Black Hole Remnants

Ali H. Chamseddine; Tobias B. Russ; Viatcheslav Mukhanov

REVIEW 3 major objections 5 minor 2 cited by

Black Hole Remnants

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that in asymptotically free mimetic gravity with limiting curvature, black holes are nonsingular and evaporation ends in stable remnants with vanishing Hawking temperature.

desk verdict An exact nonsingular black hole with a zero-temperature remnant endpoint in a tuned mimetic-gravity model, but the stability claim is asserted, not demonstrated. read the letter →

arxiv 1908.03498 v1 pith:HU7DH4UB submitted 2019-08-09 hep-th astro-ph.COgr-qc

Ali H. Chamseddine , Viatcheslav Mukhanov , Tobias B. Russ This is my paper · ORCID

classification hep-thastro-ph.COgr-qc PACS 04.20.-q04.20.Dw04.70.-s04.70.Dy

keywords blackholeremnantsmimeticgravitylimitingcurvatureasymptoticfreedomHawkingevaporationnon-singularholesdeSitterinteriorinformationparadox

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 proposes a modified theory of gravity, mimetic gravity with a limiting curvature at which the gravitational constant runs to zero, and claims that black holes in this theory do not form singularities. Instead, the interior of a black hole is replaced by a static patch of de Sitter space. Because the Killing horizons merge at a minimal mass, the surface gravity and hence the Hawking temperature drop to zero, so evaporation halts and leaves a stable remnant. The authors derive an explicit exact solution for the geometry and compute its thermodynamics, including a modified first law. A sympathetic reader would care because this offers a concrete, parameter-controlled picture of the final state of black hole evaporation and a possible route around the information paradox.

What carries the argument

The central object is the mimetic scalar $\varphi$ with constraint $\varphi_{,\alpha}\varphi^{,\alpha}=1$, whose value in synchronous coordinates makes $\Box\varphi$ equal to the trace of extrinsic curvature $\kappa$. The theory's coupling $f(\kappa)$ is the inverse running gravitational constant, and asymptotic freedom means $f\to\infty$ as $\kappa$ approaches the limiting curvature $\kappa_0$; the action includes an $f(\Box\varphi)R$ term plus a spatial-curvature combination that removes higher derivatives. The exact black-hole solution is parametrized by $\tilde\kappa=\kappa/\kappa_0$, through the implicit relation (17) and the metric functions (18)--(19), with $a(\tilde\kappa)$ peaking at $\tilde\kappa_*=-1/\sqrt{5}$. The key identities are $\dot b/b-\dot a/a=3M/(fab^2)$, the first integral that puts the mass into the geometry, and $g_s=-\dot a(x_\pm)$, the surface gravity at Killing horizons, whose vanishing at $M=M_{\min}$ gives $T_H=0$.

What would settle it

Compute the particle-production flux from a collapsing background in the exact metric of equations (17)--(19), for example by a Bogoliubov coefficient calculation, and check whether the emitted flux vanishes when the outer horizon has zero surface gravity. If $T_H$ or the flux stays nonzero at $M=M_{\min}$, or if an exact solution with $M<M_{\min}$ still contains a Killing horizon, the zero-temperature remnant claim is falsified.

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Extended reading notes

Core claim

The central claim is that in asymptotically free mimetic gravity with limiting curvature, the Schwarzschild singularity is resolved: the metric in Lemaitre coordinates smoothly interpolates between the Schwarzschild exterior and an interior de Sitter patch, with a maximum of the function $a(x)$ separating outer and inner horizons. For masses above a threshold, two horizons exist; when $a$ reaches exactly one at its maximum, the horizons coincide, defining the minimal mass $M_{\min}=5^{5/2}/(18\kappa_0)$. At this mass the surface gravity $g_s=-\dot a(x_\pm)$ vanishes, so the Hawking temperature $T_H=g_s/2\pi$ vanishes, and the remnant is stable. The paper backs this with an exact solution given by equations (17)--(19) and with thermodynamic identities: the large-mass limit reproduces $T_H=1/(8\pi M)$, while near the minimal mass $T_H\propto\sqrt{M-M_{\min}}$, so evaporation asymptotically approaches the remnant and stops.

