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arxiv: 2506.10046 · v3 · submitted 2025-06-11 · 🌀 gr-qc · hep-th

3-dimensional charged black holes in f({Q}) gravity

G. G. L. Nashed , Emmanuel N. Saridakis This is my paper

Pith reviewed 2026-05-19 09:57 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords f(Q) gravityblack holescharged solutions3D gravitynon-metricityBTZ black holethermodynamicsgeodesics
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The pith

A novel charged black hole in three-dimensional f(Q) gravity has no continuous limit to the BTZ solution of general relativity.

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

This paper constructs exact charged and uncharged black hole solutions in (2+1)-dimensional f(Q) gravity by adopting a cubic form for the function f of the non-metricity scalar. Among them is a new asymptotically anti-de Sitter charged solution that cannot be obtained as a continuous deformation of the Banados-Teitelboim-Zanelli black hole known from general relativity. The higher-order non-metricity terms are responsible for this new geometry. The solutions display multiple horizons, a less severe central singularity, thermodynamically stable behavior, and stable photon orbits whose dynamics are influenced by the cubic corrections.

Core claim

The authors derive classes of spherically symmetric charged black hole solutions in f(Q) gravity. They find a novel charged solution that is asymptotically Anti-de Sitter but cannot be continuously deformed into a GR counterpart, arising purely from the higher-order nature of the non-metricity corrections. Geometrical properties include multiple horizons and a softer central singularity. Thermodynamic quantities confirm thermal stability, and geodesic analysis reveals stable photon orbits affected by the corrections.

What carries the argument

The cubic polynomial form of f(Q) in terms of the non-metricity scalar Q, which modifies the field equations to permit new black hole solutions not accessible in general relativity.

If this is right

  • The new solutions possess multiple horizons.
  • The central singularity is softer than in general relativity.
  • Thermodynamic quantities such as Hawking temperature, entropy, and heat capacity confirm thermal stability.
  • Geodesic motion features stable photon orbits, with the cubic corrections modifying the effective potentials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar non-metricity-driven deviations from general relativity may appear in other modified gravity models or in higher dimensions.
  • These solutions could serve as simplified test cases for studying quantum corrections to black hole thermodynamics in lower-dimensional gravity.
  • The orbital dynamics changes suggest possible observable signatures in accretion or light deflection around such black holes.

Load-bearing premise

The assumption that a cubic polynomial form for f(Q) is sufficient to produce exact solutions while preserving the required symmetries and asymptotic behavior.

What would settle it

Demonstrating a continuous parameter path that connects the novel charged solution to the BTZ black hole while preserving asymptotic AdS behavior would falsify the claim that the solution arises purely from higher-order non-metricity corrections.

Figures

Figures reproduced from arXiv: 2506.10046 by Emmanuel N. Saridakis, G. G. L. Nashed.

Figure 1
Figure 1. Figure 1: (a) The temporal component of the metric given by Eq. (29). (b) The spatial component of the metric given by Eq. (29). in GR. By taking the limit ϕ → 0, we obtain the AdS uncharged black hole given by (25). Moreover, we mention that although the metric (29) behaves asymptotically as AdS charged solution, which has different gtt and g rr components, the two solutions have coinciding Killing and event horizo… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The temperature given by Eq. (32).(b) The entropy given by Eq. (33). (c) The heat capacity given by Eq. (34). (d) The pressure given by Eq. (39). In all graphs we have used Λ = −1011 , m = 1, c2 = 102 , r0 = 0.1. In order to determine the entropy of the BTZ black hole, the area law can be applied as S = A 4 where A is the horizon area defined by A = R 2π 0 √gϕϕdϕ [PITH_FULL_IMAGE:figures/full_fig_p007… view at source ↗
Figure 3
Figure 3. Figure 3: (a) The potential (46) with L = 1 and ǫ = 0,ǫ 6= 0. (b) The second derivative of the potential (46) with L = 1 and ǫ 6= 0. D. Multi horizons We close our analysis by examiining the case where the charged black-hole solution exhibits multiple horizons. In order to achieve this we use the solution (29) where ϕ takes negative values. In this case the behavior of grr is shown in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 4
Figure 4. Figure 4: (a) The metric (29) of the multi-horizon solution. (b) The Hawking temperature of the multi-horizon solution. (c) The heat capacity of the multi-horizon solution. In all graphs we have used Λ = −0.1, ϕ = −0.1, m = 0.01. IV. DISCUSSION AND CONCLUSIONS In this study we have constructed and analyzed novel exact charged black-hole solutions in (2+1) dimensions within the framework of f(Q) gravity. Our approach… view at source ↗
read the original abstract

We present new exact charged black hole solutions in (2+1) dimensions within the framework of $f({Q})$ gravity, where ${Q}$ denotes the non-metricity scalar. By considering a cubic $f({Q})$ form we derive classes of charged and uncharged spherically symmetric solutions, and we identify conditions under which these reduce to the well-known Banados-Teitelboim-Zanelli (BTZ) black hole. Notably, our analysis reveals a novel charged solution that is asymptotically Anti-de Sitter (AdS) but cannot be continuously deformed into a GR counterpart, and thus arising purely from the higher-order nature of the non-metricity corrections. Furthermore, we explore the geometrical properties of the solutions, demonstrating the existence of multiple horizons and a softer central singularity compared to GR. Additionally, we calculate various thermodynamic quantities, such as Hawking temperature, entropy, and heat capacity, with results confirming thermal stability. Finally, a detailed study of the geodesic motion and the effective potentials unveils stable photon orbits, as well as the effect of cubic non-metricity corrections on orbital dynamics.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript derives new exact charged and uncharged spherically symmetric black hole solutions in (2+1)-dimensional f(Q) gravity by adopting a cubic polynomial form for f(Q). It identifies conditions under which some solutions reduce to the BTZ black hole, while highlighting a novel asymptotically AdS charged solution that cannot be continuously deformed to a GR counterpart and thus arises purely from higher-order non-metricity corrections. The work further examines geometric features including multiple horizons and a softer central singularity, computes thermodynamic quantities (Hawking temperature, entropy, heat capacity) confirming thermal stability, and analyzes geodesic motion with stable photon orbits influenced by the cubic corrections.

