Radiative properties of a nonsingular black hole: Hawking radiation and gray-body factor

Asier Alonso-Bardaji; David Brizuela; Marc Schneider

REVIEW 3 major objections 5 minor 1 cited by

Radiative properties of a nonsingular black hole: Hawking radiation and gray-body factor

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

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Pith's one-line read A nonsingular black-hole metric radiates colder and less gray than Schwarzschild, and evaporates toward a stable remnant.

desk verdict Solid analytic computation of Hawking temperature and gray-body factors for a regular black hole, with a remnant claim that depends on an undefended modeling choice. read the letter →

arxiv 2504.13050 v2 pith:4QW522FZ submitted 2025-04-17 gr-qc hep-th

Asier Alonso-Bardaji , David Brizuela , Marc Schneider This is my paper · ORCID

classification gr-qchep-th MSC 83C5783C4783C45 PACS 04.70.Dy04.62.+v04.60.Pp

keywords regularblackholesHawkingradiationgray-bodyfactorloopquantumgravityblack-holeremnantsinformation-lossparadoxRegge-Wheelerpotentialthermodynamics

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 studies the radiative properties of a specific spherically symmetric, singularity-free black-hole spacetime whose classical singularity is replaced by a minimal surface. It computes the Hawking temperature and the gray-body factor for a massless scalar field, and finds that, for a fixed horizon radius, the regular hole is colder than Schwarzschild and its emission is closer to a perfect black-body spectrum. The central dynamical conclusion is that if the parameter $r_0$ (the minimal-surface radius) stays fixed while the hole radiates, the horizon radius asymptotically approaches $r_0$, the temperature and entropy drop to zero, and the evaporation time diverges: the hole ends as a stable remnant rather than disappearing. This matters because the same two observables, the temperature and the gray-body factor, could in principle reveal such quantum-gravity corrections and give a concrete route to resolving the information-loss paradox.

What carries the argument

The carrying object is the two-parameter regular metric (1), defined by the horizon radius $r_g$ and the minimal-surface radius $r_0$ (equivalently $\lambda=r_0/r_g$). The radiative argument runs through two named quantities: the horizon temperature obtained by the Hamilton-Jacobi tunneling method, and the Regge-Wheeler potential $V_l(r)=\left(1-\frac{r_g}{r}\right)\left(\frac{l(l+1)}{r^2}+\frac{2r_g+r_0}{2r^3}-\frac{3r_g r_0}{2r^4}\right)$, whose difference from Schwarzschild factors as $V_l(r)-V_l^{\mathrm{Schw}}(r)=\frac{r_0}{2r^5}(r-r_g)(r-3r_g)$. This factorized identity is what makes the potential lower precisely in the barrier region that controls scattering, so the gray-body factor moves toward unity. The evaporation and remnant story is carried by the temperature formula and the choice to keep $r_0$ fixed, together with the Hawking-energy definition of mass and the Stefan-Boltzmann law for the radiating power.

What would settle it

Track a small evaporating black hole's temperature against its horizon radius: the constant-$r_0$ branch predicts a temperature peak at $r_g=3r_0/2$ followed by a drop to zero as $r_g\to r_0$, whereas the constant-$\lambda$ branch predicts a monotonically rising rescaled Schwarzschild temperature; observing either behavior, or the absence of a stable remnant population at the predicted remnant scale, would settle which branch (if either) is realized.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the metric $ds^2 = -(1-r_g/r)dt^2 + (1-r_0/r)^{-1}(1-r_g/r)^{-1}dr^2 + r^2 d\Omega^2$, with $0<r_0<r_g$, describes a geodesically complete black-to-white-hole spacetime whose Hawking temperature is $T = \hbar/(4\pi r_g)\sqrt{1-r_0/r_g}$. Compared with Schwarzschild of the same $r_g$, the temperature is lower, and the Regge-Wheeler potential is shallower between the horizon and $r=3r_g$, so the reflection coefficient is reduced and the gray-body factor, the transmission probability, is closer to unity. The paper further claims that when $r_0$ is held constant during evaporation the temperature first rises, peaks at $r_g=3r_0/2$, then falls to zero as $r_g\to r_0$, with entropy going to zero and the evaporation time diverging, so the endpoint is a remnant of vanishing temperature and entropy that may resolve the information-loss problem. If instead the dimensionless ratio $\lambda=r_0/r_g$ is held fixed, the correction merely rescales Planck's constant and the evolution reproduces Schwarzschild's complete evaporation.

