Black Hole Entropy and the Dimensional Continuation of the Gauss-Bonnet Theorem

Claudio Teitelboim; Jorge Zanelli; M\'aximo Ba\~nados

REVIEW

Black Hole Entropy and the Dimensional Continuation of the Gauss-Bonnet Theorem

Máximo Bañados, Claudio Teitelboim, Jorge Zanelli

General Relativity and Quantum Cosmology · Submitted 25 Sep 1993 · 8 references

Pith's verdict: A short, honest letter that pins the 1/4 in the area law to the Euler class of a disk, with the Lovelock generalization quoted rather than claimed as new.

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arXiv IDgr-qc/9309026 (version 1)

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CitationClaudio Teitelboim, Jorge Zanelli, M\'aximo Ba\~nados. "Black Hole Entropy and the Dimensional Continuation of the Gauss-Bonnet Theorem." arXiv:gr-qc/9309026 (1993). https://pith.science/paper/PAFSQ2JD

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Editor's note

read the letter

The thing to know: this is a ten-page letter that gives a clean geometric reason for the quarter in the area law. The entropy is the Euler class of a small disk at the horizon (2π) times the area of the S^{d-2} there. That identification, plus the statement that the deficit angle of a cusp in the R² factor and the sphere area are canonically conjugate, is the actual new content. The Lovelock entropy formula, Eq. (15), is Jacobson-Myers [4] and the authors say so in the text. Credit where due: they do not dress up a known result as their own, and the conjugacy observation is a genuinely nice piece of physics — it turns the smoothness condition α = 0 into the equation that fixes the temperature, which is a cleaner statement than the usual Gibbons-Hawking bookkeeping. The normalization is self-consistent: with I_H = (1/2)∫√g R, S = 2πA is the same statement as A/4G in the 1/16π normalization. I checked that and it holds. On the soft spots, I part company with the reader on the main one. The factorization in Eq. (7) is asserted in one sentence, and that reads as a gap, but it is not one. Working out the warped product explicitly, the mixed Ricci term is a total derivative whose boundary piece cancels the sphere part of the extrinsic curvature, and the sphere-curvature term is killed by the shrinking disk area. What survives is exactly Euler class times area. So the compression is a letter-format choice, not a hole in the argument. Same for Eq. (8): the boundary-term cancellation is cited to Brown-York [6], which is published and does contain the point. The one thing I would actually flag is reference [8], an unpublished self-citation used to support the ℜ²×S^{d-2} topology assumption. That assumption is standard and the citation is honest, but it is the only place the paper leans on something a referee cannot check. Minor. Who should read it: anyone working on Lovelock gravity, horizon thermodynamics, or the topological reading of entropy. It is short enough to hand to a graduate student and worth the afternoon. Send it out.

Referee report

Significance

The result is a clean, parameter-free geometric reinterpretation of a familiar coefficient: the 1/4 in the area law is exhibited as the Euler class of the two-dimensional disk transverse to the horizon, with no adjustable constants anywhere in the derivation. The observation that the deficit angle at a point in R^2 and the area of the attached sphere are canonically conjugate, Eq. (8), is a genuinely useful structural statement, since it turns the smoothness condition at the Euclidean horizon into an equation of motion and gives a uniform derivation of the temperature. The extension to Lovelock theory, Eq. (15), organizes the entropy of the most general second-order theory as a sum of continued Euler classes and connects transparently to the published Jacobson-Myers formula. The paper is short and its claims are checkable by direct computation, which is a strength; the main weakness is that the two steps that carry the argument are stated rather than shown.

Major objections (1)

[Eq. (7) and Eq. (15)] The factorization lim_{eps->0} I_H[Delta_eps x S^{d-2}] = 2 pi x (area of S^{d-2})_origin is the load-bearing step of the whole paper, and it is supported by the single sentence 'A short analysis reveals that the first term in (6) factorizes into the product of the Euler class (1) for the triangle and the area of the S^{d-2} at the origin.' No demonstration is given that the mixed components of the Ricci tensor between the R^2 and S^{d-2} factors, and the sphere-curvature contribution, either cancel against the extrinsic-curvature boundary term or are suppressed as eps -> 0. A direct warped-product check for ds^2 = g_ab dx^a dx^b + r(x)^2 dOmega^2 suggests the factorization is in fact correct: the mixed term is a total derivative whose boundary piece cancels the sphere part of K on the boundary of Delta_eps x S^{d-2}, and the sphere-curvature term is multiplied by the vanishing disk area. But that check is not in the manuscript, and the reader is asked to take it on faith. The point matters more for Eq. (15), which is the paper's advertised extension: the statement that 'the preceding analysis goes through step by step' for every Lovelock term is exactly the claim that the same cancellation occurs for each I^p_L, and it is asserted without any indication of how the antisymmetrized curvature product factorizes on a warped product. I would ask the authors to add a short appendix, or a few explicit lines, showing the cancellation for the Einstein term and indicating why it persists for general p. This is a completeness gap rather than an error, and it is fixable within the Letter format.

