pith. sign in

arxiv: 2606.21729 · v1 · pith:6Y7WQW6Qnew · submitted 2026-06-19 · 🌌 astro-ph.GA

A new solution for the mass growth of Black Holes consistent with thermodynamics

Mar\'ia F. La Rotta Wilches , Roger Coziol This is my paper

Pith reviewed 2026-06-26 13:25 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords black hole mass growthaccretion rateradiative efficiencyEddington ratiothermodynamic state functionno-hair theoremsupermassive black holeshigh redshift
0
0 comments X

The pith

Optimizing the black hole accretion rate produces a universal growth factor γ(t) that makes mass a thermodynamic state function.

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

The paper derives a new solution for black hole mass growth in which the factor γ(t) proportional to accretion rate over mass changes over time as radiative efficiency and Eddington ratio adjust to the accretion flow. By optimizing the accretion rate the resulting γ(t) becomes independent of redshift and applies to any black hole. A sympathetic reader would care because this framing reconciles the rapid early growth of supermassive black holes seen at high redshift with the thermodynamic laws that govern black hole mechanics.

Core claim

By allowing radiative efficiency ε and Eddington ratio λ_Edd to vary with accretion flow and then optimizing the accretion rate Ṁ_BH, the growth mass factor γ(t) becomes universal, applying to any black hole at any redshift. This indicates that black hole mass is a thermodynamic state function, in agreement with the no-hair theorem and the four laws of black hole mechanics.

What carries the argument

The time-dependent growth mass factor γ(t) ∝ Ṁ_BH/M_BH obtained by optimizing the accretion rate after letting radiative efficiency and Eddington ratio respond to changes in accretion flow.

If this is right

  • The growth solution applies equally to black holes at any redshift without additional parameters.
  • Black hole mass behaves as a thermodynamic state function rather than depending on formation history.
  • The result is consistent with the no-hair theorem.
  • The result satisfies the four laws of black hole mechanics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same optimization procedure might generate testable predictions for black hole mass distributions in high-redshift galaxies observed by JWST.
  • If γ(t) is universal, growth models could be linked more directly to thermodynamic identities without separate formation channels for different epochs.
  • The approach invites direct comparison of the derived γ(t) against measured accretion rates in local and distant active galactic nuclei.

Load-bearing premise

That varying radiative efficiency and Eddington ratio with accretion flow and then optimizing the accretion rate produces a growth factor that is truly universal without hidden redshift-dependent parameters or selection effects.

What would settle it

Observation of black hole masses at different redshifts whose growth histories cannot be described by a single optimized γ(t) function without introducing extra redshift-dependent terms.

Figures

Figures reproduced from arXiv: 2606.21729 by Mar\'ia F. La Rotta Wilches, Roger Coziol.

Figure 1
Figure 1. Figure 1: Optimized growth functions for five SMBHs at high z listed by Aggarwal (2025). Panels (a) and (b) show, respectively, their masses at the time of observation, MBH, and accretion rates, M˙ BH, as functions of the cosmic time. Panel (c) shows the growth efficiency curve describing the optimized evolutionary path for each SMBH. The vertical lines in (c) locate the critical mass, Mc, which is the mass when the… view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of optimization parameters of the black hole growth for the whole sample of high-redshift galaxies separated in five redshift bins ∆z = 0.5 from z = 5.5 (left) to z = 8.0 (right). Optimal time, τ , Radiative efficiency, ϵ, mass of the seed M0, optimal mass, MBH, maximal accretion rate, M˙ max and critical mass Mc, The dark bars on each plot show the median of the distributions. Each BH has its… view at source ↗
Figure 3
Figure 3. Figure 3: Mass growth efficiency curve (MGEC) for SMBHs. Panel a: The time scale was calibrated based on the observations, where τ is the optimal time. Panel b: This figure is similar as Fig.4 in Arnoldi et al. (2025) replacing the temperatures by the masses. The mass scale is relative to the mass at the optimal time. The 107 SMBHs (identified in the label) trace exactly the same curve. Finally in Figure 2c the opti… view at source ↗
read the original abstract

The James Webb Space Telescope (JWST) has recently revealed evidence of supermassive black holes (SMBHs) forming in the cores of high redshift galaxies that grew in mass extremely rapidly during the first billion years of the universe. In this study we present a new solution where the growth mass factor of BHs, $\gamma(t) \propto \dot M_{BH}/M_{BH}$, varies with time, following the changes of radiative efficiency, $\epsilon$, and Eddington ratio, $\lambda_{Edd}$, caused by the variation in accretion flow. By optimizing the accretion rate, $\dot M_{BH}$, we obtained a solution that is universal, that is, which applies to any BH at any redshift. This suggests that the mass of the BH is a thermodynamics state function, in good agreement with the no hair theorem and the four laws of mechanics of BHs.

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.

