On the Thermodynamic Geometry of BTZ Black Holes

Bhupendra Nath Tiwari; Gautam Sengupta; Tapobrata Sarkar

REVIEW

On the Thermodynamic Geometry of BTZ Black Holes

Tapobrata Sarkar, Gautam Sengupta, Bhupendra Nath Tiwari

High Energy Physics - Theory · Submitted 9 Jun 2006 · 17 references

Pith's verdict: An honest, modest extension of the BTZ Ruppeiner-geometry program: the CS flatness check is real, but the nonzero-curvature claim from fluctuation-corrected entropy is an interpretation the paper does not defend.

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CitationBhupendra Nath Tiwari, Gautam Sengupta, Tapobrata Sarkar. "On the Thermodynamic Geometry of BTZ Black Holes." arXiv:hep-th/0606084 (2006). https://pith.science/paper/Q3VHS67R

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Quick read on this one: it's a small, honest paper that does two things. First, it checks that the Ruppeiner geometry of the BTZ black hole stays flat when you add the gravitational Chern-Simons term, using Solodukhin's modified mass and angular momentum (eq. 21). That's new relative to [10], which the authors credit for the ordinary rotating case, and it's a sensible check. Second, it redoes the Das-Majumdar-Bhaduri fluctuation-corrected entropy for BTZ-CS and recovers the -3/2 log S coefficient, which is a nice consistency result and the part I'd actually cite. The algebra I spot-checked (eqs. 16, 17, the Hessian determinant) is fine, and the specific-heat positivity check is done properly, so local stability is not hand-waved. Credit where due: the CS flatness computation is clean and the coefficient reproduction is a real, if small, result. The soft spot is the headline claim (abstract:s3). The paper takes S = S0 - (1/2)ln(CT^2) and computes the Hessian of that corrected entropy, then calls the resulting Ricci scalar a physical measure of interactions. But the log term is itself obtained by integrating over the same Gaussian fluctuations that the Ruppeiner metric encodes, so this is a second-order object, and the paper never argues that one iteration is the leading term in a systematic expansion. The polynomial corrections that [7] says accompany the log are dropped without comment. That doesn't make the arithmetic wrong, but it makes 'R probes fluctuation effects' an interpretation rather than a result. Related: the J=0 remark in section 2.2 is not well posed. At J=0 the two-dimensional geometry's curvature is a limit of a family, not the curvature of the actual one-dimensional non-rotating state space, so 'interacting at zero angular momentum' isn't a physical statement. The authors do acknowledge that some calculations are too long to reproduce and that numerics are used, and that the near-extremal regime is out of scope; those are fair caveats, but they leave the central claim resting on four plots with no closed form. Who should read it: people already working on Ruppeiner geometry of black holes, and anyone who wants the BTZ-CS entropy coefficient confirmed. It deserves a referee, mainly to press on the fluctuation-correction identification and the J=0 sentence.

Referee report

Major objections (6)

