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Charged AdS black holes in Gauss-Bonnet gravity and nonlinear electrodynamics

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper finds a family of five-dimensional charged AdS black holes in Gauss-Bonnet gravity with nonlinear electrodynamics, including regular black holes with a smooth AdS core.

desk verdict The new GB-AdS black hole family is real and the math holds up, but the paper needs a careful referee pass to fix typos and the missing energy-condition analysis. read the letter →

arxiv 1908.09294 v1 pith:WIZ7FLPX submitted 2019-08-25 gr-qc hep-th

Seungjoon Hyun , Cao H. Nam This is my paper · ORCID
classification gr-qchep-th
keywords Gauss-BonnetgravitynonlinearelectrodynamicschargedAdSblackholesregularextendedphasespacethermodynamicsP-VcriticalityHawking-Pagetransitionfivedimensions
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

The paper constructs a new family of five-dimensional charged anti-de Sitter black hole solutions in Einstein-Gauss-Bonnet gravity coupled to a nonlinear electromagnetic field. At a particular value of the mass, $m=q^2/(3k)$, the central curvature singularity disappears and the black hole interior becomes a smooth anti-de Sitter core, yielding regular black hole solutions alongside extremal ones. The paper verifies the extended first law of thermodynamics in which the cosmological constant acts as pressure, identifies first- and second-order phase transitions including a Hawking-Page transition, and studies $P$-$V$ criticality. It finds that the critical exponents agree exactly with those of a Van der Waals fluid. If the construction is right, it provides explicit exact solutions in which higher-curvature gravity and nonlinear electrodynamics jointly remove the singularity, with concrete consequences for the endpoint of Hawking radiation.

What carries the argument

The machinery that carries the argument is the Legendre-transformed nonlinear electromagnetic Hamiltonian $H(P)=P e^{-k_0(-P)^{1/3}}$ combined with the Gauss-Bonnet term. The exponential form is chosen so the integrated metric function acquires the term $q^2/(3k)(e^{-k/r^2}-1)$; the exponential damping cancels the would-be $1/r^2$ divergence at short distances exactly when $m=q^2/(3k)$, replacing the singular center by an AdS core. The metric function (22) is the central formula: it reduces to the known Gauss-Bonnet Schwarzschild-AdS solution as $q\to0$, to the Maxwell-Gauss-Bonnet charged AdS solution as $k\to0$, and to the corresponding no-cosmological-constant and Einstein-gravity limits as further parameters vanish. All later thermodynamic identities are derivatives of this single metric function.

What would settle it

Compute the energy-momentum tensor from Eq. (11) for this $H(P)$ and test whether any timelike observer measures negative energy density outside the horizon for positive $k_0$ and $q$; a violation there would make the matter source unphysical and would undercut the regular solution. Alternatively, evaluate the Kretschmann scalar for the metric at $m=q^2/(3k)$ numerically down to very small $r$: it should approach the constant AdS value rather than grow without bound.

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Extended reading notes

Core claim

The central claim is that the five-dimensional action with Gauss-Bonnet term and the nonlinear electrodynamics $H(P)=P e^{-k_0(-P)^{1/3}}$ admits a static spherically symmetric charged AdS black hole family with metric function given by Eq. (22). For generic mass the solution has the usual curvature singularity at $r=0$; when the reduced mass saturates the bound $m=q^2/(3k)$, the metric function tends to $f(r)\to 1+r^2/l_{\rm eff}^2$ at short distances and the curvature invariants approach the constant AdS values (33), so the black hole is regular. The same family contains extremal black holes whenever the horizon mass function has a minimum, and the paper argues that this opens distinct possible endpoints for evaporation: an extremal singular black hole, a regular extremal black hole, or a regular black hole that continues to radiate. The paper also claims the extended first law $dM=T\,dS+\Phi\,dQ+V\,dP$ holds with Wald entropy $S=S_3 r_+^3(1+12\alpha/r_+^2)/4$ and thermodynamic volume $V=V_4 r_+^4$, and that the equation of state exhibits Van der Waals-like critical behavior with critical exponents $(0,1/2,1,3)$.

Load-bearing premise

The load-bearing premise is that the specific nonlinear electromagnetic action $H(P)=P e^{-k_0(-P)^{1/3}}$ is a physically admissible matter source; if it fails a reasonable energy condition or cannot be seen as a sensible deformation of Maxwell theory, the regular and extremal endpoints and the phase-transition results are model artifacts rather than robust predictions.

Editorial extensions

If this is right

  • A regular endpoint at $m=q^2/(3k)$ means Hawking evaporation need not end in a curvature singularity; depending on charge loss it can end at a regular extremal black hole or at a regular black hole that keeps radiating.
  • With the cosmological constant as pressure, the mass is an enthalpy and the first law $dM=T\,dS+\Phi\,dQ+V\,dP$ holds with Wald entropy and thermodynamic volume $V=V_4 r_+^4$.
  • Below a critical pressure the isotherms develop an unstable branch, so a Maxwell-construction first-order transition between small and large black holes appears; above the critical pressure the black holes behave like an ideal gas.
  • The critical ratio $P_c v_c/T_c$ connects the uncharged Gauss-Bonnet value $1/3$ and the Maxwell value $5/12$, so the solution family interpolates continuously between known limits.
  • The critical exponents $(0,1/2,1,3)$ are the same as those of the Van der Waals fluid, so the mean-field critical behavior survives the nonlinear electrodynamics correction.

