Revisiting Thermodynamics of the Hayward Black Holes and Exploring Binary Merger Bounds

Ankur Srivastav; Neeraj Kumar; Phongpichit Channuie

arxiv: 2604.13972 · v1 · submitted 2026-04-15 · 🌀 gr-qc

Revisiting Thermodynamics of the Hayward Black Holes and Exploring Binary Merger Bounds

Neeraj Kumar , Ankur Srivastav , Phongpichit Channuie This is my paper

Pith reviewed 2026-05-10 12:32 UTC · model grok-4.3

classification 🌀 gr-qc

keywords Hayward black holesblack hole thermodynamicslogarithmic entropybinary merger boundssecond lawregular black holeshead-on collisions

0 comments

The pith

Assuming black hole thermodynamics laws hold for Hayward black holes produces a new entropy with a logarithmic correction that bounds the final mass after equal-mass head-on mergers.

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

The paper revisits the thermodynamics of Hayward black holes in asymptotic flat spacetime by assuming that the standard laws of black hole thermodynamics remain valid. Under this assumption a novel entropy formula emerges that includes a logarithmic correction term together with one extra term. The authors then apply the second law of thermodynamics with this entropy to derive bounds on the mass parameter of the single black hole formed in the head-on collision of two equal-mass Hayward black holes. They also examine how the Hayward parameter affects the phase structure and both thermal and merger properties. This connects a regular black hole model to concrete merger outcomes while staying within classical thermodynamics.

Core claim

By requiring that the laws of black hole thermodynamics remain valid for Hayward black holes in asymptotic flat spacetime, a novel entropy formula appears naturally with a logarithmic correction term along with one extra term. The validity of the second law during the head-on collision of two equal-mass black holes then yields bounds on the final black hole mass parameter, with the bounds depending on the Hayward parameter.

What carries the argument

The novel entropy formula containing a logarithmic correction and one extra term, derived directly from enforcing thermodynamic laws.

Load-bearing premise

The assumption that the laws of black hole thermodynamics remain valid for Hayward black holes in asymptotic flat spacetime.

What would settle it

A calculation or simulation in which the new entropy formula produces a net entropy decrease during an equal-mass merger for parameters allowed by the model.

Figures

Figures reproduced from arXiv: 2604.13972 by Ankur Srivastav, Neeraj Kumar, Phongpichit Channuie.

**Figure 1.** Figure 1: Hawking temperature vs horizon radius for Hayward black holes for l = (0, 0.1, 0.2). can still be assumed in the present scenario and the first law may be given as, dM = T dS . (8) Under these assumptions, the entropy associated with this regular black hole spacetime does not follow Bekenstein-Hawking formula (S = A/4). This is because such an entropy formula would be incompatible with eq.(8). Thus, one … view at source ↗

**Figure 3.** Figure 3: Heat Capacity vs Horizon Radius for l = (0, 0.1, 0.2). physically allowed singular point where the heat capacity diverges for non-zero values of l. This point corresponds to the maxima in the temperature plot, given in fig.(1). This behaviour is similar to the Kerr-Newman black hole [22]. We plot the heat capacity with horizon radius for different values of parameter l in Fig.(3). For the limit l → 0 (t… view at source ↗

**Figure 4.** Figure 4: Thermal Potential F vs Temperature T for l = (0, 0.1, 0.2). nature. It implies that there is no first order phase transition in this system. Thus, one has to look for higher order phase transition depicted in the heat capacity plot. However, starting with this multivalued potential expression in eq.(17), one cannot proceed further to analyse higher order phase transitions from standard single-valued free… view at source ↗

**Figure 5.** Figure 5: Final Mass Parameter M vs Hayward Parameter l for Mi = 1. esting features. The point where the curve meets the Mf -axis represents a minimum bound on the final mass parameter for the Schwarzschild black hole case. As one increases the Hayward parameter values, the bound becomes more stringent (that is, a 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Final Mass Parameter M vs Hayward Parameter l for Mi = 1, 2, 3. 5 Discussions Among regular black hole solutions, the Hayward black holes hold special importance from the dynamical perspective. Even with one asymptotic charge these black holes have very interesting classical and thermal characteristics. Classically, their causal structure is similar to the charged black holes as these have an inner and a… view at source ↗

read the original abstract

In this article, we revisit the thermodynamics of Hayward black holes [1] in asymptotic flat spacetime and obtain the bounds on the final mass post merger after head-on collision event of two equal mass black holes. We revisit thermal properties of these black holes from a perspective that the laws of black hole thermodynamics remain valid. Under this condition a novel entropy formula appears naturally with a logarithmic correction term along with one extra term. We discuss phase structure of the black holes. Next, we obtain the bounds on the final black hole mass parameter using the validity of the second law of black hole thermodynamics with the new entropy formula. We discuss the impact of the Hayward parameter on thermal and merger properties of these black holes.

