{"id":"3a654557-502f-4159-9c6d-61fe5db99c4a","arxiv_id":"1908.07970","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Exact charged black hole solutions, static and time-dependent, are built in a class of reverse-engineered Einstein-Maxwell-dilaton theories, with explicit collapse from a smaller to a larger black hole.","lead":"This paper constructs exact static and dynamic charged black hole solutions in Einstein-Maxwell-dilaton theories in any dimension, working backwards from a chosen scalar field profile to find the scalar potential and gauge coupling. Some of these solutions evolve from a smaller charged black hole into a larger one, providing rare analytic examples of black hole collapse.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The general-dimensional electric potential (42)/(66) has a sign error relative to the D=4 result (19)/(62); with the published sign the Maxwell equation fails for D≥5, so the 'general dimensions' exact-solution claim is not correct as written.","rationale":"The reader's weakest_assumption focused on the dynamical scalar ansatz φ(r,u)=2k0 arcsinh[(a(u)/r)^Δ], worrying that the true time-dependent profile might need a different radial form. That is not the load-bearing weakness: an exact solution is allowed to have this specific form provided it solves the full field equations, and spot checks such as a large-r expansion of the D=4 equations are consistent with the paper's consistency conditions. The more concrete, internal problem is the sign of the electric potential in general dimensions. The D=4 formulas (19)/(31) and the general-D formulas (42)/(49)/(66) disagree on the sign of the γ1 term, and substitution into the Maxwell equation shows that the general-D expressions as printed do not satisfy that equation. Since the abstract claims exact solutions in general dimensions, this affects the central claim directly. The issue is likely a typographical missing minus sign rather than a fatal obstruction, so the appropriate outcome is conditional acceptance pending correction and independent verification. A secondary concern, noted by the reader, is the imported thermodynamic volume formula; a numerical consistency check of the first law for the D=4, k0=1, γ2=0 example indicates dM ≠ T dS + Φ_e dQ_e + V_th dP with the published V_th, so the thermodynamics section also needs re-examination. However, the sign error in the electric potential is the most load-bearing because it undermines the exactness of the general-dimensional solutions as written.","tokens_in":17775,"tokens_out":47626,"duration_ms":450421,"concrete_test":"Take the D=5, γ2=0 solution given by Eq. (49) (with static a=q), substitute Z=3γ1(r^4+2q^4)/r^4 and σ=√(r^4+q^4)/r^2 from Eq. (48), and check whether the published ξ satisfies ξ' = QZ/(σ r^3). Direct differentiation shows the left side is −3γ1Q(3r^4+4q^4)/(2r^5√(r^4+q^4)) while the right side is +3γ1Q(r^4+2q^4)/(r^5√(r^4+q^4)). If the equality fails (as it does), repeat with ξ→−ξ; the resulting solution should satisfy the Maxwell equation, and Eq. (42) should be amended to reduce to Eq. (19) in D=4.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"With the static ansatz A = ξ dt, the Maxwell equation (5) integrates to ξ' = Q Z/(σ r^{D−2}) for the charge parameter Q. In D=4 the paper's own solution gives ξ = −γ1 Q σ/r for γ2=0 (Eqs. (31)/(19)), whose r-derivative is +γ1 Q/r^2, matching (5). The advertised general-D formula (42), and its dynamic counterpart (66), instead state ξ = +(D−2)γ1 Q σ/(2 r^{D−3}) + ..., with no minus sign. In the D=5 special case (49) with γ2=0 this gives ξ = +3γ1 Q√(r^4+q^4)/(2 r^4). Differentiating yields ξ' = −3γ1 Q(3r^4+4q^4)/(2 r^5√(r^4+q^4)), while (5) requires +3γ1 Q(r^4+2q^4)/(r^5√(r^4+q^4)). These are not equal, so the published D≥5 solutions do not satisfy Maxwell's equations. The error is internal: section 3's D=4 formulas are correct, but the general-D reduction does not reproduce them. A missing minus sign on the γ1 term would fix Maxwell's equation, so the construction is probably salvageable, but the central claim 'general dimensions' is not established by the formulas as printed.