Notes on solution phase space and BTZ black hole

Wei Guo

REVIEW 2 major objections 5 minor 129 references

Notes on solution phase space and BTZ black hole

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The solution phase space method reproduces BTZ black hole thermodynamics exactly.

desk verdict A useful, self-contained SPSM tutorial for BTZ/KdS3 with standard results, but a sign error in the Appendix E integrability check invalidates the proof as printed even though the conclusion survives. read the letter →

arxiv 2411.14247 v1 pith:VA23HK6K submitted 2024-11-21 gr-qc

Wei Guo This is my paper · ORCID

classification gr-qc

keywords solutionphasespacemethodcovariantBTZblackholeconservedchargesfirstlawofthermodynamicsSmarrrelationinnerhorizonKerr-deSitterspacetime

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 sets out to show that the solution phase space method, a covariant phase space technique that evaluates conserved charges by integrating a surface charge density over any closed spacelike surface, works cleanly for the simplest rotating black hole, the BTZ solution in three dimensions. The aim is to demonstrate, step by step, that the mass and angular momentum of the BTZ black hole are the charges associated with time translation and axial rotation, while the entropy is the charge of a rescaled horizon Killing vector. The explicit results are $M=m$, $J=j$, and $S_{(\pm)}=4\pi r_{\pm}$, reproducing known thermodynamics, and the same pattern extends by analytic continuation to three-dimensional Kerr-de Sitter spacetime. A sympathetic reader should care because the paper tests whether a general charge-formula framework survives contact with the simplest nontrivial case, and it supplies the detailed calculations needed to apply the method elsewhere.

What carries the argument

The load-bearing object is the solution phase space method: on the submanifold of solutions parametrized by the solution parameters $(m,j)$, one varies the metric by $\hat\delta g_{\mu\nu}=(\partial g_{\mu\nu}/\partial m)\delta m+(\partial g_{\mu\nu}/\partial j)\delta j$, builds the symplectic current of the covariant phase space formalism, and extracts a surface charge density $k_\xi^{\mu\nu}$ such that integrating $k_\xi$ over any closed compact codimension-2 surface gives the charge variation $\delta Q_\xi$. The central identity is the charge density (28), whose $tr$-component for the BTZ metric yields $\delta Q_{\partial_t}=\delta m$, $\delta Q_{\partial_\phi}=-\delta j$, and $\delta Q_{\zeta_H}=2\pi(\delta m-\Omega_H\delta j)/\kappa$; because the horizon Killing vector $\xi_H$ itself fails the integrability condition, the method instead uses the rescaled vector $\zeta_H=2\pi\xi_H/\kappa$, which is integrable and whose charge is identified with the entropy.

What would settle it

Deform the integration surface to a closed spacelike surface that passes through the ergosphere rather than a constant-radius circle and recompute $Q_{\partial_t}$, $Q_{\partial_\phi}$, and $Q_{\zeta_H}$; if any charge changes, the claimed surface-independence of the solution phase space method fails for the BTZ black hole.

Watch

Extended reading notes

Core claim

The paper's claim is that in the two-parameter BTZ family, the surface charges computed from the covariant phase space are $Q_{\partial_t}=m$, $Q_{\partial_\phi}=-j$, and $Q_{\zeta_H^{(\pm)}}=4\pi r_{\pm}$, and that with the standard identifications $M=Q_{\partial_t}$, $J=-Q_{\partial_\phi}$, $S_{(\pm)}=Q_{\zeta_H^{(\pm)}}$, these satisfy the first law $\delta M=T_{(\pm)}\delta S_{(\pm)}+\Omega_{(\pm)}\delta J$ and the Smarr relation $M=\frac12 T_{(\pm)}S_{(\pm)}+\Omega_{(\pm)}J$ for both the outer and inner horizons. The inner horizon result is obtained by assigning the horizon Killing vector $\xi_H=\partial_t+\Omega_{(\pm)}\partial_\phi$ a rescaled partner $\zeta_H=\frac{2\pi}{\kappa}\xi_H$ with surface gravity $\kappa_{(\pm)}=\pm(r_+^2-r_-^2)/(l^2 r_\pm)$, the negative sign for the inner horizon being justified by thermal instability and cosmic censorship considerations; this rescaling is what makes the charge integrable. The identical computation, after $l\to il$ and $m\to -m$, yields the mass, angular momentum, entropy $4\pi r_+$, first law, and Smarr relation for three-dimensional Kerr-de Sitter spacetime, with temperature and angular velocity carrying opposite signs relative to their geometric definitions.

