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Black Hole Thermodynamics and Negative Entropy in deSitter and Anti-deSitter Einstein-Gauss-Bonnet gravity

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M. Cvetic, S. Nojiri, S.D. Odintsov

High Energy Physics - Theory · Submitted 6 Dec 2001 · 25 references

Pith's verdict: Competent Gauss-Bonnet thermodynamics with a genuinely useful counterterm mass formula, but the negative-entropy headline rests on an unclosed gap between the counterterm mass and the free energy.

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arXiv IDhep-th/0112045 (version 4)
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CitationM. Cvetic, S.D. Odintsov, S. Nojiri. "Black Hole Thermodynamics and Negative Entropy in deSitter and Anti-deSitter Einstein-Gauss-Bonnet gravity." arXiv:hep-th/0112045 (2001). https://pith.science/paper/LG4PWXWP
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The paper

The paper claims

In five-dimensional Einstein-Gauss-Bonnet gravity, the thermodynamic entropy of Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes can come out negative or zero for a range of the higher-derivative coupling, and the paper,

The reading

The paper asks what happens to black hole thermodynamics when the gravitational action carries a Gauss-Bonnet term, the curvature-squared combination that keeps the field equations second order and arises as the next-to-leading correction in string effective actions. Working in five dimensions with an electromagnetic field, the authors solve for charged Schwarzschild-anti-de Sitter black holes, compute the Hawking temperature, the regularized action and the free energy, and draw the Hawking-Page phase diagram between the black hole and pure anti-de Sitter space. Because the Gauss-Bonnet coefficient and the electric charge are both free, the boundary between the two phases is a critical line in the parameter plane rather than a single critical point. The authors then derive the surface counterterms needed to make the action finite on asymptotically de Sitter and anti-de Sitter backgrounds in higher-derivative gravity, use them to obtain a conserved black hole mass, and feed that mass into $S = 8$ to get the entropy. In a range of couplings the resulting entropy is negative or zero, and the authors propose that this marks a new kind of instability in which a black hole with negative entropy in one asymptotic geometry gives way to one with positive entropy in the other.

What carries the argument

Two pieces of machinery carry the argument. The first is the Gauss-Bonnet solution itself: with $a = c$ and $b = -4c$ in the curvature-squared action, the metric function $e^{2} = e^{-2}$ solves a quadratic whose roots give the horizon radii, the Hawking temperature $4 T_H = (e^{2})'$ at $r = r_H$, and a free energy regularized by subtracting the $= Q = 0$ background. The second is the surface-counterterm construction: auxiliary fields $A$, $B_{}$, $C_{}$ linearize the $R^2$ action, a Dirichlet-type boundary condition is imposed on the scalar contractions, and the resulting boundary term plus a Gibbons-Hawking term and two counterterms with coefficients $1$ and $2$ yields a finite stress tensor and a conserved mass $M = (3l^2/16) V_3 (1/^2 - 40a/l^2 - 8b/l^2 - 4c/l^2)(k^2 + 16 l^2)$. That mass is what the entropy integral consumes, so the counterterm coefficients set the sign of the entropy.

Extended reading notes

Core claim

The central claim is that in higher-derivative gravity the entropy of a Schwarzschild-de Sitter or Schwarzschild-anti-de Sitter black hole need not be positive. Using surface counterterms derived for general $R^2$ gravity on these backgrounds, the authors obtain a conserved mass $M$, identify it with the thermodynamic energy $E$, and integrate $dS = dE/T$ to get the entropy. For the Gauss-Bonnet case the result is $S = (V_3/8)((1-12)/(1-4)) (4 r_H^3 + 24 k r_H) + S_0$ with $= c^2/l^2$, which turns negative for $1/12 < < 1/4$ and, when $S_0 = 0$, for small black holes with $r_H^2 < -12$ in the $ < 0$ branch. Crucially, the sign flips the other way between the two backgrounds: the parameter range that makes the anti-de Sitter entropy negative makes the de Sitter entropy positive, and vice versa. The authors read this as a possible new type of phase transition, or 'phase transmutation', between de Sitter and anti-de Sitter black holes, passing through a state of zero mass, zero free energy and zero entropy.

Load-bearing premise

The entropy is obtained by integrating $dE/T$ with the integration constant set to zero and with the counterterm-derived conserved mass identified as the thermodynamic energy, so the reported sign of the entropy rests on that normalization choice and on that identification.

Editorial extensions

If this is right

  • The Hawking-Page diagram for charged anti-de Sitter black holes in Gauss-Bonnet gravity acquires a critical line, not a critical point, because the Gauss-Bonnet coefficient and the charge can be traded against each other.
  • Surface counterterms exist for higher-derivative gravity on de Sitter as well as anti-de Sitter backgrounds, so a conserved black hole mass and a finite free energy can be defined in both cases.
  • The entropy sign is background-dependent: parameters that make the anti-de Sitter black hole entropy negative leave the de Sitter entropy positive, so a single theory can host a stable phase in one asymptotic geometry and an unstable one in the other.
  • At the critical coupling where the counterterm mass vanishes, black holes of any horizon radius become massless, which the authors suggest would allow black holes to be produced copiously.
  • Bounds on de Sitter entropy proposed for Einstein gravity, such as the conjecture that de Sitter entropy bounds asymptotically de Sitter entropy, cannot hold in higher-derivative gravity once the entropy can be zero or negative.

