Entropy of Lovelock Black Holes

Robert C. Myers; Ted Jacobson

REVIEW

Entropy of Lovelock Black Holes

Ted Jacobson, Robert C. Myers

High Energy Physics - Theory · Submitted 6 May 1993 · 29 references

Pith's verdict: The standard derivation of Lovelock black hole entropy: a general intrinsic-curvature formula from the Hamiltonian first law, with the Killing-horizon assumption openly flagged.

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arXiv IDhep-th/9305016 (version 1)

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CitationRobert C. Myers, Ted Jacobson. "Entropy of Lovelock Black Holes." arXiv:hep-th/9305016 (1993). https://pith.science/paper/5IFXABAX

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Editor's note

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This is the paper that gave Lovelock black holes their entropy formula, and it holds up. The result is Eq. (4): S = sum over m of 4 pi m c_m times the integral over a horizon cross-section of L_{m-1}(h̃). Not area over four. The derivation runs the Sudarsky-Wald Hamiltonian machinery on the Lovelock constraints, evaluates the boundary term at the bifurcation surface, and reads the entropy off the variation. The boundary calculation is explicit — Eq. (3) and the projection identities that follow are done, not waved at. The cross-check against the earlier Euclidean results of Myers-Simon and Whitt is a genuine independent check, and the charged extension is clean. The observation that the entropy is the original action evaluated in D-2 dimensions with shifted coefficients is the kind of thing you remember afterwards. The even-D topological integration constant is a choice, but it is flagged and justified by matching the Euclidean answer, so it is not smuggled in. The reader's scores look about right to me; soundness at 7 rather than 8 is fair, and the weakest assumption they name is the right one. The authors say it themselves: Hawking's proof that a stationary horizon is a Killing horizon is not obviously generalizable to Lovelock gravity, and they assume geodesic completeness of the generators. That is the load-bearing caveat, and it is stated in the text, not buried. Two steps are compressed rather than shown — the identity D_a N = kappa v_a at B, and the recognition that the boundary variation integrates to the variation of the intrinsic Lovelock density. Both are standard and correct at a bifurcation surface, but a reader who wants to reproduce them will do some work. The unpublished citations ([14] Simon-Whiting, [18] Wald) are supporting, not load-bearing. The closing wager about whether entropy stays local in general higher-curvature theories is honest about where the argument stops. This is for anyone working on higher-curvature thermodynamics or on entropy as a horizon functional; it is the reference you cite, and it deserves referee time if it were submitted today.

Referee report

Significance

This is a foundational result in higher-curvature gravity. It supplies a parameter-free, purely geometric formula for black hole entropy that is local in the intrinsic geometry of the horizon, and it does so by an independent Hamiltonian route rather than by evaluating the Euclidean action on a known solution. The cross-check against the Myers–Simon Euclidean results for the static and charged solutions is a genuine independent-method confirmation, and the extension to Maxwell and Yang–Mills fields is given explicitly. The paper is also candid about its assumptions: the Killing-horizon and past-completeness requirements are stated in the text, and the topological integration constant in even D is flagged and justified by matching the Euclidean result. The closing observation that the entropy formula is meaningful on a slice of a time-dependent horizon, and the question of a non-equilibrium second law, set an agenda that has been widely taken up.

Major objections (1)

[Text immediately following Eq. (3)] The passage from δH^(m)_B to the statement that δS^(m) is the variation of S^(m) = 4πm c_m ∮ L_{m−1}(h̃) is compressed into the single sentence beginning 'The next step is to recognize that...'. This is the load-bearing identification of the paper: the projection identities, the relation D_aN = κv_a, and the vanishing of K_ab on B are all shown explicitly, but the final step — that the antisymmetrized δR̃ contributions integrate to a total derivative on the closed cross-section and that the surviving δh̃ terms reproduce the variation of the Lovelock density L_{m−1}(h̃) — is left to the reader. I ask for two or three lines showing that integration by parts, or at minimum the explicit m = 2 case S = A/4G + 8π c_2 ∮ √h̃ R̃, which also fixes the normalization of the coefficient 4πm c_m. This is a transparency request rather than a challenge to correctness: the Euclidean comparison for the known solutions supports the result, and I have checked the m = 2 coefficient independently.

Minor objections (10)

