Rotating and non-rotating AdS black holes in $f({\cal T})$ gravity non-linear electrodynamics

Gamal G.L. Nashed; Salvatore Capozziello

REVIEW 4 major objections 5 minor 103 references

Rotating and non-rotating AdS black holes in $f({\cal T})$ gravity non-linear electrodynamics

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper derives exact d-dimensional AdS black holes in quadratic f(T) teleparallel gravity with nonlinear electrodynamics, with a milder central singularity than in General Relativity and entropy not proportional to horizon area.

desk verdict New family of AdS black holes in f(T)+NLED, but the singularity and entropy claims need correction and the diagonal vielbein needs a good-tetrad check. read the letter →

arxiv 1908.07381 v2 pith:P4POE2EP submitted 2019-08-17 gr-qc hep-th

Salvatore Capozziello , Gamal G.L. Nashed This is my paper · ORCID

classification gr-qchep-th PACS 04.50.Kd97.60.Lf98.80.-k

keywords f(T)gravityteleparallelnonlinearelectrodynamicsAdSblackholesexactsolutionsholethermodynamicstorsionentropy

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 derives exact d-dimensional anti-de Sitter black hole solutions in quadratic f(T) teleparallel gravity coupled to nonlinear electrodynamics, with $f(T)=a_0+a_1T+a_2T^2$, where $T$ is the torsion scalar. These solutions generalize the previously known charged f(T) black holes with linear electrodynamics by adding higher-order charge terms controlled by a single nonlinearity parameter $q_1$, and they reduce to the earlier solutions when $q_1$ vanishes. The central singularity is milder than the corresponding singularity in General Relativity. The entropy, computed from the f(T) area formula, is not proportional to the horizon area and can be negative unless $q_1$ is constrained. Rotating versions are obtained by a global coordinate transformation, and the paper studies heat capacity, phase transitions, and Gibbs free energy.

What carries the argument

The argument rides on three pieces: the quadratic action $f(T)=a_0+a_1T+a_2T^2$; the diagonal vielbein ansatz (17) in cylindrical coordinates, which fixes the metric (18); and nonlinear electrodynamics in the dual Legendre representation, with the chosen potential $\aleph(r) = -P \,\mathrm{sech}^2(q_1/((d-3)m r^{d-3}))/r^{2(d-2)}$. The hyperbolic-secant form is what produces the higher-order charge terms and the deviation from linear electrodynamics; taking $q_1=0$ eliminates them. The field equations then fix $A(r)$, $g(r)$, and the gauge potential $q(r)$ self-consistently. The thermodynamic conclusion follows from the f(T) entropy formula $S=(1/4)A f_T$ with $f_T=a_1+2a_2T$, which makes the entropy depend on the torsion corrections rather than on area alone.

What would settle it

Take the same quadratic f(T) action and the same nonlinear electrodynamics potential, but solve the field equations in a non-diagonal cylindrical tetrad of the type recommended for f(T) gravity by refs [59,60], and check whether the metric function (33) still satisfies them; if the equations change, the claimed solution, milder singularity, and non-area entropy are frame artifacts.

Watch

Extended reading notes

Core claim

The paper's central claim is that new exact charged black hole solutions exist in quadratic teleparallel equivalent gravity with nonlinear electrodynamics. For the metric (18) in d dimensions, the field equations yield the metric function $A(r)$ with the asymptotic expansion $A(r) \approx \Lambda_{\mathrm{eff}} r^2 - M/r^{d-3} + Q^2/r^{2(d-3)} + Q_{14}/r^{3d-8} + Q_{24}/r^{4(d-3)}$, where the monopole and quadrupole terms are accompanied by higher-order terms sourced by the nonlinear electromagnetic field. Setting the parameter $q_1=0$ returns the linear-electrodynamics solution of ref [1], so the family is a one-parameter nonlinear electrodynamics deformation. Near $r=0$ the invariants $(K, R_{\mu\nu}R^{\mu\nu})$ behave as $r^{-4(d-2)}$ and $(R, T)$ as $r^{-2(d-2)}$, instead of the steeper GR/TEGR falloffs $r^{-2d}$ and $r^{-d}$, so the singularity is milder. The entropy, defined as $S=(1/4)A f_T$, is not proportional to the area and can be negative for some $q_1$ unless a constraint is imposed. The rotating counterpart is generated by a global coordinate transformation, preserving local geometry but changing global properties.

