REVIEW
Regular black holes do not violate the first law of thermodynamics
General Relativity and Quantum Cosmology · Submitted 6 Oct 2026 · 19 references
Pith's verdict: A mostly correct and genuinely clarifying paper that deserves refereeing, but Eq. (3) as typeset has a real sign error that must be fixed: the printed positive exponent makes the mass function diverge instead of giving the Bardeen/Hayward metric used in the rest of the paper.
T0 · AI review
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Referee report
The paper
The paper claims
This paper proves that regular black holes in nonlinear electrodynamics obey the standard first law within any fixed theory in D≥4, and that the reported factor Ξ is only the Jacobian of a projection across theories, not a failure of…
The reading
What carries the argument
The machinery is a strict separation of three parameter spaces that earlier work had mixed: the two-parameter solution space of a fixed theory, labelled by the integration constants $m_0$ and $Q_{\rm ADM}$, on which the first law is proved; the one-parameter subfamily of regular solutions with $m_0=0$, on which the mass is determined by the charge; and the two-parameter family of regular metrics parametrized by $(M,Q)$, whose two members generically belong to two different theories. The load-bearing calculation is the integral representation $M_{\rm ADM}=m_h+\Omega_{D-2}\int_{r_h}^{\infty}\rho\,r^{D-2}dr$, which yields $(\partial M_{\rm ADM}/\partial S)_{Q_{\rm ADM}}=T_h$ and $(\partial M_{\rm ADM}/\partial Q_e)_S=\Phi_h$, together with the projection of the extended first law — in which the couplings $\sigma,\beta$ are treated as thermodynamic variables — onto the metric family; that projection produces the prefactor $m_h/M_{\rm ADM}=1/\Xi$ and the one-form of Eq. (17). The non-integrability of $\Xi^{-1}dM_{\rm ADM}$ is then settled by the direct mixed-derivative computation of Eq. (18), whose right-hand side is strictly positive for $Q>0$ and $D\ge4$.
Extended reading notes
Core claim
The central claim is that the thermodynamics of regular black holes is standard once the variations are taken inside a fixed theory. For the theory defined by the Hamiltonian density $H(P)$ of Eq. (2), whose couplings $\sigma,\beta,\gamma$ are fixed and contain no solution parameters, the paper proves analytically that $dM_{\rm ADM}=T_h\,dS+\Phi_h\,dQ_e$ holds on the whole two-parameter family of static solutions in any dimension $D\ge4$, with the Hawking temperature, the area entropy, and the electrostatic potential of the horizon; the derivation uses only the Einstein equations, the area entropy, and the theory-fixed energy density, and the authors note it reproduces the general first law for nonlinear electrodynamics already derived elsewhere. The reported factor $\Xi$ appears only when the mass is differentiated along the one-parameter subfamily of regular metrics at fixed $Q$ — a variation that moves through the space of theories, changes the coupling constants and the conserved charge $Q_e$, and so is not a variation within one theory at all. The paper identifies $\Xi^{-1}$ with $m_h/M_{\rm ADM}$, the fraction of the ADM mass inside the horizon, gives its general form $\Xi=(\partial\hat m(r_h;M,Q)/\partial M)^{-1}$ for an arbitrary regular mass function, and proves that $\Xi^{-1}dM_{\rm ADM}$ is not a closed one-form on the two-parameter family of regular metrics, so it cannot be the differential of any local internal energy there; the horizon mass $m_h$ is a state function, but its differential is not $\Xi^{-1}dM_{\rm ADM}$, while $dM_{\rm ADM}$ itself is exact and needs no correction. As a separate point, the paper observes — and flags as a limitation — that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\ge5$, leaving the dynamical status of those backgrounds open.
Load-bearing premise
The non-integrability claim for $\Xi^{-1}dM_{\rm ADM}$ rests on the extended phase-space treatment in which the coupling constants $\sigma$ and $\beta$ are promoted to thermodynamic variables with the specific normalization $\beta=(\gamma+D-1)\sigma/2(D-2)$; if that treatment is rejected, the no-local-state-function statement is not established, although the fixed-theory first law (14) is independent of this choice.
Editorial extensions
If this is right
- For the whole family of theories (2) in any dimension $D\ge4$, the entropy remains the area entropy, the temperature remains the Hawking temperature, and no modified entropy or corrected internal energy is needed.
- The correction proposed in Ref. [11] — putting $1/\Xi$ in front of $dM_{\rm ADM}$ and calling the result a new internal energy — is shown to be non-integrable: $1/\Xi\,dM_{\rm ADM}$ is not closed, so no local energy function has that differential, whereas $dM_{\rm ADM}$ itself is exact and its first law (14) needs no correction.
- Reported temperature mismatches for regular black holes are explained as variations that leave the fixed-theory solution space; keeping $Q$ fixed instead of $Q_e$ changes the mismatch's value and even its sign, which shows the mismatch is a property of the chosen variation, not of the solutions.
- On the regular subfamily, $S$ and $Q_e$ are not independent, so no independent derivative $(\partial M_{\rm ADM}/\partial S)_{Q_e}$ exists within it; the thermodynamic temperature of a regular black hole must be read from the larger space of neighboring solutions, which is the usual situation for an equilibrium first law.
- In $D\ge5$ the $F\leftrightarrow P$ Legendre map degenerates at a radius outside the horizon for near-extremal solutions, so those backgrounds require a separate analysis of their characteristics before they can be declared acceptable dynamical solutions.