Minor objections (7)

[Eq. (9) and the following paragraph] The text says the coefficient is 'usually quoted as one fourth in units where Newton's constant is unity', but Eq. (2) carries an explicit factor 1/2 in front of the Einstein-Hilbert term, which corresponds to the normalization 8 pi G = 1, not G = 1. With that normalization Eq. (9) gives S = 2 pi A = A/(4G) with G = 1/(8 pi), so the two statements are consistent only if the convention is stated. Please state the normalization of the action once, explicitly, so that Eq. (9) and the 'one fourth' remark cannot be read as contradictory.
[Eq. (8) and the paragraph following it] The derivation of delta I = alpha delta(area) is justified by the remark that a boundary term coming from space derivatives in H cancels the variation of the surface term in the Euler class, with a pointer to Ref. [6]. Since Eq. (8) is the basis for the conjugacy statement highlighted in the abstract, one explicit sentence identifying that boundary term and its cancellation would make the Letter self-contained without reproducing the cited work.
[Eq. (10) and Eq. (15)] The normalization of the Lovelock coefficients alpha_p is never fixed. The reader has to infer alpha_1 = 1 by matching the p = 1 term of Eq. (10) to Eq. (2) and the p = 1 term of Eq. (15) to Eq. (9). Please state the normalization once, since the comparison with the Jacobson-Myers entropy depends on it.
[Topology assumption, opening paragraph and Ref. [8]] The restriction to Euclidean spacetimes of topology R^2 x S^{d-2} is taken from Ref. [8], which is listed as 'to be published'. A one-sentence justification in the text, or a pointer to where in Ref. [8] the topology is established, would let a reader assess the assumption without consulting a forthcoming paper.
[Second paragraph, on the temperature] The statement that the black hole temperature 'is determined, in the semiclassical approximation, by extremizing with respect to the area of the S^{d-2} at the horizon' is compressed to the point of being hard to follow. Spelling out how the condition alpha = 0 fixes the relation between the period beta and the horizon radius would make the temperature claim as concrete as the entropy claim.
[Eq. (4a) and Eq. (4b)] The contrast between the two limits is attributed to the integrand of the canonical action being smooth, but the text does not say why the covariant integrand is not, nor which term survives as eps -> 0. One clause naming the surviving contribution would help the reader follow the logic of Eq. (5).
[Eq. (1) and Eq. (2)] The symbol K and the boundary integral are used in both equations with different meanings, since in Eq. (2) the boundary is (d-1)-dimensional and the extrinsic curvature refers to the full spacetime. A note distinguishing the two-dimensional Euler-class boundary term from its dimensional continuation would avoid confusion.

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The paper

The paper claims

Black hole entropy is the Euler class of a small disk at the horizon, times the horizon sphere's area.

The reading

The paper asks where the black hole entropy formula comes from, and answers that it is not a dynamical coincidence but a leftover of a topological theorem. The authors work in Euclidean signature, where a black hole has the shape of a two-dimensional plane (or cone) with a sphere of dimension $d-2$ attached at every point. They show that the deficit angle of a conical cusp at any point of that plane is canonically conjugate to the area of the sphere sitting there, in the same sense that a position and its conjugate momentum are paired. Turning the crank on the Gauss-Bonnet theorem then makes the entropy equal to $2π$ times the horizon area, with the $2π$ identified as the Euler class of the two-dimensional disk around the horizon rather than as an independent coefficient. The same bookkeeping is extended to the most general metric action that keeps the field equations second order, where the entropy becomes a sum of continued Euler classes over the horizon sphere.

What carries the argument

The Gauss-Bonnet theorem in two dimensions: the integral of the scalar curvature over a manifold plus the integral of the trace of the extrinsic curvature over its boundary equals $2π$ times the Euler number. The paper's move is to read the Hilbert action as this same expression continued to $d$ dimensions, where it stops being a topological invariant but keeps the same structure, and then to apply it to a small disk around the horizon times the transverse sphere. The load-bearing step is the factorization in Eq. (7), that the limit of the Hilbert action on the small triangle times the sphere equals the Euler class of the triangle times the area of the sphere at the origin. The same factorization, applied to the Lovelock action, yields the entropy as a sum of dimensionally continued Euler classes of all even dimensions below $d-2$, reproducing the earlier Jacobson-Myers expression.