[Paragraph following Eq. (4)] The phrase 'we have included a non-trivial integration constant in Eq. (4)' is misleading: the m = D/2 term is not an integration constant in the usual sense but the Euler density of the (D−2)-dimensional cross-section, whose integral is a topological invariant. Please rephrase so that the reader sees immediately why the first law alone cannot fix this term and why the Euclidean comparison is used to fix it.
[Penultimate paragraph, coalescence remark] The sentence 'curiously, note that with c_{D/2} > 0 the topological term in Eq. (4) could lead to violations of the second law when two black holes coalesce, even in four dimensions' is a strong physical statement made in passing. Either label it explicitly as a conjecture or add the one-line argument (the Euler characteristic of the cross-section changing from 4 to 2 during merger, so that the term decreases when c_2 > 0). As written it invites the reader to supply the reasoning.
[Footnote 5 and the abstract] The derivation is carried out for asymptotically flat spacetimes with vanishing cosmological constant, and for horizons that are Killing horizons with past-complete generators admitting a bifurcation surface. The body of the paper states these assumptions clearly, but the abstract's 'general formula for the entropy of stationary black holes in Lovelock gravity theories' does not. A short qualifier in the abstract, and a sentence in the main text on whether Eq. (4) is expected to survive with a cosmological constant, would make the scope unambiguous.
[Reference [18]] The vanishing of the extrinsic curvature of the bifurcation surface B is cited to 'R.M. Wald, unpublished'. Since this property is used directly in projecting R_abcd down to the intrinsic curvature R̃_abcd, a published reference or a one-line argument would be preferable.
[Reference [14] and the final paragraph] The concluding paragraph cites 'J.Z. Simon and B.F. Whiting, in preparation' for the claim that in general higher-curvature theories the entropy is always a quantity defined locally at the horizon. Because this is unpublished and is used to frame how special the Lovelock result is, please either soften the attribution or mark it as a private communication.
[Paragraph following Eq. (4)] 'One easily confirms' that Eq. (4) reproduces the Euclidean result should be replaced by the explicit m = 2 evaluation for at least one known solution, so that the reader can check the relative normalization of the area term and the curvature term without reconstructing the coefficients.
[Notation, Eq. (1) and following] The symbols δ, δ̄, and δ̃ are introduced in quick succession with different index ranges, and Eq. (1) uses δ with 2m indices while the text uses δ̄ and δ̃. A single sentence collecting the three definitions and their index counts would help. Also, 'Eq.'s (4) and (2)' should read 'Eqs. (4) and (2)'.
[Discussion of negative entropy] The statement that the entropy of small black holes 'is bounded below for the known solutions' should specify that the bound depends on the couplings and on the solution branch, and that the generalized-second-law implications of negative entropy are not addressed here.
[Final paragraph before the acknowledgements] The sentence about one of the authors enjoying a free dinner at a restaurant of his choice is out of place in a journal article and should be removed in the published version.
[Typesetting] The affiliation block and several lines show broken spacing (for example 'University of Californ ia', 'College Par k', 'Quebe c'); please check the final version for these artifacts.

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The paper

The paper claims

In Lovelock gravity, black hole entropy is not one quarter of the horizon area but a sum of curvature invariants integrated over the horizon.

The reading

In Einstein's theory a black hole's entropy equals one quarter of its horizon area, and that coincidence is usually treated as a hint about quantum gravity. This paper asks what replaces the coincidence when gravity is not Einstein's. Working in Lovelock gravity — the most general metric theory whose field equations remain second order — the authors derive the first law of black hole mechanics from a Hamiltonian boundary term and then integrate it to obtain the entropy. The result is that entropy is still an integral over a cross-section of the horizon, but of a sum of intrinsic curvature invariants rather than area alone, assembled in the same pattern as the Lovelock Lagrangian itself one dimension lower. If the derivation holds, entropy becomes a local property of the horizon's own geometry, defined for any stationary black hole — including rotating ones whose solutions have not yet been found — instead of a number extracted from a particular solution's Euclidean action.

What carries the argument

The load-bearing device is the Hamiltonian boundary term evaluated at the bifurcation surface. The authors take the Sudarsky–Wald method: for a stationary solution, evolve along the Killing field $t^a = chi^a$ that is null on the horizon, so the volume term of Hamilton's equations drops out because the Lie derivatives of the metric and momentum vanish. What survives is a surface term $delta H_B$ at the inner boundary, the bifurcation surface $B$ where $chi^a$ vanishes and whose extrinsic curvature vanishes. Varying the curvature in the constraint $H_{(m) perp}$ produces, after two integrations by parts, a boundary integral containing $D_a N$; at $B$ this equals $kappa v_a$, so the surface gravity factors out and the antisymmetrized Kronecker deltas project everything into $B$. The projected curvature is then the intrinsic curvature of the $(D-2)$-metric on the horizon cross-section, and the surviving variation is recognised as the variation of $4pi m c_m int L_{m-1}(tilde h)$. Dividing by the temperature $kappa/2pi$ gives the entropy.

Extended reading notes

Core claim

The paper's central claim is Eq. (4): for a stationary black hole in $D$-dimensional Lovelock gravity with couplings $c_m$, the entropy is $$S = 2.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt=1.5pt

Load-bearing premise

The derivation assumes the black hole's event horizon is a Killing horizon with generators that can be extended backwards as complete geodesics, so that a bifurcation surface exists; Hawking's proof that stationary horizons are Killing horizons is established in four-dimensional Einstein gravity but is not carried over to Lovelock gravity here.

Editorial extensions

If this is right

The first law $delta M - Omega_{(alpha)} delta J_{(alpha)} = (kappa/2pi) delta S$ holds for any stationary Lovelock black hole, with $S$ given by the curvature sum rather than by area.
Entropy is a local functional of the intrinsic geometry of the horizon, so it can be written down for rotating Lovelock black holes whose metrics have not yet been constructed.
For black holes much larger than the length scale set by the couplings $c_m$, the area term dominates and the Einstein result is recovered, so the correction is a controlled small-black-hole effect.
In even dimensions the sum includes a topological term (the Euler constant of the horizon cross-section), which is a fixed constant in four dimensions but can vary in higher dimensions where horizon topology is not unique.
The derivation extends to Maxwell and Yang-Mills fields with the usual $Phi delta Q$ term, leaving the entropy formula unchanged.

