{"id":"f88c1ccf-6d17-444c-b3ee-7c425336ce7b","arxiv_id":"2504.19700","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"New families of regular black hole solutions are derived in (2+1)-dimensional f(R,T) gravity with nonlinear electrodynamics, generalizing earlier results and showing that energy-momentum is not conserved.","lead":"The paper builds new mathematical models of black holes in a simplified three-dimensional toy universe with a modified gravity called f(R,T). It finds the first regular (singularity-free at the center) black hole solutions in this framework, though distant regions show growing curvature that the authors flag as unusual.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Regularity claim hinges on an ad hoc electric field and diverging asymptotics; the central 'first regular black hole' claim needs qualification.","rationale":"The reader's verdict is CONDITIONAL with medium correctness risk, and their weakest_assumption correctly identifies the ad hoc electric field ansatz (24) as the load-bearing element. The body of the paper explicitly acknowledges the asymptotic divergence of curvature for lambda != 0 and the discrete lambda restrictions, so the central 'first regular black hole' claim is only demonstrated for a restricted parameter set with center-only regularity. These are genuine limitations, not internal inconsistencies, and they are stated in the text. My concrete test would settle whether the restricted lambda values are essential for regularity or merely a computational convenience. The attack, test, and verdict all align with the reader's assessment, so agreement is 'agree' and verdict_should_be remains CONDITIONAL. No ad hominem or theatrical language is used; the concern is routed through correctness risk rather than circularity or soundness. The paper does deliver new solution families and recovers known GR limits, which is credit where due, but the headline claim as stated in the abstract overreaches relative to the body.","tokens_in":17771,"tokens_out":1759,"duration_ms":13461,"concrete_test":"Re-derive the metric function b(r) for the beta=1 case without the hypergeometric truncation to polynomial form, i.e., for a generic negative lambda not of the form -kappa^2/(n+1/2), and numerically integrate Eq. (45) to see whether the event horizon and the regularity at r=0 persist. If horizons disappear or curvature at r=0 diverges for non-truncated lambda values, the central claim of 'first regular black hole solutions' is restricted to a measure-zero parameter set. Also check whether the Ricci and Kretschmann scalars diverge at infinity for lambda != 0 by direct asymptotic expansion of the b(r) obtained from Eqs. (45) and (48); the figures already suggest this, but a closed-form asymptotic test would confirm the abstract's overstatement.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim, 'first regular black hole solutions in (2+1)-dimensional f(R,T) gravity', rests on the specific electric field ansatz (24), E(r)=q r^alpha (r^beta+a^beta)^{-(alpha+1)/beta}, which is chosen 'inspired by' Ref. [38] without derivation from a Lagrangian or gauge principle. Every metric and regularity property follows from this hand-picked profile; a different E(r) could change the existence or nature of horizons and the regularity at r=0. Moreover, the paper itself states that for all lambda != 0 the Ricci and Kretschmann scalars diverge as r -> infinity (Figs. 3 and 6), so 'regular' is only true at the center, not globally. The abstract's unqualified 'regular black hole' claim is therefore stronger than what the body demonstrates. In addition, the f(R,T) solutions are explicitly restricted to discrete negative lambda values (lambda = -kappa^2/(n+1/2) for beta=1, lambda = -kappa^2/(2n-1/2) for beta=2) to make hypergeometric functions polynomial, so the generality of the 'first regular' claim is not established. These restrictions are acknowledged in the text, but they limit the central claim to a sparse parameter set with asymptotically divergent curvature.