Static and Dynamic Charged Black Holes

H. L\"u; Hyat Huang; Zhong-Ying Fan

REVIEW 1 major objections 4 minor 1 cited by

Static and Dynamic Charged Black Holes

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Pith's one-line read Exact charged black holes, both static and time-dependent, are constructed in general-dimensional Einstein-Maxwell-dilaton theories, with the dynamical ones describing collapse from a smaller to a larger black hole.

desk verdict Four-dimensional solutions are solid, but the general-D formulas carry a sign error that invalidates the central claim as printed. read the letter →

arxiv 1908.07970 v1 pith:RQZO23BE submitted 2019-08-21 hep-th

Hyat Huang , Zhong-Ying Fan , H. Lü This is my paper · ORCID

classification hep-th

keywords Einstein-Maxwell-dilatontheorieschargedblackholesdynamicalholecollapsescalarhairEddington-Finkelsteincoordinatesreverseengineeringthermodynamicsholographicthermalization

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 aims to show that in a class of Einstein-Maxwell-dilaton theories in arbitrary spacetime dimension, there exist exact charged black hole solutions, both static and genuinely time-dependent, whose mass and electric charge are integration constants rather than fixed by the Lagrangian. The construction works backwards: the authors first choose the scalar field profile $\varphi = 2k_0\,\mathrm{arcsinh}[(q/r)^\Delta]$, then solve for the scalar potential $V(\varphi)$ and the scalar–Maxwell coupling $Z(\varphi)$ that make this profile consistent with the Einstein equations. For a subset of these static solutions, promoting the scalar charge $q$ to a function $a(u)$ of Eddington-Finkelstein time yields exact dynamical solutions whose evolution is controlled by a first-order ordinary differential equation for $a(u)$. In the four-dimensional examples the evolution describes a smaller charged black hole that is nonlinearly unstable and grows monotonically into a larger stable black hole, with the apparent horizon interpolating between the two. If correct, these are among the few exact analytic models of charged black hole formation, useful for holographic thermalization studies.

What carries the argument

The load-bearing object is the scalar-field ansatz $\varphi = 2k_0\,\mathrm{arcsinh}[(q/r)^\Delta]$. Fixing this profile determines, through the Einstein equations, the metric function $\sigma(r)=\left(1+q^{2\Delta}/r^{2\Delta}\right)^{k_0^2\Delta/(D-2)}$; the scalar potential $V(\varphi)$ and the coupling function $Z(\varphi)$ are then reverse engineered so that the equations close. The same profile with $q$ replaced by $a(u)$ is carried into Eddington-Finkelstein coordinates $ds^2 = 2\,dr\,du/\sigma - h\,du^2 + r^2\,d\Omega^2$; the $E^u_r=0$ equation fixes $\sigma(r,u)$, while consistency of the remaining equations forces either $\Delta=(D-2)/2$ for planar horizons in general dimensions or $D=4$ for independent topological parameter, leaving an ordinary differential equation for $a(u)$. This evolution equation, once integrated, is what controls the collapse from a small to a large black hole.

What would settle it

Take the claimed dynamical metric and scalar field with $h$, $\sigma$, $\varphi$ as given in the four-dimensional and general-dimensional solutions, substitute them directly into the full Euler-Lagrange equations without using the reduced evolution equation, and check that the equations reduce exactly to the stated ordinary differential equation with no residual $r$-dependence; any surviving $r$-dependent term would mean the ansatz is not an exact solution. A complementary numerical test is to evolve the same Lagrangian from initial data equal to the ansatz plus a small superimposed radial mode, such as $\varepsilon\,r^{-\Delta-1}$ times a smooth function of $u$, and compare the evolution with the closed-form solution.