Reading between the lines

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

If entropy is a sum of curvature invariants, two horizons with identical area but different intrinsic geometry carry different entropies; that makes the formula a sharp test of any proposal that counts horizon microstates geometrically.
The topological term suggests a selection rule: in dimensions above four, black hole coalescence could change the horizon topology and with it the constant piece of the entropy, which is where the authors themselves flag a possible second-law problem.
Because the entropy is fixed by the first law alone, any independent microscopic derivation in a Lovelock theory — string-inspired or otherwise — must reproduce this same curvature sum, not the area, in the small-black-hole regime where the correction is large.
The unresolved question is whether the curvature sum never decreases under arbitrary horizon evolution; a positive-energy theorem for Lovelock gravity, or an area-theorem analogue for this functional, would close that gap.

Original abstract

A general formula for the entropy of stationary black holes in Lovelock gravity theories is obtained by integrating the first law of black hole mechanics, which is derived by Hamiltonian methods. The entropy is not simply one quarter of the surface area of the horizon, but also includes a sum of intrinsic curvature invariants integrated over a cross section of the horizon.

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, 2,297 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.18 · compares with prior results 0.44 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

A general formula for the entropy of stationary black holes in Lovelock gravity theories is obtained by integrating the first law of black hole mechanics, which is derived by Hamiltonian methods. The entropy is not simply one quarter of the surface area of the horizon, but also includes a sum of intrinsic curvature invariants integrated over a cross section of the horizon.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A general formula for the entropy of stationary black holes in Lovelock gravity theories is obtained by integrating the first law of black hole mechanics, which is derived by Hamiltonian methods.	contribution 0.70	0.90	0.52	asserted 1.00
s1	The entropy is not simply one quarter of the surface area of the horizon, but also includes a sum of intrinsic curvature invariants integrated over a cross section of the horizon.	result 0.99	0.82	0.66	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v1

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

References

Works cited (29)

[14]

J.Z. Simon and B.F. Whiting, in preparation

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[18]

R.M. Wald, unpublished

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[3]

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[5]

We set c0, the cosmological constant, to zero, and consider only asym ptotically ﬂat spaces in the following. The spacetime signature is taken as (− + · · · +). Throughout the paper lower case latin letters are employed for all tenso r indices, including those on “spatial” tensors. That is, spatial tensors are just regard ed as spacetime tensors which van...

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D.L. Wiltshire, Phys. Lett. 169B (1986) 36; Phys. Rev. D38 (1988) 2445

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[8]

R.C. Myers and J.Z. Simon, Phys. Rev. D38 (1988) 2434; Gen. Rel. Grav. 21 (1989) 761; B. Whitt, Phys. Rev. D38 (1988) 3000

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[9]

G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2752; S.W. Hawking in General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge University Press, 1979)

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[10]

J.D. Brown and J.W. York, Phys. Rev. D47 (1993) 1407; 1420

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[11]

In D dimensions, a cross section of the horizon is a ( D–2)-dimensional space, and so the surface area refers to the volume of such a slice. 12

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[12]

C. Callan, R.C. Myers and M. Perry, Nucl. Phys. B311 (1988) 673; R.C. Myers, Nucl. Phys. B289 (1987) 701; E. Poisson, Class. Quan. Grav. 8 (1991) 639. M. Lu and M. Wise, “Black holes with a generalized gravitational a ction,” CALT–68–1850, hep-th/9301021 (1993); M. Visser, “Dirty black holes: Entr opy versus area,” hep- th/9303029 (1993)

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[13]

B.F. Whiting and J.W. York, Phys. Rev. Lett. 61 (1988) 1336

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D. Sudarsky and R. Wald, Phys. Rev. D46 (1992) 1453. For a closely related method see also J.D. Brown, E.A. Martinez and J.W. York, Phys. Rev. Lett. 66 (1991) 2281

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[16]

S.W. Hawking, Comm. Math. Phys. 25 (1972) 152; S.W. Hawking and G.R.F. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, 1973)

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I. R´ acz and R.M. Wald, Class. Quan. Grav. 9 (1992) 2643

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R.C. Myers and M.J. Perry, Ann. Phys. (N.Y.) 172 (1986) 304

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C. Teitelboim and J. Zanelli, Class. Quan. Grav. 4 (1987) L125; in Workshop on Constraint’s Theory and Relativistic Dynamics , eds. G. Longhi and L. Lusanna (World Scientiﬁc, 1987)

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[21]

In fact, one cannot in general solve uniquely for Kab in terms of πab and hab[20], but this ambiguity does not enter into our analysis. 13

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

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[23]

See for example: B. Carter in General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge University Press, 197 9)

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For a discussion of energy positivity in Lovelock theor ies see Ref. [6] and S. Deser and Z. Yang, Class. Quant. Grav. 6 (1989) L83

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[25]