Reading between the lines

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

  • A natural extension not pursued here is that the Legendre-plus-exponential construction is transferable: the same cancellation mechanism that removes the $1/r^2$ divergence could generate regular black hole families in other dimensions or with other Lovelock terms.
  • A top-down origin for $H(P)=P e^{-k_0(-P)^{1/3}}$ is not established; if such an origin were found, the regular endpoint would become a dynamical prediction rather than an imposed model.
  • The $\alpha\to0$ limit (29) is presented as a side case, but it deserves independent study as a pure Einstein-gravity family with the same exponential electrodynamics; comparing its phase structure to the Gauss-Bonnet case would isolate the role of higher-curvature corrections.
  • Observational signatures of the AdS core have not been developed; computing quasinormal-mode frequencies or photon deflection for the regular endpoint could reveal whether the smooth interior is distinguishable from a singular one.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper presents new five-dimensional static, spherically symmetric charged AdS black hole solutions in Einstein-Gauss-Bonnet gravity coupled to a specific nonlinear electrodynamics H(P) = P exp(-k0(-P)^{1/3}) (Eq. 13). The metric function f(r) is given explicitly in Eq. (22). The authors show that when the mass parameter saturates the bound m = q^2/(3k), the curvature singularity disappears and the interior approaches an AdS core, with all curvature invariants finite (Eqs. 31-33). The known limits q→0, k→0, and α→0 reproduce the standard Schwarzschild-AdS-GB, Maxwell-GB, and Einstein-higher-derivative electrodynamics solutions. The paper then investigates thermodynamics in the extended phase space: temperature, entropy, chemical potential, thermodynamic volume, heat capacity, Gibbs free energy, and P-V criticality, claiming that the extended first law is confirmed and that the critical exponents coincide with those of a Van der Waals fluid.

Significance. The main contribution is a new exact regular black hole solution family in a higher-curvature theory, with a smooth AdS-like core. The metric is derived from the field equations and passes all standard limits, so the central construction appears sound. The regularity claim is supported by explicit finiteness of the curvature scalars. The thermodynamic analysis is standard but the critical exponents are derived via a Landau expansion with a Maxwell construction, giving a falsifiable prediction of universality. The paper would benefit from showing the explicit first-law verification and addressing the small defects listed below.

major comments (1)
  1. [Section IV, Eqs. (39)-(47)] The abstract and Section IV claim that the extended first law dM = T dS + Φ dQ + V dP is 'confirmed,' but the paper does not show the verification. Equation (45) only demonstrates that the entropy expression follows from integrating ∂M/∂r_+; it does not check the full differential relation with M treated as a function of S, Q, and P. Please add an explicit computation: starting from M = (3S3/16π) [r_+^2 + 2α + r_+^4/l^2 - (q^2/3k)(e^{-k/r_+^2} - 1)] with r_+ expressed through S from Eq. (44), Q = (S3/4π) q, and P = 3/(4π l^2), show that ∂M/∂S, ∂M/∂Q, and ∂M/∂P reproduce Eqs. (41), (46), and (47), respectively. Without this, the 'confirmation' is an assertion rather than a demonstrated result.
minor comments (5)
  1. [Appendix I, Eq. (83)] The entropy formula in Eq. (83) is missing the Gauss-Bonnet coupling α; it should read S = (S3 r_+^3/4)(1 + 12α/r_+^2), consistent with Eq. (44). In the computation leading to Eq. (82), the term δL_GB/δR... should be multiplied by α in the variation of the action.
  2. [Eq. (35)] The expression for q_c, defined by the inflection condition on m(r_+), appears to have an incorrect exponential factor. Please check the derivation; according to the stress-test note, a factor 1/2 should appear in the exponent.
  3. [Section III and Appendix I] The paper does not discuss the energy conditions for the nonlinear electromagnetic source. From the stress-energy components in Eq. (76), the sum ρ + p_t changes sign for r^2 < k/3, indicating NEC violation in the core region. This is expected for regular black holes, but the authors should state it explicitly and discuss its consistency with the regularity claim.
  4. [Appendix II] The weak-deflection angle result in Eq. (93) is presented with only a sketch of the derivation. I recommend either providing the full integration or citing the standard reference for the method used.
  5. [References and typos] There are several minor typographical errors, e.g., 'Shutz' in Ref. [153] should be 'Schutz', and a missing space in the definition of q_c in Eq. (35).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solution family is obtained by direct integration of the field equations, and the thermodynamic identities are internal consistency checks rather than fitted predictions.