Load-bearing premise

The paper's conclusions stand or fall on whether the diagonal cylindrical coordinate frame used for the calculation is an allowed frame for f(T) gravity, since f(T) is not invariant under local frame rotations and an unallowed frame would make the resulting metric and its thermodynamic properties artifacts of that choice.

Editorial extensions

If this is right

Setting $q_1=0$ reproduces the linear-electrodynamics charged AdS black holes of ref [1], so the new family is a one-parameter nonlinear electrodynamics deformation of that baseline.
The curvature and torsion invariants diverge near $r=0$ as $r^{-4(d-2)}$ and $r^{-2(d-2)}$ rather than $r^{-2d}$ and $r^{-d}$, so the central singularity is softer in every dimension.
The entropy is proportional to $f_T$ rather than to the horizon area alone, and it can become negative unless the nonlinearity parameter $q_1$ satisfies the bound derived at the event horizon.
The heat capacity is negative below the degenerate horizon and positive above it, with an infinite discontinuity at the degenerate horizon, indicating a second-order phase transition.
The rotating AdS solutions are generated by a global coordinate transformation of the static solution, so they preserve local geometry while changing global properties such as horizon identification.

Reading between the lines

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

If the diagonal frame is legitimate, the softened singularity suggests that nonlinear electrodynamics in f(T) gravity can act as a low-energy mechanism for regularizing black hole interiors, though the invariants still diverge and the paper does not prove regularity.
In the TEGR limit ($a_1\to 1$, $a_2\to 0$), $f_T\to 1$ and the entropy formula reverts to the area law; verifying that the derived expression (43) has this limit would be a consistency test the paper leaves implicit.
The higher-order charge terms $Q_{14}$ and $Q_{24}$ are not universal features of f(T) plus nonlinear electrodynamics but are inherited from the hand-chosen hyperbolic-secant form of $\aleph$; testing other NLED functions would reveal which singularity-softening effects are robust.
Because the rotating metric comes from a global coordinate transformation, the thermodynamics of the rotating family may differ from the static one; a direct computation of rotating horizon quantities would settle whether the non-area entropy persists there.

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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

Colleague — quick read of 1908.07381.

The paper does add something real: a d-dimensional family of charged AdS black holes in quadratic f(T) gravity with nonlinear electrodynamics, parametrized by q1, which reduces to the authors' earlier Maxwell solution [1] when q1→0. That is a legitimate extension, and the asymptotic expansion for A(r) and the gauge field is explicit. The dual-formulation method is standard and the reductions to known limits are useful checks. The rotating version via coordinate transformation is the usual trick.

The soft spots are in the interpretation, not the algebra (as far as I can tell). First and most serious: the diagonal vielbein (17) is adopted without showing it is a 'good tetrad' for f(T). The paper's own introduction cites [59,60] noting diagonal ansätze are unsuitable in spherically symmetric f(T); cylindrical coordinates may be different, but the authors report only the non-vanishing components (21)-(24) and never state the off-diagonal components of (13) vanish. If those don't vanish, the solution isn't a solution of the same f(T) theory. That is a load-bearing gap and needs an explicit check.

Second, the 'milder singularity' claim contradicts the paper's own exponents: for the solution, (K, RμνRμν) ~ r^{-4(d-2)}; for GR/TEGR charged black holes they give r^{-2d}. At d=4 these are equal, and for d>4 the solution's singularities are stronger, not milder. The comparison in the text as written is wrong.