Significance. If the central claim of a non-deformable novel solution holds after verification, the paper would contribute to modified gravity by exhibiting black hole solutions intrinsically tied to non-metricity corrections in three dimensions, offering potential distinctions from Einstein gravity in lower-dimensional spacetimes along with supporting analyses of thermodynamics and orbital dynamics.

major comments (2)
  1. [Section on novel charged solution] The section presenting the novel charged AdS solution: the claim that this solution 'cannot be continuously deformed into a GR counterpart' and 'arising purely from the higher-order nature of the non-metricity corrections' is load-bearing for the paper's main novelty. The derivation solves the modified equations for nonzero cubic coefficients and notes the solution vanishes at c=0, but without explicitly substituting the limiting metric functions into the f(Q)=Q (or 3D Einstein) equations and showing they fail to hold for any choice of mass/charge parameters, it remains possible that the limit recovers a BTZ solution with adjusted parameters. This explicit check is required to support the central assertion.
  2. [Field equations and exact solutions] The derivation of the field equations and solutions: full intermediate steps from the f(Q) action through the variation to the modified Einstein equations for the cubic f(Q) and the chosen metric ansatz are not fully visible, making it difficult to verify that the presented solutions satisfy the equations. Inclusion of these steps (or an appendix) is needed to confirm the solutions are exact.
minor comments (2)
  1. [Introduction or solutions section] The explicit form of the cubic polynomial f(Q) = aQ + bQ^2 + cQ^3 (or equivalent) and the meaning of its coefficients should be stated at the outset of the solutions section for notational clarity.
  2. [Geometrical properties and geodesics] Figure captions for horizon plots or effective potential diagrams should include the specific parameter values used (e.g., values of a, b, c, mass, charge) to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We have addressed both major points by adding the requested explicit verification for the novel solution and including a full derivation appendix. These revisions strengthen the clarity and support for our central claims without altering the core results.

read point-by-point responses

  1. Referee: The section presenting the novel charged AdS solution: the claim that this solution 'cannot be continuously deformed into a GR counterpart' and 'arising purely from the higher-order nature of the non-metricity corrections' is load-bearing for the paper's main novelty. The derivation solves the modified equations for nonzero cubic coefficients and notes the solution vanishes at c=0, but without explicitly substituting the limiting metric functions into the f(Q)=Q (or 3D Einstein) equations and showing they fail to hold for any choice of mass/charge parameters, it remains possible that the limit recovers a BTZ solution with adjusted parameters. This explicit check is required to support the central assertion.

    Authors: We agree that an explicit check is necessary to rigorously support the novelty claim. We have now substituted the c=0 limit of the metric functions into the 3D Einstein equations (equivalent to f(Q)=Q) and explicitly verified that no choice of mass and charge parameters satisfies the equations, as the resulting curvature terms do not balance for the given asymptotic AdS behavior. This calculation is added as a new paragraph in the relevant section, with full algebraic details placed in Appendix B. The solution remains intrinsically tied to the cubic non-metricity corrections. revision: yes

  2. Referee: The derivation of the field equations and solutions: full intermediate steps from the f(Q) action through the variation to the modified Einstein equations for the cubic f(Q) and the chosen metric ansatz are not fully visible, making it difficult to verify that the presented solutions satisfy the equations. Inclusion of these steps (or an appendix) is needed to confirm the solutions are exact.

    Authors: We acknowledge that the intermediate steps were not fully expanded in the main text. To address this, we have added a comprehensive Appendix A that details the variation of the f(Q) action, the resulting modified field equations for the cubic polynomial form, the substitution of the spherically symmetric metric ansatz, and all algebraic manipulations leading to the exact solutions. This appendix confirms that the presented metrics satisfy the equations identically. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained from action to exact solutions

full rationale

The paper obtains its charged black hole solutions by direct substitution of a cubic f(Q) ansatz and a spherically symmetric metric into the field equations derived from the f(Q) action. The novel AdS solution and its non-deformability to the BTZ case are identified by solving these modified equations and inspecting the c→0 limit of the resulting metric functions. No step reduces a central prediction to a fitted input, self-citation chain, or definitional equivalence; the derivation remains independent of the target result and is self-contained against the modified gravity equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The cubic f(Q) introduces free parameters whose specific values are chosen to obtain exact solutions; the framework assumes the standard f(Q) action and spherical symmetry in 3D.

free parameters (1)
  • coefficients in cubic f(Q)
    The cubic polynomial form of f(Q) contains free coefficients that are tuned to yield exact charged solutions.
axioms (1)
  • domain assumption The gravitational action is an arbitrary function f of the non-metricity scalar Q in three dimensions.
    Standard starting point for f(Q) gravity theories invoked to derive the modified field equations.

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