Load-bearing premise

The remnant conclusion rests on the unproven modeling choice, stated in Sec. VI, that the minimal-surface radius $r_0$, rather than the ratio $\lambda=r_0/r_g$, stays constant while the hole radiates; the paper itself shows that the opposite choice only rescales Planck's constant and gives complete Schwarzschild-like evaporation.

Editorial extensions

If this is right

For a fixed horizon radius $r_g$, the regular geometry is colder by the factor $\sqrt{1-r_0/r_g}$; this reduces the radiated power and shifts the spectrum to lower frequencies.
Because the barrier is shallower between $r_g$ and $3r_g$, a larger fraction of the emitted modes escapes to infinity: the reflection coefficient is smaller and the gray-body factor is closer to one.
If $r_0$ is a fixed scale, a black hole that starts large first heats up as it shrinks, then cools and asymptotically settles at $r_g=r_0$ with zero temperature and zero entropy after an infinite time.
If $\lambda=r_0/r_g$ is the fixed scale instead, the same horizon radiates like a Schwarzschild hole with a rescaled Planck constant and evaporates completely.
The horizon entropy acquires corrections proportional to $\sqrt{A}$ and $\ln A$ relative to the Bekenstein-Hawking area law, providing an observational signature of the minimal scale.

Reading between the lines

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

The factorized sign of the potential difference, $\propto (r-r_g)(r-3r_g)$, looks like a generic mechanism: any regular geometry that flattens the barrier in the photon-sphere band should show the same qualitative 'colder and less gray' trade-off, so the scalar-field result may survive for higher spins and other regular metrics.
The $r_0$-fixed versus $\lambda$-fixed dichotomy offers a sharp observational discriminator the paper does not pursue: a population of stable remnants with masses near $r_0/2$ would support the first branch, while their absence would favor the second.
Computing the evaporation with the gray-body factor included in the luminosity, rather than the pure black-body Stefan-Boltzmann estimate used in Sec. VI, would refine the remnant scenario and could change the quantitative approach rate, even though the divergence of the evaporation time already comes from the black-body part.
Applying the same WKB machinery to fermions and vector perturbations (the paper treats a massless scalar, with a numerical higher-spin study referenced) would test the $l$-dependence of the 'purer spectrum' effect beyond s-waves.

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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

First thing to know: this is a mostly solid radiative computation for a specific regular black-hole metric, and the headline claim to be careful about—the stable remnant that 'resolves' information loss—hangs on an unargued modeling choice. Read the temperature and gray-body parts as reliable; read Secs. VI–VII as conditional.

The paper does the math honestly. The temperature (20) follows from the stated metric via standard surface-gravity/tunneling reasoning and matches. The potential difference (24) is a clean observation: between rg and 3rg the regular-geometry potential is shallower than Schwarzschild for equal rg, independent of l. The analytic WKB gray-body formulas for large-l and s-waves, in both the astrophysical (small λ) and near-remnant (λ→1) limits, are genuinely new and complement the numerical study in [50]. The entropy integral and evaporation-rate calculation are consistent with the temperature law. The authors clearly flag that the deformed Hamiltonian model is not directly derived from loop quantum gravity and that their WKB expansion has a limited validity range. That is the right level of epistemic hygiene.

The soft spot is real and central to the remnant. r0 is not an independent constant of the model; it is defined as λ rg, where λ is the polymerization parameter from [29,30]. The paper acknowledges that if λ is held fixed during evaporation, the evolution is qualitatively Schwarzschild (with ℏ rescaled) and there is no remnant. The remnant appears only if r0 is held fixed, which the authors simply assert as their choice, motivated by an area gap in LQG. That is a plausible conjecture, but it is not a prediction of the model, and the abstract's claim that the black hole 'naturally leads to a remnant' oversells it. The information-loss tie-in inherits this conditionality. Also, the 'less gray' statement in the abstract is regime-dependent: their own Eq. (63) shows s-wave reflection increases near the remnant, and low-frequency s-waves in the astrophysical limit have the opposite sign in Eq. (51) until ν ≳ 0.098. Minor mechanical issues: the surface gravity printed in footnote 2 looks dimensionally inconsistent, and the threshold in Sec. V A 1 appears to be missing a square root.

Bottom line: if you work on effective polymer black holes or gray-body factors of regular metrics, this paper is worth your time and worth citing for the analytic formulas. It should go to peer review; the radiative part is solid and the remnant part can be fixed by either deriving r0 constancy from the underlying model or softening the claim to 'if r0 is fundamental, then...'.