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies static, circularly symmetric geometries in (2+1)-dimensional General Relativity and in f(R,T) gravity with f(R,T)=R-2Λ+λT, both minimally coupled to nonlinear electrodynamics. The authors derive the field equations and a first integral of the gauge equation, then propose the electric-field profile E(r)=q r^α (r^β+a^β)^{-(α+1)/β} (Eq. 24), which has the Maxwell limit q/r at large r. For λ=0 they obtain the NED Lagrangian and metric function, recovering known solutions for α=3, β=2 and α=2, β=1, presenting a new solution for α=β=2 with explicitly smooth Ricci and Kretschmann scalars, and giving a general α=β+1 family. For λ≠0 they construct solutions for β=1 and β=2, selecting discrete negative values of λ to make hypergeometric functions polynomial, and analyze horizons, curvature scalars, the trace of the energy-momentum tensor, and the non-conservation of T_{μν}. The paper claims the first regular black hole solutions in (2+1)-dimensional f(R,T) gravity, but the body shows that for every λ≠0 considered the curvature scalars diverge as r→∞, so the regularity is only at the center. The central construction is explicit and reproduces known GR limits, but the headline claim and the generality of the f(R,T) solutions need to be qualified.","tokens_in":17982,"tokens_out":4738,"duration_ms":49641,"significance":"If the results are taken with the necessary qualifications, the paper provides a useful explicit family of (2+1)-dimensional regular-at-the-center black hole solutions in GR, including a genuinely new α=β=2 solution, and the first concrete f(R,T) examples of this type. The derivation is constructive: the electric-field ansatz is an input, and the NED Lagrangian and metric are solved from the field equations, with the known solutions recovered as limiting cases. The explicit curvature invariants for the new GR solution and the analysis of the non-conservation of the energy-momentum tensor are valuable and checkable. However, the significance is diminished because the abstract's unqualified 'regular black hole' claim is stronger than what the body demonstrates: for λ≠0 the curvature diverges at infinity, and the f(R,T) construction is restricted to a sparse set of discrete negative λ values. These are correctness-of-claim issues rather than flaws in the individual derivations, and they can be addressed by careful rewriting.","major_comments":[{"comment":"The central claim of 'regular black hole solutions' is stronger than the body supports. For every λ≠0 considered, the Ricci and Kretschmann scalars diverge as r→∞ (Figs. 3 and 6), and the conclusion states that 'the general trend is to intensify the strength of gravity at far distances despite describing regular black holes.' In the standard nomenclature, a regular black hole is a globally nonsingular spacetime, not merely one with finite curvature at r=0. Please qualify the abstract and conclusion accordingly (e.g., 'regular at the center' or 'finite curvature at r=0'), or provide an argument that the asymptotic divergence is an artifact that can be removed by a physical renormalization.","section":"Abstract and Section III.C, Figs. 3 and 6"},{"comment":"The entire construction, including the regularity at r=0 and the existence of a single event horizon, rests on the ad hoc electric-field profile (24), which is chosen 'inspired by' Ref. [38] without derivation from a NED Lagrangian or a gauge principle. The paper should state explicitly that the regularity is a consequence of this ansatz and discuss what general conditions on E(r) near r=0 (e.g., a power-law falloff) would guarantee center regularity, so that the robustness of the result with respect to other profiles can be assessed.","section":"Section III.A, Eq. (24)"},{"comment":"The f(R,T) solutions are constructed only for discrete negative values of λ chosen to terminate the hypergeometric functions: λ=-κ²/(n+1/2) for β=1 and λ=-κ²/(2n-1/2) for β=2, with the additional bound -6<λ/κ²<0 for β=2. Therefore the phrase 'first regular black hole solutions in (2+1)-dimensional f(R,T) gravity' is demonstrated for a sparse set of parameter values, not for generic nonzero λ. The paper should clearly frame these as particular solutions rather than a general class, which is especially important because the abstract and introduction do not mention this restriction.","section":"Section III.C, Eqs. (44)-(48)"},{"comment":"The Maxwell-limit condition is applied to the electric field profile rather than to the full matter Lagrangian. For λ≠0 the NED Lagrangian does not approach the Maxwell Lagrangian at infinity; instead, the curvature scalars and the trace T grow with r. This distinction should be made explicit in the abstract and in the discussion of the asymptotic behavior, since it directly bears on whether the solutions can be considered asymptotically AdS or asymptotically Maxwell in the usual sense.","section":"Section III.C and Section IV"}],"minor_comments":[{"comment":"The label 'Kretchmann scalar' should be corrected to 'Kretschmann scalar' in both figure panels and in the surrounding text.","section":"Figures 3 and 6"},{"comment":"The condition α>1 and β>0 is stated only after Eq. (32), but the expression for