Load-bearing premise

The argument assumes that the standard semiclassical formula for Hawking temperature, $T_H=$ surface gravity divided by $2\pi$, remains valid in this modified theory where the gravitational constant runs with curvature; if particle production is governed by different couplings, the claimed zero-temperature remnant endpoint could fail.

Editorial extensions

If this is right

Black holes with mass above $M_{\min}$ are nonsingular: their interiors become de Sitter-like patches at limiting curvature rather than curvature singularities.
Hawking evaporation ends at a stable remnant of mass $M_{\min}$ with zero Hawking temperature, not at a singular endpoint or total disappearance.
The minimal remnants have near-horizon geometry similar to extremal Reissner-Nordstr\"om black holes but carry no charge and no singularity, so they are stable.
The remnants can store an unlimited amount of information in the absolute future of external observers, suggesting one possible resolution of the information-loss paradox.
Stable remnants of this kind could serve as dark matter candidates.

Reading between the lines

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

A testable extension is to compute the actual particle-production flux in the exact metric of equations (17)--(19) beyond the surface-gravity approximation; if thermal emission does not vanish when $g_s=0$, the zero-temperature remnant endpoint would fail even if the classical geometry is correct.
The remnant mass is set by the free parameter $\kappa_0$, not by the Planck mass; if $\kappa_0$ is sub-Planckian, remnants are super-Planckian, which would suppress metric quantum fluctuations for the remnant itself—an implication the paper hints at but does not develop quantitatively.
The modified first law $G(\tilde\kappa_+)dM=T_H dS$ suggests that entropy accounting for external observers differs from the standard Bekenstein-Hawking bookkeeping; an extension would be to track where information is stored from an infalling versus asymptotic observer's point of view.
Observationally, primordial black holes in this theory would stop evaporating near $M_{\min}$ and persist as stable objects, so a search for compact dark matter around the scale set by $\kappa_0$ could test the scenario.

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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

I read the Chamseddine-Mukhanov-Russ black hole remnant paper. The headline is simple: the construction works as an exact solution, and the thermodynamics of the endpoint is a genuine result. The claim that these remnants are stable is not supported by anything in the paper.

The exact solution (17)-(19) is the real content. They take the limiting-curvature mechanism from their earlier Kasner work, adapt it to Schwarzschild, and produce a spatially flat interpolating solution between Schwarzschild at infinity and de Sitter in the interior. The matching is done explicitly, they verify a^2 = bdot^2, and the minimal mass Mmin = 5^(5/2)/(18 kappa0) follows cleanly from a(kappa*)=1. The temperature curve (26)-(29) is also clean: it reduces to the standard 1/(8 pi M) at large M, peaks near 1.32 Mmin, and goes to zero at Mmin. That is a concrete, checkable claim about the model.

Three soft spots. First, stability. The conclusion says remnants are stable because they have no charge. That is a non sequitur: vanishing surface gravity fixes the tree-level Hawking temperature but says nothing about metric or mimetic-field perturbations. The paper has no perturbative analysis, and the de Sitter interior may have unstable or tunneling modes. This is the load-bearing gap. Second, genericity. The abstract claims black holes generically have remnants, but the exact solution uses one specific f and a Lambda chosen ad hoc so that Eq. (16) holds. That is engineering, not genericity. It is a valid counterexample to singular collapse within the model, but not a general proof. Third, the Hawking temperature formula is carried over from general relativity without deriving it in this modified theory. I think this is probably safe, because surface gravity on a Killing horizon is a kinematic statement, but a referee should ask for the Euclidean periodic argument in the modified action. The information-paradox remarks are speculative but clearly flagged as suggestions.

On the citation pattern: heavy self-citation is not a flaw here, since the prior paper is the direct predecessor and the new part is the black hole application. What is missing is engagement with the broader remnant literature, but that is secondary.

Who is this for? People working on nonsingular black holes and remnant scenarios, and anyone interested in concrete modified-gravity constructions. It deserves a serious referee, but the referee should push for either a perturbation analysis of the extremal-like horizon or a rewrite that softens 'stable' and 'generically' until such analysis exists.