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Extended reading notes

Core claim

The central claim is that the Lagrangian with the specifically derived $V(\varphi)$ and $Z(\varphi)$ admits exact electrically charged black holes whose metric function $h(r)$ is given in terms of hypergeometric functions and whose mass parameter and electric charge are free integration constants; in the planar case the scalar hair parameter $q$ is also an integration constant. When the static solutions are rewritten in Eddington-Finkelstein-like coordinates and $q$ is promoted to $a(u)$ with the scalar retaining the form $\varphi(r,u)=2k_0\,\mathrm{arcsinh}[(a(u)/r)^\Delta]$, the full field equations still close, provided the consistency conditions $\Delta=\tfrac12(D-2)$ for vanishing topological parameter or $D=4$ for independent $k$ hold, and $a(u)$ obeys a second-order ordinary differential equation that integrates once to a first-order evolution equation. In the four-dimensional $k_0=1$ case this evolution has two fixed points $q_-$ and $q_+$, with $q_-$ unstable and $q_+$ stable; the solution interpolates from a smaller charged black hole in the past to a larger one in the future, and the Vaidya mass increases monotonically throughout the process. The paper thus claims the first exact charged generalization of the neutral dynamical collapse solutions of the earlier literature in general dimensions.

Load-bearing premise

The dynamical claims assume the time-dependent scalar field keeps exactly the same radial profile $\varphi=2k_0\,\mathrm{arcsinh}[(a(u)/r)^\Delta]$, with only the scalar charge $a(u)$ changing; if the true time-dependent field needed a different radial dependence, the reduction to an ordinary differential equation would not work.

Editorial extensions

If this is right

The static solutions satisfy the first law $dM = T\,dS + \Phi_e\,dQ_e + V_{\mathrm{th}}\,dP$ with a standard thermodynamic volume, so they are genuine black hole solutions with well-defined thermodynamics.
For planar horizons the dynamical solutions exist in every dimension and reduce in the neutral limit to the known exact hairy collapse solutions, so the charged construction extends that family without losing exactness.
In four dimensions, spherical charged black holes can collapse: the smaller state $a=q_-$ is stable against linear perturbations but nonlinearly unstable, and the evolution ends at the larger state $a=q_+$.
Because $a(u)$ approaches the static endpoint exponentially with relaxation time $\tau = 2q_+/[3\alpha(q_+^2-q_-^2)]$, the solutions provide explicit time scales for holographic thermalization-like processes.
In the $\gamma_2=0$ or $D=3$ cases the initial state is a charged naked singularity with vanishing Vaidya mass, so the construction also models horizon formation from a singular seed.

Reading between the lines

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

The reverse-engineering recipe is more general than the paper's examples: any scalar profile of the single-parameter form $\varphi(r,q)$ with a fixed radial shape could plausibly be promoted to a dynamical ansatz by $q\to a(u)$, with the consistency conditions acting as a selection rule; applying it to rotating or non-spherically symmetric seeds is a natural test.
The monotonic mass increase $dM/du = \dot a^2/(4\pi) \ge 0$ suggests an entropy-like or Lyapunov interpretation for the collapse, with the scalar charge acting as a clock and the final black hole as the unique attractor of this ansatz.
The charged naked-singularity initial state in the $\gamma_2=0$ case offers a concrete arena to test cosmic censorship in these theories, since the exact solution provides both the initial data and the explicit collapse endpoint.

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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

Short version: this paper has a real result and a real bug. The D=4 charged black holes and their dynamical promotion in section 5.2 are constructed carefully and pass the first-law and Maxwell checks. The reverse-engineering method is transparent, and the paper is honest about what is fixed by ansatz. However, the advertised general-D static solution (42) and the corresponding dynamical solution (66) do not satisfy the Maxwell equation as written. The gamma1 term has a plus sign where the D=4 result in (19) has a minus sign. With the published sign, differentiating (49) in D=5 gives a negative xi-prime, while the Maxwell equation (5) requires a positive right-hand side (for gamma1>0). I checked the D=5 example explicitly; it fails. So the 'general dimensions' claim is not correct as printed.