[§3.1, Eqs. (25)-(28)] The K -> 0 limit of the BTZ-CS thermodynamic quantities needs to be exhibited explicitly, because as printed it does not appear to hold. Solving Eq. (22) for M gives a quadratic whose two roots are M = [W ± sqrt((1-K^2)(W^2 - K^2 J^2))]/K^2 with W = S^2/(8 pi^2) + K J; the branch that reduces to M = S^2/(16 pi^2) + 4 pi^2 J^2/S^2 as K -> 0 is the one with the minus sign, whereas Eq. (25) is written with the plus sign, which diverges as W/K^2 in that limit. Likewise, Eq. (26) carries an overall factor K^2 in the numerator, which as printed drives T -> 0 as K -> 0 instead of to Eq. (16). Since the specific heat of Eq. (28) and hence the corrected entropy and the plots in Figs. 3-4 are built from these expressions, please re-derive Eqs. (25)-(28) and show the K -> 0 reduction to Eqs. (16)-(17) explicitly.
[§2.2, Figs. 1 and 2] The sign of the induced curvature is not stable between the two truncations the paper compares. Fig. 1, using only the leading logarithm of Eq. (18), gives R of order -4.66 x 10^-4, while Fig. 2, using the full Eq. (14), gives R of order +1.5 x 10^-4 for the same mass M = 100. Both are truncations of the same expansion, so the sign of R is evidently not controlled at the order retained; the text notes the positive-definite result in Fig. 2 as 'interesting' without commenting on the sign change. This matters for the interpretation, because a curvature whose sign depends on which subleading terms are kept cannot yet be read as a measure of attractive versus repulsive statistical interactions. Please state which truncation is the controlled one, or show that the sign is stable when the polynomial corrections that accompany the logarithm in Ref. [7] are included.
[§2.2, paragraph after Eq. (18)] The paper reports that increasing M decreases the magnitude of R while preserving the shape of the graph, and the plotted values are already of order 10^-4 at M = 100. If R scales to zero with increasing M, then the induced curvature vanishes precisely in the thermodynamic limit in which the canonical and microcanonical entropies coincide and the correction itself disappears. That is consistent with the correction being a finite-size effect, but it undercuts the language of 'interacting statistical system' used in the text. Please give the M-scaling of R explicitly (an analytic fit or a table of values at several M would suffice) and discuss what physical statement the nonzero curvature is meant to support.
[§2.2, Eq. (14) and its use] The Ruppeiner metric is defined as the Hessian of an equilibrium entropy, but Eq. (14) is itself obtained by integrating over Gaussian fluctuations about that equilibrium. Taking the Hessian of the fluctuation-corrected entropy therefore describes the fluctuations of a distribution whose entropy already includes fluctuation effects, and the paper does not argue that this one-step iteration is the leading term of a systematic expansion of a physical curvature. This is a correctness risk rather than a stylistic point, and it can be tested: compute R from S0 - (1/2) ln(C T^2) and compare it with the leading correction to R obtained by expanding the Ruppeiner curvature functional about the flat S0 metric, and check whether the two agree in sign and magnitude. If they do not, the nonzero-R result should be presented as a property of the truncated formula rather than of the black hole.
[§2.2, sentence on J = 0] The statement that 'even at zero angular momentum, there is a small but finite value of the curvature scalar' and that this 'indicates that even at zero angular momentum, the statistical system is interacting' is not well defined as written. At J = 0 the two-dimensional state space degenerates and the non-rotating BTZ black hole has a one-dimensional thermodynamic state space, whose Ruppeiner geometry is trivially flat; the value quoted is the J -> 0 limit of a two-dimensional family, not the curvature of the actual non-rotating state space. Either compute the genuinely one-dimensional case and state that it is flat, or remove the sentence and the contrast with the non-interacting non-rotating BTZ black hole.
[§2.2 and §3.1, Figs. 1-4] The entire nonzero-curvature claim is numerical, and the paper states in the introduction that 'Some of the calculations are unfortunately too long to reproduce here, and whereever necessary, we have used numerical techniques.' No closed-form expression for R is given, no code or notebook is provided, and no numerical precision or error estimate is quoted. For a result whose content is a set of four plots, this is a reproducibility problem. Please supply at least the explicit metric components g_ij for the corrected entropy, or a closed form for R in the limits considered, or a supplementary notebook; and state the numerical precision and the software used.

Minor objections (10)