L(r) in Eq. (29) contains factors like (α-1) in denominators; the domain of validity should be stated together with the formula.","section":"Eq. (29) and surrounding text"},{"comment":"The discussion of the Maxwell case around Eq. (14) would benefit from an explicit statement that the normalization of the integration constant e is chosen to reproduce E=q/r for all λ, since this choice is central to the subsequent ansatz.","section":"Section II.B"},{"comment":"The parameter list for the ansatz (24) is given as α, β, and a, but the abstract and introduction refer to the 'Maxwell limit condition'; a very short explanation of why the chosen form preserves the q/r falloff would improve readability.","section":"Section III.A"}],"recommendation":"major_revision","confidential_remarks":"The paper is a standard construction of solutions in modified gravity, and the derivation appears internally consistent. The main issue is the mismatch between the unqualified 'first regular black hole' claim and the body's own admission of curvature divergence at infinity for all nonzero λ. This can be fixed by rewriting the abstract, introduction, and conclusion to say 'regular at the center' and 'particular solutions for selected λ values.' I would also encourage the editor to ask the authors to place their f(R,T) examples in the context of the known (2+1)-dimensional NED regular black holes of Refs. [38,39] more explicitly, so that the novelty is stated as 'center-regular solutions in f(R,T) gravity' rather than a claim about global regularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things. First, the paper is genuinely constructive: the α=β=2 GR solution (Eqs. 36–37) and the α=β+1 family (Eqs. 41–42) are new, and the β=1,2 f(R,T) solutions (Eqs. 44–45 and 47–48) are, as far as I can tell, the first in that theory in 2+1 dimensions. The field equations are derived cleanly, the λ→0 limits reduce to Cataldo-Garcia and He-Ma, and the new GR solution's Ricci and Kretschmann scalars are explicitly smooth at the origin. Credit where due.\n\nSecond, the abstract says 'regular black hole solutions' without qualification, but the body shows that for every λ≠0 both curvature scalars diverge as r→∞ (Figs. 3 and 6). So 'regular' means center-regular, not globally regular. The authors know this—they say so in Sec. IV—but the headline claim is stronger than what is demonstrated. For someone reading only the abstract, that is misleading.\n\nOther soft spots are minor or contextual. The electric field ansatz (24) is ad hoc and acknowledged; it is the standard constructive strategy for Bardeen-type solutions, so I would not hold that against the paper, though it limits physical interpretation. The f(R,T) examples are restricted to discrete negative λ values to make hypergeometric functions polynomial; the authors state this plainly, but it narrows the 'first' claim to a sparse parameter set. The fact that these solutions are not asymptotically AdS and instead grow curvature at infinity is a real physical oddity, and it reinforces the need to soften the wording.\n\nNo circularity, no data problems, and the citation pattern looks honest. This is a toy-model paper, but a competent one. The audience is the modified-gravity and exact-solutions community. I would send it to a serious referee: a revision that fixes the abstract and explicitly distinguishes center-regular from globally regular would make it publishable.","headline":"A genuinely constructive derivation of new 2+1 black hole solutions in f(R,T)+NED, but the abstract's 'regular' label overstates the λ≠0 case, where curvature diverges at infinity.","tokens_in":18645,"tokens_out":3022,"would_cite":false,"duration_ms":31528,"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":"The paper constructs the first centrally regular black hole metrics in (2+1)-dimensional f(R,T) gravity, using a tunable electric-field ansatz that reduces to Maxwell at infinity.","keywords":["regular black holes","f(R,T) gravity","nonlinear electrodynamics","(2+1)-dimensional gravity","BTZ black hole","energy-momentum non-conservation","curvature regularity","hypergeometric solutions"],"falsifier":"Compute the metric and curvature invariants for a value of $\\lambda$ outside the special discrete set, for example $\\lambda=-\\kappa^{2}/3$ in the $\\beta=1$ case, by evaluating the hypergeometric integrals numerically; if the Ricci or Kretschmann scalar diverges at $r=0$, or if $b(r)$ develops more than one zero (or none), then the claim of regular single-horizon black holes holds only for the special polynomial cases, not for the general $f(R,T)$ model.","tokens_in":17507,"feed_emoji":"🕳️","tokens_out":13274,"duration_ms":114096,"temperature":0.7,"pith_summary":"This paper aims to show that (2+1)-dimensional $f(R,T)$ gravity, with $f(R,T)=R-2\\Lambda+\\lambda T$ together with nonlinear electrodynamics, admits black hole solutions that are regular at the center: the metric has one event horizon, the curvature invariants are finite at $r=0$, and the electric field falls off as $q/r$ at infinity. The key move is to choose a parametrized electric-field profile $E(r)=q r^{\\alpha} (r^{\\beta}+a^{\\beta})^{-(\\alpha+1)/\\beta}$ rather than a Lagrangian, then solve the field equations for the NED Lagrangian and the metric function. For $\\lambda=0$ (General Relativity) the construction reproduces previously known regular solutions and yields new families, including the general $\\alpha=\\beta+1$ class. For $\\lambda\\neq 0$ it yields the first regular black hole geometries in this modified theory, though the paper notes that, for $\\lambda\\neq 0$, the Ricci and Kretschmann scalars diverge as $r\\to\\infty$, so 'regular' refers to the center rather than the full spacetime. The authors also quantify the non-conservation of the energy-momentum tensor that the $\\lambda T$ term induces, which they interpret as a possible signal of energy exchange between matter and geometry.","feed_headline":"First centrally regular black holes are found in 2+1 f(R,T) gravity","feed_subtitle":"A tunable electric field produces the metrics; the λT term also breaks energy-momentum conservation.","key_machinery":"The load-bearing object is the generalized electric field $E(r)=q r^{\\alpha} (r^{\\beta}+a^{\\beta})^{-(\\alpha+1)/\\beta}$, proposed in analogy with an earlier (2+1)-dimensional regular solution; it is chosen so that $E\\to q/r$ at infinity while remaining finite at $r=0$. This ansatz converts the coupled gravity-matter equations into a linear differential equation for the NED Lagrangian $L(r)$ (Eq. 23) and a single integral for the metric function $b(r)$ (Eq. 21), which is what makes analytical solutions possible. The hypergeometric functions ${}_{2}F_{1}$ that appear are the mechanism by which the infinite family of GR solutions is organized, and the $\\alpha=\\beta+1$ subfamily selects Lagrangians that recover Maxwell's theory in the asymptotic limit. In the $f(R,T)$ case, restricting $\\lambda$ to special negative values turns those hypergeometric functions into polynomials, which is the technical condition that allows $b(r)$ to be integrated explicitly.","core_discovery":"Starting from the field equations of $f(R,T)$ gravity with $f(R,T)=R-2\\Lambda+\\lambda T$ coupled to nonlinear electrodynamics, the paper constructs static, circularly symmetric solutions of the form $ds^{2}=-b(r)dt^{2}+b(r)^{-1}dr^{2}+r^{2}d\\theta^{2}$. With the electric field $E(r)=q r^{\\alpha} (r^{\\beta}+a^{\\beta})^{-(\\alpha+1)/\\beta}$, the independent equations reduce to a linear differential equation for the NED Lagrangian $L(r)$ and an integral for $b(r)$. For $\\lambda=0$ this yields an infinite family of regular black holes that contains the known (2+1)-dimensional regular solutions of earlier studies as special cases, plus genuinely new solutions, in particular the $\\alpha=\\beta=2$ model whose Ricci and Kretschmann scalars are smooth everywhere. For $\\lambda\\neq 0$ the same ansatz produces, for $\\beta=1$ and $\\beta=2$ with special discrete negative values of $\\lambda$, the first regular black hole solutions in (2+1)-dimensional $f(R,T)$ gravity, each with a single event horizon whose position depends on $\\lambda$. The paper further shows that the $\\lambda T$ term makes $\\nabla_{\\mu} T^{\\mu\\nu}$ nonzero and that both the trace $T$ and the curvature scalars grow without bound at large $r$ in the $f(R,T)$ cases, so the regularity is central rather than asymptotic; the deviation from energy-momentum conservation is quantified explicitly.","pith_inferences":["Editorial extension: If the $\\lambda T$ term generically drives curvature to diverge at infinity, then the class of 'regular' $f(R,T)$ black holes may be better described as singularity-free at the horizon and center but asymptotically singular; a natural next test is whether any special $\\lambda$ (or a different $f(R,T)$ form) restores asymptotically anti-de Sitter behavior.","Editorial extension: Since the electric-field profile is assumed rather than derived, one could try to construct the underlying NED Lagrangian $L(F)$ by eliminating $r$ from Eq. (24) and the associated $L(r)$; if no closed-form $L(F)$ exists, the ansatz may be a coordinate artifact rather than a physical matter model.","Editorial extension: The