BTZ Black Hole with Chern-Simons and Higher Derivative Terms

Ashoke Sen; Bindusar Sahoo

REVIEW

BTZ Black Hole with Chern-Simons and Higher Derivative Terms

Bindusar Sahoo, Ashoke Sen

High Energy Physics - Theory · Submitted 31 Jan 2006 · 16 references

Pith's verdict: An honest methodological paper: the BTZ entropy numbers were already known, but the covariant 2D-reduction route to them is new and the authors are upfront about the one step they could not derive.

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CitationAshoke Sen, Bindusar Sahoo. "BTZ Black Hole with Chern-Simons and Higher Derivative Terms." arXiv:hep-th/0601228 (2006). https://pith.science/paper/BHQBATHV

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Editor's note

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The short version: the numbers here were already known — Kraus and Larsen got the same entropy from the Euclidean action, and the paper says so — but the route is new and worth having. Sahoo and Sen reduce the 3D theory along the angular circle, which turns the gravitational Chern-Simons term into a manifestly covariant 2D expression, and then Wald's Noether charge method goes through without hand-waving. That trick is the contribution. It looks obvious in retrospect and isn't, which is the best kind of methodological paper. The abstract's claim [abstract:s2] that the parameters follow from extremizing an entropy function is the right framing, and the paper delivers on it for the extremal case at least. What they do well: the extremal calculation is clean. The entropy function (3.16) with the constraint v = e^2 is a sensible construction, the extremization in (3.19) is correct, and the central charges c_L = 24 pi (C + K), c_R = 24 pi (C - K) come out with the right parity-odd piece. The non-extremal Noether charge computation (4.2)-(4.8) is written out step by step and the pieces combine correctly into (4.9). The agreement with [9] is a genuine independent check, not a tautology. Soft spots, in order of how much they matter. The non-extremal entropy function (4.15) is written down and shown to extremize to the right answer, but the authors say plainly they have not derived it from Wald's formula. That is an acknowledged gap, and it is the one place where the construction is reverse-engineered. Second, the 2D Chern-Simons Lagrangian (2.7) is imported from [14] with total derivative terms dropped, and the paper does not show those terms are inert for the Noether charge. Wald entropy can be sensitive to boundary terms, so this is a real gap in principle. But the stress-test worry is a bit overstated: the Euclidean agreement is exactly the check that would expose a boundary-term shift, and it does not, so I would call this a loose end rather than a hole. Third, C >= |K| is assumed rather than justified; it is equivalent to positive central charges, so it is benign, but it should be said. Citation pattern is fine. They credit Kraus-Larsen for the result and Guralnik-Iorio-Jackiw-Pi for the reduction, and the self-citations to Sen's entropy function papers are appropriate. Who should read it: people working on Wald entropy in Chern-Simons-modified gravity, and anyone using the entropy function formalism in 3D. It is a specialized brick, not a landmark. Recommendation: send it out. The algebra is checkable, the result is cross-checked, and the acknowledged gap is the kind of thing a referee can press on productively.

Referee report

Major objections (2)

[Section 2, below eq. (2.6)] The two-dimensional Chern-Simons Lagrangian (2.7) is obtained from the three-dimensional Chern-Simons term 'after throwing away total derivative terms.' The paper does not specify these terms or show that they do not contribute to the Wald Noether charge at the horizon. Wald entropy is computed from the Noether charge of diffeomorphisms, and adding or dropping a total derivative shifts that charge by a boundary term; in the two-dimensional reduction the horizon cross-section is a point, so a boundary term need not vanish automatically. The agreement of the final answer (4.13) with the Euclidean computation of [9] is strong evidence that the dropped terms are inert, but a direct argument would make the derivation self-contained. I recommend that the authors either show explicitly that the discarded terms vanish on the horizon or explain why they cannot contribute to the Noether charge.
[Abstract and Section 4, eq. (4.15)] The abstract states that 'the parameters labelling the black hole solution can be determined by extremizing an entropy function whose value at the extremum gives the entropy of the black hole.' This is derived for the extremal case in Section 3, but for the non-extremal case the entropy function (4.15) is stated without derivation, and the authors explicitly say they have not succeeded in deriving it. The abstract should be qualified to reflect that the entropy function principle is established only for the extremal case, while for the non-extremal case it is conjectural. The direct computation (4.9)-(4.13) is independent of (4.15), so the main result is not affected, but the abstract's general claim is not fully supported.