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18156,"tokens_out":11652,"duration_ms":108313,"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":[{"comment":"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).","section":"Sec. 4.1, Eq. (42); Sec. 5.3, Eq. (66)"}],"minor_comments":[{"comment":"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.","section":"Sec. 3, Eq. (13)"},{"comment":"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\".","section":"Sec. 4.2, Eq. (47)"},{"comment":"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.","section":"Sec. 5.5 and Sec. 3.2.1"},{"comment":"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.","section":"Sec. 5.1, Eq. (59)"}],"recommendation":"major_revision","confidential_remarks":"The general-dimensional sign error appears to be a typographical error rather than a fundamental flaw, since the D=4 sector and the associated first-law checks are correct. However, the manuscript in its current form does not establish the central claim of exact static and dynamic solutions in general dimensions. The paper is likely appropriate for publication in a journal such as Physical Review D or JHEP after the sign/argument errors in Eqs. (42) and (66) are fixed and the Maxwell equation is explicitly verified for the corrected solutions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThis 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.\n\nWhat 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.\n\nWhat 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.\n\nBottom 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.","headline":"Four-dimensional solutions are solid, but the general-D formulas carry a sign error that invalidates the central claim as printed.","tokens_in":18628,"tokens_out":8472,"would_cite":false,"duration_ms":101699,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Einstein-Maxwell-dilaton theories","charged black holes","dynamical black hole collapse","scalar hair","Eddington-Finkelstein coordinates","reverse engineering","black hole thermodynamics","holographic thermalization"],"falsifier":"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.","tokens_in":17563,"feed_emoji":"🕳️","tokens_out":5504,"duration_ms":252404,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exact charged black holes that grow from smaller to larger","feed_subtitle":"Static and time-dependent solutions in Einstein-Maxwell-dilaton gravity, with mass and charge as free constants.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the first exact analytic black-hole collapse solution that the charged dynamical solutions generalize.","marker":"[6]"},{"why":"Provides the general-dimension scalar hairy black hole framework whose special examples are generalized here.","marker":"[17]"},{"why":"Introduces the static and dynamic hairy planar black hole construction that the paper extends to the charged case.","marker":"[19]"},{"why":"Gives the neutral exact formation solutions and the evolution-equation technique used to analyze $a(u)$.","marker":"[20]"},{"why":"Supplies the apparent-horizon and global-structure analysis method used for the dynamical charged black holes.","marker":"[25]"}],"fun_headline_variants":["Exact charged black holes that grow over time","Charged black holes evolve from small to large exactly","Exact solutions show charged black holes growing","Small to large: exact charged black hole growth","Time-dependent charged black holes: exact growth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact charged black holes that grow over time","Charged black holes evolve from small to large exactly","Exact solutions show charged black holes growing","Small to large: exact charged black hole growth","Time-dependent charged black holes: exact growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":1195,"prompt_tokens":925,"completion_tokens":270,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":199}},"tokens_in":541,"tokens_out":270,"duration_ms":3573,"temperature":1.0,"reasoning_tokens":199,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:52:49.156651+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Exact Black Hole Formation in Asymptotically (A)dS and Flat Spacetimes","cited_arxiv_id":"1403.6874","evidence_quote":"Supplies the first exact analytic black-hole collapse solution that the charged dynamical solutions generalize."},{"cited_title":"Exact formation of hairy planar black holes","cited_arxiv_id":"1512.09145","evidence_quote":"Gives the neutral exact formation solutions and the evolution-equation technique used to analyze $a(u)$."},{"cited_title":"Global Structure of Exact Scalar Hairy Dynamical Black Holes","cited_arxiv_id":"1601.07246","evidence_quote":"Supplies the apparent-horizon and global-structure analysis method used for the dynamical charged black holes."}],"review_version":1}