spectrum exactly and showing it splits into a left and a right family whose frequencies match two conformal field theories at temperatures $T_L$ and $T_R$. Using the damped-oscillator interpretation of quasinormal modes, the paper reads the magnitude of each mode frequency as the energy of one absorbed quantum; the first law in each sector then gives an entropy change of $2\pi$ times an integer. Because the outer-horizon entropy is the sum of the two sector entropies, the outer-horizon area changes in integer multiples of $8\pi l_{\mathrm{Pl}}^2$.

What carries the argument

The central object is the exact hypergeometric quasinormal-mode solution of the scalar wave equation in subtracted geometry, combined with the damped-oscillator map between a purely imaginary mode frequency and a real oscillator frequency. Subtracted geometry is a deformation of the Kerr metric that keeps the horizon areas and surface gravities unchanged while making all quasinormal modes analytically computable. The radial solution has exponents fixed by the horizon temperatures $T_\pm$, and the quantization condition comes from a connection formula for the hypergeometric function whose $\Gamma$-function poles produce two evenly spaced families. These families are matched to two CFT sectors with temperatures $T_L$ and $T_R$, and the sector first laws convert an added quantum into a discrete entropy change.

What would settle it

Compute the exact gravitational quasinormal-mode spectrum for the same subtracted-geometry background: if metric perturbations do not produce the same evenly spaced families proportional to $T_L$ and $T_R$, the area quantum would not be $8\pi l_{\mathrm{Pl}}^2$. Alternatively, an exact strong-field first-law calculation in which adding one quantum changes $S_+$ by anything other than an integer multiple of $2\pi$ would refute the claim.

Watch

Extended reading notes

Core claim

For the subtracted-geometry metric (2), the radial wave equation for a minimally coupled massless scalar reduces to a hypergeometric equation, and the boundary condition that the wave be bounded at infinity fixes the mode frequencies by the vanishing coefficient of the $r^l$ branch. The resulting families are $\omega_L/T_L=-4\pi i(n+l)$ and $\omega_R/T_R=-4\pi i(n+l)+k\Omega_+/T_+$, with $n\in\mathbb{Z}_+$; the paper identifies these with the left and right sectors of a $CFT_L\otimes CFT_R$ pair. The absorption cross-section of the black hole, up to the factor $|\Gamma(a)\Gamma(b)|^2$, matches the CFT absorption cross-section at fixed $k=0$. Adding a quantum of energy $\Delta M=|\omega_L|$ or $|\omega_R|$ at fixed $J$, $Q^{(e)}$, and $Q^{(m)}$, and applying the sector first laws, yields $\Delta S_L$ or $\Delta S_R$ equal to $2\pi$ times an integer; since $S_+=S_L+S_R$, the outer-horizon area is quantized in units of $8\pi l_{\mathrm{Pl}}^2$.

Load-bearing premise

The argument assumes that the quasinormal-mode frequencies of a minimally coupled massless scalar field, interpreted through the damped-oscillator map, give the true quantum excitation energies of the black hole, so that the magnitude of the imaginary frequency is the energy of a single absorbed quantum.

Editorial extensions

If this is right

Every absorption of a single quantum changes the outer-horizon entropy by $2\pi$ times an integer, independent of the charge and rotation parameters of the subtracted black hole.
The area quantum $8\pi l_{\mathrm{Pl}}^2$ applies to the whole subtracted-geometry family, including the Kerr limit $\Pi_c=1,\Pi_s=0$ and the Kerr-Newman-like limit.
The exact quasinormal-mode spectrum reproduces the absorption cross-section of a pair of conformal field theories at temperatures $T_L$ and $T_R$, supporting the description of horizon microstates as a $CFT_L\otimes CFT_R$ system.
At fixed angular momentum and charges, the energy levels of the black hole are evenly spaced in both the left and right sectors.

Reading between the lines

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

An extension left implicit is that if the $8\pi l_{\mathrm{Pl}}^2$ spacing follows only from the shared horizon data, the same area ladder would apply to the asymptotically flat Kerr and Kerr-Newman black holes, whose exact quasinormal spectra are not yet analytically known.
A direct way to test the load-bearing assumption would be to compute the gravitational (metric) quasinormal modes of subtracted geometry rather than minimally coupled scalars; agreement would strengthen the claim, while disagreement would pinpoint where the scalar spectrum fails to represent the true degrees of freedom.
The sector-by-sector first-law analysis keeps $J$, $Q^{(e)}$, and $Q^{(m)}$ fixed; relaxing those constraints could show whether the area quantum changes with angular momentum or charge, distinguishing this spectrum from other proposed area spectra.