Referee Report

3 major / 5 minor

Summary. The paper studies the radiative properties of the regular black-hole metric (1), originally proposed in Refs. [29,30]. It computes the Hawking temperature (20) using the tunneling method, derives the scalar Regge-Wheeler potential (23), and obtains analytic gray-body factors via a first-order WKB approximation in the astrophysical limit lambda<<1 and the near-remnant limit lambda->1, for both large angular momentum and s-waves. It then defines an entropy via the first law and integrates a Stefan-Boltzmann evaporation law. Assuming r0 is held constant during evaporation, the horizon approaches r0 asymptotically with vanishing temperature and entropy and infinite evaporation time, and the authors present the resulting remnant as a possible resolution of the information-loss problem.

Significance. The radiative part of the paper is clean and internally coherent. I checked the temperature against the surface gravity of the metric, the potential difference (24) against the potential (23), the entropy integral (65), and the evaporation rate (68); these are consistent. The paper contains explicit analytic expressions, has no fitted parameters, and plainly states its assumptions, which is a strength. The comparison of equal-horizon-radius, equal-ADM-mass, and equal-temperature cases in Appendix A is also useful. The main weakness is that the remnant and information-loss conclusion depends on an assumption that is not derived from the model, and this weakens the headline claim rather than the radiative computation itself.

major comments (3)

[Sec. VI (paragraph after Eq. (64)) and Sec. II A] The remnant and information-loss conclusion is not a prediction of the model as currently derived. In Sec. II A the model is parameterized by rg and the dimensionless polymerization parameter lambda, with r0 := lambda rg. The authors themselves note that if lambda is held fixed during evaporation, the temperature correction is merely a rescaling of hbar and the evolution is qualitatively Schwarzschild, ending at rg -> 0. They then choose r0 constant without deriving this from the Hamiltonian dynamics of Refs. [29,30] or from an independent quantum-gravity input. Since lambda is the Hamiltonian-deformation parameter, holding lambda fixed is the natural reading, and under that reading no remnant and no information-loss resolution follow. Because the abstract and Sec. VII advertise the stable remnant as a possible resolution to the information-loss issue, this is load-bearing. Please either derive the constancy of r0 from the underlying theory (for example from an area gap, as in Ref. [46]) or explicitly reframe the evaporation section as a speculative scenario that is not a prediction of the model. If the latter course is chosen, the abstract and conclusions should be softened accordingly.
[Sec. IV A (text after Eq. (24))] The statement that a shallower potential leads to 'a smaller value of the gray-body factor' is inconsistent with the definition of the gray-body factor as the transmission coefficient sigma = |T|^2 in Eq. (27). A lower potential barrier increases transmission and therefore increases sigma, which is exactly what the later WKB results in Sec. V find (the lambda-correction reduces the reflection coefficient). This is a sign/terminology error that should be corrected to 'larger gray-body factor' or 'smaller reflection coefficient'.
[Footnote 2 (after Eq. (19))] The quoted surface gravity appears to be in tension with the temperature obtained in Eq. (20). For the metric (1), a direct computation of kappa^2 for the Killing field partial_t gives (rg - r0)/(4 rg^3), which yields T = hbar sqrt(1 - r0/rg)/(4 pi rg), consistent with Eq. (20). The footnote instead writes (rg - r0)/(4 rg r0^2). Please check this formula and correct the typo or the derivation, since readers may otherwise find an apparent inconsistency between the temperature used in the paper and the associated surface gravity.

minor comments (5)

[Sec. V (Eqs. (44)-(45))] The integrals defining gamma_l^{black} and gamma_l^{gray} do not state their integration limits explicitly; they should be specified as nu in (0, infinity), or whatever range is intended.
[Sec. V B 2 (text after Eq. (63))] The sentence claiming that no particular nu-value is needed to perceive the depletion in reflectivity is confusing, because the next sentence states that the higher-order terms become nu-dependent and a threshold nu^2 < 8l(l+1)/135 appears. Please clarify the order in l at which each statement holds.
[References] Reference [54] duplicates reference [44] (both are Hawking's 1975 particle-creation paper); please merge or disambiguate them.
[General notation] The use of the same symbol lambda for the dimensionless ratio r0/rg and for the polymerization parameter, with the relation lambda := lambda^2/(1+lambda^2), is prone to confusion. Since several equations use lambda, consider denoting the polymerization parameter by, for example, lambda_bar throughout, or otherwise explicitly distinguishing the two at every occurrence.
[Eq. (39)] The definitions of f_o^l(nu) and f_c^l(nu) are cumbersome; a brief statement of their role (the lambda-independent and lambda-correction exponentials) before or after Eq. (40)-(41) would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction in the radiative derivation; the remnant claim is an explicit modeling assumption, and the only self-citation is in model provenance, not in the computation.