Load-bearing premise

The argument assumes that the surface-charge integral for the rescaled horizon Killing vector on the inner horizon is the inner horizon's thermodynamic entropy, a step that relies on taking the inner horizon's surface gravity to be negative.

Editorial extensions

If this is right

For the BTZ black hole, the method assigns $M=m$, $J=j$, and $S_{(\pm)}=4\pi r_{\pm}$, so it reproduces the accepted mass, angular momentum, and Bekenstein-Hawking entropy without any surface at infinity or at a bifurcation horizon.
The first law $\delta M=T\delta S+\Omega\delta J$ and the Smarr relation $M=\frac12 TS+\Omega J$ follow directly from the computed charges, for both the outer and inner horizons.
Because the charges are independent of the choice of integration surface, the same setup applies to three-dimensional Kerr-de Sitter spacetime by analytic continuation, giving $M=m$, $J=j$, and $S=4\pi r_+$.
The inner horizon carries the same formal thermodynamics with a negative surface gravity, so the method yields an inner-horizon first law rather than breaking down there.
The method remains in principle available in the extremal case where no bifurcation surface exists, since the integration surface need not be a bifurcation horizon.

Reading between the lines

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

If the inner-horizon charge $S_{(-)}=4\pi r_-$ is taken to be a genuine thermodynamic entropy, the natural next test is whether it satisfies a second-law-type inequality or a holographic counting in the same way the outer horizon entropy does; the paper does not address this.
The surface-independence of the charges suggests an immediate cross-check: computing the same integrals on a deformed surface that crosses the ergosphere would test whether the method's core promise is realized in a non-trivial geometry.
The same algorithm could be run for the charged BTZ solution to see whether the electric charge and potential slot into the first law with the same pattern; the paper lists this as a natural continuation.
If the negative surface gravity assignment is rejected, the inner-horizon results still stand as geometric identities, but they would lose their thermodynamic reading; that distinction is a decision point for future work.

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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, here's my read on arXiv:2411.14247. The paper is exactly what it says: a detailed, pedagogical run of the solution phase space method for the BTZ black hole (outer and inner horizons) and the KdS3 spacetime, reproducing the standard M=m, J=j, S=4πr±, first law and Smarr relation. The novelty is nil, and the author says so. The value is in the explicit computations: surface charge densities, integrability discussion, and the clean extension to KdS3 by analytic continuation. As a tutorial it is genuinely useful; I'd point a student to it to see SPSM in action.

That said, there is a real flaw in Appendix E. The integrability condition is δ2Q_{η1} − δ1Q_{η2}=0. With η=A∂t+B∂φ and Q_{∂φ}=−j, the correct expansion gives −(∂A/∂j + ∂B/∂m) δmδj, not the (∂A/∂j − ∂B/∂m) δmδj printed in Eq (E4). The paper's intermediate line also drops the minus sign before the second pair of terms. For the horizon Killing vector ζH=2π/κ ξH, the correct combination vanishes because ∂A/∂j = −∂B/∂m, so the conclusion that ζH is integrable is right. But as written, the check in the appendix does not vanish, and the proof step fails. For a paper whose raison d'être is showing every step, this is more than a typo; it is the central demonstration of integrability. It is fixable, but it needs fixing.

The other soft spots are minor. The KdS3 derivation is sketched rather than shown, and the author acknowledges that. The inner horizon assignments S(−)=4πr− and negative κ(−) rely on an imported identification from Castro-Rodriguez and a cosmic censorship reference, so they are interpretive steps rather than derivations; readers should be aware. These do not undermine the outer horizon BTZ result.

The citation pattern is fine: broad coverage, proper credit to Hajian-Sheikh-Jabbari for the method and to the earlier BTZ/KdS3 literature. No self-citation issue.