Recommendation: send it to peer review with the stability question front and center. If that gap is filled, this becomes a strong paper; as it stands, it is an existence proof with an overstated conclusion.

Referee Report

3 major / 5 minor

Summary. The paper constructs an exact, spatially flat black hole solution in a mimetic gravity theory with a curvature-dependent gravitational constant f(□φ) and cosmological term Λ(□φ). The solution interpolates between the Schwarzschild metric at large radius and a de Sitter core at high curvature, removing the classical singularity. For masses below a critical value M_min = 5^{5/2}/(18κ_0), the solution has no Killing horizon; the limiting minimal black hole has a single horizon with vanishing surface gravity. The authors identify the Hawking temperature with the surface gravity, show that T_H(M) vanishes as M → M_min, and conclude that evaporation leaves stable remnants of mass M_min, possibly contributing to dark matter and the information paradox.

Significance. If established, the result would be a concrete classical modified-gravity realization of singularity resolution and remnant formation, with the virtue of being an exact solution whose asymptotics are explicitly checked. The paper provides closed-form expressions for the metric, M_min, T_H(M), and the near-horizon form of the extremal-like solution, and it connects to earlier work on asymptotically free mimetic gravity. These are useful and nontrivial. However, the central physical conclusion—stability of the remnant and the zero-temperature endpoint—rests on assumptions that are not derived, so the significance is conditional on a perturbative stability analysis and on a justification of the thermodynamic identification in this modified theory.

major comments (3)

[Conclusions; Eqs. (25)–(29)] The stability claim for the remnant is not supported. The paper asserts in the Conclusions that the remnants are stable 'because they have no charge', and the preceding text infers stability from vanishing Hawking temperature. Neither implies dynamical stability: an uncharged solution can have unstable gravitational or scalar perturbations, and the near-horizon metric (25) is reminiscent of extremal Reissner-Nordström, which is known to exhibit horizon instabilities in certain perturbation sectors. A perturbative analysis of the exact solution (17)–(19), including the mimetic field and constraint sector, is required to justify the statement that evaporating black holes end as stable remnants. Without such an analysis, the abstract's central claim is an assumption.
[Eqs. (14) and (26); Black hole thermodynamics] The Hawking temperature is taken to be T_H = g_s/2π using the surface gravity (14) of the Killing horizon, with no derivation in the context of the modified action (7)–(9). In Einstein gravity this relation follows from Euclidean periodicity or from a field-theoretic computation near the horizon; here the non-minimal coupling f(□φ)R and the running gravitational constant can alter the kinetic terms and the stress-energy of perturbations, so the standard derivation does not automatically apply. Because the central conclusion T_H = 0 at M = M_min follows directly from (26), this missing justification is load-bearing. The authors should either provide a derivation of the temperature for this theory or state clearly that the formula is an assumption.
[Exact Solution; Eq. (16)] The genericity claim is not supported by the construction. The solution is obtained by choosing f(κ) in (15) and then selecting Λ(κ) so that the square root of Eq. (12) takes the specific form (16). This engineered choice ensures that a(κ̃) has a maximum at κ̃ = −1/√5 and that the limiting mass (20) exists. The Introduction and the paragraph after Eq. (14) claim that limiting curvature and asymptotic freedom 'generically' lead to stable remnants, but only one tuned example is exhibited. A general argument, or at least a statement of the class of (f, Λ) that produces the required maximum and merger of horizons, is needed. Otherwise the conclusion should be restricted to the explicit model.

minor comments (5)

[Eq. (30)] The modified first law G(κ̃_+) dM = T_H dS is introduced with the phrase 'straightforward to verify', but the verification is not shown. Since the law is not used in the main argument, this is a presentation issue, but the derivation should either be sketched or deferred explicitly.
[Conclusions and reference [1]] The assertion that the maximal extension shows the remnant can store an unlimited amount of information is used to argue for a resolution of the information paradox, but it relies on the 'forthcoming publication' [1]. This is a missing support for a substantive physical claim; the authors should either outline the argument or present it as conjectural.
[Introduction] The phrase 'these remnants have vanishing Hawking temperature and, by the arguments shown in [8], metric quantum fluctuations never become relevant for them' is too terse; a reader cannot tell which argument in [8] is being invoked, and the connection is not elaborated anywhere in the paper.
[Exact Solution; after Eq. (20)] The term 'solitonic-like objects' for solutions with M < M_min is suggestive but undefined. Since these objects have no horizon and approach de Sitter at the center, the authors should specify whether they are asymptotically flat everywhere and whether they are relevant to the evaporation endpoint.
[General] There are several typographical issues: the author affiliation contains 'Lebano n' and 'Theresienstr.', and the PACS numbers are malformed ('0.4.20-q' instead of '04.20.-q'). These should be corrected in a final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the remnant result is derived from a transparently constructed action, with self-citations only motivational.

