The first laws of thermodynamics of the (2+1)-dimensional BTZ black holes and Kerr-de Sitter spacetimes

Quick read on hep-th/0601147. The part that is actually new is small and correct: an explicit 2+1 check of the variable-Lambda first law, for BTZ (negative Lambda) and Kerr-de Sitter (positive Lambda). I went through the algebra and it holds. The mass formulas M = S²/(16 pi² l²) +/- 4 pi² J²/S², the temperatures, Omega, and Theta = -S²/(8 pi² l³) all come out as stated, and 0 = T S + Omega J + Theta l follows cleanly from Euler's theorem on a mass function that is homogeneous of degree zero in (S, J, l). That is a tidy, self-contained little result and a useful reference formula for anyone doing BTZ thermodynamics. Credit where due: the computation is clean and the authors are upfront about what they are borrowing from Caldarelli-Cognola-Klemm and Creighton-Mann for the 4D case. One correction to the reader's report: the homogeneity the reader worries about is not an extra dynamical assumption that could fail. It is forced by dimensions, since M, S, J, l have length weights D-3, D-2, D-2, 1. So the general-D integral formula is a one-line scaling argument, not a hidden premise. That makes the gap smaller than the reader thinks, but it also makes it less excusable that the paper does not just write the line down. Instead the abstract says the formulas 'come into existence in any dimensions also' and the text defers the general case to ref. [10], 'Wu S Q, work in preparation' — the authors' own unpublished work. That is the main soft spot, and it is a scope problem, not an error. Second soft spot, and the authors flag it themselves: in the Kerr-dS case r+ is 'the only cosmological horizon, instead of a black-hole horizon.' So the abstract's 'rotating black holes in ... de Sitter backgrounds' overstates what is shown; this is cosmological-horizon thermodynamics. The sign flip in T and Omega relative to the geometric kappa and Omega_+ is noted but never explained, which leaves the dS section feeling asserted rather than derived. Neither problem is fatal, and both are fixable in a paragraph. Who should read it: people working on 2+1 thermodynamics or on horizon thermodynamics with a running Lambda, who want the explicit formulas. It deserves a referee look, mainly to force the dimensional-analysis line into the text and to tighten the abstract's dS wording.

The paper's headline claim [abstract:s1] is that the first-law formulas extend to 'any dimensions.' The paper demonstrates only D=3 explicitly (BTZ, Eqs. 16-18; Kerr-dS, Eqs. 25-26). For general D it says 'Combining with the previous results obtained in the 4-dimensional case, [9] we deduce that the general formulas ... should be recast into the forms of Eqs. (3) and (4) in the case of rotating black holes in anti-de Sitter backgrounds in three, four, five and higher dimensions.' The 4D input is ref. [9] (published), but the general-D study is ref. [10], listed as 'Wu S Q, work in preparation' — an unpublished self-citation. The paper does not present the argument that makes Eq. 4 follow in general D. That argument is short: M has length dimension D-3, S and J have D-2, l has 1, so scaling all lengths by lambda gives M -> lambda^{D-3}M, S -> lambda^{D-2}S, J -> lambda^{D-2}J, l -> lambda l; Euler's theorem then gives (D-2)(TS + Omega J) + Theta l = (D-3)M, which is Eq. 4. The reader's framing — that the paper must show M(S,J,l) is homogeneous of an assumed degree — is close but slightly off: the homogeneity is automatic from dimensional analysis, not an extra dynamical assumption that could fail. So the concern is a completeness/scope gap (the general-D claim is not self-contained), not a correctness error. The dS side has a separate issue: the paper itself notes r+ is 'the only cosmological horizon, instead of a black-hole horizon,' so the 2+1 dS example illustrates cosmological-horizon thermodynamics, not black-hole thermodynamics, despite the abstract's 'rotating black holes in ... de Sitter backgrounds.'

The reader's verdict (CONDITIONAL, HIGH confidence) is right, and the reader's two substantive observations are both real: the general-D claim rests on unpublished ref. [10], and the Kerr-dS example is a cosmological horizon rather than a black hole horizon. I part company only on the diagnosis of the general-D gap. The reader frames it as a missing homogeneity proof for M(S, J, l). In fact the homogeneity is fixed by dimensional analysis: M, S, J, l have length dimensions D-3, D-2, D-2, 1, so Euler's theorem yields Eq. 4 directly. The paper does not spell this out, which is a presentation gap, but the result is not at risk from a hidden dynamical assumption. The dS issue is acknowledged in the text ('the radius r+ is the only cosmological horizon, instead of a black-hole horizon'), so the authors are aware; what remains unresolved is whether the abstract's 'rotating black holes in ... de Sitter backgrounds' is an overclaim for the 2+1 case. The D=3 algebra (Eqs. 11-18, 25-26) checks out, including Theta = -S^2/(8 pi^2 l^3) and the identity 0 = TS + Omega J + Theta l. So the paper's demonstrated content is sound; the headline general-D claim is plausible and, I believe, correct, but not self-contained. CONDITIONAL remains the right verdict.

The D=3 results are correct and directly checkable, and the Euler-theorem structure behind the integral formula is clean; the paper supplies explicit formulas that a reader can verify line by line. The value of the paper is, however, narrower than the abstract suggests: the 'any dimensions' extension is not derived in the manuscript and rests on an unpublished self-citation, and the de Sitter example is a cosmological horizon rather than a black-hole horizon. The paper is a useful short letter on the D=3 cases, but its headline generality is not established within its own pages.