This is not a subtle interpretation issue. The D=4 formula is correct, and the general-D formula should reduce to it, but instead it differs by a sign on the dominant term. A single missing minus sign likely fixes it, since the rest of the structure lines up. But as it stands, sections 4 and 5.3 cannot be trusted, including the thermodynamics in 4.3 and the evolution analysis in 5.5 for D>4.

What the paper does well: the D=4 charged planar and spherical solutions with arbitrary topology, the explicit Z and V functions, and the dynamical example in section 5.2 showing collapse from a smaller to a larger charged black hole. That is a genuine extension of the neutral program in [19,20], and the presentation is clear about the reverse engineering.

What is soft, aside from the sign error: the thermodynamic volume is imported from [31] rather than derived, which is minor; and the construction is reverse-engineered by design, so existence of static solutions is tautological. Those are not fatal—the explicit solutions still satisfy the equations—but they limit impact.

Bottom line: this deserves a serious referee, but the referee should catch the sign error. The D=4 material is worth keeping; the general-D sections need a corrected derivation. I would not cite the paper in its current form, but if the authors fix the sign and recheck the reductions, the corrected version would be useful for holographic thermalization studies.

Referee Report

1 major / 4 minor

Summary. The paper constructs static and dynamical charged black hole solutions in a class of Einstein-Maxwell-dilaton theories with a non-minimally coupled scalar and gauge field. The authors use a reverse-engineering procedure: they impose the scalar ansatz φ=2k0 arcsinh[(q/r)^Δ], then derive the scalar potential V(φ) and gauge coupling Z(φ) that admit this form. They present explicit static solutions in D=4 with planar, spherical, and hyperbolic topologies, verify the first law of thermodynamics, and generalize to arbitrary D. A subset of these solutions is then promoted to exact time-dependent solutions in Eddington-Finkelstein coordinates, with the scalar charge q promoted to a function a(u) obeying a nonlinear ODE. The dynamical solutions are interpreted as describing collapse from a smaller charged black hole (or a charged naked singularity) to a larger stable hairy black hole.

Significance. If correct, the construction is a useful addition to the limited set of exact dynamical black hole solutions in Einstein-Maxwell-dilaton theories. The paper is transparent about the reverse-engineering logic, provides explicit examples in D=4 and D=5, and checks the first law for the static solutions. The dynamical promotion with a consistency condition on the scalar ansatz is a technically nontrivial step, and the evolution analysis includes analytic control of the apparent horizon growth. However, the manuscript's central claim of exact solutions in general dimensions is currently not supported by the printed equations: the general-dimensional electric potential fails the Maxwell equation as written. The D=4 sector appears correct, and the error is likely fixable, but the general-D static and dynamic solutions require correction and re-verification before the central claim can be accepted.

major comments (1)

[Sec. 4.1, Eq. (42); Sec. 5.3, Eq. (66)] The advertised general-D electric potential does not satisfy the Maxwell equation (5) and does not reduce to the correct D=4 result (19). For γ2=0, Eq. (42) gives ξ = (D−2)γ1Qσ/(2r^{D−3}) with a plus sign, whereas the correct D=4 expression is ξ = −γ1Qσ/r (Eq. (19)). Substituting the D=5 solution (49) with γ2=0, one obtains ξ = 3γ1Q√(r^4+q^4)/(2r^4), whose derivative is dξ/dr = −3γ1Q(r^4+2q^4)/(r^5√(r^4+q^4)), while the Maxwell equation (5) with Z from (48) requires +3γ1Q(r^4+2q^4)/(r^5√(r^4+q^4)). The sign mismatch is internal: the D=4 formulas in Sec. 3 are correct, but the general-D formula (42) does not reduce to them. An additional discrepancy appears in the γ2 term: the hypergeometric argument in (42) is +(q/r)^{2Δ}, while the correct D=4 expression (19) uses −(q/r)^{2Δ}; in D=5 this also violates the Maxwell equation. The same issues propagate into the dynamical solutions of Sec. 5.3 via Eq. (66). Since the general-dimensional static and dynamic solutions are the central claim of the paper, the formulas as printed do not establish the result; they need to be corrected (a missing minus sign on the γ1 term and the correct hypergeometric argument) and then re-checked against Eq. (5).

