Causality Constraints on Black Hole Thermodynamics in Nonlinear Electrodynamics

Kaho Yoshimura; Maxime M\'edevielle; Toshifumi Noumi; Yoshihiko Abe

REVIEW 3 major objections 3 minor 86 references

Causality Constraints on Black Hole Thermodynamics in Nonlinear Electrodynamics

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that forbidding superluminal light propagation forces the mass-to-charge ratio of extremal black holes to grow with charge and the entropy-to-mass-squared ratio to fall with mass, for arbitrary nonlinear electrodynamics…

desk verdict Entropy-density monotonicity is new and the result survives a sign fix, but the paper as printed has sign errors in the mass/Smarr formulas that block the proof. read the letter →

arxiv 2505.23483 v2 pith:C3FY3FGY submitted 2025-05-29 hep-th gr-qc

Yoshihiko Abe , Maxime Médevielle , Toshifumi Noumi , Kaho Yoshimura This is my paper

classification hep-thgr-qc

keywords nonlinearelectrodynamicsblackholethermodynamicscausalityconstraintssubluminalpropagationweakgravityconjectureentropydensitymodifiedSmarrrelationextremalholes

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 tries to show that a single causality assumption—that electromagnetic fluctuations never propagate faster than light—fixes the sign of two thermodynamic quantities for every black hole in any nonlinear electrodynamics. If the assumption, a convexity condition on the Lagrangian, holds, then the mass-to-charge ratio of extremal black holes grows monotonically with charge, and the entropy-to-mass-squared ratio, called the entropy density, falls monotonically as mass grows at fixed mass-to-charge ratio. The argument does not rely on a derivative expansion, so it extends earlier four-derivative results to all orders, and it applies to dyonic black holes as well. A reader should care because it turns a microphysical causality condition into universal, model-independent constraints on black hole thermodynamics.

What carries the argument

The load-bearing objects are the Euler operators $\theta_F=F\partial_F$ and $\theta_G=G\partial_G$ acting on the Lagrangian, together with the modified Smarr relation $M=2ST+Q_m\Phi_m+\frac{1}{4\sqrt{2}\pi^2}\int_S^\infty \frac{dS'}{S'}S'^{3/2}(\theta_F+\theta_G-1)L$. The causality condition $(\theta_F+\theta_G)L\ge L$ makes the integral correction nonpositive, and that sign turns the standard thermodynamic identities into the two monotonicities. For dyonic black holes, the electromagnetic rotation $P_{\mu\nu}=\cos\alpha\,F_{\mu\nu}+\sin\alpha\,H_{\mu\nu}$ preserves the combination $(\theta_F+\theta_G-1)L$, so the same derivation applies after rotation.

What would settle it

Take any explicit Lagrangian that satisfies $(\theta_F+\theta_G)L\ge L$, solve the zero-temperature condition $T(S,Q_m)=0$ exactly, and numerically integrate Eq. (3.18); a theory for which the extremal mass-to-charge ratio decreases with charge, or for which $ds_\mu/dM>0$, would refute the claimed theorem. Alternatively, a gravitational calculation of wave propagation around the black hole that exhibits superluminal modes even when the convexity condition holds would break the premise, since the paper's inequalities follow only if the condition survives in black hole backgrounds.

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Extended reading notes

Core claim

The paper establishes that in four-dimensional Einstein gravity coupled to a general Lagrangian $L(F,G)$ with $L(0,0)=0$, absence of superluminal propagation, stated as $(\theta_F+\theta_G)L(F,G)\ge L(F,G)$, implies two inequalities. For extremal magnetic black holes, $d\mu_{\rm ext}(Q_m)/dQ_m\ge 0$ (Eq. (3.18)); and for any fixed mass-to-charge ratio $\mu$, the entropy density $s_\mu(M)=S(M,M/\mu)/M^2$ satisfies $ds_\mu(M)/dM\le 0$ (Eq. (3.21)). The proof combines the modified Smarr relation with thermodynamic derivative identities, and the dyonic case is mapped onto the magnetic case by an electromagnetic rotation that leaves the causality condition invariant.