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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

Here's the short version: the paper almost delivers a clean derivation of the 8π area quantum for the full subtracted-geometry family, but the central step as printed has a factor-of-two inconsistency that the authors notice and then talk around rather than fix. I checked the stress-test note's factor-of-two concern against the text and it lands.

The genuinely new thing is applying Maggiore's damped-oscillator argument to the full family with the CFT_L × CFT_R structure. The QNM spectrum itself is taken from [13] (with the factor-of-2 correction to [12] noted), and the step from that spectrum to area quantization is new. The paper does well: the QNM computation is exact and analytic, the CFT thermodynamics is laid out transparently, and there are no fitted parameters. If the result holds, it's a nice confirmation of the Bekenstein-Mukhanov-Hod program.

Where it goes wrong: in the Perturbations section, the first law for the left CFT is applied with dE_L = dM = |ω_L|. But the paper itself states E_L = M/2, an exact relation for the equilibrium family. For nearby equilibrium states, dE_L = dM/2, not dM. The 'on average' comment about Smarr doesn't repair this, because Smarr isn't where the E_L = M/2 factor comes from, and a differential relation between state functions can't be made to disappear by averaging. With the printed assignment you get ΔS_L = 4π(n+l), so ΔA = 16π l_P^2. With the consistent assignment dE_L = dM/2 and ΔM = |ω_L|, you get ΔS_L = 2π(n+l), giving exactly the advertised 8π. So the result is likely right, but the argument needs a corrected line, not an aside.

Two softer caveats: using scalar-field QNMs as proxies for the true gravitational/microstate spectrum is a load-bearing assumption (Maggiore's interpretation is standard but still an assumption). And the leap from subtracted geometry to Kerr rests on an environment-independence claim that's plausible but unproven.

Who's this for? It's a letter for people working on area quantization, black hole CFTs, and exact QNM spectra. It's the kind of paper you'd want to discuss in a reading group precisely because the flaw is instructive.

My recommendation: yes, send it to a serious referee. The referee should ask for the energy bookkeeping to be fixed. Once that's done, this becomes a clean, citable result.

Referee Report

3 major / 5 minor

Summary. The paper studies four-dimensional black holes in subtracted geometry arising in N=2 supergravity. It derives the exact quasinormal mode (QNM) spectrum for minimally coupled massless scalar perturbations, obtaining two families of frequencies that the authors associate with a left and a right conformal field theory. Adopting Maggiore's interpretation of QNMs as damped oscillators, the paper identifies |ω| with the energy of a single absorbed quantum and uses the CFT first law to argue that the outer horizon area is quantized in units of 8π l_P^2. The claim is intended to hold for the full family of subtracted geometries, with the asymptotically flat Kerr and Kerr-Newman cases included in the limit.

Significance. If the result is correct, it provides an analytic, parameter-free derivation of Bekenstein-Mukhanov area quantization with a specific factor 8π in a string-theoretic setting, going beyond heuristic arguments. The exact analytic QNM spectrum and the detailed matching to CFT absorption cross-sections are genuine strengths, and the paper contains no fitted parameters. However, the central derivation contains an internal factor-of-two inconsistency in the energy-bookkeeping step, and it relies on an unproven identification of the scalar-field QNM spectrum with the full quantum excitation spectrum of the black hole. These issues must be resolved before the claimed area quantum can be regarded as established.

major comments (3)