[§2.2, stability condition] The text states that the Hessian determinant 1/(S^2 l^2) - 64 pi^4 J^2/S^6 is positive 'provided J S^2 < 1'. That condition does not follow from the displayed expression; with l = 1 the determinant is positive when S^2 > 8 pi^2 |J|. Please correct the inequality.
[§2.2, Eq. (18)] Eq. (18) is presented as the leading logarithmic correction of Ref. [12], but it also follows in one line from Eq. (14) with C = S and T = S/(8 pi^2). Showing that step would make clear that the -3/2 coefficient is a consequence of Eq. (14) rather than an independent input, and would sharpen the comparison with Carlip's Cardy-formula result.
[Abstract and §3.1] There is a terminology inconsistency about which entropy is being corrected. The text below Eq. (14) says S is 'the corrected microcanonical entropy' and S0 'the entropy calculated in the canonical ensemble', whereas the abstract and §3.1 refer to 'the canonical entropy' being corrected. Please use one convention consistently.
[Abstract] The abstract says the geometry is flat 'in the canonical ensemble', but the flatness computation uses the uncorrected classical entropy S0, not the canonical-ensemble-corrected entropy of Eq. (14). The phrase should be removed or qualified.
[§3, flatness of the BTZ-CS metric] The flatness result would be much more transparent if the paper noted that Eq. (21) is a constant invertible linear map on (M, J), so the Hessian transforms as g = Lambda^T g0 Lambda with constant Lambda and flatness is inherited from the ordinary BTZ result of Ref. [10]. As written, the calculation is asserted ('the expressions leading to the calculation of the Ricci scalar are not important') rather than shown.
[§3.1, bound on K] The statement that 'the bound on K from the Ruppeiner geometry is |K| <= l' is not an independent derivation: the divergence at |K| = 1 is a direct consequence of the square roots of (1 ± K) in Eq. (22). It should be presented as a consistency check of the bound obtained in Ref. [15], not as a new determination.
[§4] The sentence 'We have explicitly verified this', referring to higher-derivative corrections, is not supported by any calculation in the paper. Either present the verification or soften the claim to an expectation.
[Figs. 1 and 2] The vertical ranges of both figures span less than 0.1 percent of the plotted value (for example 0.00015074 to 0.00015077 in Fig. 2), which visually amplifies what is essentially a constant. Please state the absolute scale in the caption and, where possible, plot on a range that makes the physical variation visible.
[References] Reference [13] duplicates Ref. [3] (Weinhold, J. Chem. Phys. 63 (1975) 2479) and should be removed or merged.
[Throughout] There are numerous typographical errors, including 'logartithmic' (abstract), 'posess', 'resect', 'equlibrium', 'seciﬁc', 'correspoding', 'intetesting', 'expeted', 'geomteries', 'gravtitation al', 'fundmaental', 'contarst', 'acknowldges', 'generaly', 'ﬂuctutaions', and 'α” corrections' (presumably α'). A careful copy-edit is needed.

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The paper

The paper claims

The thermodynamic geometry of BTZ black holes stays flat when a Chern-Simons term is added, but turns curved once thermal fluctuations are folded into the entropy.

The reading

The paper asks what shape the space of thermodynamic states has for a class of three-dimensional (BTZ) black holes. The tool is the Ruppeiner metric, which measures how entropy curves as the black hole's mass and angular momentum are varied, so that a flat metric means the system behaves like a non-interacting gas and a curved one means there are interactions. The authors confirm that the metric is flat for the ordinary rotating BTZ black hole and show it remains flat when a gravitational Chern-Simons term is added, which modifies the entropy by a term proportional to the inner horizon radius. They then add the standard correction that finite-size thermal fluctuations impose on any canonical-ensemble entropy, $S = S_0 - (1/2) A ln(C T^2)$, and find that this correction produces a nonzero curvature scalar. If correct, the flatness of these black hole state spaces is a property of the leading entropy only, and the curvature becomes a diagnostic of fluctuation effects rather than of the classical entropy function alone.

What carries the argument

The Ruppeiner metric, $g_{ij} = - partial_i partial_j S(U, N^a)$, the Hessian of entropy with respect to the extensive variables, with the two coordinates here being angular momentum $J$ and mass $M$. Its Ricci scalar is the diagnostic: zero means no thermodynamic interaction, nonzero means the state space is curved. The second piece of machinery is the fluctuation-corrected entropy $S = S_0 - (1/2) ln(C T^2)$, where $C$ is the specific heat at fixed $J$ and $T$ the Hawking temperature, both expressible in terms of $S$ and $J$. Because the corrected entropy is a function of $S_0$ and $J$, substituting it into the Hessian and computing the curvature is mechanical but long, which is why the paper presents the curvature as numerical plots rather than closed forms.

