arxiv: 2602.13694 · v1 · submitted 2026-02-14 · 🌀 gr-qc · hep-th

Recognition: 3 theorem links

· Lean Theorem

Spherically symmetric black holes in Gravity from Entropy and spontaneous emission

Udaykrishna Thattarampilly , Yunlong Zheng , Vishnu Kakkat

Authors on Pith no claims yet

Pith reviewed 2026-05-15 22:35 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black holesspherically symmetric solutionsgravity from entropyevaporation ratemodified gravityHawking radiationr^{-4} corrections

0 comments

The pith

Gravity from Entropy modifies Schwarzschild black holes with r^{-4} corrections and produces a constant background evaporation rate for large masses.

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

The paper derives static spherically symmetric black hole solutions inside the Gravity from Entropy framework and shows that the classical Schwarzschild geometry receives perturbative corrections that scale as r to the minus four. These corrected geometries remain compatible with present strong-field astrophysical observations. For time-dependent black holes the same framework generates a mass-loss law that approaches a nonzero constant value of negative beta over twenty-four when the black hole mass becomes very large, which the authors describe as an intrinsic entropic leakage of the vacuum. At intermediate mass scales the mass-loss rate recovers the familiar inverse-square dependence on mass that is normally associated with Hawking radiation, but here it emerges from a purely classical response of the modified background geometry.

Core claim

In the Gravity from Entropy framework the static spherically symmetric vacuum solutions are Schwarzschild geometries corrected by terms proportional to r to the minus four. Higher-order geometric stresses then drive a mass-evolution profile for dynamical black holes in which the evaporation rate approaches minus beta over twenty-four in the large-mass limit and recovers the standard Hawking scaling of mass-dot proportional to mass to the minus two at intermediate scales.

What carries the argument

the modified vacuum field equations of the Gravity from Entropy framework, whose higher-order geometric stresses generate both the r^{-4} corrections to the static geometry and the beta-dependent mass-loss law for dynamical black holes

Load-bearing premise

The Gravity from Entropy framework supplies the correct modified vacuum field equations whose solutions produce the stated r^{-4} corrections and the beta-dependent evaporation law.

What would settle it

An observation that the evaporation rate of sufficiently large black holes does not approach a nonzero constant independent of mass, or that the intermediate-mass scaling deviates from the M to the minus two law, would falsify the predicted mass-evolution profile.

Figures

Figures reproduced from arXiv: 2602.13694 by Udaykrishna Thattarampilly, Vishnu Kakkat, Yunlong Zheng.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Prediction for shadow diameter of Sagittarius A* [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

We investigate static and dynamical spherically symmetric black hole solutions within the Gravity from Entropy (GfE) framework. We derive and solve the modified vacuum field equations for a static, spherically symmetric spacetime, revealing that the classical Schwarzschild geometry receives perturbative corrections scaling as $r^{-4}$. We establish that the GfE framework is consistent with current strong-field astrophysical observations. Higher-order geometric stresses inherent to the GfE vacuum drive a consistent mass-evolution profile. In the limit of large black hole mass, the theory predicts a constant background evaporation rate $ -\beta/24$, suggesting an inherent "entropic leakage" of the vacuum. At intermediate scales, the framework replicates the standard Hawking radiation mass-loss law as $\dot{M} \propto M^{-2}$ through a purely classical response of the modified background.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives r^{-4} corrections to Schwarzschild in the GfE framework and claims a constant large-mass evaporation rate of -β/24 alongside recovered Hawking scaling, but the step linking static geometry to dynamical mass loss stays implicit.

read the letter

The paper's main claim is that the Gravity from Entropy framework produces a static spherical solution with Schwarzschild plus O(r^{-4}) corrections, and that the extra geometric stresses then drive black-hole mass loss at a constant rate -β/24 when the mass is large, switching to the usual M^{-2} scaling at intermediate masses. That dual-regime profile is the concrete new piece. The static part is worked out in enough detail to compare with strong-field observations, which is useful and keeps the modified geometry under control. The idea that higher-order terms can source an effective outward flux without invoking quantum fields is a clean way to embed an entropic effect inside a classical modified-gravity setting. The authors also keep the framework consistent with existing data, so the corrections do not immediately conflict with current tests. That is the part that earns credit. The soft spot is the dynamical step. The abstract states the mass-loss rates but does not display the intermediate equations that promote the perturbative metric to a time-dependent mass function or show how the stress tensor is contracted to give exactly -β/24 independent of M. Without those lines it is hard to judge whether the constant term follows from the field equations or is inserted by the choice of effective conservation law. The recovery of M^{-2} scaling is presented as automatic, yet any mismatch in the time coordinate or in the definition of the flux could change the functional form. β remains a free parameter, so the constant rate is not yet a sharp, falsifiable number. The work is aimed at people already following entropic or thermodynamic modifications of gravity. A reader who knows the earlier GfE papers will see the black-hole application as a natural extension. It is worth sending to a referee who can check the explicit field equations and the reduction to the mass-loss law; the claims are specific enough that a careful review would either confirm the derivation or force the authors to tighten the dynamical part.

