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Note on WAdS$_3$ black holes in extended BHT gravity

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Pith's one-line read For warped AdS3 black holes in extended new massive gravity, this paper derives explicit ADM mass, angular momentum, and equal dual CFT central charges $c_L = c_R = 192\nu^3\ell\Xi/G$, using the covariant phase space formalism and the…

desk verdict Useful Wald-formalism re-derivation, but the ENMG WAdS3 central charge is internally inconsistent and dimensionally wrong as printed. read the letter →

arxiv 2411.18062 v2 pith:4PWR3SXD submitted 2024-11-27 hep-th gr-qc

Davood Mahdavian Yekta This is my paper · ORCID
classification hep-thgr-qc MSC 83C5783D0581T40 PACS 04.70.Bw04.60.-m11.25.Hf
keywords warpedAdS3blackholenewmassivegravityextendedNoetherchargecovariantphasespacecentralchargesthermodynamicsmethodentropyproduct
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

This paper establishes the full set of conserved charges, entropies, and dual CFT central charges for warped anti-de Sitter black holes in three-dimensional new massive gravity (NMG) and its extended version (ENMG). Using the covariant phase space formalism, it integrates Noether-charge variations to obtain the ADM mass and angular momentum, first for the NMG WAdS3 solution and then for BTZ and WAdS3 black holes in ENMG. It then applies the thermodynamics method, in which equal outer and inner horizon temperatures and entropies imply $c_L=c_R=6\,d/dN\,(S_+S_-/4\pi^2)$, to extract equal left- and right-moving central charges, $c_L=c_R=192\nu^3\ell\Xi/G$ for the ENMG WAdS3 case. A sympathetic reader would care because these explicit formulas are the input needed to check whether the holographic dictionary, matching black hole entropy to a dual CFT through the statistical entropy formula, holds for these higher-derivative backgrounds.

What carries the argument

The load-bearing object is the Noether charge $(n-2)$-form $Q_\xi$ associated with the Killing vector $\xi = \partial_t + \Omega_H\partial_\varphi$; its integral over a spatial boundary gives the conserved charges and, through $S = (2\pi/\kappa)\oint Q_\xi$, the entropy. The second device is the thermodynamics method: when outer and inner horizon quantities satisfy $T_+ S_+ = T_- S_-$, the central charges follow from $c_L = c_R = 6\,d/dN\,(S_+S_-/4\pi^2)$ with $N=J$. The paper also relies on a first-order Lagrangian formulation with frame fields and auxiliary fields, which turns the charge variation into an integral over the metric functions $F$, $K$, $N$ and their derivatives.

What would settle it

An independent calculation of the asymptotic symmetry group of the ENMG warped AdS3 background would settle the central-charge claim: if the central charges read off from this algebra differ from $192\nu^3\ell\Xi/G$, the thermodynamics-method result is wrong. A cheaper check is to insert the paper's entropies and temperatures into the statistical entropy formula $S = \pi^2/3\,(c_L T_L + c_R T_R)$ with $c_L = c_R = 192\nu^3\ell\Xi/G$ and compare with the entropy $S_\pm$ of Eq. (60) for several values of $r_+$, $r_-$; any discrepancy shows the central charge formula is being misapplied.

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Extended reading notes

Core claim

The central discovery is that the warped AdS3 black hole, a circle fibration over AdS2 with warping parameter $\nu$, in extended new massive gravity carries the ADM mass $M = \Xi/(G\ell)\,\big(\nu(r_+ + r_-) - \sqrt{(\nu^2+3)r_+r_-}\big)$, angular momentum $J = \Xi/(4G)\,\big[(\nu(r_+ + r_-) - \sqrt{(\nu^2+3)r_+r_-})^2 - \nu^2(r_+-r_-)^2\big]$ with $\Xi$ defined in Eq. (51), and entropies $S_\pm = 16\pi\nu^2\ell\Xi/G\,\big(2\nu r_\pm - \sqrt{(\nu^2+3)r_+r_-}\big)$. These quantities satisfy the first law and Smarr relation on both the outer and inner horizons, and the entropy product $S_+S_-$ is universal, independent of the mass. The paper further claims that the thermodynamics method yields equal dual CFT central charges $c_L = c_R = 192\nu^3\ell\Xi/G$, with an analogous BTZ result in ENMG. The computations reproduce prior results for NMG WAdS3 and for BTZ, which the paper presents as consistency checks.

Load-bearing premise

The argument depends on assuming that the formula $c_L=c_R=6\,d/dN\,(S_+S_-/4\pi^2)$, borrowed from earlier work, gives the correct central charges for this higher-derivative, non-maximally-symmetric warped black hole once $T_+S_+=T_-S_-$ holds; the paper checks that condition but does not independently confirm the formula for the ENMG WAdS3 background.