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 builds a new entropy for Hayward black holes from the first law assumption and derives merger bounds from it, but the construction makes the bounds dependent on that assumption without an independent geometric check.

read the letter

The main takeaway is that the authors assume the laws of black hole thermodynamics hold for Hayward black holes in flat space, integrate dM = T dS with the usual temperature to produce an entropy containing a log term and one extra piece, and then apply the second law to bound the final mass after two equal-mass black holes merge head-on. They do well in making the phase structure explicit and in tracing how the Hayward parameter changes both the thermodynamics and the merger limits. The formulas are given directly, which makes it easy to follow the dependence. The soft spot is that the entropy is defined so the first law is satisfied by construction, which makes the subsequent bounds a logical consequence of the starting assumption rather than a robust prediction from general relativity. For these metrics sourced by nonlinear electrodynamics, the Wald entropy is expected to be area-based, and the paper would be stronger with a direct comparison or a check for extra work terms. The stress-test note on this point holds up based on the abstract. This paper is for specialists in regular black hole thermodynamics. A reader already working on similar models will find the bounds and parameter effects useful for their own calculations. It deserves peer review because the topic is active and the derivations are specific enough for referees to assess the validity of the assumption and the resulting expressions.

Referee Report

3 major / 3 minor

Summary. The manuscript revisits the thermodynamics of Hayward black holes in asymptotically flat spacetime under the assumption that the standard laws of black hole thermodynamics remain valid. This assumption yields a novel entropy formula containing a logarithmic correction term plus one additional term. The authors analyze the phase structure and then apply the second law to this entropy to derive bounds on the final mass parameter after the head-on collision of two equal-mass Hayward black holes, discussing the influence of the Hayward parameter throughout.

Significance. If the entropy derivation can be independently justified, the resulting merger bounds would provide concrete, falsifiable constraints on regular black hole models that could be tested against numerical relativity simulations of head-on collisions. The phase-structure analysis would also contribute to the catalog of thermodynamic properties for non-singular black holes. At present the significance is limited by the lack of cross-checks against standard entropy prescriptions for the underlying nonlinear-electrodynamics source.

major comments (3)

[thermal properties section] § on thermal properties / entropy derivation: the entropy is obtained by direct integration of dM = T dS with T = κ/2π taken from the surface gravity of the Hayward metric. No explicit comparison is made to the Wald-Noether charge entropy appropriate for Einstein gravity coupled to nonlinear electrodynamics, which is expected to remain proportional to the horizon area (or a simple function thereof). This omission is load-bearing because the subsequent merger bounds rest entirely on the constructed S.
[merger bounds section] § on binary merger bounds: the inequality S(M_f, g) ≥ 2 S(M_i, g) is applied to the newly derived entropy to bound the post-merger mass parameter. Because the entropy was defined precisely so that the first and second laws hold, the resulting bounds follow directly from the initial assumption rather than constituting an independent prediction; a concrete test (e.g., comparison with the area-law entropy or with numerical merger data) is required to establish their robustness.
[phase structure section] § on phase structure: critical points and stability conclusions are drawn using the modified entropy; these should be re-derived with the standard area-based entropy to isolate which features are genuinely new versus artifacts of the integration procedure.

minor comments (3)

[introduction] The Hayward parameter is introduced without a brief reminder of its physical interpretation (regularizing the central singularity) in the opening paragraphs; a short sentence would aid readability.
[merger bounds section] Notation for the final mass parameter after merger should be defined once and used consistently; occasional switches between M_f and M_final appear.
[thermal properties section] Reference [1] for the original Hayward metric is cited but the precise form of the metric function f(r) is not restated; including it would make the surface-gravity calculation self-contained.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each of the major comments in detail below, indicating the changes we will implement in the revised manuscript.

read point-by-point responses

Referee: [thermal properties section] § on thermal properties / entropy derivation: the entropy is obtained by direct integration of dM = T dS with T = κ/2π taken from the surface gravity of the Hayward metric. No explicit comparison is made to the Wald-Noether charge entropy appropriate for Einstein gravity coupled to nonlinear electrodynamics, which is expected to remain proportional to the horizon area (or a simple function thereof). This omission is load-bearing because the subsequent merger bounds rest entirely on the constructed S.