[Perturbations] The energy bookkeeping in this section is internally inconsistent. The text sets dE_L = dM = |ω_L| when applying the first law (6), whereas the thermodynamic section establishes E_L = E_R = M/2, which for any state function implies dE_L = dM/2. If one uses dE_L = |ω_L| together with the spectrum (8), one obtains ΔS_L = |ω_L|/T_L = 4π(n+l), hence ΔA = 16π l_P^2, contradicting the abstract's claimed unit of 8π. The consistent choice dE_L = dM/2 with ΔM = |ω_L| gives ΔS_L = 2π(n+l) and ΔA = 8π l_P^2, which is what the sentence 'the change in SL or SR is 2π × (some integer)' in the same section implies, but it is not the derivation printed. The argument that Smarr relations need hold only on average does not repair the step, because the first law is applied to an infinitesimal single-quantum absorption, and an equality between differentials cannot be altered by averaging. The derivation must be corrected to maintain a consistent energy assignment.
[Quasi-normal modes and Oscillators] The exact QNM spectrum is derived for minimally coupled massless scalars in subtracted geometry. The subsequent identification of these modes with the quantum excitation spectrum of the black hole assumes that the gravitational or microstate spectrum shares the same frequencies. This is not established; indeed, for Kerr the scalar and gravitational QNM spectra are known to differ. Since the magnitude |ω| is used as the energy of a single absorbed quantum in the first law, this identification is load-bearing for the area quantum. The paper should either provide a supporting argument that the scalar sector captures the relevant degrees of freedom, or explicitly state that the 8π result is conditional on this identification.
[Introduction and Section 2] The claim that the results have general applicability to the asymptotically flat black holes is justified only by the statement that black hole properties should not depend on their environment. However, the QNM spectrum is environment-dependent: the wave equation in subtracted geometry differs from that in the original asymptotically flat geometry, even though the horizon areas and surface gravities coincide. Equality of horizon thermodynamics does not guarantee equality of the perturbation spectra. The manuscript should clarify whether the 8π quantization is a prediction for subtracted geometry only or is also claimed for the asymptotically flat family, and if the latter, supply evidence for spectral universality.

minor comments (5)

[Title and Introduction] The title contains an extraneous space in 'Subtra cted', and the introduction has 'conventionly' for 'conventionally' and 'It hs been' for 'It has been'.
[Quasi-normal modes] The text says 'the polar functions are just the associated Legnedre polynomials'; 'Legnedre' should be 'Legendre'.
[General] The manuscript does not number its displayed equations, which makes precise referencing (e.g., 'the first law (6)') impossible. Numbering would greatly improve the clarity of the derivation.
[Thermodynamics] The parameter m is used for the metric but is not explicitly defined before its first use; a brief definition would help the reader.
[Footnote 5] The footnote states that the results are true for the whole family of charged subtracted geometries, but the derivation shown for the quantization step does not explicitly track the charge dependence of the QNM spectrum; a sentence explaining why the charge parameters drop out would be useful.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: the area-quantization claim follows algebraically from the independently re-derived QNM spectrum and the first law; self-citations are present but not load-bearing. An internal factor-of-two bookkeeping inconsistency is a correctness concern, not circularity.

full rationale

The paper derives the quasinormal-mode spectrum in §QUASI-NORMAL MODES by solving the radial equation and imposing boundedness at infinity, obtaining ω_L/T_L = −4πi(n+l) and ω_R/T_R = −4πi(n+l)+kΩ_+/T_+. This derivation is self-contained: it starts from the subtracted-geometry metric, gives the explicit hypergeometric solutions and connection formula, and then states the resulting mode condition. The footnote 'These results agree with the calculation in [13]' indicates that the prior self-citation is a check, not the source of the load-bearing spectrum. The thermodynamics (Smarr relations and first laws) are also stated from the geometry and from previous work, but the needed formulas are quoted and used algebraically; no parameters are fitted to force the area quantization. The step from |ω|/T = 4π(n+l) and the first law to ΔS_L = 2π×integer is a direct substitution, and the conclusion ΔA = 8πl_P^2×(integer) is the paper's advertised result. This is not circular because the area unit is not used to determine the QNM frequencies; it emerges from them. The main weakness is an internal consistency issue flagged by the authors in §PERTURBATIONS: they set dE_L = dM = |ω_L| while earlier establishing E_L = M/2, so that a straight first-law computation gives ΔS_L = 4π(n+l) and hence ΔA = 16π(n+l), not 8π(n+l). The authors' appeal to Smarr relations holding 'only on average' does not repair an infinitesimal first-law step. This is a numerical/correctness problem, not a circularity: the prediction is not equivalent to the input; rather, the printed derivation does not uniquely select the advertised 8π unit without an additional assumption about the energy split. The self-citations to [10]–[15] and [18] are to the established subtracted-geometry framework and thermodynamics, which are external to the present quantization claim and are independently checkable. Therefore the circularity score is low, reflecting the presence of self-citations but the absence of any load-bearing circular reduction.