Load-bearing premise

Everything rests on the convexity condition $(\theta_F+\theta_G)L\ge L$, which was derived without dynamical gravity; the paper concedes in Footnote 2 that with dynamical gravity the subluminal condition is not well defined because the light cone changes under metric field redefinition.

Editorial extensions

If this is right

The extremality bound becomes monotone at all orders: in any theory satisfying the convexity condition, the extremal mass-to-charge ratio cannot decrease as charge increases.
The entropy-to-mass-squared ratio, interpreted as an entropy density, is a monotonically decreasing function of mass at fixed mass-to-charge ratio.
The constraints do not require parity invariance, so they cover parity-violating nonlinear electrodynamics, unlike the earlier four-derivative analysis.
Because the electromagnetic rotation maps dyonic solutions to magnetic ones while preserving the causality condition, the same two inequalities hold for generic charged black holes, not just purely magnetic ones.
The result upgrades the earlier four-derivative signs $\Delta\mu_{\rm ext}<0$ and $\Delta S>0$ from perturbative statements to exact, all-order statements under the same causality premise.

Reading between the lines

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

Editorial inference: if one treats the entropy density as a measure of stored information per unit mass squared, the monotonic decrease gives a gravitational analogue of a c-theorem—larger black holes at fixed charge-to-mass ratio carry less entropy per unit mass squared.
Editorial inference: because the derivation uses only the Smarr relation and the convexity sign, the same inequalities should hold for other static black holes whose thermodynamics admit a modified Smarr relation of the same form, such as higher-form charged solutions, provided a similar convexity condition holds.
Editorial inference: a concrete numerical check is to construct exact extremal solutions for the Euler-Heisenberg and Born-Infeld Lagrangians, both of which satisfy the convexity condition, and verify the two inequalities away from the perturbative regime.

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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

The genuinely new result is the entropy-density monotonicity ds_μ/dM ≤ 0 for full nonlinear electrodynamics, plus the dyonic version. The extremal mass-to-charge monotonicity was already in the authors' earlier paper [39]; the entropy part extends the four-derivative checks to all orders. The four-derivative check and the dyonic extension look right, and the electromagnetic rotation identity (4.10) is neat.

But as printed, the thermodynamic scaffolding has sign errors that block the proof. Eq. (2.28) has a plus sign where a minus is needed: for Maxwell it gives M = √(2S) − Q²/(2√(2S)), the wrong Reissner-Nordstrom mass and the opposite sign of their own Eq. (3.4). The modified Smarr relation (2.35) and the version used later, Eq. (3.2), differ by a sign. The correct relation is (3.2), with the minus before the integral. Used correctly, it gives (3.18) and (3.21): at T = 0, dμ/dQ = I/Q² with I = ∫(θ_F+θ_G−1)L ≥ 0, and ds/dM = −I/(T M³) ≤ 0. Follow the printed (2.35), and the extremal inequality flips. So the final claims are right, but the manuscript does not contain a consistent derivation as written; the sign typos in Sec. 2.3 and (2.24)/(2.28) must be fixed.

The second limitation is the causality input itself. Eq. (3.1) comes from a fixed-background analysis. Footnote 2 acknowledges that with dynamical gravity the lightcone is not invariant under metric field redefinition, so the subluminal condition is not well defined. If (3.1) does not survive in black hole backgrounds, both monotonicities fail. This is a substantive caveat but it is stated honestly.

The citation pattern is clean and Ref. [39] is appropriately credited. No fitting or hidden parameters; the derivation is analytic and model-independent within the stated assumptions.

Verdict: this is a solid paper in need of a correction. The new entropy-density inequality is a real contribution for the WGC/black-hole thermodynamics crowd. I would send it to review, with the sign issue fixed before acceptance. Once fixed, I'd be happy to cite it.