Extended reading notes

Core claim

The central claim is that in Einstein's theory the black hole entropy is the Euler class of a small disk centered at the horizon multiplied by the area of the $S^{d-2}$ at that point, so that $S = 2π × (area of S^{d-2})_{horizon}$. The route is dimensional continuation: the Hilbert action in $d$ dimensions has the same algebraic shape as the two-dimensional Euler class, so one may continue the Gauss-Bonnet theorem upward in dimension, at the price of losing its topological invariance. Applied to the region near the horizon, the Hilbert action on a small triangle of the $ℜ^2$ factor times the sphere tends to the Euler class of the triangle, which is $2π$ because a triangle is topologically a disk, times the area of the sphere at the origin. Combined with the canonical action, which vanishes on the black hole because the constraints hold and the spatial metric is time independent, this leaves the disk contribution as the whole value of the action on the extremum, and hence as the entropy. The authors also show that allowing a cusp of deficit angle $α$ at the origin makes the action change by $δI = α δ(area of S^{d-2} at cusp)$, which is the statement that deficit angle and sphere area are canonically conjugate. Extremizing the covariant action then forces $α = 0$, the condition for the Euclidean manifold to be metrically smooth, whereas extremizing the canonical action alone would force $α = 2π$ and introduce a source at the origin.

Load-bearing premise

The argument rests on the claim that the Hilbert action on a small disk around the horizon times the transverse sphere factorizes cleanly into the Euler class of the disk times the sphere's area, a step the paper asserts with the single remark that a short analysis reveals it and does not derive.

Editorial extensions

If this is right

The overall coefficient in front of the horizon area is fixed by topology rather than by matching to a dynamical calculation, so the quarter-area law acquires a geometric origin.
The black hole temperature is determined semiclassically by extremizing the action with respect to the horizon sphere's area, since the deficit angle is the conjugate variable to that area.
In Lovelock gravity the entropy is not simply proportional to area but is a sum of continued Euler classes over the horizon sphere, Eq. (15), with the cosmological constant term contributing nothing.
Covariance of the action requires adding the small-disk Hilbert term to the canonical action, which relocates the fixed point of the Euclidean time circle without introducing a source.

Reading between the lines

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

If the entropy is a topological quantity in this sense, one might expect it to be robust under smooth deformations of the horizon geometry that preserve the disk's Euler class, which suggests a check: deform the horizon sphere non-uniformly and see whether the semiclassical entropy is unchanged to leading order.
The conjugation statement pairs a purely two-dimensional intrinsic quantity (deficit angle) with an extrinsic one (sphere area), which reads like a hint that the $ℜ^2$ and $S^{d-2}$ factors of the Euclidean solution play complementary roles rather than being independent spectators.
The Lovelock sum in Eq. (15) suggests a diagnostic for candidate higher-curvature theories: any entropy formula that cannot be written as a sum of continued Euler classes over the horizon sphere sits outside this scheme, whether or not it satisfies the second-order field equation condition.
The paper's insistence that the partition function needs the small-disk term because it is the trace of a transition amplitude hints at a relation between the Euclidean topology of the time circle and the counting of states that is not spelled out here.

Original abstract

The Euclidean black hole has topology $\Re^2 \times {\cal S}^{d-2}$. It is shown that -in Einstein's theory- the deficit angle of a cusp at any point in $\Re^2$ and the area of the ${\cal S}^{d-2}$ are canonical conjugates. The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the ${\cal S}^{d-2}$ there.These results are obtained through dimensional continuation of the Gauss-Bonnet theorem. The extension to the most general action yielding second order field equations for the metric in any spacetime dimension is given.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Euclidean black hole has topology $\Re^2 \times {\cal S}^{d-2}$. It is shown that -in Einstein's theory- the deficit angle of a cusp at any point in $\Re^2$ and the area of the ${\cal S}^{d-2}$ are canonical conjugates. The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the ${\cal S}^{d-2}$ there.These results are obtained through dimensional continuation of the Gauss-Bonnet theorem. The extension to the most general action yielding second order field equations for the metric in any spacetime dimension is given.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Euclidean black hole has topology $\Re^2 \times {\cal S}^{d-2}$.	background 0.99	0.16	0.03	asserted 1.00
s1	It is shown that -in Einstein's theory- the deficit angle of a cusp at any point in $\Re^2$ and the area of the ${\cal S}^{d-2}$ are canonical conjugates.	result 0.99	0.77	0.18	asserted 1.00
s2	The black hole entropy emerges as the Euler class of a small disk centered at the horizon multiplied by the area of the ${\cal S}^{d-2}$ there.These results are obtained through dimensional continuation of the Gauss-Bonnet theorem.	result 0.99	0.89	0.67	asserted 1.00
s3	The extension to the most general action yielding second order field equations for the metric in any spacetime dimension is given.	contribution 0.96	0.86	0.14	asserted 0.98