S.W. Hawking, Comm. Math. Phys. 43 (1975) 199

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W.G. Unruh and R.M. Wald, Phys. Rev. D25 (1982) 942

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W.H. Zurek and K.S. Thorne, Phys. Rev. Lett. 54 (1985) 2171; K.S. Thorne, W.H. Zurek and R.H. Price, in Black Holes: The Membrane Paradigm , eds. K.S. Thorne, R.H. Price and D.A. MacDonald (Yale University Press, New Haven,

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p. 280; R.M. Wald, in Highlights in Gravitation and Cosmology , eds. B.R. Iyer et al (Cambridge University Press, Cambridge, 1988); V.P. Frolo v and D.N. Page, “Proof of the Generalized Second Law for Quasistationary Se miclassical Black Holes,” Preprint Alberta–Thy–10–93, gr–qc/9302017. 14

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Generalized entropy and higher derivative Gravity
hep-th 2013-10 conditional novelty 7.0 of 10

For curvature-squared gravity, holographic entanglement entropy is Wald's entropy plus a term quadratic in the extrinsic curvature of the entangling surface, with coefficient the second derivative of the Lagrangian wi...
Entanglement Entropy for Singular Surfaces
hep-th 2012-06 conditional novelty 7.0 of 10

Sharp corners, cones, and creases in the boundary of an entangled region produce new universal logarithmic terms in entanglement entropy, with coefficients fixed by central charges in even dimensions.
Holographic Calculations of Renyi Entropy
hep-th 2011-10 accept novelty 7.0 of 10

Renyi entropy for a spherical entangling surface equals a temperature integral of thermal entropy on a hyperbolic cylinder, and in holographic models it is a nonlinear function of the CFT central charges.
A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices
gr-qc 2011-05 conditional novelty 7.0 of 10

The generalized second law of thermodynamics is proven for semiclassical quantum fields crossing any causal horizon, conditional on four axioms about the horizon algebra that are verified for free fields.
On Holographic Entanglement Entropy and Higher Curvature Gravity
hep-th 2011-01 conditional novelty 7.0 of 10

For Lovelock gravity, holographic entanglement entropy is computed by extremizing the Jacobson-Myers functional of intrinsic surface curvature, not Wald's horizon entropy formula.
Holographic c-theorems in arbitrary dimensions
hep-th 2010-11 conditional novelty 7.0 of 10

A holographic flow function built from tuned higher-curvature gravity decreases monotonically and equals the A-type trace anomaly coefficient in even dimensions and a universal entanglement-entropy coefficient in all ...
Viscosity Bound Violation in Higher Derivative Gravity
hep-th 2007-12 accept novelty 7.0 of 10

In Gauss-Bonnet gravity duals the shear viscosity to entropy density ratio is (1/4π)(1-4λ_GB), which drops below the conjectured 1/4π bound for positive coupling.
Curvature Singularities: the Good, the Bad, and the Naked
hep-th 2000-02 accept novelty 7.0 of 10

Singular five-dimensional gravity solutions are physically admissible only if the scalar potential stays bounded above along the solution, a criterion that matches known AdS/CFT examples and rules out several proposed...
Crossover from Schwarzian dynamics to rotational soft modes in near-extremal Lovelock black holes
hep-th 2026-11 conditional novelty 6.0 of 10

In seven-dimensional third-order Lovelock gravity, the log T coefficient of near-extremal entropy runs from 3/2 to 9 across the rotational crossover and never reaches 19/2.
Entropically favored scalar hair on AdS black branes
hep-th 2026-10 conditional novelty 6.0 of 10

A quadratic scalar-Gauss-Bonnet coupling produces source-free black-brane scalar hair in AdS5, and the hairy phase is entropically favored over the symmetric phase.
Constraints on the number of inner horizons from gravitational Noether charges
gr-qc 2026-10 accept novelty 6.0 of 10

In Lovelock gravity, under a TCC-like curvature condition and positive Wald entropy, a static black hole has at most one non-degenerate inner Killing horizon.
Thermodynamics of Lorentzian Taub-NUT spacetimes in Einstein-Gauss-Bonnet AdS gravity
hep-th 2026-09 conditional novelty 6.0 of 10

Lorentzian Taub-NUT-AdS spacetimes in Einstein-Gauss-Bonnet gravity admit a first law δM = TδS + ψδN with an explicitly given NUT charge and conjugate potential.
Timelike Entanglement First Law and Linearized Field Equations in Higher Curvature Gravity
hep-th 2026-07 unverdicted novelty 6.0 of 10

Timelike entanglement first law holds in Lovelock gravity about AdS, with both entropy and modular Hamiltonian variations carrying the same coupling factor that renormalizes Newton's constant in the linearized equations.
Off-shell Hessian thermodynamic stability of higher-curvature black holes
gr-qc 2026-06 unverdicted novelty 6.0 of 10

An off-shell Hessian criterion H = S'_W(r_h) T'(r_h) governs thermodynamic stability of higher-curvature black holes, recovering the temperature-slope rule on physical branches and producing mean-field critical exponents.
Black Hole Entropy Beyond the Wald Term in Nonminimally Coupled Gravity: A Covariant Phase Space Decomposition
gr-qc 2026-05 unverdicted novelty 6.0 of 10