minor comments (4)

[Sec. 3, Eq. (13)] The hypergeometric function is written as 1F2[...] in Eq. (13), but the standard notation used throughout the paper is 2F1[...]; this appears to be a typographical inconsistency.
[Sec. 4.2, Eq. (47)] The parenthesis in the electric potential expression is unbalanced: "σ√(1− (q/r)^{2(D−3)}− 1" should read "στ√(1− (q/r)^{2(D−3)}) − 1".
[Sec. 5.5 and Sec. 3.2.1] The symbol β is reused in Eq. (68) for the evolution-equation parameter after being defined as β=k/q^2 in Sec. 3.2.1; the authors note this, but the notation remains confusing and should be changed or explicitly distinguished.
[Sec. 5.1, Eq. (59)] The expression "1/2(D−2)" in Eq. (59) is ambiguous; it should be written as (D−2)/2 to make clear the condition Δ=(D−2)/2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reverse-engineered construction is explicitly declared, and the dynamical solutions are derived from the field equations.

full rationale

The paper openly states its reverse-engineering strategy in the abstract and introduction: it chooses a scalar ansatz and then determines the scalar potential V(φ) and coupling Z(φ) so that the ansatz solves the EMD equations. This is a transparent construction method, not a disguised fit or a prediction equivalent to an input. The mass and electric charge are shown to be integration constants by direct substitution into the equations of motion, and the thermodynamics first law is verified explicitly. The dynamical extension is similarly non-circular: the scalar ansatz is promoted to φ(r,u) = 2k0 arcsinh[(a(u)/r)^Δ], the consistency conditions are solved to fix Δ and k0, and the remaining equation yields a nontrivial second-order ODE for a(u) whose static limit a = q reproduces the static solutions. The cited prior works [19,20,25,31] are used for techniques and analogous results, but the central claims are checked in-paper and no load-bearing uniqueness theorem or self-citation chain is invoked to forbid alternatives. Any possible sign discrepancy in the higher-dimensional electric potential would be a correctness issue, not a circularity issue.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The construction relies on chosen ansatz parameters (k0, Δ) and theory couplings (α, γ1, γ2, g) that are postulates rather than derived; the thermodynamic volume is imported from prior work. No new particles or fields are added.

free parameters (6)

k0 (dilaton amplitude)
Appears in the scalar ansatz φ=2k0 arcsinh[(q/r)^Δ] and in the resulting V and Z; chosen by hand, with examples k0=1,√3,√5,√7 in D=4.
Δ (dilaton exponent) = fixed to (D-2)/2 for k=0 dynamical solutions
Exponent in the scalar ansatz; arbitrary in static constructions, but the general-dimension dynamical solution forces Δ=(D-2)/2 (Eq. 64).
α (scalar potential coupling)
Coupling constant in V(φ); constrained to α≥0 for positive static mass, but otherwise free.
γ1, γ2 (gauge coupling parameters)
Parameters in Z(φ); sign of γ2 controls the evolution regimes; positivity of Z(φ) constrains them.
β = k/q² (topological coupling) = k/q²
For non-vanishing k, the topological parameter is fixed as k=βq², making k a derived quantity rather than an independent integration constant.
g (AdS/cosmological scale)
Sets the asymptotic AdS radius; treated as an input parameter of the Lagrangian.

assumptions (4)

ad hoc to paper Scalar ansatz φ=2k0 arcsinh[(q/r)^Δ] is imposed from the start (Eq. 9).
The reverse-engineering procedure guarantees the solution exists for the constructed V and Z, but the ansatz itself is a postulate.
ad hoc to paper Dynamical ansatz preserves the scalar form with q→a(u) (Eq. 57).
Key assumption for the dynamical solutions; consistency then forces Δ=(D-2)/2 for k=0 or D=4 for spherical.
domain assumption Thermodynamic volume formula from [31] applies to these hairy black holes.
Used to verify the first law; imported from prior literature, not derived in this paper.
domain assumption Einstein-Maxwell-dilaton Lagrangian (1) is the framework.
The central claim is existence of solutions within this class of theories.

