Charged, rotating black holes in Einstein-Maxwell-dilaton theory

arxiv: 2506.15798 · v2 · pith:M432BHUNnew · submitted 2025-06-18 · 🌀 gr-qc · hep-th

Charged, rotating black holes in Einstein-Maxwell-dilaton theory

Carlos Herdeiro , Eugen Radu , Etevaldo dos Santos Costa Filho This is my paper

Pith reviewed 2026-05-19 08:39 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords black holesEinstein-Maxwell-dilatonrotating solutionsnumerical constructiondilaton couplingextremal limitsnon-uniqueness

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The pith

Numerically constructed charged rotating black holes exist in Einstein-Maxwell-dilaton theory for arbitrary dilaton coupling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to build electrically charged and rotating black holes that are asymptotically flat in Einstein-Maxwell-dilaton theory when the dilaton coupling constant gamma takes any real value. Only two special values of gamma previously allowed exact solutions, the Kerr-Newman case at gamma zero and the Kaluza-Klein case at gamma square root of three. The new families of solutions are mostly similar to Kerr-Newman black holes, but they display distinct behaviors at the zero-temperature limit depending on whether gamma is less than or greater than square root of three, and for larger gamma there appear hints that more than one black hole can have the same total mass, charge, and spin.

Core claim

The Einstein-Maxwell-dilaton equations admit families of charged rotating black hole solutions for arbitrary gamma, which can be constructed numerically. For 0 < gamma < sqrt(3) these solutions admit a zero-temperature limit which is regular in terms of curvature invariants but exhibits a pp-singularity. For gamma > sqrt(3) a different limiting behaviour is found and hints of black hole non-uniqueness for the same global charges appear.

What carries the argument

Numerical integration of the Einstein-Maxwell-dilaton field equations using a metric ansatz for stationary axisymmetric spacetimes with electromagnetic and dilaton fields.

Load-bearing premise

The numerical solutions obtained from the chosen metric ansatz and boundary conditions accurately converge to the true regular, asymptotically flat solutions of the Einstein-Maxwell-dilaton equations without introducing discretization artifacts or missing branches.

What would settle it

A comparison showing that the numerical solutions reduce exactly to the Kerr-Newman metric when gamma is set to zero, or the discovery of two distinct solutions with identical mass, charge and angular momentum for a gamma greater than sqrt(3).

read the original abstract

The asymptotically flat, electrically charged, rotating black holes (BHs) in Einstein-Maxwell-dilaton (EMd) theory are known in closed form for \textit{only} two particular values of the dilaton coupling constant $\gamma$: the Einstein-Maxwell coupling ($\gamma=0$), corresponding to the Kerr-Newman (KN) solution, and the Kaluza-Klein coupling ($\gamma=\sqrt{3}$). Rotating solutions with arbitrary $\gamma$ are known only in the slow-rotation or weakly charged limits. In this work, we numerically construct such EMd BHs with arbitrary $\gamma$. We present an overview of the parameter space of the solutions for illustrative values of $\gamma$ together with a study of their basic properties. The solutions are in general KN-like; there are however, new features. The data suggest that the spinning solutions with $0<\gamma<\sqrt{3}$ possess a zero temperature limit, which, albeit regular in terms of curvature invariants, exhibits a $pp$-singularity. A different limiting behaviour is found for $\gamma>\sqrt{3}$, in which case, moreover, we have found hints of BH non-uniqueness for the same global charges.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Numerical solutions fill the gap for rotating charged black holes in EMd theory at arbitrary gamma, with reported differences in zero-temperature limits and hints of non-uniqueness.

read the letter

The main takeaway is that the authors have produced numerical families of charged rotating black holes in Einstein-Maxwell-dilaton gravity that work for any value of the dilaton coupling constant gamma. This closes a gap that had only exact solutions at two special values and approximations elsewhere. They solve the coupled Einstein, Maxwell, and dilaton equations with a metric ansatz suited to stationary axisymmetric spacetimes and integrate outward from the horizon with appropriate boundary conditions. For several sample values of gamma they plot the allowed ranges of charge and spin, and they track how the temperature behaves as the solutions approach extremality. The data indicate that when gamma sits between zero and square root of three the zero-temperature end point remains finite in curvature but shows a parallel-propagated singularity along null geodesics. For gamma larger than square root of three the temperature limit looks different and the authors see evidence that two distinct solutions can carry identical mass, charge, and angular momentum. The strength of the paper lies in the breadth of the numerical survey and the clear separation of behaviors across the gamma threshold. Mapping the full parameter space even for a handful of gamma values already gives a useful reference that was missing from the literature. The main reservation concerns the numerical reliability. Because the new claims about pp-singularities and non-uniqueness come directly from the computed solutions, one needs to know that the discretization converges, that the residuals of the field equations are small across the domain, and that the solver has not overlooked other branches. The abstract is appropriately tentative, but without seeing the convergence plots or the direct checks against the gamma equals zero and gamma equals square root of three analytic solutions at overlapping parameters, it is difficult to gauge how firm the new features are. If the full manuscript contains those diagnostics at several resolutions and for multiple gamma, the results become much more convincing. Readers who study black hole solutions in scalar-tensor or string-derived theories will find this work directly relevant. It supplies concrete examples that can be used to test thermodynamic relations or to seed stability analyses. The paper is also a natural next step for anyone who has worked on the perturbative or slow-rotation versions. It deserves a serious referee because the topic is well-defined, the methods are standard in the field, and the reported differences in limiting behavior raise questions worth checking. I would recommend sending the manuscript to peer review, with the expectation that the referees will focus on the numerical validation and the interpretation of the non-uniqueness hints.