Black hole entropy in diffeomorphism-invariant nonminimal gravity decomposes as S_H = S_W + S_1 + ΔS, with the extra terms required for bumblebee and Weyl-vector Gauss-Bonnet solutions but not for regular Kalb-Ramond ...
Zeroth law of black hole thermodynamics for higher derivative Proca theories
hep-th 2025-09 conditional novelty 6.0 of 10

The zeroth law of black hole thermodynamics holds to all perturbative orders for higher curvature Einstein-Proca effective field theories.
Hydrodynamical transports in generic AdS Gauss-Bonnet-scalar Gravity
hep-th 2025-07 conditional novelty 6.0 of 10

Analytic shear and bulk viscosity formulas are derived for a five-dimensional Einstein-Scalar-Maxwell-Gauss-Bonnet holographic model, with the shear viscosity cross-checked by Kubo methods.
Higher-curvature corrections and near horizon symmetries
hep-th 2025-06 conditional novelty 6.0 of 10

Near-horizon supertranslation charges in Lovelock gravity are nested curvature invariants weighted by a supertranslation function, and in four dimensions they reduce to the Jackiw-Teitelboim action.
Time dependence of complexity for Lovelock black holes
hep-th 2019-08 conditional novelty 6.0 of 10

For Lovelock black holes, the Complexity=Action growth rate at late times is a coupling-independent multiple of the mass, and the Schwarzschild limit is recovered only up to a constant under the authors' boundary-term...
Quasi-local energy and ADM mass in pure Lovelock gravity
gr-qc 2019-08 accept novelty 6.0 of 10

In pure Lovelock gravity, the large-surface limit of the variation of Brown-York quasi-local energy equals the variation of the ADM mass, giving an explicit new mass formula.
Complexity Growth Rate in Lovelock Gravity
hep-th 2018-03 conditional novelty 6.0 of 10

In Lovelock gravity, the late-time growth rate of holographic complexity for charged black holes equals the difference of internal energies at the inner and outer horizons, reducing to 2(M - M_min)/π in the uncharged limit.
The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom
gr-qc 2015-11 accept novelty 6.0 of 10

By formulating gauge fields on a principal bundle, the paper derives a general first law of black hole mechanics in which horizon contributions appear as potential times perturbed charge, including a generally non-van...
Universal Terms of Entanglement Entropy for 6d CFTs
hep-th 2015-03 conditional novelty 6.0 of 10

The universal logarithmic entanglement entropy of 6d CFTs is derived holographically and field-theoretically, with exact agreement on the C^2 and C k^2 terms and agreement on k^4 terms only after assuming a conical-me...
Holographic Entanglement Entropy for the Most General Higher Derivative Gravity
hep-th 2014-11 conditional novelty 6.0 of 10

A generalized Wald entropy plus an anomaly-of-entropy term gives holographic entanglement entropy for gravity actions containing derivatives of the curvature, and the extra term resolves the Hung-Myers-Smolkin mismatch.
Holographic Holes and Differential Entropy
hep-th 2014-08 accept novelty 6.0 of 10

The differential entropy of a family of boundary intervals equals the gravitational entropy of a corresponding bulk surface, for a broad class of holographic backgrounds including time-varying ones.
Multiple Reentrant Phase Transitions and Triple Points in Lovelock Thermodynamics
hep-th 2014-06 conditional novelty 6.0 of 10

Hyperbolic AdS black holes in 3rd-order Lovelock gravity show a small/large/small/large reentrant transition at fixed pressure, and at a tuned coupling ratio an isolated critical point appears with critical exponents ...
On entanglement entropy functionals in higher derivative gravity theories
hep-th 2014-05 conditional novelty 6.0 of 10

For general R² and quasi-topological gravity, the surface equation from the proposed entropy functional does not match the one from the generalized gravitational entropy method.
Holographic Holes in Higher Dimensions
hep-th 2014-03 accept novelty 6.0 of 10

For bulk surfaces with planar symmetry, the Bekenstein-Hawking entropy equals a 'differential entropy' built from boundary entanglement entropies of overlapping strips, and this identity extends to higher dimensions, ...
Generalized second law at linear order for actions that are functions of Lovelock densities
gr-qc 2013-06 conditional novelty 6.0 of 10

For gravitational actions that are arbitrary functions of Lovelock densities, a specific horizon entropy increases locally under linearized perturbations of a regular Killing horizon, both classically and for semiclas...
Entanglement entropy in higher derivative holography
hep-th 2013-05 conditional novelty 6.0 of 10

Applying the Lewkowycz-Maldacena generalized gravitational entropy method to four-derivative gravity, the authors show Gauss-Bonnet gravity admits an area functional matching prior proposals, while general four-deriva...
On Spacetime Entanglement
hep-th 2013-04 accept novelty 6.0 of 10

Holographic calculations in the Randall-Sundrum II braneworld show that the entanglement entropy of a large smooth region is the Bekenstein-Hawking area term plus curvature-squared corrections that match Wald entropy ...
Do gauge fields really contribute negatively to black hole entropy?
hep-th 2012-06 conditional novelty 6.0 of 10