full rationale

The radiative core of the paper is self-contained. The Hawking temperature (20) is derived from the given metric (1) through the Hamilton-Jacobi tunneling calculation: the radial momentum k− in Eq. (16) has a simple pole at the horizon, and the Sokhotski-Plemelj evaluation in Eq. (17) yields Im(S0), from which T = ℏ/(4πrg)√(1−r0/rg) follows. The gray-body factor is likewise derived from the Regge-Wheeler potential (23) and the standard WKB transmission/reflection formulas (33)-(35), with no fitted parameter and no quantity being renamed from an input. All r0→0 limits reduce to the standard Schwarzschild results, providing an external consistency check. The authors' prior work [29,30] supplies the metric itself and its loop-quantum-gravity-inspired interpretation; this is a normal follow-up self-citation rather than a circular step, because the temperature, potential, and transmission coefficients computed here are not assumed in those references and are not equivalent to the metric input. The remnant and information-loss conclusions in Secs. VI-VII are explicitly conditional: the paper states that 'we will consider that r0 is kept constant during the evaporation process' and explicitly acknowledges that the alternative choice λ = constant merely rescales ℏ and reproduces Schwarzschild-like complete evaporation. That is an un-derived modeling choice and a limitation, not a circularity, since the entropy, temperature, and infinite evaporation time are then derived explicitly from the stated assumption rather than being inserted as the conclusion. The LQG-motivated identification of r0 as a fundamental scale is supported by external citation [46], not by the present computation. Overall, no load-bearing step reduces by construction to its own inputs; the only minor issue is the heavy reliance on the authors' own prior construction for the background metric, which does not affect the internal validity of the radiative calculation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central claims rest on two inputs the reader cannot verify here: the model parameter r0/λ, whose value is not fixed by data or by LQG, and the choice r0 = const during evaporation, which alone generates the remnant. The tunneling and WKB methods are standard and reused as-is; no parameter is fitted to data. The radiative part of the computation is largely self-contained, while the remnant thermodynamics is assumption-driven.

free parameters (1)

r0 (equivalently λ = r0/rg) = not fitted; free model input with λ in (0,1)
Every radiative quantity depends on r0/rg (Eqs. 20, 23, 30, 39, 51, 63), yet no value is fixed by data or derived from LQG. The remnant result additionally requires choosing r0 constant during evaporation instead of λ constant (Sec. VI), a choice the paper flags.

assumptions (4)

domain assumption Metric (1) is the correct effective spacetime; the deformed-Hamiltonian framework of Refs. [29,30,41,42] describes quantum-corrected spherical black holes.
The whole computation runs on this background. The paper states the model 'is not directly derived from the theory of loop quantum gravity' (Sec. I), so the LQG interpretation is an external motivation, not a derivation.
standard math The tunneling (Hamilton-Jacobi) method yields the Hawking temperature; equivalence to the Bogolyubov computation.
Invoked in Sec. III via Refs. [58,65-68]. Internally corroborated because Eq. (20) matches the surface gravity of the metric.
domain assumption Leading-order WKB eikonal (parabolic-barrier) formulas (33)-(35) give accurate reflection and transmission coefficients.
Standard in the literature (Konoplya, Iyer-Will, Fabbri), but validity is only checked a posteriori via Eq. (32), and no comparison is made to the numerical gray-body data of Ref. [50].
domain assumption During evaporation r0 stays constant; horizon energy is the Hawking energy E_BH = rg/2; mass loss follows the Stefan-Boltzmann law (67).
Needed for the remnant, entropy (65), and infinite evaporation time. Sec. VI explicitly notes λ = constant would instead give full evaporation, and Eq. (67) neglects the gray-body factor, so the evaporation time is only a lower bound (the paper says so).

invented entities (2)

Minimal transition surface T at r = r0 connecting trapped and anti-trapped regions
purpose: Replaces the singularity; it is the geometric source of the lower temperature and shallower potential.
Inherited from the authors' earlier model (Refs. [29,30]); no independent observational evidence is presented, and no concrete value of r0 is predicted, so the spectral and remnant handles are in-principle only.
Stable remnant with rg = r0, T = 0, S = 0
purpose: Candidate end state retaining information and offering a resolution of the information-loss issue.
It arises only under the r0-const assumption, is approached after infinite evaporation time (Sec. VI), its mass scale r0 is not fixed by the paper, and its entropy is set to zero by choice of integration constant in Eq. (65).