Extended reading notes

Core claim

Two claims carry the paper. First, the Ruppeiner geometry of the BTZ black hole is flat, and stays flat when the gravitational Chern-Simons term is included, once the mass and angular momentum are correctly shifted by that term, $M = M_0 - (K/l^2) J_0$ and $J = J_0 - K M_0$. Second, and this is the result the authors single out, adding the thermal-fluctuation correction to the canonical entropy destroys that flatness: the Ricci scalar of the corrected Ruppeiner metric is nonzero, and the authors compute it numerically for the rotating BTZ case (with only the leading $-3/2 A ln S_0$ correction, and again with the full $- (1/2) ln(C T^2)$ form) and for the BTZ Chern-Simons case as a function of the coupling $K$. A by-product is that the leading logarithmic correction for the Chern-Simons black hole reduces to $S = S_0 - (3/2) ln S_0$ in the large-entropy limit, the same coefficient Carlip obtained, and that the curvature scalar diverges at $|K| = 1$ in units where the cosmological length is one, recovering the stability bound $|K| leq l$.

Load-bearing premise

The paper applies a metric defined for equilibrium entropy to an entropy that has been corrected by a small-fluctuation expansion, and treats the resulting curvature as physically meaningful even though the expansion is only trustworthy far from extremality and to the order retained.

Editorial extensions

If this is right

Flat Ruppeiner geometry is not a robust property of BTZ black holes: it survives the Chern-Simons deformation but not the inclusion of finite-size thermal fluctuations.
The $-3/2$ coefficient of the logarithmic entropy correction is reproduced for the Chern-Simons black hole, extending the evidence that this coefficient is universal across black hole families.
The divergence of the curvature scalar at $|K| = 1$ (with $l = 1$) reproduces Solodukhin's stability bound $|K| \leq l$ from a purely thermodynamic route, since the entropy becomes unphysical beyond it.
Because the entropy form is unchanged by higher-derivative corrections apart from a shift in the central charge, the authors expect the same qualitative geometry for BTZ Chern-Simons black holes with such corrections, and say they verified this.

Reading between the lines

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

The paper notes a small but finite curvature at zero angular momentum once the leading logarithmic correction is included, while describing the non-rotating BTZ black hole as non-interacting even with fluctuations. That tension is worth chasing: it may indicate that the fluctuation-corrected curvature is an artifact of treating a truncated expansion as an exact entropy function.
If the curvature scalar really tracks fluctuation strength, it should scale with the inverse system size in a controlled way; checking whether $R$ falls off as the mass grows (the authors report that it does, at fixed shape) would turn the plots into a quantitative statement rather than a sign.
The divergence at $|K| = 1$ suggests a general pattern: thermodynamic curvature can locate the boundary of a parameter regime where the entropy stops making sense, an approach that could be applied to other couplings in higher-derivative gravity.
Since thermal-fluctuation and higher-derivative corrections can enter with opposite signs, the curvature scalar is a natural place to look for a cancellation between the two, which the authors flag as future work.

Original abstract

We investigate the Ruppeiner geometry of the thermodynamic state space of a general class of BTZ black holes. It is shown that the thermodynamic geometry is flat for both the rotating BTZ and the BTZ Chern Simons black holes in the canonical ensemble. We further investigate the inclusion of thermal fluctuations to the canonical entropy of the BTZ Chern Simons black holes and show that the leading logartithmic correction due to Carlip is reproduced. We establish that the inclusion of thermal fluctuations induces a non zero scalar curvature to the thermodynamic geometry.