The negative 'contact term' that Maxwell fields were thought to add to black hole entropy is a gauge-fixing artifact; quantized on the physical phase space in two dimensions, the gauge field's entropy is positive, fin...
Higher-derivative scalar-vector-tensor theories: black holes, Galileons, singularity cloaking and holography
hep-th 2012-06 conditional novelty 6.0 of 10

Generalized Kaluza-Klein reduction of Einstein-Gauss-Bonnet gravity yields exact higher-derivative scalar-tensor black holes whose horizons can cloak naked singularities, with holographic thermodynamics and transport ...
Lovelock theories, holography and the fate of the viscosity bound
hep-th 2010-10 conditional novelty 6.0 of 10

In Lovelock gravity duals, causality violation and instabilities arise in the bulk interior, and the shear viscosity to entropy ratio can be driven arbitrarily small by including higher Lovelock terms, so no dimension...
Smarr Formula and an Extended First Law for Lovelock Gravity
hep-th 2010-05 conditional novelty 6.0 of 10

A Smarr formula and an extended first law are derived for static asymptotically AdS Lovelock black holes, with new geometric potentials conjugate to the Lovelock couplings.
Enthalpy and the Mechanics of AdS Black Holes
hep-th 2009-04 accept novelty 6.0 of 10

For static AdS black holes, the mass is the enthalpy of the spacetime: the first law gains a V dP term where V is a finite effective volume set by surface integrals of the Killing potential.
Black Hole Entropy and Holographic Entanglement Entropy in DGP Brane Gravity
hep-th 2026-10 conditional novelty 5.0 of 10

In DGP braneworlds, black hole entropy equals holographic entanglement entropy precisely when the brane extrinsic curvature satisfies a condition that is automatic for static black holes, and the same condition govern...
A comparison of two constructions for dynamical corrections to Wald entropy
hep-th 2026-07 conditional novelty 5.0 of 10

For small perturbations, the dynamical entropy S_dyn and Wall's entropy S_Wall satisfy S_dyn = (1-v∂_v)S_Wall, and this paper gives a local algorithm that reconstructs S_Wall from S_dyn once a bifurcation-surface cond...
Dynamical Entropy Is a Noether Charge
hep-th 2026-07 reject novelty 5.0 of 10

A Noether-charge derivation of the known HWZ dynamical entropy for generic null surfaces, whose local second-law proof is invalid as printed because Eq. (19) has the wrong sign in K.
Hawking area law in quantum gravity
gr-qc 2026-04 unverdicted novelty 5.0 of 10

If Hawking's area law is taken as exact, nonlocal and Stelle quantum-gravity theories are forced to drop R^2 and (Riemann)^2 terms (or use singular Ricci-flat black holes), and the standard entropy-area law follows as...
D-dimensional black holes in extended Gauss-Bonnet gravity
gr-qc 2025-08 conditional novelty 5.0 of 10

Charged and rotating anti-de Sitter black hole solutions are constructed in extended Gauss-Bonnet gravity for dimensions three, four, five and higher.
Topological dark energy from black-hole formations and mergers through the gravity-thermodynamics approach
gr-qc 2024-12 reject novelty 5.0 of 10

Black hole formation and merger, through topology changes of the cosmic apparent horizon, generate an effective dark energy whose equation of state is phantom-like or quintessence-like depending on the sign of the Gau...
Black hole entropy and Lorentz-diffeomorphism Noether charge
gr-qc 2015-07 accept novelty 5.0 of 10

Black hole entropy in frame-based gravity is the Noether charge of a combined diffeomorphism and local Lorentz symmetry, which avoids the singular frame required by diffeomorphisms alone.
Corner contributions to holographic entanglement entropy
hep-th 2015-05 accept novelty 5.0 of 10

The ratio of the corner coefficient sigma in entanglement entropy to the stress-tensor central charge C_T equals pi squared over 24 for all holographic models studied and for free scalars and fermions, suggesting a un...
Aspects of Quadratic Gravity
hep-th 2015-05 conditional novelty 5.0 of 10

Pure R^2 gravity propagates only a massless scalar on flat space, with the graviton returning on curved backgrounds or when an Einstein term is added, and a string compactification is proposed as a UV origin.
On generalized gravitational entropy, squashed cones and holography
hep-th 2013-08 conditional novelty 5.0 of 10

The Fursaev-Patrushev-Solodukhin squashed cone regularization, combined with the Lewkowycz-Maldacena generalized gravitational entropy procedure, reproduces the known universal entanglement entropy terms for spherical...
Comments on Holographic Entanglement Entropy and RG Flows
hep-th 2012-02 conditional novelty 5.0 of 10

A slab-geometry entanglement-entropy c-function is proved monotonic along holographic RG flows in Einstein gravity under the null energy condition, but not in Gauss-Bonnet gravity, and it can jump discontinuously at a...
A Lovelock black hole bestiary
hep-th 2011-03 accept novelty 5.0 of 10

By reading black hole solutions off the shape of one Lovelock polynomial, the authors classify every branch by asymptotics, horizon topology, and thermodynamic stability.
Holographic Entanglement Entropy in Lovelock Gravities
hep-th 2011-01 conditional novelty 5.0 of 10