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

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arXiv listing abstract v1

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References

Works cited (8)

[4]

T. Jacobson and R. Myers, Phys. Rev. Lett. 70, 3684 (1993)

View on Pith 1993
[6]

See J.D. Brown and J.W. York, Phys. Rev. D. 47, 1420 (1993), for a lucid discussion of this point. These authors assume α = 0 from the outset and do not obtain it as an equation of motion. They do not discuss the role of the Gauss-Bonn et theorem

View on Pith 1993
[8]

M. Ba˜ nados, C. Teitelboim and J.Zanelli, CECS/IAS-preprint (199 2), gr-qc/9307033, to be published in Phys. Rev. D. 10

View on Pith Pith review arXiv
[1]

D. Lovelock, J. of Math. Phys. 12, 498 (1971)

View on Pith 1971
[2]

B. Zumino, Phys. Rep. 137, 108 (1986)

View on Pith 1986
[3]

C. Teitelboim and J. Zanelli, Class. and Quant.Grav. 4, L125 (1987); and in Constraint Theory and Relativistic Dynamics , G. Longhi and L. Lussana, eds. (World Scientiﬁc, Singapore, 1987)

View on Pith 1987
[5]

T. Regge and C. Teitelboim, Ann.Phys. (N.Y.) 88, 286 (1974)

View on Pith 1974
[7]

Note that the term which must be added to the canonical action d oes not refer to any history in time t. This is a deep property of the black-hole. Steven Carlip (private co mmu- nication) describes it by saying that the action of the small triangle r efers to “intrinsically time independent degrees of freedom”

View on Pith

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How later papers used it

47 later papers cite it

Evidence: substantial. 46 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 71 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 8 papers with this label

To calculate the on-shell action we will use the method of [ref] which we refer to as the BTZ prescription.

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Followed the common procedure of this paper of cutting the manifold near the horizon and evaluating the action of the interior and exterior regions separately as the cut surface approaches the horizon.

Thermodynamic ensembles with cosmological horizons (2022) · 0.92 · paraphrased: the licence does not allow quoting

How the literature uses it

42 arXiv papers cite this in 63 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 · 46 sentences without a confident label

Builds on it 1 paper

This is a semiclassical extension of the microcanonical action formula $I_E^mc=-S_BH$ obtained by [ref] , and may be interpreted as a path integral derivation of the generalized entropy of causal diamonds.

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We use Gauss-Bonnet theorem on the $ρ-τ_E$ surface in order to calculate the contribution from the conical singularity and we have: [equation] where $δ^2(P)$ represents the Dirac delta function on the $ρ-τ_E$ plane, and $R_h$ is the Riemann curvature on the horizon before the replica trick [ref] .

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Gravitational R\'enyi entropy from corner terms · Introduction · 0.81

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] [4]

T. Jacobson and R. Myers, Phys. Rev. Lett. 70, 3684 (1993)

View on Pith 1993

[2] [6]

See J.D. Brown and J.W. York, Phys. Rev. D. 47, 1420 (1993), for a lucid discussion of this point. These authors assume α = 0 from the outset and do not obtain it as an equation of motion. They do not discuss the role of the Gauss-Bonn et theorem

View on Pith 1993

[3] [8]

M. Ba˜ nados, C. Teitelboim and J.Zanelli, CECS/IAS-preprint (199 2), gr-qc/9307033, to be published in Phys. Rev. D. 10

View on Pith Pith review arXiv

[4] [1]

D. Lovelock, J. of Math. Phys. 12, 498 (1971)

View on Pith 1971

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Note that the term which must be added to the canonical action d oes not refer to any history in time t. This is a deep property of the black-hole. Steven Carlip (private co mmu- nication) describes it by saying that the action of the small triangle r efers to “intrinsically time independent degrees of freedom”

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