Third, the thermodynamics has internal inconsistencies: (42) gives S ∝ r_b^2 f_T, but (43) has a different normalization with Ω_{d-2} r_b^{d-2}/6 and terms that don't look like f_T expanded; there are also typographical problems in (51) with an undefined c. The q1→0 limit also looks singular for the mass parameter M, which conflicts with the claim that this limit returns to the Maxwell solution.

The circularity from choosing the sech ansatz (30) and the constraints is mild; that's how these constructions work, and it doesn't invalidate the solution.

Bottom line: the derivation is plausible and the literature will want to know whether the diagonal tetrad passes a full check. A serious referee should demand that check, plus corrections to the singularity comparison, entropy normalization, and q1 limit. I'd send it to review with a request for major revision, not desk-reject it. For a reading group, maybe, mainly if someone is actively working on f(T) black holes. I wouldn't cite it until the tetrad and thermodynamics issues are cleared up.

Referee Report

4 major / 5 minor

Summary. The paper studies d-dimensional static charged AdS black holes in f(T) = a0 + a1 T + a2 T^2 teleparallel gravity coupled to nonlinear electrodynamics. Starting from a diagonal vielbein in cylindrical coordinates, the authors derive a formal solution (31) and then impose constraints N = 0 and 2 a2 c1^2 c2 = a1 P to obtain the simplified metric (33), whose asymptotic expansion contains an effective cosmological term, a mass term, a monopole term Q^2, and higher-order terms Q14 and Q24. The electric field (34) reduces to the Maxwell-field result of Ref. [1] when q1 -> 0. The paper also constructs a rotating counterpart by the coordinate transformation (35), computes invariants near r = 0, and studies entropy, Hawking temperature, heat capacity, and Gibbs free energy, concluding that the entropy is not proportional to the horizon area and can be negative.

Significance. Charged AdS black hole families in quadratic f(T) gravity coupled to nonlinear electrodynamics are of interest for modified-gravity phenomenology, and this paper provides explicit formulas together with a direct reduction to the known Maxwell solution of Ref. [1]. If the central solution is genuine, the q1 parameter gives a tunable nonlinear-electrodynamics correction and the thermodynamic analysis is a concrete extension of earlier work. However, the physical claims are presently conditional: the use of a diagonal vielbein is not justified in a theory that is not local-Lorentz invariant, the singularity comparison is internally inconsistent, and the thermodynamic formulas contain coefficient and topology inconsistencies. These issues affect the central claims of the paper, and the manuscript therefore requires substantial revision.

major comments (4)

[Section III, Eq. (17)] The entire static solution is derived from the diagonal vielbein (17), but the paper does not demonstrate that this is a 'good tetrad' for the f(T) field equations. The introduction explicitly cites Refs. [59,60] for the statement that diagonal ansatze are unsuitable in spherically symmetric f(T) setups and Ref. [61] for a non-diagonal good tetrad. Equations (21)-(24) report only diagonal components of (13); the off-diagonal components are never shown to vanish for the resulting A(r), g(r), q(r), and \aleph(r). Since f(T) gravity is not local-Lorentz invariant, a different vielbein for the same metric satisfies different equations, so unless the full set of components of (13) is checked, the metric (18) with (33) cannot be claimed to be a solution of the theory. The rotating vielbein (38) inherits the same problem.
[Section III, paragraph after Eq. (34)] The claim that the central singularity is 'much milder' than in GR/TEGR contradicts the paper's own exponents. For the new solution the invariants are reported to behave as (K, R_{\mu\nu}R^{\mu\nu}) ~ r^{-4(d-2)} and (R, T) ~ r^{-2(d-2)}, while the GR/TEGR Maxwell values are given as r^{-2d} and r^{-d}. At d = 4 these exponents coincide, and for d > 4 one has 4(d-2) > 2d and 2(d-2) > d, so the new singularity is in fact more singular, not milder. The comparison should be redone with the correct d-dimensional GR exponents and the claim corrected or removed.
[Eqs. (42) and (43)] The entropy calculation has an internal coefficient mismatch. For d = 4, Eq. (42) gives S = \pi r_b^2 f_T, whereas the leading term of Eq. (43) is (\Omega_2/6) r_b^2 a1 = (2\pi/3) r_b^2 a1. In addition, Eq. (43) uses \Omega_{d-2} r_b^{d-2} as the horizon area, but the line element (18) is written in cylindrical coordinates with infinite \xi directions, so the horizon area is not the volume of a unit (d-2)-sphere unless compactification is assumed. The positivity constraint on q1 and the negative-entropy conclusion depend on this approximate formula and need to be re-examined.
[Section III.A, Eqs. (30)-(33)] The physical content attributed to the nonlinear electrodynamics is largely inherited from the ansatz rather than derived. The dual function \aleph(r) is fixed by hand to the sech form (30), and the constraints N = 0 and 2 a2 c1^2 c2 = a1 P are imposed to obtain the simplified solution; the higher-order terms Q14 and Q24 and the q1 corrections in (34) are therefore direct consequences of this input. The abstract's statement that these higher-order terms have 'their source' in the NLED field should be qualified, since the theory does not single out the chosen \aleph(r).