The generalized Wald-entropy formula for holographic entanglement entropy in Lovelock gravity reproduces the conformal-anomaly coefficients of the dual CFT for balls and cylinders, including a nontrivial cubic-Lovelock check.
Black Holes in Pure Lovelock Gravities
hep-th 2006-04 accept novelty 5.0 of 10

Static black hole solutions and their temperatures, entropies, and stabilities are derived for pure Lovelock gravity with a cosmological constant, including de Sitter Nariai-type bounds and anti-de Sitter branches abs...
Embedding into flat spacetime and black hole thermodynamics
gr-qc 2019-08 conditional novelty 4.0 of 10

Black hole temperature and area-law entropy are reproduced from a higher-dimensional flat spacetime embedding, mapping static observers to accelerating observers and counting scalar edge states.
Black hole chemistry: thermodynamics with Lambda
hep-th 2016-08 accept novelty 3.0 of 10

Treating the cosmological constant as pressure in black hole thermodynamics yields an extended dictionary with enthalpy, thermodynamic volume, and chemical-like phase transitions including Van der Waals behavior, reen...

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How later papers used it

125 later papers cite it

Evidence: substantial. 119 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

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Applied the same reasoning to this paper's Lovelock gravity results and modified the equation for δM to replace GδM with δ(GM).

Smarr Formula and an Extended First Law for Lovelock Gravity (2010) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

120 arXiv papers cite this in 193 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 · 150 sentences without a confident label

Builds on it 1 paper

Applying this same reasoning to the results of [ref] in Lovelock gravity, we modify the equation for $δM$ to replace $GδM$ with $δ(GM)$ giving

Smarr Formula and an Extended First Law for Lovelock Gravity · Varying Newton's constant · 0.99

Uses its method or tool 4 papers

We adopt the approach of [ref] , which relates entanglement entropy to the area of a co-dimension two bulk hypersurface that extremizes the Jacobson-Myers functional [ref] .

Entanglement entropy and its linear response following a global quench in holographic Gauss-Bonnet gravity · Introduction · 1.00
The Regge-Teitelboim type construction of the ADM mass in [ref] yields an expression of the form

Mass and Free Energy of Lovelock Black Holes · ADM Mass · 0.81
The Hamiltonian formulation of Lovelock gravity given in [ref] was used in [ref] to derive a first law and the expression ([ref]) for the gravitational entropy in Lovelock theories.

Smarr Formula and an Extended First Law for Lovelock Gravity · Hamiltonian formulation · 0.95

Compares with it 5 papers

Choosing $T_m$ as Eq. ([ref]) and substituting it into the above equations, we get [equation] The central term reads [equation] from which we can read off the central charge $C$ and zero mode $Q_0$ as [equation] Using the Cardy formula, we obtain the entropy [equation] which is exactly the Wald entropy of the Gauss-Bonnet black holes [ref] .

Surface term, Virasoro algebra, and Wald entropy of black holes in higher-curvature gravity · Entropy of a Gauss-Bonnet black hole · 0.89
A comparison to the results found in Riemannian theories with quadratic curvature Lagrangians [ref] , as well as in the general PG models [ref] , reveals how black holes react to different dynamical frameworks.

Thermodynamics of Riemannian Kerr-AdS black holes in Poincar\'e gauge theory · Introduction · 0.99
The entropy calculation presented here is compatible with these results and the Wald entropy [ref] emerges at leading order (with correct prefactor for the analytic continuation $β→±i/2$).

Black hole entropy from loop quantum gravity in higher dimensions · Comments · 0.87

Cites it as support 2 papers

In the case of General Relativity, as pointed out by [ref] , the gravitational charge does acquire such a topological contribution (the Euler number of $B$) when the Euler density is added to the Einstein-Hilbert Lagrangian in $4$-dimensions even though the equations of motion are unaffected (also see <ref>).

The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom · The first law for gauge-invariant Lagrangians · 0.96
This additional inequality ensures that the coefficient of the Gauss-Bonnet term is strictly positive, and can be motivated to hold based on scattering amplitudes [ref] , the second law of thermodynamics [ref] , weak gravity conjecture [ref] , consistency of EFT strings [ref] , and holography [ref] .

Towards a complete classification of 6D supergravities · Consistency conditions · 0.96

Cites it as background 25 papers

For example, in the gravity theory with higher-curvature corrections, the entropy is described by Wald entropy, which contains properly the contributions of higher-curvature corrections [ref] .

Surface term, Virasoro algebra, and Wald entropy of black holes in higher-curvature gravity · Introduction · 0.85
First, there is Wald's entropy formula [ref] , which can be applied for any covariant gravity action, and then an earlier result derived specifically for Lovelock gravity [ref] .

On Holographic Entanglement Entropy and Higher Curvature Gravity · A few preliminaries · 0.96
* The Wald entropy can be constructed for arbitrary $α$ with $=0$. * In [ref] the authors construct entropy functions for theories where the higher derivative Lagrangian is written solely in terms of the Ricci scalar (the so-called $f(R)$ theories), for a finite range of $α$, and arbitrary $$ and $ω$. * In recent years, [ref] first constructed an entropy functional that was valid for higher derivative interactions of the Lovelock form (see below) for small amplitude departures from equilibrium $≪1$.