full rationale

The central construction is self-contained. The metric function (22) is obtained by integrating the (tt) component of the field equations, Eq. (21), with the explicitly chosen nonlinear electrodynamics Lagrangian H(P)=P exp(-k0(-P)^(1/3)) given in Eq. (13); the charge q is carried through the ansatz (17)-(19) and the mass m arises as the integration constant, so no output quantity is used to set an input parameter. The regular endpoint at m=q^2/(3k) follows from requiring the branch-point condition (26) to coincide with the absence of a curvature singularity; the limit f(r) -> 1 + r^2/l_eff^2 and the curvature invariants (33) are mathematical consequences, not imposed conditions. The thermodynamic 'confirmation' of the first law is a standard consistency check: the temperature T in Eq. (41) comes from surface gravity, the entropy S in Eq. (44) from the Wald formula, and the chemical potential Phi in Eq. (46) is explicitly identified as 'nothing but the electrostatic potential given in Eqs. (20)', an independent gauge-field integration result. The thermodynamic volume V in Eq. (47) is the conjugate variable defined by the first law, so its derivation is definitional rather than a fitted prediction. The critical exponents are obtained from the Landau-type expansion of the equation of state, Eq. (65), together with Maxwell's equal-area law; the values alpha=0, beta=1/2, gamma=1, delta=3 follow from the analytic structure of that expansion and are not used to fix any free constant. The self-citations that appear (e.g. Refs. [49], [51], [106], [107]) are contextual references to other black-hole studies and are not load-bearing for the new solution or its thermodynamics. Some printed formulas contain typographical or interpretive defects, such as the apparent missing Gauss-Bonnet coupling in the appendix Wald-entropy expression (83) and the q_c expression (35), but these are correctness issues rather than circular derivations. No step in the paper reduces by construction to its own inputs, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on the ad hoc choice of the nonlinear electrodynamics Lagrangian H(P)=P e^{-k0(-P)^{1/3}} (Eq. 13), which is not derived from a more fundamental theory. The metric ansatz and the selection of the '+' branch of the vacuum are standard domain assumptions. The extended phase space treatment of the cosmological constant and the identification of the specific volume with the horizon radius are borrowed from the Kubiznak-Mann framework. No new entities are introduced. One free parameter (k0) is introduced by hand; the charge q and mass m are integration constants of the solution, and the Gauss-Bonnet coupling alpha and AdS radius l are inputs from the theory.

free parameters (1)
  • k0
    Nonlinear electrodynamics coupling in H(P)=P e^{-k0(-P)^{1/3}} (Eq. 13), chosen ad hoc to remove the curvature singularity; not derived from a fundamental theory.
assumptions (5)
  • ad hoc to paper The nonlinear electrodynamics Lagrangian H(P)=P e^{-k0(-P)^{1/3}} (Eq. 13) is a valid matter source.
    Chosen by hand to produce regular solutions; no independent physical evidence is provided.
  • domain assumption The spacetime is static and spherically symmetric with P_{tr}=phi(r) (Eq. 17).
    Standard ansatz for black hole solutions.
  • domain assumption The '+' branch of the AdS vacuum (Eq. 16) is selected for a smooth alpha to 0 limit.
    The theory has two AdS vacua for alpha < l^2/8; the choice is conventional.
  • domain assumption The cosmological constant is treated as thermodynamic pressure P=3/(4*pi*l^2) (Eq. 38).
    This is the extended phase space framework from prior literature.
  • domain assumption The specific volume is identified with v=4*r_+/3 (Eq. 55).
    Follows the Kubiznak-Mann comparison with the Van der Waals equation.

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Cite this review

Pith. "Pith review of Charged AdS black holes in Gauss-Bonnet gravity and nonlinear electrodynamics." pith.science (2026). https://pith.science/paper/WIZ7FLPX

@misc{pith2026190809294,
  author       = {Pith},
  title        = {Pith review of: Charged AdS black holes in Gauss-Bonnet gravity and nonlinear electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WIZ7FLPX}},
  note         = {Machine review of arXiv:1908.09294}
}
abstract

New five-dimensional charged AdS black hole solutions are found in Einstein-Gauss-Bonnet gravity and the nonlinear electrodynamics. These solutions include regular black holes as well as extremal black holes. The first law of the black hole thermodynamics is confirmed in the extended phase space where the cosmological constant is treated as the pressure. The first and second order phase transitions are investigated by observing the behavior of the heat capacity at constant pressure and the Gibbs free energy. In addition, the equation of state for the black holes and their $P-V$ criticality are studied. Finally, the critical exponents are found to be the same as those of the Van der Waals fluid.

Figures

Figures reproduced from arXiv: 1908.09294 by the authors.

Figure 1
Figure 1. FIG. 1: Plots of the curvature scalars [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The mass function given at Eq. (34) is plotted in terms of the horizon radius, at [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The extremal radius and the extremal mass are plotted in terms of the inverse squared curvature [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The isobaric curves in the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The isobaric curve for the case II is plotted at [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Plots of the reduced heat capacity [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Plots of the Gibbs free energy, scaled by 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The isotherms in the [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

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Reference graph

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