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

No data-fitting or hand-chosen constants enter the derivation; m, a, Π_c, Π_s are general solution parameters. No new particles, forces, or dimensions are introduced. The two CFTs are imported from prior literature.

assumptions (5)

domain assumption The exact scalar quasinormal mode spectrum for subtracted geometry is as given by the hypergeometric quantization condition (Eqs. 7-9), quoted from [12,13].
The paper does not re-derive the radial equation or the connection formula; it relies on prior results by the same authors.
domain assumption A purely damped quasinormal mode with imaginary frequency corresponds to a natural oscillation frequency equal to |ω| (Maggiore's damped oscillator model, Eq. 14).
This is a heuristic from nuclear and black hole physics that maps QNM frequencies to quantum energy levels; it is used to set the added energy to |ω_L| or |ω_R|.
standard math The first law and Smarr relations for the CFT sectors (Eqs. 5-6) apply, with E_L = E_R = M/2.
These are inherited from the black hole thermodynamics of [18] and are used to translate energy changes into entropy changes.
ad hoc to paper The properties of a black hole are independent of its environment, so the subtracted geometry result applies to the asymptotically flat black holes.
Stated in the introduction without proof; it is the bridge from the conical subtracted geometry to physically relevant black holes.
domain assumption The perturbation is at fixed J, Q(e), Q(m) and k=0, so the work terms in the first law vanish.
The paper restricts to k=0 and says removal of this constraint will be discussed elsewhere.

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Cite this review

Pith. "Pith review of Quantization of Black Hole Entropy for Black Holes in Subtracted Geometry." pith.science (2026). https://pith.science/paper/VL2RVCDQ

@misc{pith2026250414247,
  author       = {Pith},
  title        = {Pith review of: Quantization of Black Hole Entropy for Black Holes in Subtracted Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VL2RVCDQ}},
  note         = {Machine review of arXiv:2504.14247}
}

read the original abstract

We carefully examine the exact analytic spectrum of quasinormal modes of general black holes in the so-called subtracted geometry of maximally supersymmetric supergravity. These black holes have the same area and surface gravity at both the outer and inner horizons as the original asymptotically flat black holes. We proceed to explore the relationship with conformal field theories that describe horizon physics of these black holes. As a consequence, we show that the horizon area of these black holes is quantized in units of 8{\pi}l_{Planck}^2.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [13]

Quasi-Normal Modes for Subtracted Rotating and Magnetised Geometries

M. Cvetiˇ c, G. W. Gibbons, and Z. H. Saleem, Quasinor- mal modes for subtracted rotating and magnetized geome- tries, Phys. Rev. D 90, 124046 (2014), arXiv:1401.0544 [hep-th]

work page Pith review arXiv 2014

[2] [12]

Exact quasi-normal modes for the near horizon Kerr metric

M. Cvetiˇ c and G. W. Gibbons, Exact quasinormal modes for the near horizon Kerr metric , Phys. Rev. D 89, 064057 (2014), arXiv:1312.2250 [gr-qc]

work page Pith review arXiv 2014

[3] [1]

S. W. Hawking, Particle Creation by Black Holes , Com- mun. Math. Phys. 43, 199 (1975), [Erratum: Com- mun.Math.Phys. 46, 206 (1976)]

work page 1975

[4] [2]

J. D. Bekenstein, Black holes and the second law , Lett. Nuovo Cim. 4, 737 (1972)

work page 1972

[5] [3]

J. D. Bekenstein, Black holes and entropy , Phys. Rev. D 7, 2333 (1973)

1973

[6] [4]

L. Boltzmann, ¨Uber die Beziehung eines allgemeine mechanischen Satzes zum zweiten Haupsatze der W¨ armetheorie, Sitzungberichte der Kaiserlichen Akademie der Wissenschaften, Mathematische- Naturwissen Classe. Abt II 67, 373 (1877)

work page

[7] [5]

J. D. Bekenstein, The quantum mass spectrum of the Kerr black hole , Lett. Nuovo Cim. 11, 467 (1974)

work page 1974

[8] [6]

V. F. Mukhanov, Are Black Holes Quantized? , JETP Lett. 44, 63 (1986)

work page 1986