Towards a second law for Lovelock theories · Introduction · 0.83

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] [14]

J.Z. Simon and B.F. Whiting, in preparation

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[2] [18]

R.M. Wald, unpublished

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[3] [1]

J.M. Bardeen, B. Carter and S.W. Hawking, Comm. Math. Phys. 31 (1973) 161

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See for example: N.D. Birrell and P.C.W. Davies, Quantum ﬁelds in curved space . (Cambridge University Press, 1982). 11

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[5] [3]

See for example: M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory. (Cam- bridge University Press, 1987)

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[6] [4]

D. Lovelock, J. Math. Phys. 12 (1971) 498; 13 (1972) 874

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We set c0, the cosmological constant, to zero, and consider only asym ptotically ﬂat spaces in the following. The spacetime signature is taken as (− + · · · +). Throughout the paper lower case latin letters are employed for all tenso r indices, including those on “spatial” tensors. That is, spatial tensors are just regard ed as spacetime tensors which van...

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[8] [6]

D.G. Boulware and S. Deser, Phys. Rev. Lett. 55 (1985) 2656; J.T. Wheeler, Nucl. Phys. B268 (1986) 737; B273 (1986) 732

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[9] [7]

D.L. Wiltshire, Phys. Lett. 169B (1986) 36; Phys. Rev. D38 (1988) 2445

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[10] [8]

R.C. Myers and J.Z. Simon, Phys. Rev. D38 (1988) 2434; Gen. Rel. Grav. 21 (1989) 761; B. Whitt, Phys. Rev. D38 (1988) 3000

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G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2752; S.W. Hawking in General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge University Press, 1979)

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[12] [10]

J.D. Brown and J.W. York, Phys. Rev. D47 (1993) 1407; 1420

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[13] [11]

In D dimensions, a cross section of the horizon is a ( D–2)-dimensional space, and so the surface area refers to the volume of such a slice. 12

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[14] [12]

C. Callan, R.C. Myers and M. Perry, Nucl. Phys. B311 (1988) 673; R.C. Myers, Nucl. Phys. B289 (1987) 701; E. Poisson, Class. Quan. Grav. 8 (1991) 639. M. Lu and M. Wise, “Black holes with a generalized gravitational a ction,” CALT–68–1850, hep-th/9301021 (1993); M. Visser, “Dirty black holes: Entr opy versus area,” hep- th/9303029 (1993)

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B.F. Whiting and J.W. York, Phys. Rev. Lett. 61 (1988) 1336

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[16] [15]

D. Sudarsky and R. Wald, Phys. Rev. D46 (1992) 1453. For a closely related method see also J.D. Brown, E.A. Martinez and J.W. York, Phys. Rev. Lett. 66 (1991) 2281

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[17] [16]

S.W. Hawking, Comm. Math. Phys. 25 (1972) 152; S.W. Hawking and G.R.F. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, 1973)

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I. R´ acz and R.M. Wald, Class. Quan. Grav. 9 (1992) 2643

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[19] [19]

R.C. Myers and M.J. Perry, Ann. Phys. (N.Y.) 172 (1986) 304

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[20] [20]

C. Teitelboim and J. Zanelli, Class. Quan. Grav. 4 (1987) L125; in Workshop on Constraint’s Theory and Relativistic Dynamics , eds. G. Longhi and L. Lusanna (World Scientiﬁc, 1987)

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[21] [21]

In fact, one cannot in general solve uniquely for Kab in terms of πab and hab[20], but this ambiguity does not enter into our analysis. 13

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[22] [22]

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

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[23] [23]

See for example: B. Carter in General Relativity, An Einstein Centenary Survey , eds. S.W. Hawking and W. Israel (Cambridge University Press, 197 9)

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[24] [24]

For a discussion of energy positivity in Lovelock theor ies see Ref. [6] and S. Deser and Z. Yang, Class. Quant. Grav. 6 (1989) L83

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[25] [25]

S.W. Hawking, Comm. Math. Phys. 43 (1975) 199

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[26] [26]

W.G. Unruh and R.M. Wald, Phys. Rev. D25 (1982) 942

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[27] [27]

W. Zurek, Phys. Rev. Lett. 49 (1982) 1683

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[28] [28]

W.H. Zurek and K.S. Thorne, Phys. Rev. Lett. 54 (1985) 2171; K.S. Thorne, W.H. Zurek and R.H. Price, in Black Holes: The Membrane Paradigm , eds. K.S. Thorne, R.H. Price and D.A. MacDonald (Yale University Press, New Haven,

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[29] [1986]

p. 280; R.M. Wald, in Highlights in Gravitation and Cosmology , eds. B.R. Iyer et al (Cambridge University Press, Cambridge, 1988); V.P. Frolo v and D.N. Page, “Proof of the Generalized Second Law for Quasistationary Se miclassical Black Holes,” Preprint Alberta–Thy–10–93, gr–qc/9302017. 14

View on Pith 1988