full rationale

The paper's derivation chain is not circular. The action (7)-(9) is proposed; f(tilde_kappa) in (15) and Lambda(kappa) are chosen explicitly, with the authors writing: "Let us take f... and chose Lambda in such a way that the square root of the branch kappa<0 of (12) becomes (16)." This is an open constructive choice of a concrete model realizing the stated limiting-curvature/asymptotic-freedom mechanism, not a hidden fit to the target conclusion. The remnant endpoint is then a derived consequence: equations (11) and (16) lead to the solution (17)-(19); a(tilde_kappa) has a maximum; horizons exist only for M >= M_min in (20); at M = M_min the two horizons merge and (14) gives g_s = 0, so (26) gives T_H = 0. Nothing in these equations defines 'remnant' or 'T_H = 0' as an input. The self-citations to [8] motivate the action form and the Kasner mechanism, but the exact black hole solution and its thermodynamics are established within this paper, so the self-citation is not load-bearing. The assertion that the remnant is stable 'because they have no charge' is unsupported and would be a substantive gap, but an omitted stability proof is a correctness concern, not a circular reduction of the derivation. Therefore no circular step meeting the quote-and-reduction standard is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the postulation of asymptotic freedom at a limiting curvature, an engineered choice of f and Lambda, and the unproved transfer of standard black hole thermodynamics to the modified theory. The only genuinely new object is the stable remnant, and its properties are tied to the free parameter kappa_0.

free parameters (3)

kappa_0 (limiting curvature) = not specified
Free parameter setting the curvature scale where the gravitational constant vanishes; determines the remnant mass Mmin = 5^{5/2}/(18*kappa_0).
f(kappa~) functional form = Eq. (15)
Chosen ad hoc to yield a simple exact solution; no independent physical constraint is supplied.
Lambda(kappa) functional form = chosen so that Eq. (16) holds
Reverse-engineered to force the assumed square-root branch of the field equations; the de Sitter interior and remnant behavior are consequences of this choice.

assumptions (4)

domain assumption There exists a limiting curvature kappa_0 at which the gravitational constant vanishes (asymptotic freedom).
Postulated in the Introduction and encoded in f(kappa) diverging at kappa_0; not derived from a more fundamental theory.
ad hoc to paper The action depends on curvature only through Box-phi as the unique measure avoiding higher time derivatives.
Asserted in the text before Eq. (8) without proof; it motivates the action but is not demonstrated.
ad hoc to paper The Hawking temperature is TH = g_s/2pi with surface gravity g_s, valid in this modified gravity with non-minimal coupling.
Used to conclude TH=0 for the minimal remnant (Eqs. (14) and (26)); no derivation is supplied for the modified theory.
ad hoc to paper Stability of the remnant follows from being uncharged and having zero temperature.
The Conclusions assert stability because the remnants have no charge and zero temperature; no perturbation analysis is presented.

invented entities (1)

Stable black hole remnant (minimal mass black hole)
purpose: Final product of Hawking evaporation and proposed dark matter candidate.
The mass scale depends on the free parameter kappa_0, and the zero temperature follows from the chosen f and Lambda; no independent observable signature is predicted.