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Pith. "Pith review of Radiative properties of a nonsingular black hole: Hawking radiation and gray-body factor." pith.science (2026). https://pith.science/paper/4QW522FZ

@misc{pith2026250413050,
  author       = {Pith},
  title        = {Pith review of: Radiative properties of a nonsingular black hole: Hawking radiation and gray-body factor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QW522FZ}},
  note         = {Machine review of arXiv:2504.13050}
}

read the original abstract

We study the radiative properties of a spherical and singularity-free black-hole geometry recently proposed in the literature. Contrary to the Schwarzschild spacetime, this geometry is geodesically complete and regular, and, instead of the singularity, it presents a minimal surface that connects a trapped (black-hole) with an antitrapped (white-hole) region. The geometry is characterized by two parameters: the Schwarzschild radius and another parameter that measures the area of the minimal surface. This parameter is related to certain corrections expected in the context of loop quantum gravity to the classical general-relativistic dynamics. We explicitly compute the spectrum of the Hawking radiation and the gray-body factor. Since the gravitational potential is shallower than in Schwarzschild, the emission spectrum turns out to be colder and purer (less gray). From this, we sketch the evaporation history of this geometry and conclude that, under certain assumptions, instead of completely evaporating, the black hole naturally leads to a remnant, which provides a possible resolution to the information-loss issue.

Figures

Figures reproduced from arXiv: 2504.13050 by the authors.

**Figure 2.** FIG. 2. Regge-Wheeler potential with [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. We illustrate the splitting of the modes in the external region (shaded in gray) of the black hole. The ingoing modes [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Regge-Wheeler potential for [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. For [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6. The decrease of the black-hole size during the evaporation process (red) and its classical trajectory (gray) as a function [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: FIG. 7. Left: Regge-Wheeler potential with [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

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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

Evaporating cosmologically coupled black holes
astro-ph.CO 2026-07 conditional novelty 6.0 of 10

If a black hole's mass grows with cosmic expansion, Hawking evaporation is slowed or reversed, weakening gamma-ray bounds on primordial black holes.

Reference graph

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Pith tools

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

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These turning points are given by the roots r =ρrg that solve ¯Ωl(r(r∗);ω)|r=ρrg = 0

Then, to describe the full scattering process, the next step is to match the ψl at the turning points. These turning points are given by the roots r =ρrg that solve ¯Ωl(r(r∗);ω)|r=ρrg = 0. Due to the r0-dependent terms, this equation implies finding the roots of a fifth-order polynomial, instead of a fourth-order polynomial like in the Schwarzschild space...

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Large l analysis We follow the analysis in Ref. [71] and first determine the gray-body factor for large angular mode numberl. In this limit, our second-order approximation should be especially good because the higher l becomes, the more the height of the potential increases, which in turn allows for larger frequencies. As a first step, we express the posi...

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S-wave analysis To isolate the influence of λ, we now consider s-waves withl = 0. This waves are ignorant of the centrifugal barrier and experience the reduced potential v0(x) = 1− 1 x 2 +λ 2x3 − 3λ 2x4 . (46) In Fig. 2, we show the form of the potential V0(r/rg) [equivalentlyv0(x)] for different values of λ =r0/rg. We recall that for x< 3, a higher value...

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Large l analysis To understand the reflection coefficient for the almost-remnant state for large angular mode number l, we repeat the previous steps in Sec. V A 1. That is, we expand the location of the maximum (B2) for large l, xmax = 3 2 + 1 12l(l + 1)− ϵ 4l(l + 1) +O 1p (l(l + 1))3 ! , (54) which agrees exactly with the already found expression for the...

work page

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classical

S-wave analysis For s-waves in the almost-remnant scenario, we find that the experienced potential simplifies dramatically. For the exact remnant configuration, Ω 0(x,ν ) reduces to the simple form Ω0(x;ν) = 1− 1 x 2 3 2x3 −ν2, (59) which, by definition, has no dependence on λ. As argued above, we are interested in the approach to the remnant case. Theref...

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