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We investigate the Ruppeiner geometry of the thermodynamic state space of a general class of BTZ black holes. It is shown that the thermodynamic geometry is flat for both the rotating BTZ and the BTZ Chern Simons black holes in the canonical ensemble. We further investigate the inclusion of thermal fluctuations to the canonical entropy of the BTZ Chern Simons black holes and show that the leading logartithmic correction due to Carlip is reproduced. We establish that the inclusion of thermal fluctuations induces a non zero scalar curvature to the thermodynamic geometry.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate the Ruppeiner geometry of the thermodynamic state space of a general class of BTZ black holes.	contribution 0.99	0.90	0.06	asserted 0.58
s1	It is shown that the thermodynamic geometry is flat for both the rotating BTZ and the BTZ Chern Simons black holes in the canonical ensemble.	result 1.00	0.88	0.45	asserted 1.00
s2	We further investigate the inclusion of thermal fluctuations to the canonical entropy of the BTZ Chern Simons black holes and show that the leading logartithmic correction due to Carlip is reproduced.	result 0.71	0.89	0.11	asserted 1.00
s3	We establish that the inclusion of thermal fluctuations induces a non zero scalar curvature to the thermodynamic geometry.	result 1.00	0.94	0.34	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Figures

**Figure 1.** The Ricci scalar R of the Ruppeiner metric for the BTZ black hole, as a function of the angular momentum J, with only the logarithmic correction to the entropy (eq. 18) being taken into account. The mass M has been set to 100. of the curvature scalar. This indicates that even at zero angular momentum, the statistical system is interacting, once small fluctuations are included. This should be contrasted with the non-rotating BTZ black hole which is a non-interacting system even when small fluctuations are included. We have checked that increasing the value of M, the value of the…

**Figure 2.** The Ricci scalar R of the Ruppeiner metric for the BTZ black hole, as a function of the angular momentum J, including small fluctuations (eq. 14). The mass M has been set to 100. CS black hole.In [14], [15], the entropy of BTZ-CS black holes have been analysed, and these authors have derived an expression for the entropy, which differes from the entropy of the “usual” BTZ black hole, eq. (7). The modified entropy for the BTZ-CS black hole is S = 4π r+ − K l r− (19) where K is the Chern-Simons coupling. The extra term in eq. (19) as compared to eq. (7) is the contribution f

**Figure 3.** The Ricci scalar R of the Ruppeiner metric for the BTZ-CS black hole, as a function of the Chern-Simons coupling K, with only the logarithmic correction to the entropy being taken into account. The mass M has been set to 100 and we have set J = 1 to ensure that we are far from extremality. We now calculate the Ruppeiner geometry correspoding to the modified entropy of the BTZ-CS black hole with thermal fluctuations. As in the last section, we first present the numerical result for the Ricci scalar using the leading order correction of eq. (30). This is depicted in fig. (3). In this…

**Figure 4.** The Ricci scalar R of the Ruppeiner metric for the BTZ-CS black hole, as a function of the Chern-Simons coupling K. The mass M has been set to 100 and we have set J = 1 to ensure that we are far from extremality. angular momentum, and studied its behaviour. The plots in this case are qualitatively the same as in fig. (1) and fig. (2) and we do not discuss them further. 4 Discussions and Conclusions In this article, we have mainly investigated the thermodynamic geometry of a class of BTZ black holes, both with and without the Chern-Simons term. We have shown that the Ruppeiner…

Figures reproduced from arXiv:hep-th/0606084 by the authors.

References

Works cited (17)

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[7]

K. Huang, “Statistical Mechanics,” Publishers John Wiley, 1963, L. D. Landau, E. M. Lifshitz, “Statistical Physics,” Publishers Perg amon, 1969

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[1]

R. M. Wald, “The Thermodynamics of Black Holes,” Living Rev. Rel. 4 (2001) 6

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H. B. Callen, Thermodynamics and an Introduction to Thermostatistics , Pub. Wiley, New York (1985); L.Tisza Generalized Thermodynamics, Pub. MIT Press, Cambridge, MA (1966)

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F. Weinhold, J.Chem. Phys. 63, 2479 (1975), ibid J. Chem. Phys 63 , 2484 ( 1975)

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G. Ruppeiner, Rev. Mod. Phys 67 (1995) 605, Erratum 68 (1996) 313

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S. Ferrara, G. W. Gibbons, R. Kallosh, “Black Holes and Critical Po ints in Moduli Space,” Nucl. Phys. B500 (1997) 75, hep-th 9702103

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S. Carlip, “ Logarithmic corrections to black hole entropy from t he Cardy formula,” Class. Quant. Grav. 17 (2000) 4175, gr-qc/0005105