Referee Report

3 major / 1 minor

Summary. The paper claims that by allowing the radiative efficiency ε and Eddington ratio λ_Edd to vary with accretion flow and then optimizing the black-hole accretion rate Ṁ_BH, a universal growth factor γ(t) ≡ Ṁ_BH/M_BH is obtained that applies to any black hole at any redshift. This is presented as evidence that black-hole mass is a thermodynamic state function, consistent with the no-hair theorem and the four laws of black-hole mechanics, motivated by JWST observations of rapid high-redshift SMBH growth.

Significance. If the optimization step can be shown to yield a strictly universal γ(t) free of hidden redshift-dependent parameters or circular enforcement of universality, the result would supply a thermodynamically grounded, parameter-light description of black-hole mass assembly that unifies observations across cosmic time.

major comments (3)
  1. [Abstract] Abstract: the claim that optimizing Ṁ_BH after varying ε and λ_Edd produces a γ(t) that is strictly universal (identical functional form for any BH at any z) is not supported by an explicit derivation showing that the procedure introduces no implicit z-dependent boundary conditions or selection effects from the accretion-flow models.
  2. [Abstract] Abstract: the thermodynamic-state-function interpretation is asserted once γ(t) is obtained, yet no derivation connects the optimized γ(t) to the first, second, or third laws of black-hole mechanics, nor demonstrates that mass satisfies the state-function property required by those laws.
  3. [Abstract] Abstract: the optimization is described as yielding universality, but the manuscript provides no explicit functional form for the resulting γ(t), no error-propagation analysis, and no quantitative comparison against redshift-binned observational samples that would test whether the claimed universality survives selection biases.
minor comments (1)
  1. [Abstract] The proportionality symbol in γ(t) ∝ Ṁ_BH/M_BH is introduced without stating the exact functional dependence that emerges after optimization.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive report and the recommendation for major revision. We address each major comment below, agreeing that additional explicit derivations and analyses are needed to support the claims made in the abstract. Revisions will be made to strengthen these aspects without altering the core results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that optimizing Ṁ_BH after varying ε and λ_Edd produces a γ(t) that is strictly universal (identical functional form for any BH at any z) is not supported by an explicit derivation showing that the procedure introduces no implicit z-dependent boundary conditions or selection effects from the accretion-flow models.

    Authors: We agree that the abstract does not contain the full derivation. The manuscript obtains universality through optimization of Ṁ_BH after allowing ε and λ_Edd to vary with accretion flow properties, with the resulting γ(t) independent of specific redshift inputs. To address the concern directly, the revised version will include an expanded derivation (new appendix) that explicitly demonstrates cancellation of any z-dependent terms and confirms no hidden boundary conditions or selection effects from the standard accretion models employed. revision: yes

  2. Referee: [Abstract] Abstract: the thermodynamic-state-function interpretation is asserted once γ(t) is obtained, yet no derivation connects the optimized γ(t) to the first, second, or third laws of black-hole mechanics, nor demonstrates that mass satisfies the state-function property required by those laws.

    Authors: The current manuscript asserts the state-function interpretation from the universality of γ(t) and its consistency with the no-hair theorem, but does not provide the requested explicit connections. We will add a dedicated subsection in the revision that derives the link: showing how the optimized γ(t) satisfies path-independence (state function), connects to the first law via energy balance in the BH mechanics, the second law via the area theorem, and the third law via limiting behavior, all while remaining consistent with the four laws. revision: yes

  3. Referee: [Abstract] Abstract: the optimization is described as yielding universality, but the manuscript provides no explicit functional form for the resulting γ(t), no error-propagation analysis, and no quantitative comparison against redshift-binned observational samples that would test whether the claimed universality survives selection biases.

    Authors: We acknowledge that the manuscript does not present an explicit functional form for γ(t), nor the requested error-propagation analysis or quantitative observational tests. The revised manuscript will include the derived functional form of γ(t), an error-propagation analysis on the optimized parameters, and a comparison against redshift-binned observational samples to assess robustness against selection biases. revision: yes

Circularity Check

1 steps flagged

Optimization of Ṁ_BH to enforce universality reduces the 'new solution' to a fitted construction

specific steps
  1. fitted input called prediction [abstract]
    "By optimizing the accretion rate, $\dot M_{BH}$, we obtained a solution that is universal, that is, which applies to any BH at any redshift."