Black hole thermodynamics comes in two forms: a differential first law, $dM = T$ $dS + Omega$ $dJ$, and an integrated Smarr-type relation that ties the mass to $TS$ and $Omega J$ with a coefficient that depends on dimension. The integrated version breaks once a cosmological constant is present, and in $2+1$ dimensions it fails outright: for the BTZ black hole one finds $TS + Omega J = 2r_+^2/l^2$, which is neither the mass nor zero. The paper's proposal is to stop treating the cosmological constant, equivalently the length scale $l$, as a fixed constant and let it vary, adding a work term $Theta$ $dl$ with a conjugate generalized force. With that single extra term, both the differential and the integrated laws come out right for the rotating BTZ black hole and, after a sign flip, for the $2+1$-dimensional Kerr-de Sitter spacetime. If the proposal holds, the thermodynamic bookkeeping of black holes no longer depends on whether the surrounding geometry is flat, anti-de Sitter, or de Sitter.

What carries the argument. The carrying device is the Christodoulou-type mass formula, which rewrites the black hole mass as an explicit function $M(S, J, l)$ of entropy, angular momentum, and the cosmological length scale. Once the mass is a function of those three variables, the temperature, angular velocity, and generalized force are just its partial derivatives, and the integral mass formula follows from the same function by Euler-type scaling. In three dimensions the scaling degree is zero, so the mass drops out of the integral relation entirely and leaves $0 = TS + Omega J + Theta l$; that vanishing left-hand side is what exposes the missing cosmological-constant term in the earlier flat-space-style formula $M = (1/2)TS + Omega J$.

Core claim. The paper claims that promoting the cosmological constant to a thermodynamic state variable is exactly what is needed to make the first law of black hole thermodynamics work in curved backgrounds, in both its differential and integrated forms. Concretely, the authors write the BTZ mass as a Christodoulou-type function of entropy, angular momentum, and the length scale $l$, namely $M = S^2/(16pi^2 l^2) + 4pi^2 J^2/S^2$, and read off $T = partial M / partial S$, $Omega = partial M / partial J$, and a new generalized force $Theta = partial M / partial l = -S^2/(8pi^2 l^3) = -2r_+^2/l^3$. With these, the differential law becomes $dM = T$ $dS + Omega$ $dJ + Theta$ $dl$, and the integral law becomes $0 = TS + Omega J + Theta l$ in three dimensions, which is the $D = 3$ specialization of the general coefficient $(D-3)/(D-2)$ on the mass. The same structure is then checked for the $2+1$-dimensional Kerr-de Sitter spacetime with positive cosmological constant, where the mass formula is $M = S^2/(16pi^2 l^2) - 4pi^2 J^2/S^2$, the generalized force keeps the same form, and the temperature and angular velocity derived thermodynamically carry a minus sign relative to their geometric definitions. The authors conclude that the flat-space formulas extend directly to rotating black holes in anti-de Sitter and de Sitter backgrounds, and assert that the same holds in any dimension.

The premise it rests on. The load-bearing premise is that the mass function $M(S, J, l)$ scales with its arguments in exactly the degree needed for the integral relation to hold in every dimension, so that the three-dimensional and four-dimensional patterns carry over unchanged; the paper checks the two low-dimensional cases explicitly but never verifies that scaling degree for general $D$.

The mismatch in three dimensions between mass and $TS + Omega J$ is resolved rather than tolerated: the leftover piece is absorbed by the generalized force conjugate to $l$.

The same bookkeeping applies to positive cosmological constant, giving a first law for the $2+1$-dimensional Kerr-de Sitter cosmological horizon, with temperature and angular velocity opposite in sign to their geometric values.

The generalized force keeps the form $Theta = -S^2/(8pi^2 l^3)$ in both the anti-de Sitter and de Sitter cases, so one formula covers both signs of the cosmological constant.

The authors identify the charged BTZ black hole as the next case to which the same variable-cosmological-constant treatment should be applied.

The general-dimension integral formula $(D-3)/(D-2), M = TS + Omega J + Theta l/(D-2)$ reduces correctly to the three-dimensional result the paper derives explicitly.

If the cosmological constant is a genuine state variable, then black hole thermodynamics in anti-de Sitter space acquires a pressure-volume equation of state, and the $Theta$ term derived here is the piece that would make such an equation of state consistent with the Smarr relation.

Because the three-dimensional integral formula collapses to $0 = TS + Omega J + Theta l$, the mass cancels out entirely; that makes $2+1$ dimensions an unusually clean testbed, since any proposed correction term must sum to exactly zero rather than to a dimension-dependent fraction of the mass.

The sign flip in the de Sitter case, where thermodynamically defined temperature and angular velocity are the negatives of their geometric counterparts, suggests that the cosmological horizon is being treated as a system with negative temperature in the standard convention, which is a checkable point against horizon thermodynamics in de Sitter space.

The method suggests a general recipe: write the mass as a function of entropy, charges, and the cosmological length, then demand that the integral relation hold as an identity in the scaling degree. Any black hole family where that identity fails would be a counterexample worth cataloguing.

This review was written by a machine. No person has reviewed this paper on Pith yet. A scientist listed on Pith can review it for a fee, and their review is published here whatever it concludes.

Request a human review