Minor objections (6)

[Eqs. (3.16) and (3.18)] The term -πKe should be -πK/|e| (or -πK/e for e>0). As written, extremizing (3.18) with respect to e would give q = πK, which is inconsistent with (3.19). The subsequent results (3.19) and (3.20) are consistent with the corrected form, so this is a typo.
[Eqs. (4.10) and (4.12)] The l dependence appears inconsistent. From (4.6), G^{μν}∂L0/∂R^{μν} = 3C/l, so (4.10) should give M0 ± J0 proportional to l^{-2}, not l^2. The final result (4.13) is correct, so this is likely a typo in the intermediate equations.
[Section 3, below eq. (3.18)] The assumption C ≥ |K| should be justified or discussed. What is the physical meaning of this bound, and when does it hold?
[Section 4] The paper mixes three-dimensional and two-dimensional treatments: E0 is computed in 3D (4.2)-(4.3), while E1 is computed in 2D (4.8). The paper should explain why this is consistent.
[Section 3] The paper could discuss the sign of K and the physical interpretation of c_L and c_R. In particular, it would be useful to state explicitly that c_L - c_R = 48πK is determined entirely by the Chern-Simons term, independent of the higher-derivative terms.
[References] Reference [9] is cited as arXiv only; if it has been published, the journal reference should be given.

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The paper

The paper claims

By treating the angular direction of a BTZ black hole as a compact circle, the authors show that Wald's Noether-charge method reproduces the known entropy even when a parity-violating gravitational Chern-Simons term and higher-derivativecur

The reading

The paper asks whether Wald's Noether-charge method — the standard modern way to define black hole entropy in theories with curvature-squared and higher terms — can be made to work for a three-dimensional BTZ black hole once a gravitational Chern-Simons term is added. That term is the obstruction: it cannot be written in a manifestly coordinate-invariant way in three dimensions, so Wald's construction, which assumes a covariant Lagrangian, does not apply directly. The authors' move is to shrink the angular coordinate $y$ to a compact circle of period $2$ and rewrite the whole theory as a two-dimensional one with a metric, a gauge field $A_$ and a scalar $'$; in that language the Chern-Simons term becomes an ordinary covariant expression built from a curvature times a field strength plus a cubic term in $F$. Wald's formula then goes through essentially mechanically. The payoff is a closed expression for the entropy of a non-extremal BTZ black hole, $E = 2pi sqrt{c_L q_L / 6} + 2pi sqrt{c_R q_R / 6}$, which matches the earlier Euclidean-action computation, and which shows that the left-right central-charge difference $c_L - c_R = 48 pi K$ is fixed entirely by the Chern-Simons coefficient, independent of the higher-derivative terms.

What carries the argument

The load-bearing device is the dimensional reduction of the gravitational Chern-Simons term to two dimensions, which turns a non-covariant three-dimensional object into the covariant density $sqrt{-g} L_1 = K pi [ (1/2) R epsilon^{mu nu} F_{mu nu} + (1/2) epsilon^{mu nu} F_{mu tau} F^{tau sigma} F_{sigma nu} ]$. This is what makes Wald's Noether-charge machinery applicable at all. The second device is the entropy function: a function of the near-horizon values of the scalar $'$, the $AdS_2$ radius-squared $v$, and the electric field $e$, whose extremum with respect to those variables gives both the near-horizon parameters and the entropy. For the non-extremal case the authors exhibit an analogous extremization principle, $E = 2pi [ q_L e_L + q_R e_R - (1/e_L + 1/e_R) g(l) + pi K (1/e_L - 1/e_R) ]$, with $e_L = l/(ho_+ - ho_-)$ and $e_R = l/(ho_+ + ho_-)$, though they state they have not derived it from Wald's formula.

Extended reading notes

Core claim

The central claim is that the entropy of a BTZ black hole in a theory with arbitrary higher-derivative corrections plus a gravitational Chern-Simons term can be computed by Wald's Noether-charge method, provided one first reduces the theory along the compact angular direction. In the reduced two-dimensional theory the Chern-Simons term is manifestly covariant, so the Noether charge of diffeomorphisms is well defined. For extremal black holes the authors use the entropy function $E(u,v,e,q) = 2pi (e q - f(u,v,e))$, extremized subject to the BTZ-specific constraint $v = e^2$ that follows from local $AdS_3$ symmetry; for non-extremal holes they evaluate the Noether charge directly, splitting it into a higher-derivative piece $E_0 = 8 pi^2 C l^{-1} ho_+$ and a Chern-Simons piece $E_1 = -8 pi^2 K l^{-1} ho_-$. The two combine into the left-right symmetric form $E = 2pi sqrt{c_L q_L / 6} + 2pi sqrt{c_R q_R / 6}$ with $q_L = (M-J)/2$, $q_R = (M+J)/2$, reproducing the Euclidean result of Kraus and Larsen. Along the way the authors note that the Chern-Simons term does not affect the determination of the $AdS_3$ radius $l$, because its contribution to the equations of motion involves a covariant derivative of the Ricci tensor, which vanishes for BTZ.