minor comments (5)

[Eq. (21)] The angular component is garbled as '... = \zeta_{\xi_{d-n-2}}^{\xi_{d-n-2}} = f_{TT}[...]'; the equation should be typeset cleanly with all indices explicit.
[Section III.A, text near Eq. (22)] The sentence 'Eq. (22) is a second-order algebraic equation and it gives T = T0 = const' is confusing because the later non-constant-T solution also uses Eq. (22) through the relation (32); the logical structure of the two branches should be clarified.
[Section IV, Eqs. (35)-(40)] The rotating solution is presented as a coordinate transformation of the static vielbein, but the paper does not state explicitly how the field equations behave under this transformation in f(T) theory; in particular the sentence at the end of Section IV that 'the torsion components are vanishing' is unexplained and should be removed or expanded.
[Eq. (49)] The heat capacity is written with symbols \alpha and c that are not defined consistently in the surrounding text; the calculation should be checked for notational consistency.
[Abstract and general text] There are several typographical and grammatical issues, including 'this inanition' in the abstract, which should be corrected in a careful revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the solution is an honest integration of the explicitly chosen f(T)+NLED field equations, with the main caveat being frame validity rather than circular reasoning.

full rationale

The derivation chain is self-contained in the relevant sense. The only non-derived input is the NLED ansatz for the auxiliary function in Eq. (30), but the paper states it openly as a choice ('let us fix the arbitrary function to have the form') and then solves the stated field equations (21)-(24) for A, g, q. The constraints N=0 and 2 a2 c1^2 c2 = a1 P are algebraic parameter restrictions that make the integrals tractable; they are not fitted to the output metric, and the asymptotic coefficients Q14, Q24 are functions of the already chosen constants. The comparison with the cited paper for q1=0 is a direct limit check, not a load-bearing self-citation. The entropy non-proportionality follows from the adopted external formula S = (1/4) A f_T, Eq. (42), not from a circular redefinition. The serious concern raised in the introduction, that diagonal vielbeins may be bad tetrads in f(T) gravity (Refs. [59,60]), is a mathematical correctness question about whether all components of Eq. (13) vanish for the ansatz (17), not a circularity; it does not reduce the paper's output to its input by construction.