Editorial extensions

If this is right

  • The explicit ENMG WAdS3 mass and angular momentum in Eq. (52), together with the entropy, satisfy the first law $dM = T_\pm dS_\pm + \Omega_\pm dJ$ and the Smarr relation $M = T_\pm S_\pm + 2\Omega_\pm J$.
  • Because $T_+S_+ = T_-S_-$ holds, the entropy product $S_+S_-$ is mass-independent and equals $2(8\pi)^2\nu^3\ell\Xi\,J/G$, a universal feature for these diffeomorphism-invariant theories.
  • The equal central charges $c_L = c_R$ imply the dual CFT is anomaly-free, so the left and right sectors carry the same number of degrees of freedom, unlike theories with gravitational anomaly where $c_L \neq c_R$.
  • Using the left and right temperatures, the mass and angular momentum of the WAdS3 black hole can be written in terms of left and right CFT energies as $M = \tfrac12\sqrt{2\nu c_L E_L/6}$ and $J = \ell^2/(2(\nu^2+3))\,(E_L - E_R)$, analogous to TMG and MMG but with ENMG-modified coefficients.
  • For the BTZ limit in ENMG, the central charges reduce to $c_L=c_R=\frac{3\ell}{2G}(1 - \frac{1}{\tilde m^2} - \frac{1}{2\tilde m^4})$, consistent with earlier asymptotic-symmetry and c-theorem analyses.

Reading between the lines

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

  • A natural extension the author leaves implicit: the same workflow should assign charges and central charges to any stationary solution of the higher-order Born-Infeld-like extensions, with each curvature order entering only through the prefactor $\Xi$.
  • If the central charges in Eq. (62) are confirmed by an independent asymptotic-symmetry computation, it would validate the thermodynamics method on a non-maximally-symmetric higher-derivative background; a mismatch would show the method fails exactly where $T_+S_+=T_-S_-$ is not sufficient.
  • The explicit charge formulas offer a direct way to test extremality and cosmic censorship bounds for warped black holes in these theories by checking that $r_+ \ge r_-$ and that the $J/M$ ratios respect the claimed thermodynamics.
  • Since the paper's computations are background-independent at the level of the covariant phase space, the same Noether-charge integrals could be applied to other warped or Lifshitz-type solutions in NMG and ENMG, where earlier boundary-based methods gave unsatisfactory results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript derives conserved charges for warped AdS3 black holes in new massive gravity (NMG) and in extended new massive gravity (ENMG) using the Wald covariant phase space formalism in a first-order Chern-Simons-like formulation, and then computes dual CFT central charges by the thermodynamics method. The main new results are the ENMG mass and angular momentum in Eq. (52) and the central charges in Eq. (62). The paper also presents a parallel computation for BTZ black holes and claims consistency with known results in the literature.

Significance. If the results held as stated, the paper would provide the first Wald-formalism derivation of conserved charges for WAdS3 black holes in ENMG and a thermodynamics-method central charge for a higher-derivative, non-maximally-symmetric background. The computations are parameter-free and the supplementary material gives explicit auxiliary-field coefficients, which is a useful resource. However, the central-charge sector as printed is internally inconsistent: the entropy formula, the entropy product, and the central charge do not mutually agree and carry incorrect dimensions. Until that inconsistency is resolved, the main new claim cannot be considered established.

major comments (3)
  1. [Sec. V.B, Eqs. (60)-(62)] The entropy and central-charge sector is internally inconsistent. With Eq. (52), r± are dimensionless, M has dimension 1/ℓ, and J is dimensionless (G has dimension ℓ in three dimensions), so the entropies S± and the product S+S− must be dimensionless. The prefactor 16πν²ℓΞ/G in Eq. (60) has dimension of length, since Ξ∝ℓ, so Eq. (60) cannot be the entropy. Moreover, substituting Eq. (60) into S+S− and using Eq. (52) gives S+S− = 1024π²ν⁴ℓ²ΞJ/G (or 1024π²ν⁴ΞJ/G if the ℓ in Eq. (60) is removed), whereas the paper states S+S− = 128π²ν³ℓΞJ/G. The two results differ by a factor 8νℓ or 8ν/ℓ, not by a numerical constant. Consequently Eq. (62), c = 192ν³ℓΞ/G, does not follow from the paper's own Eqs. (60) and (52), and it is dimensionful (Ξ∼ℓ) rather than dimensionless. The authors should correct the entropy formula, the product, and the central charge so that they are mutually consistent and dimensionally sound.
  2. [Sec. IV.B, Eq. (51)] The central derivation of the ENMG charges is not shown. The passage from Eq. (37) and the interior products in Eqs. (48)-(50) to the compact expressions in Eq. (51) is stated as 'substituting the above relations' without displaying the intermediate algebra. The supplementary material lists the Pi and Yi coefficients but not the contractions and simplifications that produce Eq. (51). Since Eq. (52) is used in Section V.B to obtain the central charge, this is a load-bearing step and should be presented in enough detail to be verified by a reader.
  3. [Sec. V.B, Eq. (1)] The thermodynamics-method formula Eq. (1) is applied to the WAdS3 ENMG background using only the check T+S+ = T−S−. That condition is necessary but not sufficient for Eq. (1). For this higher-derivative, non-maximally-symmetric background, the conjecture should be benchmarked independently. A concrete test would be to compare the corrected central charge with an asymptotic-symmetry-group computation or with the Cardy formula using TL, TR and the energies in Eq. (63).
minor comments (4)
  1. [Throughout] The manuscript contains several typographical errors, including 'taht', 'vinishing', 'balck', 'exppressions', and 'killing'; these should be corrected in a revision.
  2. [Sec. III, Eq. (23)] The condition on ν² is typeset ambiguously as 'ν2 = ˜m2 + 3 /20'; if the intended expression is (m̃²+3)/20, it should be written explicitly to avoid confusion.
  3. [Data Availability Statement] The statement 'No new data were created or analysed in this study' is slightly misleading because the supplementary material contains substantial new algebraic expressions; the wording should be adjusted to reflect that the explicit coefficient functions are provided as supplementary material.
  4. [Sec. V.B, Eq. (63)] The expression M = (1/2)√(2ν)√(c_L E_L/6) is dimensionally sensitive to the (incorrect) dimensionful c_L given in Eq. (62); after the central charge is corrected, Eq. (63) should be rechecked for dimensional consistency.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Wald charges and entropies are computed self-containedly, and the central-charge result applies an external formula without fitted inputs.