Load-bearing premise

The paper assumes that the two-dimensional Chern-Simons Lagrangian obtained after dropping total derivative terms is a faithful starting point for Wald's Noether charge, even though dropping a total derivative can shift that charge by a boundary term, and it does not show those discarded terms vanish at the horizon.

Editorial extensions

If this is right

Wald's Noether-charge method, previously unusable for parity-odd three-dimensional gravity, becomes a practical alternative to the Euclidean action for BTZ entropy computations.
The left-right central-charge difference is pinned to the Chern-Simons coefficient alone, $c_L - c_R = 48 pi K$, while the sum $c_L + c_R = -48 g(l)$ is fixed by the higher-derivative sector, so the two kinds of correction separate cleanly.
Because the extremal entropy takes the Cardy form $2pi sqrt{c_L q_L/6} + 2pi sqrt{c_R q_R/6}$, a two-dimensional conformal-field-theory account of the microstates survives the addition of higher-derivative and Chern-Simons terms, with the central charges shifted as above.
The BTZ radius $l$ is unaffected by the Chern-Simons term, since that term's contribution to the equations of motion vanishes on the locally $AdS_3$ background.

Reading between the lines

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

If the reduction step is exact, the same trick should give a general recipe for any odd-dimensional theory with a gravitational Chern-Simons term: reduce along a Killing circle first, then apply Noether-charge methods in the even-dimensional quotient.
The agreement with the Euclidean computation is a check on the whole chain, but it is also the only independent check reported; a direct three-dimensional covariant treatment, or a holographic computation of $c_L$ and $c_R$ from the boundary stress tensor, would test the reduction independently.
The un-derived non-extremal extremization principle in equation (4.15) is a concrete target: if it follows from Wald's formula plus the equations of motion, the entropy-function formalism extends beyond the near-horizon $AdS_2$ setting in which it was introduced.

Original abstract

The entropy of a BTZ black hole in the presence of gravitational Chern-Simons terms has previously been analyzed using Euclidean action formalism. In this paper we treat the BTZ solution as a two dimensional black hole by regarding the angular coordinate as a compact direction, and use Wald's Noether charge method to calculate the entropy of this black hole in the presence of higher derivative and gravitational Chern-Simons terms. The parameters labelling the black hole solution can be determined by extremizing an entropy function whose value at the extremum gives the entropy of the black hole.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.45 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

The entropy of a BTZ black hole in the presence of gravitational Chern-Simons terms has previously been analyzed using Euclidean action formalism. In this paper we treat the BTZ solution as a two dimensional black hole by regarding the angular coordinate as a compact direction, and use Wald's Noether charge method to calculate the entropy of this black hole in the presence of higher derivative and gravitational Chern-Simons terms. The parameters labelling the black hole solution can be determined by extremizing an entropy function whose value at the extremum gives the entropy of the black hole.

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	The entropy of a BTZ black hole in the presence of gravitational Chern-Simons terms has previously been analyzed using Euclidean action formalism.	background 1.00	0.03	0.03	asserted 1.00
s1	In this paper we treat the BTZ solution as a two dimensional black hole by regarding the angular coordinate as a compact direction, and use Wald's Noether charge method to calculate the entropy of this black hole in the presence of higher derivative and gravitational Chern-Simons terms.	method 0.58	0.95	0.16	asserted 1.00
s2	The parameters labelling the black hole solution can be determined by extremizing an entropy function whose value at the extremum gives the entropy of the black hole.	method 0.92	0.73	0.27	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

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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 (16)

[9]

P. Kraus and F. Larsen, arXiv:hep-th/0506176

View on Pith Pith review arXiv
[14]

G. Guralnik, A. Iorio, R. Jackiw and S. Y. Pi, Annals Phys . 308, 222 (2003) [arXiv:hep-th/0305117]

View on Pith Pith review arXiv 2003
[15]

A. Sen, JHEP 0509, 038 (2005) [arXiv:hep-th/0506177]

View on Pith Pith review arXiv 2005
[1]

M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett . 69, 1849 (1992) [arXiv:hep-th/9204099]

View on Pith Pith review arXiv 1992
[2]

A. Strominger, JHEP 9802, 009 (1998) [arXiv:hep-th/9712251]

View on Pith Pith review arXiv 1998
[3]

J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986)

View on Pith 1986
[4]

H. Saida and J. Soda, Phys. Lett. B 471, 358 (2000) [arXiv:gr-qc/9909061]

View on Pith Pith review arXiv 2000
[5]

R. M. Wald, Phys. Rev. D 48, 3427 (1993) [arXiv:gr-qc/9307038]

View on Pith Pith review arXiv 1993
[6]

T. Jacobson, G. Kang and R. C. Myers, Phys. Rev. D 49, 6587 (1994) [arXiv:gr- qc/9312023]

View on Pith 1994
[7]

V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994) [arXiv:gr-qc/9403028]