Referee Report

3 major / 0 minor

Summary. The manuscript investigates static and dynamical spherically symmetric black hole solutions in the Gravity from Entropy (GfE) framework. It derives modified vacuum field equations yielding Schwarzschild geometry with perturbative r^{-4} corrections, asserts consistency with strong-field astrophysical observations, and claims that higher-order geometric stresses produce a mass-evolution profile with constant evaporation rate −β/24 for large black hole masses (suggesting entropic vacuum leakage) while recovering the Hawking scaling Ṁ ∝ M^{-2} at intermediate scales via classical response of the modified background.

Significance. If the derivations hold, the work would introduce a novel classical entropic mechanism for black hole evaporation independent of quantum fields, with the constant background rate implying inherent vacuum properties and the recovery of the standard law providing a consistency check. This could impact models of black hole thermodynamics and modified gravity, though its broader significance hinges on independent validation of the GfE framework and the role of the free parameter β.

major comments (3)

The explicit form of the GfE-modified vacuum field equations is not supplied, preventing verification of the claimed static spherically symmetric solution with r^{-4} corrections or the higher-order geometric stresses invoked for the dynamical mass loss.
No intermediate equations are provided linking the static metric perturbations (r^{-4} terms) to a time-dependent mass function or to the contraction of the effective stress tensor that yields the exact constant rate −β/24 (M-independent) for large M; this step is load-bearing for the central evaporation claim.
The replication of the Hawking Ṁ ∝ M^{-2} scaling at intermediate masses is asserted to arise from a 'purely classical response of the modified background,' but without the explicit effective conservation law or coordinate choice used to integrate the flux, the functional dependence cannot be confirmed and risks circularity with the choice of β.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript investigating spherically symmetric black holes in the Gravity from Entropy framework. We address each major comment point by point below and have revised the manuscript to improve clarity and completeness of the derivations.

read point-by-point responses

Referee: The explicit form of the GfE-modified vacuum field equations is not supplied, preventing verification of the claimed static spherically symmetric solution with r^{-4} corrections or the higher-order geometric stresses invoked for the dynamical mass loss.

Authors: We agree that the modified vacuum field equations should be presented more explicitly for verification. Although derived in Section 2, we will add the full set of GfE-modified Einstein equations in spherical symmetry to the revised manuscript, explicitly showing the perturbative r^{-4} corrections to the Schwarzschild metric and the resulting higher-order geometric stresses. revision: yes
Referee: No intermediate equations are provided linking the static metric perturbations (r^{-4} terms) to a time-dependent mass function or to the contraction of the effective stress tensor that yields the exact constant rate −β/24 (M-independent) for large M; this step is load-bearing for the central evaporation claim.

Authors: We acknowledge the need for explicit intermediate steps. In the revised version, we will insert a new subsection deriving the time-dependent mass function from the perturbed metric, including the contraction of the effective stress-energy tensor and the integration leading to the constant rate −β/24 for large M, thereby clarifying the entropic leakage mechanism. revision: yes
Referee: The replication of the Hawking Ṁ ∝ M^{-2} scaling at intermediate masses is asserted to arise from a 'purely classical response of the modified background,' but without the explicit effective conservation law or coordinate choice used to integrate the flux, the functional dependence cannot be confirmed and risks circularity with the choice of β.

Authors: We thank the referee for highlighting this point. The effective conservation law follows from the modified Bianchi identities in the GfE framework, and we employ Schwarzschild-like coordinates for the flux integration. We will explicitly state both the conservation equation and coordinate choice in the revision, demonstrating that the M^{-2} scaling arises classically without circular dependence on β. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The abstract and available excerpts describe deriving static spherically symmetric solutions from modified GfE vacuum equations, obtaining r^{-4} perturbative corrections to Schwarzschild, then asserting that higher-order geometric stresses produce a mass-loss rate of -β/24 for large M and recover the classical M^{-2} scaling at intermediate M. No explicit equations or steps are supplied in the given text that reduce the quoted evaporation rates to the input parameter β or the framework assumptions by construction. The GfE framework is introduced as an independent source of the field equations, and the dynamical claims are presented as consequences rather than tautological re-statements of the static ansatz. Therefore the derivation chain does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the Gravity from Entropy framework and the introduction of the parameter β; no independent evidence for either is supplied in the abstract.

free parameters (1)

β
Appears directly in the constant evaporation rate −β/24; its value is not derived from first principles within the abstract.