full rationale

The paper's derivation chain is not circular. The WAdS3 mass and angular momentum in Eqs. (31) and (52) are obtained by integrating Wald charge variations (30) and (51), with no free parameter fitted to the central charges; the ENMG result is cross-checked against the independent calculation [50]. The entropies (53) and (60) come from the Noether charge via Eq. (13), independently of the central-charge formula. The central charges (56) and (62) are then computed by applying Eq. (1), an external thermodynamics-method formula cited to [38-40], after verifying T+S+ = T−S−; this is an unproved conjecture about the dual CFT, not a circular reduction, because cL and cR never enter the definitions of S±, M, or J and no fitted parameter is renamed as a prediction. The self-citations [33,37,44,60] appear only as consistency or compatibility checks and are not load-bearing for Eqs. (52), (60), or (62). A separate arithmetic/dimensional discrepancy between the entropy product stated in Section V.B and the product obtained from Eqs. (60) and (52) is a correctness or typographical concern, not a circularity, and does not change the verdict.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the thermodynamics method conjecture and the validity of the Wald formalism for these theories, plus the known WAdS3 solution. No free parameters are fitted; the parameter ν is determined by m̃ via Eq. (46).

assumptions (4)
  • domain assumption The thermodynamics method formula cL = cR = 6 d/dN (S+S-/(4π²)) correctly gives the central charges of the dual CFT.
    Equation (1) is cited from [38-40] and used without proof to compute central charges for BTZ and WAdS3 black holes in ENMG.
  • domain assumption The WAdS3 metric (22) is a solution to the ENMG field equations when (46) holds.
    The paper states this condition without derivation and uses it to compute auxiliary fields and charges.
  • domain assumption The covariant phase space method yields well-defined conserved charges at spatial infinity for these higher-derivative theories.
    The paper follows the standard Wald/Iyer-Wald formalism; for higher-curvature gravities in WAdS3 backgrounds, other methods (ADT) are known to fail, but the validity of the covariant phase space approach is assumed.
  • domain assumption The choice N = J in Eq. (1) is the correct conserved charge to use in the thermodynamics method.
    For rotating BTZ and WAdS3 black holes, the paper selects N = J; this is standard in the thermodynamics method literature.

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Pith. "Pith review of Note on WAdS$_3$ black holes in extended BHT gravity." pith.science (2026). https://pith.science/paper/4PWR3SXD

@misc{pith2026241118062,
  author       = {Pith},
  title        = {Pith review of: Note on WAdS$_3$ black holes in extended BHT gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4PWR3SXD}},
  note         = {Machine review of arXiv:2411.18062}
}
abstract

In this letter, we study a holographic diffeomorphism invariant higher-derivative extension of Bergshoeff-Hohm-Townsend (BHT) cosmological gravity in the context of Wald's formalism. We calculate the entropy, mass and angular momentum of warped anti-de Sitter (WAdS$_3$) black holes in ghost-free BHT massive gravity and its extension using the covariant phase space method. We also compute the central charges of the dual boundary conformal field theories (CFT) from the thermodynamics method.

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Forward citations

Cited by 1 Pith paper

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    Near-horizon extremal BTZ black holes in NMG and TMG carry two Virasoro algebras whose central charges equal their spatial-infinity values.

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