This paper is for students and researchers who want a worked SPSM example. It deserves a serious referee because it is a solid pedagogical computation and the error is correctable. My recommendation: send to peer review, and require a corrected Appendix E before acceptance.

Referee Report

2 major / 5 minor

Summary. This paper applies the solution phase space method (SPSM) to the rotating BTZ black hole in three-dimensional Einstein gravity. After reviewing the covariant phase space formalism and the SPSM, the author computes the surface charge densities for the Killing vectors ∂_t, ∂_φ and the rescaled horizon Killing vector ζ_H = (2π/κ) ξ_H, and obtains Q_{∂_t}=m, Q_{∂_φ}=-j, and Q_{ζ_H}(±)=4πr_±. Setting M=Q_{∂_t}, J=-Q_{∂_φ}, S(±)=Q_{ζ_H}(±) yields the standard BTZ mass, angular momentum, and entropy; the first law δM=TδS+ΩδJ and the Smarr relation M=(1/2)TS+ΩJ are then derived for both the outer and inner horizons. The same method is extended, by analytic continuation, to three-dimensional Kerr-de Sitter spacetime. The paper is explicitly pedagogical and benchmarks every charge against known BTZ thermodynamics.

Significance. The paper does not claim a new physical effect; its value is technical and pedagogical. Its strengths are the explicit surface-charge densities in Appendix D, the careful treatment of the integration surface and reference point, and the demonstration that the SPSM reproduces the known BTZ charges and first law, including the inner-horizon version, without taking the integration surface to the bifurcation surface. The KdS3 results provide a useful cross-check. Because the computations are shown in enough detail to be checked by hand, the paper is a potentially useful reference for readers wishing to learn the SPSM. The principal caveat is that the integrability proof in Appendix E contains a sign error, although the final charges are correct.

major comments (2)

[Appendix E, Eq. (E4)] The integrability calculation in Eq. (E4) has a sign error. For ξ=A∂_t+B∂_φ, the SPSM charge variation is the one-form δQ_ξ=Aδm−Bδj, so closedness requires ∂A/∂j = −∂B/∂m, i.e. A_j+B_m=0. The printed expression (A_j−B_m)δmδj is therefore not the correct integrability condition. With the printed formula, the stated check for ζ_H does not vanish (for example at l=1, m=2, j=1, A_j−B_m≈2.54), whereas the correctly signed combination does vanish. The appendix thus fails as written to prove the integrability of ζ_H. This is a load-bearing step for the definition S=Q_{ζ_H}, although the final integrated charges are unaffected because δQ_{ζ_H} integrates exactly to 4πr_±.
[Sec. IV.D, Eqs. (66)-(68)] The KdS3 computation is presented only as an analytic continuation, and the charge variations in Eqs. (66)-(68) are asserted without showing the corresponding surface-charge densities or their integrals. Since the paper's stated goal is to give the most detailed possible derivation, this is a gap: either include the KdS3 analogs of Eqs. (46)-(48) or of the Appendix D results, or state explicitly that the continuation argument is the entire derivation.

minor comments (5)

[Throughout] There are several typographical errors, including 'spactime' in the abstract, 'horzion' in Sec. IV, 'obtianed' in Sec. V, and 'dimenstions' in the Smarr-relation footnote; these should be corrected.
[Sec. II.A] The acronym is given as CPSF in the introduction and Sec. II.A, but 'CPSM' appears later in Sec. II.B and Sec. IV.A; the notation should be made consistent.
[Appendix E, Eqs. (E1)-(E2)] The notation in the integrability condition is under-specified: the symbols η, ξ, δ_1η and δ_2η are not precisely defined before Eq. (E4), which makes the sign error harder for a reader to detect.
[Sec. IV.D, Eqs. (75)-(76)] The thermodynamic potentials for KdS3 are assigned minus signs relative to the geometric quantities (T_H=−κ/2π, Ω_H=−Ω_+); a sentence explaining why this sign assignment is required for the first law would improve clarity.
[Sec. IV.C, Eq. (59)] The interpretation of Q_{ζ_H}(−)=4πr_− as the thermodynamic entropy of the inner horizon is an identification imported from Refs. [33,120]; the paper should state more explicitly that this is an assumption about inner-horizon mechanics rather than a derivation within the present computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BTZ charges and first law are computed from the metric via the SPSM surface-charge integral; the entropy normalization is standard Wald charge, not a fitted input.