axioms (1)

domain assumption The Gravity from Entropy framework yields the correct modified vacuum field equations for static spherically symmetric spacetimes.
All reported solutions and evaporation profiles are obtained by solving these equations.

pith-pipeline@v0.9.0 · 5445 in / 1520 out tokens · 27564 ms · 2026-05-15T22:35:14.637248+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

the gravitational action is then identified with the Araki quantum relative entropy... L = -Tr log(˜G ˜g^{-1})... ˜G = ˜g - β/2 ˜R
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

a4 = -β M²/12, b4 = β M²/3... rh = rS + β/(48 rS)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

˙M ≈ -β/24 - β 0.17/M²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages · 23 internal anchors

[1]

Bekenstein, Bekenstein-Hawking entropy, Scholarpedia 3, 7375 (2008)

J. Bekenstein, Bekenstein-Hawking entropy, Scholarpedia 3, 7375 (2008)

work page 2008
[2]

J. D. Bekenstein, Generalized second law of thermody- namics in black hole physics, Phys. Rev. D9, 3292 (1974)

work page 1974
[3]

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]

The Holographic Principle

G. ’t Hooft, The Holographic principle: Opening lecture, Subnucl. Ser.37, 72 (2001), arXiv:hep-th/0003004

work page internal anchor Pith review Pith/arXiv arXiv 2001
[5]

The World as a Hologram

L. Susskind, The World as a hologram, J. Math. Phys. 36, 6377 (1995), arXiv:hep-th/9409089

work page internal anchor Pith review Pith/arXiv arXiv 1995
[6]

Entanglement Renormalization and Holography

B. Swingle, Entanglement Renormalization and Hologra- phy, Phys. Rev. D86, 065007 (2012), arXiv:0905.1317 [cond-mat.str-el]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[7]

Aspects of Holographic Entanglement Entropy

S. Ryu and T. Takayanagi, Aspects of Holographic Entan- glement Entropy, JHEP08, 045, arXiv:hep-th/0605073

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Holographic Entanglement Entropy: An Overview

T. Nishioka, S. Ryu, and T. Takayanagi, Holographic En- tanglement Entropy: An Overview, J. Phys. A42, 504008 (2009), arXiv:0905.0932 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[9]

Quantum corrections to holographic entanglement entropy

T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11, 074, arXiv:1307.2892 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv
[10]

Notes on Some Entanglement Properties of Quantum Field Theory

E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys.90, 045003 (2018), arXiv:1803.04993 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[11]

Sorce, Notes on the type classification of von Neu- mann algebras, Rev

J. Sorce, Notes on the type classification of von Neu- mann algebras, Rev. Math. Phys.36, 2430002 (2024), arXiv:2302.01958 [hep-th]

work page arXiv 2024
[12]

Ben-Dayan, The quantum focusing conjecture and the improved energy condition, JHEP02, 132, arXiv:2310.14396 [hep-th]

I. Ben-Dayan, The quantum focusing conjecture and the improved energy condition, JHEP02, 132, arXiv:2310.14396 [hep-th]

work page arXiv
[13]

Thermodynamical Aspects of Gravity: New insights

T. Padmanabhan, Thermodynamical Aspects of Grav- ity: New insights, Rept. Prog. Phys.73, 046901 (2010), arXiv:0911.5004 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[14]

Ambjørn, J

J. Ambjørn, J. Jurkiewicz, and R. Loll, Emergence of a 4d world from causal quantum gravity, Phys. Rev. Lett. 93, 131301 (2004). [15]Approaches to Quantum Gravity: Toward a New Under- standing of Space, Time and Matter(Cambridge Univer- sity Press, 2009)

work page 2004
[15]

Black holes, gravitational waves and fundamental physics: a roadmap

L. Baracket al., Black holes, gravitational waves and fun- damental physics: a roadmap, Class. Quant. Grav.36, 143001 (2019), arXiv:1806.05195 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[16]

Thermodynamics of Spacetime: The Einstein Equation of State

T. Jacobson, Thermodynamics of space-time: The Ein- stein equation of state, Phys. Rev. Lett.75, 1260 (1995), arXiv:gr-qc/9504004

work page internal anchor Pith review Pith/arXiv arXiv 1995
[17]

E. P. Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP04, 029, arXiv:1001.0785 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv
[18]

Bianconi, Gravity from entropy, Phys

G. Bianconi, Gravity from entropy, Phys. Rev. D111, 066001 (2025), arXiv:2408.14391 [gr-qc]

work page arXiv 2025
[19]

Araki, Relative Entropy of States of Von Neumann Algebras, Publ

H. Araki, Relative Entropy of States of Von Neumann Algebras, Publ. Res. Inst. Math. Sci. Kyoto1976, 809 (1976)

work page 1976
[20]