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Cite this review

Pith. "Pith review of Static and Dynamic Charged Black Holes." pith.science (2026). https://pith.science/paper/RQZO23BE

@misc{pith2026190807970,
  author       = {Pith},
  title        = {Pith review of: Static and Dynamic Charged Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RQZO23BE}},
  note         = {Machine review of arXiv:1908.07970}
}

read the original abstract

We consider a class of Einstein-Maxwell-dilaton theories in general dimensions and construct both static and dynamic charged black holes. We adopt the reverse engineering procedure and make a specific ansatz for the scalar field and then derive the necessary scalar potential and the non-minimal coupling function between the scalar and the Maxwell field. The resulting static black holes contain mass and electric charge as integration constants. We find that some of the static solutions can be promoted to become dynamical ones in the Eddington-Finkelstein-like coordinates. The collapse solutions describe the evolution from a smaller charged black hole to a larger black hole state, driven by the scalar field.

Figures

Figures reproduced from arXiv: 1908.07970 by the authors.

**Figure 2.** Left panel: The evolution of the scalar function [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Phase Transition and Critical Phenomena of Charged Einstein-Maxwell-Scalar Black Holes
gr-qc 2025-05 conditional novelty 5.0 of 10

Charged EMs black holes show van der Waals-type phase transitions with mean-field critical exponents, and the phase transition disappears above a threshold scalar charge.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [31]

Feng and H

X.H. Feng and H. L¨ u, Butterﬂy velocity bound and reverse isoperimetric in- equality, Phys. Rev. D 95, no. 6, 066001 (2017) doi:10.1103/PhysRevD.95.066001 [arXiv:1701.05204 [hep-th]]. 26

arXiv 2017

[2] [1]

Maldacena, The large N limit of superconformal ﬁeld theories and supergrav- ity, Adv

J.M. Maldacena, The large N limit of superconformal ﬁeld theories and supergrav- ity, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]

arXiv 1998

[3] [2]

Gubser, I.R

S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non- critical string theory, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]

arXiv 1998

[4] [3]

Witten, Anti-de Sitter space and holography, Adv

E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. 23

arXiv 1998

[5] [4]

Balasubramanian, A

V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski- Vakkuri, B. M¨ uller, A. Sch¨ afer, M. Shigemori, and W. Staessens, Thermalization of strongly coupled ﬁeld theories, Phys. Rev. Lett. 106, 191601 (2011) [arXiv:1012.4753 [hep-th]]

arXiv 2011

[6] [5]

Balasubramanian, A

V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski- Vakkuri, B. M¨ uller, A. Sch¨ afer, M. Shigemori, and W. Staessens,Holographic thermal- ization, Phys. Rev. D 84, 026010 (2011) [arXiv:1103.2683 [hep-th]]

arXiv 2011

[7] [6]

Exact Black Hole Formation in Asymptotically (A)dS and Flat Spacetimes

X. Zhang and H. L¨ u, Exact black hole formation in asymptotically (A)dS and ﬂat spacetimes, Phys. Lett. B 736, 455 (2014) [arXiv:1403.6874 [hep-th]]

work page Pith review arXiv 2014

[8] [7]

Critical Behavior in a Massless Scalar Field Collapse with Self-interaction Potential

X. Zhang and H. L¨ u, Critical behavior in a massless scalar ﬁeld collapse with self- interaction potential, Phys. Rev. D 91, no. 4, 044046 (2015) [arXiv:1410.8337 [gr-qc]]

work page Pith review arXiv 2015