Reading between the lines

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

  • The negative-entropy region is only as robust as the choice $S_0 = 0$; a nonzero integration constant would shift the zero of entropy and could remove the negative region for small black holes while leaving the large-black-hole statement intact.
  • An independent check would be to compute the Gauss-Bonnet black hole entropy by the Noether-charge (Wald) method, which does not rely on integrating $dE/T$, and compare signs; agreement would turn the observation from a convention-dependent statement into a physical one.
  • If the proposed de Sitter/anti-de Sitter transmutation is real, it should show up as a discontinuity or a runaway in the finite-temperature effective potential of the dual boundary theory, which is in principle a computable quantity rather than a speculation.
  • Because the effect is driven by the curvature-squared term competing with the Einstein term, it should scale with the ratio of the Gauss-Bonnet coupling to the cosmological scale, suggesting a sharp threshold rather than a gradual onset that a numerical scan of the phase diagram could confirm.
Original abstract

We investigate the charged Schwarzschild-Anti-deSitter (SAdS) BH thermodynamics in 5d Einstein-Gauss-Bonnet gravity with electromagnetic field. The Hawking-Page phase transitions between SAdS BH and pure AdS space are studied. The corresponding phase diagrams (with critical line defined by GB term coefficient and electric charge) are drawn. The possibility to account for higher derivative Maxwell terms is mentioned. In frames of proposed dS/CFT correspondence it is demonstrated that brane gravity maybe localized similarly to AdS/CFT. SdS BH thermodynamics in 5d Einstein and Einstein-Gauss-Bonnet gravity is considered. The corresponding (complicated) surface counterterms are found and used to get the conserved BH mass, free energy and entropy. The interesting feature of of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. We speculate that the appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,865 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.45 · human participants 0.01 · controlled comparison 0.22 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.16 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We investigate the charged Schwarzschild-Anti-deSitter (SAdS) BH thermodynamics in 5d Einstein-Gauss-Bonnet gravity with electromagnetic field. The Hawking-Page phase transitions between SAdS BH and pure AdS space are studied. The corresponding phase diagrams (with critical line defined by GB term coefficient and electric charge) are drawn. The possibility to account for higher derivative Maxwell terms is mentioned. In frames of proposed dS/CFT correspondence it is demonstrated that brane gravity maybe localized similarly to AdS/CFT. SdS BH thermodynamics in 5d Einstein and Einstein-Gauss-Bonnet gravity is considered. The corresponding (complicated) surface counterterms are found and used to get the conserved BH mass, free energy and entropy. The interesting feature of of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. We speculate that the appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the charged Schwarzschild-Anti-deSitter (SAdS) BH thermodynamics in 5d Einstein-Gauss-Bonnet gravity with electromagnetic field. contribution 0.99 0.89 0.03 asserted 0.92
s1 The Hawking-Page phase transitions between SAdS BH and pure AdS space are studied. contribution 1.00 0.85 0.04 asserted 0.97
s2 The corresponding phase diagrams (with critical line defined by GB term coefficient and electric charge) are drawn. contribution 0.51 0.91 0.06 asserted 1.00
s3 The possibility to account for higher derivative Maxwell terms is mentioned. future work 0.75 0.24 0.02 hedged possibility 0.57
s4 In frames of proposed dS/CFT correspondence it is demonstrated that brane gravity maybe localized similarly to AdS/CFT. contribution 0.54 0.66 0.05 asserted 0.81
s5 SdS BH thermodynamics in 5d Einstein and Einstein-Gauss-Bonnet gravity is considered. contribution 0.99 0.84 0.03 asserted 0.99
s6 The corresponding (complicated) surface counterterms are found and used to get the conserved BH mass, free energy and entropy. method 0.89 0.93 0.07 asserted 1.00
s7 The interesting feature of of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. result 0.92 0.59 0.50 asserted 0.92
s8 We speculate that the appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur. interpretation 0.99 0.16 0.41 speculative 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v4
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The phase diagrams when ǫ < 1 2 TH ✲ rH ✻ pure AdS phase AdS black hole phase r 2 H = 12ǫ ✻
Figure 2. The phase diagrams when ǫ > 1 2 18

Figures reproduced from arXiv:hep-th/0112045 by the authors.

References
How the literature uses it

112 arXiv papers cite this in 155 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 · 85 sentences without a confident label

Builds on it 1 paper
Compares with it 4 papers
  • The consistency of the procedure requires [equation] This counterterm is general and can be used to regularize the action of both branches.[A discussion of the counterterm method for GB gravity with cosmological constant also appears in the forth-coming paper [ref] .] For solutions with a well defined Einstein gravity limit, one finds as $α'→0$, that $c_1→-3/L+α'/L^3+O(α')^2,$ $c_2 →L/2+3α'/2L+O(α')^2$ that match the results in [ref] (see, also, [ref] ).

    (Un)attractor black holes in higher derivative AdS gravity · The counterterm method · 0.98

  • This also matches the Gauss-Bonnet black hole mass [ref] in the case $α̃_1=α̃_2=0$, and agrees with [ref] , with arbitrary $α_i$ coefficients (note that we have removed the $k^2$ dependent `Casimir energy' by the addition of finite counterterms, which was not done in [ref] ).

    Higher Derivative Corrections to R-charged Black Holes: Boundary Counterterms and the Mass-Charge Relation · Thermodynamics · 1.00

  • [equation] For Gauss-Bonnet gravity this agrees with the expressions for the surface gravity obtained from the explicit solutions in [ref] using an approach similar to the one here.

    Mass and Free Energy of Lovelock Black Holes · Surface gravity of Lovelock black holes · 0.99

Cites it as background 60 papers

Discussion

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