full rationale

The paper's central results are not circular. The charges Q_dt=m, Q_dphi=-j and Q_zetaH=4*pi*r_+/- are obtained by substituting the parametric metric variations into the surface charge density and integrating; the reference point Q=0 at m=j=0 is a convention, not an injection of the answers. The normalization zetaH=2*pi/kappa*xiH is fixed by the Noether-Wald identification of entropy as a charge, with integrability checked in App. E, and the final entropy value 4*pi*r_+/- emerges from the integral rather than being imposed. The first law is an algebraic consequence of the computed charge variations and the geometric definitions of T and Omega; although footnote 2 notes that the entropy variation is normalized by 1/T_H to make the first law consistent, the numerical charge and area factor are not fitted, and the Smarr relation is likewise an identity from the derived charges. The Kerr-dS results are obtained by the stated analytic continuation l->il, m->-m, and benchmarked against known results, so no input is renamed as a prediction. Refs. [33] and [120] are external, not self-citations, and no load-bearing uniqueness claim rests on the present author's prior work. The sign issue in App. E is a correctness concern about a printed check, not a circularity concern, since the correct sign still makes the claimed combination vanish and the charges are computed independently of that check.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters and no invented physical entities. It relies on standard covariant phase space formalism, the solution phase space method from [33], and the Wald Noether charge identification for entropy. The only convention chosen by the author is the reference point for charges at m=j=0. All other inputs are standard solution parameters (m, j) and theory inputs (Λ, G).

assumptions (6)

domain assumption The Einstein-Hilbert action in three dimensions with 8G=1 and L=(1/2π)(R-2Λ) is the theory under consideration.
This normalization is set in Sec. III A, Eq. (19), and determines the numerical factors in all charge formulas.
domain assumption The BTZ metric ansatz (30) with lapse N and shift N^φ is the solution under study, and ∂t and ∂φ are exact Killing symmetries.
The solution phase space method requires the existence of Killing vectors; this is stated in Sec. III B.
standard math The solution phase space method of Hajian and Sheikh-Jabbari [33] correctly defines conserved charges via Eq. (15), including the integrability criterion used in Appendix E.
The paper builds directly on this method and cites [33] for the formalism; it does not re-derive the method from first principles.
standard math Black hole entropy equals the Noether charge associated with the rescaled horizon Killing vector ζH=2π/κξH, following Wald's formalism.
Used in Sec. IV A, Eqs. (37)-(42); this is the standard Wald entropy result from [22, 98].
ad hoc to paper The charge reference point is set to Q=0 at m=j=0, corresponding to the massless BTZ black hole.
Chosen in Sec. II B, Eq. (16) to fix the integration constant. This convention sets the absolute values of M and J.
domain assumption For the inner horizon, the surface gravity κ(-) is taken to be negative, attributed to thermal instability and cosmic censorship.
Used in Eq. (36); the sign convention is imported from reference [109] rather than derived in this paper.

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Pith. "Pith review of Notes on solution phase space and BTZ black hole." pith.science (2026). https://pith.science/paper/VA23HK6K

@misc{pith2026241114247,
  author       = {Pith},
  title        = {Pith review of: Notes on solution phase space and BTZ black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VA23HK6K}},
  note         = {Machine review of arXiv:2411.14247}
}

read the original abstract

In this paper, we use the solution phase space approach based on the covariant phase space formalism to compute the conserved charges of the BTZ black hole, namely mass, angular momentum, and entropy. Furthermore, we discuss the first law of the BTZ black hole and the Smarr relation. For completeness, outer horizon and inner horizon cases have been all included. Additionally, the results of the three-dimensional Kerr-dS spacetime have also been obtained. Our results are consistent with previous investigations. Considering the simplicity of the circumstances, we have presented the most detailed possible information, with the aim of facilitating research in related fields.

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