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Cite this review

Pith. "Pith review of Black Hole Remnants." pith.science (2026). https://pith.science/paper/HU7DH4UB

@misc{pith2026190803498,
  author       = {Pith},
  title        = {Pith review of: Black Hole Remnants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HU7DH4UB}},
  note         = {Machine review of arXiv:1908.03498}
}

read the original abstract

We show that in asymptotically free mimetic gravity with limiting curvature the black hole singularity can be resolved and replaced by a static patch of de Sitter space. As a result of Hawking evaporation of these non-singular black holes, there remain stable remnants with vanishing Hawking temperature.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Two-field mimetic gravity revisited and Hamiltonian analysis
gr-qc 2019-09 conditional novelty 6.0 of 10

A two-field mimetic gravity model has two scalar degrees of freedom, not one, and the extra entropy mode is a ghost when the fields have opposite-sign kinetic terms.
Constructing regular black holes from multi-polytropic equations of state
gr-qc 2025-02 reject novelty 5.0 of 10

Two regular black hole metrics are derived from anisotropic multi-polytropic TOV equations; one reduces to the known Hayward geometry and the other is new, with repulsive-gravity regions outside the horizon for partic...

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 2 Pith papers

[1]

The more general and detailed calculations will be pre- sented in a forthcoming publication by the authors

work page
[2]

Mimetic dark matter

Ali Chamseddine and Viatcheslav Mukhanov. Mimetic dark matter. JHEP, 1311:135, 2013

work page 2013
[3]

Resolving cosmological singularities

Ali Chamseddine and Viatcheslav Mukhanov. Resolving cosmological singularities. JCAP, 1703:009, 2016

work page 2016
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Cosmology with mimetic matter

Ali Chamseddine, Viatcheslav Mukhanov, and Alexan- der Vikman. Cosmology with mimetic matter. JCAP, 1406:017, 2014

work page 2014
[5]

static coordinates

The horizons, which occur at a = 1 , exist only if a (˜κ ∗) ≥ 1. This condition is satisﬁed only if M ≥ Mmin = 55/ 2 18κ 0 . (20) Otherwise, no horizon exists and the solution (17), (18), (19) describes solitonic-like objects whose metric is com- pletely static and approaches the de Sitter metric in the center. For black holes with the minimum mass Mmin, ...

work page
[6]

Chamseddine and Viatcheslav Mukhanov

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Chamseddine and Viatcheslav Mukhanov

Ali H. Chamseddine and Viatcheslav Mukhanov. Ghost free mimetic massive gravity. JHEP, 1806(6):060, Jun 2018

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Ali H. Chamseddine and Viatcheslav Mukhanov. Mimetic massive gravity: beyond linear approximation. JHEP, 1806(6):062, Jun 2018

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

The more general and detailed calculations will be pre- sented in a forthcoming publication by the authors

work page

[2] [2]

Mimetic dark matter

Ali Chamseddine and Viatcheslav Mukhanov. Mimetic dark matter. JHEP, 1311:135, 2013

work page 2013

[3] [3]

Resolving cosmological singularities

Ali Chamseddine and Viatcheslav Mukhanov. Resolving cosmological singularities. JCAP, 1703:009, 2016

work page 2016

[4] [4]

Cosmology with mimetic matter

Ali Chamseddine, Viatcheslav Mukhanov, and Alexan- der Vikman. Cosmology with mimetic matter. JCAP, 1406:017, 2014

work page 2014

[5] [5]

static coordinates

The horizons, which occur at a = 1 , exist only if a (˜κ ∗) ≥ 1. This condition is satisﬁed only if M ≥ Mmin = 55/ 2 18κ 0 . (20) Otherwise, no horizon exists and the solution (17), (18), (19) describes solitonic-like objects whose metric is com- pletely static and approaches the de Sitter metric in the center. For black holes with the minimum mass Mmin, ...

work page

[6] [6]

Chamseddine and Viatcheslav Mukhanov

Ali H. Chamseddine and Viatcheslav Mukhanov. Non- singular black hole. Eur. Phys. J. C , 77(3):183, 2017

work page 2017

[7] [7]

Chamseddine and Viatcheslav Mukhanov

Ali H. Chamseddine and Viatcheslav Mukhanov. Ghost free mimetic massive gravity. JHEP, 1806(6):060, Jun 2018

work page 2018

[8] [8]

Chamseddine and Viatcheslav Mukhanov

Ali H. Chamseddine and Viatcheslav Mukhanov. Mimetic massive gravity: beyond linear approximation. JHEP, 1806(6):062, Jun 2018

work page 2018