    The universality of γ(t) is presented as the output of the optimization; the procedure is constructed to produce a single functional form independent of redshift, making the universality a direct consequence of the optimization choice rather than an emergent prediction.

full rationale

The abstract states that a universal γ(t) is obtained specifically by optimizing Ṁ_BH after allowing ε and λ_Edd to vary. This matches the fitted-input-called-prediction pattern: the optimization procedure is the mechanism that produces the claimed universality, so the result is forced by the fitting step rather than independently derived. No other load-bearing circular steps are identifiable from the provided text without the full equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the ledger therefore records the minimal set of assumptions visible in the abstract. The optimization step and the mapping from ε and λ_Edd to γ(t) are treated as ad-hoc modeling choices whose justification is not supplied.

free parameters (1)
  • time dependence of ε and λ_Edd
    The functional forms that make γ(t) universal are chosen or fitted so that the solution holds at all redshifts.
axioms (1)
  • ad hoc to paper Black hole mass can be treated as a thermodynamic state function once γ(t) is optimized.
    The abstract asserts agreement with the four laws but does not derive the state-function property from those laws.

pith-pipeline@v0.9.1-grok · 5684 in / 1350 out tokens · 30904 ms · 2026-06-26T13:25:45.500030+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    2025, MNRAS, 536, 4, 3177

    Aggarwal, Y. 2025, MNRAS, 536, 4, 3177

    2025

  2. [2]

    B., Casey, C

    Akins, H. B., Casey, C. M., Lambrides, E., et al. 2025, ApJ, 991, 1, 37

    2025

  3. [3]

    L., Peralta-Maraver, I., & Payne, N

    Arnoldi, J.-F., Jackson, A. L., Peralta-Maraver, I., & Payne, N. L. 2025, Proc. Natl. Acad. Sci., 122, e2513099122

    2025

  4. [4]

    N., Kirhakos, S., Saxe, D

    Bahcall, J. N., Kirhakos, S., Saxe, D. H., & Schneider, D. P. 1997, ApJ, 479, 642

    1997

  5. [5]

    M., Carter, B., & Hawking, S

    Bardeen, J. M., Carter, B., & Hawking, S. W. 1973, Commun. Math. Phys., 31, 161

    1973

  6. [6]

    Bekenstein, J. D. 1972, Nuovo Cimento Lett., 4, 737

    1972

  7. [7]

    Bekenstein, J. D. 1980, Phys. Today, 33, 24

    1980

  8. [8]

    J., et al

    Conselice, C. J., et al. 2025, ApJ, 983, 30 L´ opez-Corredoira, M., Melia, F., Wei, J.-J., & Gao, C.-Y. 2024, ApJ, 970, 63

    2025

  9. [9]

    A., & Moreno del Rio, F

    Ortega-Minakata, R. A., & Moreno del Rio, F. 2017, MNRAS, 466, 921

    2017

  10. [10]

    A., Coziol, R., & Torres-Papaqui, J

    Cutiva-Alvarez, K. A., Coziol, R., & Torres-Papaqui, J. P. 2023, MNRAS, 521, 3058

    2023

  11. [11]

    M., & Furlanetto, S

    Hegde, S., Wyatt, M. M., & Furlanetto, S. R. 2024, JCAP, 2024, 025

    2024

  12. [12]

    S., & Noda, H

    Inayoshi, K., Kimura, S. S., & Noda, H. 2025, PASJ, 77, 811

    2025

  13. [13]

    1967, Phys

    Israel, W. 1967, Phys. Rev., 164, 1776

    1967

  14. [14]

    2025, Open J

    Jahnke, K. 2025, Open J. Astrophys., 8, 9

    2025

  15. [15]

    D., Finkelstein, S

    Kocevski, D. D., Finkelstein, S. L., & Bromm, V. 2025, ApJ, 988, 110

    2025

  16. [16]

    2026, MNRAS, 545, staf1949

    Jockel, C., Kawaguchi, K., Fujibayashi, S., & Shibata, M. 2026, MNRAS, 545, staf1949

    2026

  17. [17]

    R., & Pringle, J

    King, A. R., & Pringle, J. E. 2006, MNRAS, 373, L90

    2006

  18. [18]

    S., & Glover, S

    Klessen, R. S., & Glover, S. C. O. 2023, ARA&A, 61, 65

    2023

  19. [19]

    Li, Z., Inayoshi, K., Chen, K., Ichikawa, K., & Ho, L. C. 2025, ApJ, 980, 36

    2025

  20. [20]

    2024, ApJ, 970, 63

    Lopez-Corredoira, M., Melia, F., Wei, J.-J., & Gao, C.-Y. 2024, ApJ, 970, 63

    2024

  21. [21]

    2024, ApJ, 963, 129

    Matthee, J., et al. 2024, ApJ, 963, 129

    2024

  22. [22]

    Tan, J. C. 2025, MNRAS, 544, 4317

    2025

  23. [23]

    Matthews, K., Neugebauer, G., & Scoville, N. Z. 1988, ApJ, 325, 74

    1988

  24. [24]

    S., et al

    Tanaka, T. S., et al. 2025, ApJ, 995, 21

    2025

  25. [25]

    A., Aird, J., & Coil, A

    Terrazas, B. A., Aird, J., & Coil, A. L. 2025, ApJ, 993, 187

    2025

  26. [26]

    2025, Nat

    Tripodi, R., et al. 2025, Nat. Commun., 16, 9830

    2025

  27. [27]

    2024, ApJ, 963, 74

    Wang, B., et al. 2024, ApJ, 963, 74

    2024