discrete-$\\lambda$ restriction is a technical convenience, not a physical principle; using numerical continuation for continuous $\\lambda$ (including $\\lambda>0$) would show whether the single-horizon and central-regularity properties persist or are an artifact of polynomial hypergeometrics.","Editorial extension: The same two-step procedure, prescribe a regular electric field and then solve for $L(r)$ and $b(r)$, could be repeated in 3+1 dimensions with a magnetic charge, which would test whether the central regularity and the far-region curvature growth are robust across dimensions."],"forward_implications":["If the construction is correct, modified gravity in 2+1 dimensions can host horizon-having black holes with finite curvature at the origin, so regularity does not require going to higher dimensions or to full quantum gravity.","The $\\alpha=\\beta+1$ family supplies a catalog of nonlinear electrodynamics models with the correct Maxwell limit, reproducing earlier models at $\\beta=1,2,4$ and adding new ones that can be studied for geodesics, quasinormal modes, and thermodynamics.","Because the $\\lambda T$ term breaks energy-momentum conservation by a computable amount that grows with $|\\lambda|$, these solutions give a concrete arena to test whether matter creation or energy-exchange effects are compatible with horizon physics.","The explicit divergence of curvature and of the trace $T$ at large $r$ for $\\lambda\\neq 0$ implies that the 'regular' label applies only to the center; any physical application of these $f(R,T)$ solutions must account for the strong-gravity behavior at infinity.","The paper's results point to the possibility that analogous regular solutions exist in (3+1)-dimensional $f(R,T)$ gravity, a direction the authors explicitly flag as worthy of future investigation."],"supporting_citations":[{"why":"It supplies the earlier (2+1)-dimensional regular black hole solution whose electric-field form inspires Eq. (24); the paper's $\\alpha=3$, $\\beta=2$ case reduces to it.","marker":"[38]"},{"why":"It provides the earlier regular (2+1)-dimensional GR solutions recovered by the $\\alpha=\\beta+1$ family at $\\beta=1$, $2$ and $4$, forming the main comparison set.","marker":"[39]"},{"why":"It introduces the BTZ black hole whose charged version is recovered in the Maxwell limit and whose dependence on the negative cosmological constant is used throughout.","marker":"[40]"},{"why":"It gives the uncharged static BTZ solution recovered when $\\lambda=2\\kappa^{2}$ in the Maxwell case.","marker":"[41]"},{"why":"It defines the $f(R,T)$ action and the field equations, including the non-conservation identity that the paper's energy-momentum analysis relies on.","marker":"[48]"}],"fun_headline_variants":["First regular black holes in 2+1 f(R,T) gravity","Tunable fields yield regular black holes in 2+1 f(R,T)","2+1 f(R,T) gravity hosts first regular black holes","Regular black holes emerge in 2+1 f(R,T) gravity","Non-conserved energy-momentum joins regular black holes in 2+1 f(R,T)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the hand-picked electric field $E(r)=q r^{\\alpha} (r^{\\beta}+a^{\\beta})^{-(\\alpha+1)/\\beta}$, proposed by analogy with an earlier solution rather than derived from a Lagrangian; the $f(R,T)$ analysis further restricts $\\lambda$ to special discrete negative values that make hypergeometric functions polynomial.","fun_headline_variants_meta":{"raw":{"variants":["First regular black holes in 2+1 f(R,T) gravity","Tunable fields yield regular black holes in 2+1 f(R,T)","2+1 f(R,T) gravity hosts first regular black holes","Regular black holes emerge in 2+1 f(R,T) gravity","Non-conserved energy-momentum joins regular black holes in 2+1 f(R,T)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00125,"raw_usage":{"total_tokens":5150,"prompt_tokens":996,"completion_tokens":4154,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":4051}},"tokens_in":612,"tokens_out":4154,"duration_ms":29251,"temperature":1.0,"reasoning_tokens":4051,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:45:41.775335+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the metric and curvature invariants for a value of $\\lambda$ outside the special discrete set, for example $\\lambda=-\\kappa^{2}/3$ in the $\\beta=1$ case, by evaluating the hypergeometric integrals numerically; if the Ricci or Kretschmann scalar diverges at $r=0$, or if $b(r)$ develops more than one zero (or none), then the claim of regular single-horizon black holes holds only for the special polynomial cases, not for the general $f(R,T)$ model.","supporting_citations":[],"review_version":1}