Referee Report

3 major / 3 minor

Summary. The paper studies four-dimensional Einstein gravity coupled to a general nonlinear electrodynamics Lagrangian L(F,G) and derives, under the convexity/subluminality condition (θ_F+θ_G)L ≥ L, two monotonicity statements for magnetic and, via an electromagnetic rotation, dyonic black holes. The first is that the extremal mass-to-charge ratio μ_ext(Q_m) is non-decreasing in the charge, and the second is that the entropy-to-mass-squared ratio s_μ(M)=S(M,M/μ)/M^2 is non-increasing in the mass at fixed μ. The proofs are based on a modified Smarr relation, the first law, and the scaling identity θ_S L = -2(θ_F+θ_G)L. The paper also checks the new statements against four-derivative corrections, recovering known signs for the weak-gravity extremality shift and the microcanonical entropy increase.

Significance. If the corrected derivation is accepted, the paper gives model-independent, all-order statements for black hole thermodynamics in nonlinear electrodynamics under a single convexity assumption. It cleanly extends earlier four-derivative results and connects them to the weak gravity conjecture, and the electromagnetic-rotation argument for dyonic black holes is elegant. The authors also provide explicit four-derivative checks and derive the necessary scaling identities in the appendix. However, the printed thermodynamic derivation in Sec. 2.3 contains systematic sign errors, so the central claims are not reproducible from the displayed equations; the final inequalities appear to be the correct consequences of the intended sign convention, but a substantial revision is needed to make the proof self-consistent.

major comments (3)

[Sec. 2.3, Eqs. (2.25), (2.28), (2.32)-(2.35)] The sign in front of the integral in the mass formula is reversed as printed. The correct expression is M(S,Q_m)=√(2S) - (1/(8√2π^2))∫_S^∞ (dS'/S') S'^{3/2}L, with a minus sign rather than the printed plus sign; equivalently Eq. (2.25) should have a minus sign before its integral. With the printed plus sign, the Maxwell term L=-2π^2Q_m^2/S^2 gives M=√(2S)-Q_m^2/(2√2√S), which is the opposite of the Reissner-Nordström formula used in Eq. (3.4). The same global sign error propagates into Eqs. (2.32), (2.33), (2.34), and (2.35); in particular the modified Smarr relation (2.35) has a plus sign, whereas the version actually used in Sec. 3, Eq. (3.2), has a minus sign. Since Eqs. (3.18) and (3.21) are derived from the minus-sign Smarr relation, the central monotonicity claims are not consequences of the displayed equations as they stand. The derivation should be redone with one consistent sign convention, after which the final inequalities appear to follow.
[Sec. 4.2, Eqs. (4.28)-(4.30)] The dyonic extension inherits the same sign inconsistency. Equation (4.28) reproduces the plus-sign form of the modified Smarr relation from Eq. (2.35), but the monotonicity formulas (4.29) and (4.30) require the minus-sign form used in Eqs. (3.18) and (3.21). As printed, the dyonic derivation is therefore not self-consistent either. The authors should re-derive the dyonic Smarr relation after fixing the sign convention in Sec. 2.3, or explicitly state which version of the Smarr relation is being used in each step.
[Abstract, Sec. 3, footnote 2] The abstract and introduction state that the monotonicities are shown 'by requiring the absence of superluminal propagation', but footnote 2 concedes that the subluminality condition (3.1) was derived in the absence of dynamical gravity and that the lightcone is not invariant under metric field redefinitions when gravity is dynamical. Since the black hole spacetimes studied here are solutions of the coupled gravity-matter system, condition (3.1) is an external assumption rather than a causality constraint derived for the present setup. The paper should either justify the regime in which gravitational corrections are negligible or soften the causal claim in the abstract and conclusions.

minor comments (3)

[Appendix A, Eq. (A.6)] In the third relation of Eq. (A.6), the left-hand side should read θ_Qm L rather than just θ_Qm, to match the notation used in Eq. (2.29).
[Sec. 3.2, Eq. (3.12)] The term 'entropy density' for s_μ(M)=S(M,M/μ)/M^2 is used throughout, but the only justification is the brief holographic remark after Eq. (3.12). A short sentence explaining why this object is naturally a density would improve the presentation.
[Sec. 3.2, Eq. (3.17)] The statement that the last term in Eq. (3.17) vanishes at T=0 assumes that the entropy of extremal black holes is a smooth function of Q_m; this smoothness assumption should be stated explicitly in the text before Eq. (3.18).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new thermodynamic monotonicities are derived from an external causality condition and the modified Smarr relation, with no fitted parameter and no load-bearing self-citation.