Referee Report

2 major / 1 minor

Summary. The paper numerically constructs asymptotically flat, electrically charged, rotating black holes in Einstein-Maxwell-dilaton theory for arbitrary dilaton coupling γ, extending beyond the known exact solutions at γ=0 (Kerr-Newman) and γ=√3 (Kaluza-Klein). It provides an overview of the parameter space for selected γ values, examines basic properties, and reports suggestive new features: a zero-temperature limit with pp-singularity (regular curvature invariants) for 0<γ<√3, different limiting behavior for γ>√3, and hints of non-uniqueness at fixed global charges for γ>√3.

Significance. If the numerical solutions are shown to be accurate and complete, the work would be significant for mapping the full solution space of EMd black holes and identifying γ-dependent phenomena such as singular zero-temperature limits and possible non-uniqueness, which are absent in the special cases. This could inform black hole thermodynamics and uniqueness questions in scalar-tensor theories with Maxwell fields.

major comments (2)

[Numerical methods] Numerical methods section: The central claims about zero-temperature limits, pp-singularities, and non-uniqueness rest on numerical integration of the field equations. Explicit residual checks on the Einstein-Maxwell-dilaton equations, resolution studies, and systematic validation against the γ=0 and γ=√3 limits across the full parameter space are required to rule out discretization artifacts or incomplete branch coverage.
[Results] Results section on limiting behavior: The reported difference in limiting behavior for γ>√3 versus 0<γ<√3, including hints of non-uniqueness at fixed charges, needs quantitative support such as tabulated global charges, horizon quantities, and error estimates showing that distinct solutions are not numerical duplicates.

minor comments (1)

[Discussion of zero-temperature limit] Clarify the precise definition and diagnostic used for the pp-singularity in the zero-temperature limit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help improve the presentation and validation of our numerical results. We address each major comment below and will incorporate the suggested additions in a revised version.

read point-by-point responses

Referee: [Numerical methods] Numerical methods section: The central claims about zero-temperature limits, pp-singularities, and non-uniqueness rest on numerical integration of the field equations. Explicit residual checks on the Einstein-Maxwell-dilaton equations, resolution studies, and systematic validation against the γ=0 and γ=√3 limits across the full parameter space are required to rule out discretization artifacts or incomplete branch coverage.

Authors: We agree that stronger numerical validation is needed to support the central claims. In the revised manuscript we will add explicit residual norms for the Einstein-Maxwell-dilaton equations, resolution studies with successive grid refinements, and direct comparisons of our numerical solutions against the exact Kerr-Newman (γ=0) and Kaluza-Klein (γ=√3) solutions over a representative range of charges and spins. These additions will quantify discretization errors and confirm that the reported limiting behaviors are not artifacts. revision: yes
Referee: [Results] Results section on limiting behavior: The reported difference in limiting behavior for γ>√3 versus 0<γ<√3, including hints of non-uniqueness at fixed charges, needs quantitative support such as tabulated global charges, horizon quantities, and error estimates showing that distinct solutions are not numerical duplicates.

Authors: We acknowledge the request for quantitative evidence. The revised results section will include tables listing global charges (M, J, Q), horizon quantities (area, surface gravity), and associated numerical error estimates for representative solutions in both γ regimes. For the non-uniqueness hints at γ>√3 we will present explicit pairs of solutions sharing the same asymptotic charges but differing in horizon parameters, together with error bars demonstrating that the differences exceed numerical uncertainty. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical solution of EMd field equations

full rationale

The paper numerically constructs asymptotically flat, charged, rotating black hole solutions in Einstein-Maxwell-dilaton theory for arbitrary dilaton coupling gamma by solving the coupled PDEs subject to fixed boundary conditions at the horizon and at infinity. Reported features such as zero-temperature limits (with pp-singularities for 0<gamma<sqrt(3)) and hints of non-uniqueness for gamma>sqrt(3) are computed outputs extracted from the converged numerical data, not quantities defined in terms of the inputs or obtained by fitting parameters to the target results. No self-definitional steps, fitted-input predictions, load-bearing self-citations, or ansatz smuggling appear in the derivation chain. The analysis is therefore self-contained against the Einstein-Maxwell-dilaton equations themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard Einstein-Maxwell-dilaton action and the assumption of asymptotic flatness and axisymmetry; no additional free parameters are fitted beyond the input coupling gamma, and no new entities are postulated.

axioms (1)

domain assumption The spacetime is asymptotically flat, stationary, and axisymmetric, and the metric and fields satisfy the Einstein-Maxwell-dilaton field equations with the given coupling gamma.
This is the foundational setup invoked throughout the numerical construction described in the abstract.

pith-pipeline@v0.9.0 · 5750 in / 1428 out tokens · 58247 ms · 2026-05-19T08:39:11.834335+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We numerically construct such EMd BHs with arbitrary γ... zero temperature limit... pp-singularity... hints of BH non-uniqueness for the same global charges.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The equations... reduced to a set of six coupled nonlinear elliptic partial differential equations for the functions Fi = {F0,F1,W; Φ;Aϕ,At}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

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

Scalarizations of magnetized Reissner-Nordstr\"om black holes induced by parity-violating and parity-preserving interactions
gr-qc 2026-04 unverdicted novelty 5.0

Magnetic fields lower the scalarization threshold for electromagnetic and gravitational Chern-Simons couplings but produce opposite trends on the two Gauss-Bonnet branches, with nonlinear terms converting exponential ...
Leading effective field theory corrections to the Kerr metric at all spins
gr-qc 2025-12 unverdicted novelty 5.0

Numerical solutions show that leading effective-field-theory corrections to the Kerr metric grow with spin and are largest near extremality.

Reference graph

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