Authors: We acknowledge the importance of comparing our entropy to the Wald-Noether charge entropy. The Hayward solution is sourced by nonlinear electrodynamics, for which the Wald entropy is typically the horizon area divided by 4. Our method assumes the first law holds with the surface gravity temperature and derives S by integration, yielding a logarithmic term. This is a phenomenological approach common in studies of regular black holes. In the revision, we will add a dedicated paragraph in the thermal properties section contrasting the two entropies and explaining why we proceed with the integrated form under our assumption. We note that providing a full independent derivation of the Wald entropy for this specific model would require additional calculations not central to the current manuscript. revision: yes
Referee: [merger bounds section] § on binary merger bounds: the inequality S(M_f, g) ≥ 2 S(M_i, g) is applied to the newly derived entropy to bound the post-merger mass parameter. Because the entropy was defined precisely so that the first and second laws hold, the resulting bounds follow directly from the initial assumption rather than constituting an independent prediction; a concrete test (e.g., comparison with the area-law entropy or with numerical merger data) is required to establish their robustness.

Authors: The referee is correct that the bounds are a consequence of the second law applied to the entropy that satisfies the first law by construction. The manuscript's goal is to investigate the consequences of this entropy formula for binary mergers. To enhance robustness, we will add a comparison of the resulting mass bounds with those obtained from the standard area-law entropy. We will also elaborate on potential tests against numerical relativity simulations of head-on collisions, although such simulations lie outside the scope of this theoretical work. revision: yes
Referee: [phase structure section] § on phase structure: critical points and stability conclusions are drawn using the modified entropy; these should be re-derived with the standard area-based entropy to isolate which features are genuinely new versus artifacts of the integration procedure.

Authors: We agree that re-deriving the phase structure with the area entropy will help distinguish new features. In the revised version, we will include an additional analysis or subsection using S = A/4 to compare critical points, heat capacity, and stability. This will clarify the impact of the logarithmic correction term. revision: yes

Circularity Check

2 steps flagged

Entropy obtained by integrating dM=TdS under the assumption that first law holds with unmodified T; merger bounds then follow directly from second-law inequality on that constructed S.

specific steps

self definitional [Abstract and thermodynamics section]
"Under this condition a novel entropy formula appears naturally with a logarithmic correction term along with one extra term. We obtain the bounds on the final black hole mass parameter using the validity of the second law of black hole thermodynamics with the new entropy formula."

The entropy is constructed by enforcing dM = T dS (T = κ/2π from the Hayward metric) so that the first law holds by definition. Inserting this S into the second-law inequality S(M_f) ≥ 2 S(M_i) then forces the mass bounds as a direct algebraic consequence; the bounds are not an independent test of the metric or of GR.
fitted input called prediction [Merger-bounds derivation]
"we obtain the bounds on the final black hole mass parameter using the validity of the second law of black hole thermodynamics with the new entropy formula"

The 'prediction' of allowed final-mass values is obtained by applying the second law to an entropy that was itself defined to satisfy the first law; the inequality therefore reproduces the input assumption rather than constraining the Hayward parameter from external data or from a Noether-charge entropy.

full rationale

The paper states that a novel entropy (with log term) 'appears naturally' once the laws of black-hole thermodynamics are assumed to remain valid for the Hayward metric. This S is obtained by integrating the first-law identity using the standard surface-gravity temperature. The subsequent bounds on post-merger mass are extracted by imposing S_final ≥ 2 S_initial on the same expression. Because the entropy formula is defined precisely so that the thermodynamic identity is satisfied, the inequality and resulting mass bounds are tautological consequences of the initial assumption rather than independent predictions. No Wald/Noether-charge calculation or comparison to the area law for nonlinear-electrodynamics sources is supplied to justify the extra terms.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that thermodynamic laws hold exactly for Hayward black holes and on the pre-existing Hayward regularization parameter taken from earlier literature.

free parameters (1)

Hayward parameter
The parameter that regularizes the black hole singularity; its specific value affects both the entropy expression and the merger bounds.

axioms (1)

domain assumption Laws of black hole thermodynamics remain valid for Hayward black holes
Explicitly invoked in the abstract to obtain the novel entropy formula.

pith-pipeline@v0.9.0 · 5420 in / 1447 out tokens · 73501 ms · 2026-05-10T12:32:58.435806+00:00 · methodology

discussion (0)

Reference graph

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