full rationale

The central monotonicity claims (3.18) and (3.21) are derived from the externally sourced convexity condition (3.1) (Ref. [30]) combined with the modified Smarr relation (2.35)/(3.2), which is reproduced from Ref. [29] and re-derived in the paper via the mass formula (2.28) and scaling identities (2.29). The target inequalities are not used as inputs, and no parameter is fitted to the quantities being predicted. The entropy density s_mu(M) is defined in (3.12), and its mass derivative is obtained by standard thermodynamic relations (3.19)-(3.20), then bounded using the Smarr relation; this is a direct derivation, not a renaming of the conclusion. The self-citation to the authors' previous work [39] appears only as motivation and for the extremality formulation; Eq. (3.18) is re-derived in this paper rather than assumed, and the new entropy-density result is not taken from [39]. Footnote 2 explicitly limits the causality condition to a regime where dynamical gravity is negligible; this is a stated assumption rather than a circular step. The paper's derivation is self-contained given the external convexity input, so the circularity burden is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical parameters are fitted to data. The input L(F,G) is an arbitrary function constrained only by the causality condition, so it is not a fitted parameter. The load-bearing assumptions are the external causality condition, the modified Smarr relation, the existence of static spherically symmetric black hole solutions, and integral convergence. No new particles, forces, or dimensions are introduced. The sign inconsistency between (2.28), (2.35), and (3.2) is an internal presentation issue rather than a fitted parameter.

assumptions (6)

domain assumption Causality condition (3.1): (θ_F+θ_G)L ≥ L
Taken from Schellstede, Perlick, and Lämmerzahl [30] as a proxy for absence of superluminal propagation. It is derived without dynamical gravity, and footnote 2 concedes that subluminality is not well defined when gravity is dynamical.
domain assumption Modified Smarr relation for nonlinear electrodynamics
Reproduced from Gulin and Smolić [29] and Bokulić, Jurić, and Smolić [33]. The sign convention differs between Eq. (2.35) and Eq. (3.2), so the derivation as printed is not self-consistent.
domain assumption Existence of static, spherically symmetric asymptotically flat black hole solutions with L(0,0)=0
The mass formula (2.28) and the thermodynamic treatment assume such solutions exist and that the cosmological constant is set to zero.
domain assumption First law of thermodynamics dM = T dS + Φ dQ with M(S,Q) as a thermodynamic potential
Used throughout Secs. 2 and 3 to define temperature and chemical potential and to derive the scale derivatives in Sec. 3.2.
domain assumption Smoothness of the entropy of extremal black holes as a function of charge
Used in Eq. (3.17) to drop the T-dependent term at T=0 when deriving dμ_ext/dQ_m. If S_ext(Q) is not smooth, the derivative formula may require additional contributions.
standard math Convergence of the integrals and vanishing of boundary terms at infinity
Integration by parts in the derivation of (2.32) and the Smarr relation assumes the boundary terms at S=∞ vanish.

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Cite this review

Pith. "Pith review of Causality Constraints on Black Hole Thermodynamics in Nonlinear Electrodynamics." pith.science (2026). https://pith.science/paper/C3FY3FGY

@misc{pith2026250523483,
  author       = {Pith},
  title        = {Pith review of: Causality Constraints on Black Hole Thermodynamics in Nonlinear Electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C3FY3FGY}},
  note         = {Machine review of arXiv:2505.23483}
}

read the original abstract

We study causality constraints on black hole thermodynamics in nonlinear electrodynamics, where the Lagrangian is taken to be an arbitrary function of the electromagnetic field strength tensor. By requiring the absence of superluminal propagation, we show that the mass-to-charge ratio of extremal black holes exhibits a certain monotonicity previously studied in the context of the weak gravity conjecture. Furthermore, under the same condition, we demonstrate that the entropy-to-mass-squared ratio of black holes, which we interpret as an entropy density, decreases monotonically with increasing mass, while keeping the mass-to-charge ratio fixed. This new monotonicity property extends previous studies on the positivity of four-derivative corrections to black hole entropy in the microcanonical ensemble to all orders in nonlinear electrodynamics.

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