H. Araki, Title pages, inMathematical Theory of Quantum Fields(Oxford University Press, 1999) https://academic.oup.com/book/0/chapter/422692897/chapter- pdf/52587741/isbn-9780198517733-front-matter-part- 1.pdf

work page arXiv 1999
[21]

Ohya and D

M. Ohya and D. Petz,Quantum Entropy and Its Use, Texts and monographs in physics (Springer-Verlag, 1993)

work page 1993
[22]

The Role of Relative Entropy in Quantum Information Theory

V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys.74, 197 (2002), arXiv:quant-ph/0102094

work page internal anchor Pith review Pith/arXiv arXiv 2002
[23]

Rosen, A bi-metric theory of gravitation, General Rel- ativity and Gravitation4, 435 (1973)

N. Rosen, A bi-metric theory of gravitation, General Rel- ativity and Gravitation4, 435 (1973)

work page 1973
[24]

A Bi-Metric Theory with Exchange Symmetry

S. Hossenfelder, A Bi-Metric Theory with Exchange Sym- metry, Phys. Rev. D78, 044015 (2008), arXiv:0807.2838 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[25]

Thattarampilly and Y

U. Thattarampilly and Y. Zheng, Inflation from entropy, Eur. Phys. J. C85, 1433 (2025), arXiv:2509.23987 [gr-qc]

work page arXiv 2025
[26]

Bianconi, The quantum relative entropy of the Schwarzschild black-hole and the area law, Entropy27, 266 (2025), arXiv:2501.09491 [gr-qc]

G. Bianconi, The quantum relative entropy of the Schwarzschild black-hole and the area law, Entropy27, 266 (2025), arXiv:2501.09491 [gr-qc]

work page arXiv 2025
[27]

J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50, 3874 (1994), arXiv:gr-qc/9405057

work page internal anchor Pith review Pith/arXiv arXiv 1994
[28]

C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel.7, 5 (2004), arXiv:gr-qc/0311082

work page internal anchor Pith review Pith/arXiv arXiv 2004
[29]

Abuteret al.(GRAVITY), Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole, Astron

R. Abuteret al.(GRAVITY), Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole, Astron. Astro- phys.636, L5 (2020), arXiv:2004.07187 [astro-ph.GA]

work page arXiv 2020
[30]

First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

K. Akiyamaet al.(Event Horizon Telescope), First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way, Astrophys. J. Lett.930, L12 (2022), arXiv:2311.08680 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2022
[31]

The Thermodynamics of the Gravity from Entropy Theory

G. Bianconi, The Thermodynamics of the Gravity from Entropy Theory, (2025), arXiv:2510.22545 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[32]

Black hole mass decreasing due to phantom energy accretion

E. Babichev, V. Dokuchaev, and Y. Eroshenko, Black hole mass decreasing due to phantom energy accretion, Phys. Rev. Lett.93, 021102 (2004), arXiv:gr-qc/0402089

work page internal anchor Pith review Pith/arXiv arXiv 2004
[33]

C. Gao, X. Chen, V. Faraoni, and Y.-G. Shen, Does the mass of a black hole decrease due to the accretion of phantom energy, Phys. Rev. D78, 024008 (2008), arXiv:0802.1298 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[34]

Sultana and C

J. Sultana and C. C. Dyer, Cosmological black holes: A black hole in the Einstein-de Sitter universe, Gen. Rel. Grav.37, 1347 (2005)

work page 2005
[35]

Does cosmological expansion affect local physics?

D. Giulini, Does cosmological expansion affect local physics?, Stud. Hist. Phil. Sci. B46, 24 (2014), arXiv:1306.0374 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[36]

Accelerating Black Holes

R. Gregory, Accelerating Black Holes, J. Phys. Conf. Ser. 942, 012002 (2017), arXiv:1712.04992 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[37]

J. D. Barrow, Varying G and other constants, NATO Sci. Ser. C511, 269 (1998), arXiv:gr-qc/9711084

work page internal anchor Pith review Pith/arXiv arXiv 1998
[38]

Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario

A. Cisterna and C. Erices, Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario, Phys. Rev. D89, 084038 (2014), arXiv:1401.4479 [gr-qc]. 10 Appendix A: Modified Einstein equations for static spherically symmetric space time By substituting the metric anzats in equation (10) we obtain the modified e...

work page internal anchor Pith review Pith/arXiv arXiv 2014
[39]

Contribution of lapse function to the modified equations are of higher order and ignored henceforth

Reintroducing Lapse function to Einstein equations In general the lapse function is nonzero and can contribute to the Einstein equations at first order in coupling. Contribution of lapse function to the modified equations are of higher order and ignored henceforth. We perform this two stage derivation of equations of motion, since deriving the modified eq...

work page