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F. Weinhold, J. Chem. Phys 63 (1975) 2479

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P. Kraus, F. Larsen, “Holographic Gravitational Anomalies,” hep-th/0508218

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Forward citations

Cited by 5 Pith papers

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For a four-dimensional neutral Gauss-Bonnet AdS black hole, the intrinsic curvature of the zero-charge hypersurface is positive at small volumes and low temperatures, suggesting repulsive microstructure interactions, ...
Second Order Phase Transition and Thermodynamic Geometry in Kerr-AdS Black Hole
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At fixed angular velocity the Kerr-AdS black hole's small-to-large mass transition satisfies both Ehrenfest equations, and the Ruppeiner curvature diverges at the same critical entropy, though it also diverges where t...
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The statistical entropy of a class of quarter-BPS dyons in N=4 string theories matches the extremal black hole entropy, including the first inverse-charge correction from a Gauss-Bonnet term.

How the literature uses it

45 arXiv papers cite this in 67 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 36 sentences without a confident label

Compares with it 1 paper

Importantly, the new entropy is given by the Bekenstein-Hawking entropy as [equation] which is slightly different from the shifted-entropy $S_c$ of Eq.([ref]) expressed as [ref] [equation] Here we check easily that the first-law of thermodynamics [equation] is satisfied for the CS-BTZ black hole.

Entropy of black holes in topologically massive gravity · Thermodynamics and entropy for CS-BTZ black hole · 0.91

Cites it as support 1 paper

Furthermore, it turns our that the state-space interactions generically remain finite and non-zero when small thermal fluctuations in the canonical ensemble are taken into account [ref] .

State-space Correlations and Stabilities · Introduction · 0.97

Cites it as background 26 papers

Thermodynamic geometry [ref] plays important role in understanding the stability and phase structure properties of black holes.

Thermodynamic Geometry and Topological Einstein-Yang-Mills Black Holes · Introduction · 0.98
Before analyzing the state-space properties of the eight parameter black brane configuration, let us first provide a brief introduction to the thermodynamic geometry [ref] .

State-space geometry, non-extremal black holes and Kaluza-Klein monopoles · Introduction · 1.00
Phase transitions of black holes have been extensively studied using different frameworks such as extended black hole thermodynamics [ref] , restricted phase space thermodynamics and holographic thermodynamics [ref] , geometrical interpretation of phase transition [ref] etc.

Revisiting thermodynamic topology of Hawking-Page and Davies type phase transitions · introduction · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [10]

J. E. Aman, I. Bengtsson, N. Pidokrajt, “Geometry of Black H ole Thermo- dynamics,” Gen. Rel. Grav. 35 (2003) 1733, gr-qc/0304015, G. Arcioni, E. Lozano-Tellechea, Phys.Rev. D72 (2005) 104021, hep-th/ 0412118 . 16

View on Pith Pith review arXiv 2003

[2] [7]

K. Huang, “Statistical Mechanics,” Publishers John Wiley, 1963, L. D. Landau, E. M. Lifshitz, “Statistical Physics,” Publishers Perg amon, 1969

View on Pith 1963

[3] [1]

R. M. Wald, “The Thermodynamics of Black Holes,” Living Rev. Rel. 4 (2001) 6

View on Pith 2001

[4] [2]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics , Pub. Wiley, New York (1985); L.Tisza Generalized Thermodynamics, Pub. MIT Press, Cambridge, MA (1966)

View on Pith 1985

[5] [3]

F. Weinhold, J.Chem. Phys. 63, 2479 (1975), ibid J. Chem. Phys 63 , 2484 ( 1975)

View on Pith 1975

[6] [4]

G. Ruppeiner, Rev. Mod. Phys 67 (1995) 605, Erratum 68 (1996) 313

View on Pith 1995

[7] [5]

S. Ferrara, G. W. Gibbons, R. Kallosh, “Black Holes and Critical Po ints in Moduli Space,” Nucl. Phys. B500 (1997) 75, hep-th 9702103

View on Pith 1997

[8] [6]

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