Reading between the lines
- The same projection logic should transfer to other regular metrics built by prescribing $f$ and deriving the source: whenever the source can be rewritten with fixed couplings, the factor $\Xi=(\partial\hat m(r_h;M,Q)/\partial M)^{-1}$ should again equal the inverse mass fraction inside the horizon, a relation one could check for each new regular-metric proposal.
- A practical consequence for the field is reporting hygiene: a 'first law of regular black holes' is meaningful only after declaring which space the variation lives in, and adopting the paper's three-space distinction as a convention would remove a whole class of apparent paradoxes from future papers.
- If $1/\Xi\,dM_{\rm ADM}$ is genuinely non-closed, then any corrected first law built on that prefactor is path-dependent: the value of the putative internal energy at a given $(r_h,Q)$ would depend on the history of the process, a signature that the one-form is a projection artifact rather than new physics.
- The branch-point result for $D\ge5$ suggests a concrete probe: electromagnetic perturbations on near-extremal regular backgrounds should feel the singular effective metric at $r_b>r_{\rm ext}$, which could be searched for once the dynamics across $r_b$ is understood.
Original abstract
Regular black holes in general relativity coupled to nonlinear electrodynamics have been reported to exhibit an apparent mismatch in their thermodynamics: the temperature obtained by differentiating the mass along the family of regular metrics differs from the Hawking temperature by a factor $\Xi$, and the first law seems to need a modified entropy or a corrected internal energy. We examine this question for an electrically charged family of regular black holes in $D\geq4$ dimensions whose metrics reduce to the Bardeen and Hayward geometries in four dimensions. The source is written as a Hamiltonian density $H(P)$ with fixed couplings and no solution parameters. This theory has a two-parameter family of static solutions, of which the regular metrics form a one-parameter subfamily. We prove analytically that on the whole solution space the standard first law of thermodynamics holds with the Hawking temperature, the area entropy and the horizon potential, in agreement with the general first law for nonlinear electrodynamics. The factor $\Xi$ appears when the mass is differentiated along the regular metric family at fixed length parameter, a variation that moves through the space of theories and also changes the conserved charge. We show that $\Xi^{-1}$ is the Jacobian of this projection, equal to the fraction of the mass inside the horizon, give its general form for an arbitrary regular mass function, and show that $\Xi^{-1}dM_{ADM}$ is not a closed one-form on the two-parameter metric family, so this correction cannot be the differential of a local state function there. We also point out that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\geq5$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Regular black holes in general relativity coupled to nonlinear electrodynamics have been reported to exhibit an apparent mismatch in their thermodynamics: the temperature obtained by differentiating the mass along the family of regular metrics differs from the Hawking temperature by a factor $\Xi$, and the first law seems to need a modified entropy or a corrected internal energy. We examine this question for an electrically charged family of regular black holes in $D\geq4$ dimensions whose metrics reduce to the Bardeen and Hayward geometries in four dimensions. The source is written as a Hamiltonian density $H(P)$ with fixed couplings and no solution parameters. This theory has a two-parameter family of static solutions, of which the regular metrics form a one-parameter subfamily. We prove analytically that on the whole solution space the standard first law of thermodynamics holds with the Hawking temperature, the area entropy and the horizon potential, in agreement with the general first law for nonlinear electrodynamics. The factor $\Xi$ appears when the mass is differentiated along the regular metric family at fixed length parameter, a variation that moves through the space of theories and also changes the conserved charge. We show that $\Xi^{-1}$ is the Jacobian of this projection, equal to the fraction of the mass inside the horizon, give its general form for an arbitrary regular mass function, and show that $\Xi^{-1}dM_{ADM}$ is not a closed one-form on the two-parameter metric family, so this correction cannot be the differential of a local state function there. We also point out that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\geq5$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Regular black holes in general relativity coupled to nonlinear electrodynamics have been reported to exhibit an apparent mismatch in their thermodynamics: the temperature obtained by differentiating the mass along the family of regular metrics differs from the Hawking temperature by a factor $\Xi$, and the first law seems to need a modified entropy or a corrected internal energy. | background 0.99 | 0.05 | 0.04 | asserted 0.65 |
| s1 | We examine this question for an electrically charged family of regular black holes in $D\geq4$ dimensions whose metrics reduce to the Bardeen and Hayward geometries in four dimensions. | contribution 0.90 | 0.65 | 0.03 | asserted 0.67 |
| s2 | The source is written as a Hamiltonian density $H(P)$ with fixed couplings and no solution parameters. | method 0.90 | 0.51 | 0.02 | asserted 1.00 |
| s3 | This theory has a two-parameter family of static solutions, of which the regular metrics form a one-parameter subfamily. | method 0.54 | 0.32 | 0.03 | asserted 1.00 |
| s4 | We prove analytically that on the whole solution space the standard first law of thermodynamics holds with the Hawking temperature, the area entropy and the horizon potential, in agreement with the general first law for nonlinear electrodynamics. | result 0.99 | 0.96 | 0.94 | asserted 1.00 |
| s5 | The factor $\Xi$ appears when the mass is differentiated along the regular metric family at fixed length parameter, a variation that moves through the space of theories and also changes the conserved charge. | interpretation 0.56 | 0.73 | 0.07 | asserted 1.00 |
| s6 | We show that $\Xi^{-1}$ is the Jacobian of this projection, equal to the fraction of the mass inside the horizon, give its general form for an arbitrary regular mass function, and show that $\Xi^{-1}dM_{ADM}$ is not a closed one-form on the two-parameter metric family, so this correction cannot be the differential of a local state function there. | result 0.93 | 0.96 | 0.12 | asserted 1.00 |
| s7 | We also point out that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\geq5$. | result 0.96 | 0.88 | 0.03 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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