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

The derivation rests on: the standard f(T) field equations; a diagonal vielbein chosen without a good-tetrad check; the hand-picked sech NLED function (30) borrowed from Ayon-Beato [64]; and constraints N=0 and 2a2 c1^2 c2 = a1P imposed to close the system. Free parameters include a1, a2, P, q1, m, and integration constants c1, c2, c3; the charge q is determined from them. No new physical entities are introduced. The genuinely new content is the q1-dependent extension in d dimensions of the Maxwell solutions of [1].

free parameters (5)

a1 and a2 (a0 fixed by N=0) = free model parameters
Coefficients of the quadratic f(T); the constraint N=0 sets a0 = -a1^2/(12a2), so a0 is not independent.
P = free amplitude in Eq. (30)
Amplitude of the sech-shaped NLED dual function; no independent determination.
q1 = free; range restricted only by entropy positivity
Controls deviation from Maxwell theory; all new terms in the solution vanish or reduce when q1 goes to 0.
m = free mass scale in the sech argument
Appears in the argument q1/((d-3)m r^{d-3}) of Eq. (30); not fixed by first principles.
c1, c2, c3 = integration constants
Integration constants of the field equations; the relation 2a2 c1^2 c2 = a1P is imposed to obtain Eq. (34), leaving fewer independent combinations.

assumptions (5)

domain assumption The vielbein field equations (13)-(15) derived from action (8) are correct and the diagonal vielbein (17) is a legitimate frame for them.
Sections II-III. f(T) field equations are not local-Lorentz-invariant and depend on the vielbein gauge (refs [59,60]); the paper does not establish that (17) is a good tetrad.
ad hoc to paper The NLED dual function ℵ(r) is fixed to the sech form of Eq. (30) by hand.
Section III: 'We can assume a given form for the arbitrary function ℵ(r)'; the d=4 form is borrowed from Ayon-Beato [64].
ad hoc to paper The constraints N=0 (a0=-a1^2/12a2) and 2a2 c1^2 c2 = a1P are imposed so that the simplified solution (33)-(34) exists.
Section III: 'if we assume the constraint N=0' and 'To get this result, we have put 2a2 c1^2 c2 = a1P'.
domain assumption The entropy formula S = A f_T/4 of Eq. (42), taken from ref [85], applies to this solution.
Section V. The formula is quoted from Miao-Li-Miao [85]; its normalization appears inconsistent with the d-dimensional expression (43).
domain assumption The rotating vielbein (38) satisfies the f(T) field equations because it is obtained by the coordinate transformation (35).
Section IV. The field equations are not explicitly checked; the argument relies on global-property results [69,70].

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Pith. "Pith review of Rotating and non-rotating AdS black holes in $f({\cal T})$ gravity non-linear electrodynamics." pith.science (2026). https://pith.science/paper/P4POE2EP

@misc{pith2026190807381,
  author       = {Pith},
  title        = {Pith review of: Rotating and non-rotating AdS black holes in $f(\cal T)$ gravity non-linear electrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4POE2EP}},
  note         = {Machine review of arXiv:1908.07381}
}

abstract

We derive new exact charged $d$-dimensional black hole solutions for quadratic teleparallel equivalent gravity, $f({\cal T})=a_0+a_1{\cal T}+a_2{\cal T}^2$, where $\cal T$ is the torsion scalar, in the case of non-linear electrodynamics. We give a specific form of electromagnetic function and find out the form of the unknown functions that characterize the vielbeins in presence of the electromagnetic field. It is possible to show that the black holes behave asymptotically as AdS solutions and contain, in addition to the monopole and quadrupole terms, other higher order terms whose source is the non-linear electrodynamics field. We calculate the electromagnetic Maxwell field and show that our d-dimensional black hole solutions coincide with the previous obtained one \cite{2017JHEP...07..136A}. The structure of the solutions show that there is a central singularity that is much mild in comparison with the respective one in General Relativity. Finally, the thermodynamical properties of the solutions are investigated by calculating the entropy, the Hawking temperature, the heat capacity, and other physical quantities. The most important result of thermodynamics is that the entropy is not proportional to the area of the black hole. This inanition points out that we must have a constrain on the quadrupole term to get a positive entropy otherwise we get a negative value.

Figures

Figures reproduced from arXiv: 1908.07381 by the authors.

**Figure 2.** FIG. 2. Schematic plots of the degenerate horizons of soluti [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. Schematic plot of the heat capacity shows the locally [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Pith tools

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