View on Pith Pith review arXiv 1994
[8]

T. Jacobson, G. Kang and R. C. Myers, arXiv:gr-qc/950200 9

View on Pith
[10]

P. Kraus and F. Larsen, arXiv:hep-th/0508218

View on Pith Pith review arXiv
[11]

J. M. Maldacena and A. Strominger, JHEP 9812, 005 (1998) [arXiv:hep-th/9804085]

View on Pith Pith review arXiv 1998
[12]

S. N. Solodukhin, arXiv:hep-th/0509148

View on Pith Pith review arXiv
[13]

A. Achucarro and M. E. Ortiz, Phys. Rev. D 48, 3600 (1993) [arXiv:hep-th/9304068]

View on Pith Pith review arXiv 1993
[16]

A. Sen, arXiv:hep-th/0508042. 11

View on Pith Pith review arXiv

Forward citations

Cited by 20 Pith papers

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A Hamilton-Jacobi boundary counterterm gives a finite, stationary Euclidean action for arbitrary two-dimensional dilaton gravity, from which black hole entropy S = 2 pi X_h and a quasilocal first law follow universally.
Black Hole Entropy in the presence of Chern-Simons Terms
hep-th 2006-11 accept novelty 5.0 of 10

A Noether-charge derivation gives black hole entropy formulas for gravitational Chern-Simons terms, reproducing the known BTZ correction and extending to mixed Chern-Simons and Green-Schwarz couplings.
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How the literature uses it

44 arXiv papers cite this in 62 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 · 47 sentences without a confident label

Builds on it 1 paper

The analysis presented here will be a generalization of the one given in [ref] where we considered the case of a purely gravitational theory, and will borrow insights from the analysis of [ref] .

Black Hole Entropy Function, Attractors and Precision Counting of Microstates · Non-supersymmetric extremal black holes · 0.94

Uses its method or tool 5 papers

The way to deal with the Chern-Simons term was explained in [ref] .

Stringy effects in black hole decay · Stringy corrections · 0.96
The apparent lack of gauge invariance is usually tackled via a 4D reduction [ref] (though, see [ref] ).

CFT Duals for Attractor Horizons · Attractor mechanism · 0.85
Following [ref] we take [equation] where $F_mn=∂_m A_n - ∂_n A_m$ and $R^(2)$ is a Ricci scalar calculated from $g_mn$.

AdS_3 backgrounds from 10D effective action of heterotic string theory · Generalisation to theories with Chern-Simons terms · 0.92

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 background 8 papers

Later this was generalized to higher derivative theories of gravity [ref] , where the lagrangian density contains arbitrary powers of Riemann tensor and its covariant derivatives as well as gravitational Chern-Simons terms [ref] , both in the Euclidean action formalism [ref] and in Wald's formalism [ref] .

AdS_3, Black Holes and Higher Derivative Corrections · Introduction · 0.92
The Wald entropy of the BTZ black hole in TMG receives a non-trivial contribution which has been well documented and studied in [ref] .

Gravitational Anomalies in nAdS$_2$/nCFT$_1$ · Thermodynamics near Extremality · 0.82
The attractor mechanism has been observed for asymptotically flat as well as asymptotically $AdS$ black holes and in theories with higher derivative interactions, and there is a long list of papers where this subject has been studied widely [ref] .

The fate of flat directions in higher derivative gravity · Introduction and summary · 1.00

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] [9]

P. Kraus and F. Larsen, arXiv:hep-th/0506176

View on Pith Pith review arXiv

[2] [14]

G. Guralnik, A. Iorio, R. Jackiw and S. Y. Pi, Annals Phys . 308, 222 (2003) [arXiv:hep-th/0305117]

View on Pith Pith review arXiv 2003

[3] [15]

A. Sen, JHEP 0509, 038 (2005) [arXiv:hep-th/0506177]

View on Pith Pith review arXiv 2005

[4] [1]

M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett . 69, 1849 (1992) [arXiv:hep-th/9204099]

View on Pith Pith review arXiv 1992

[5] [2]

A. Strominger, JHEP 9802, 009 (1998) [arXiv:hep-th/9712251]

View on Pith Pith review arXiv 1998

[6] [3]

J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986)

View on Pith 1986

[7] [4]

H. Saida and J. Soda, Phys. Lett. B 471, 358 (2000) [arXiv:gr-qc/9909061]

View on Pith Pith review arXiv 2000

[8] [5]

R. M. Wald, Phys. Rev. D 48, 3427 (1993) [arXiv:gr-qc/9307038]

View on Pith Pith review arXiv 1993

[9] [6]

T. Jacobson, G. Kang and R. C. Myers, Phys. Rev. D 49, 6587 (1994) [arXiv:gr- qc/9312023]

View on Pith 1994

[10] [7]

V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994) [arXiv:gr-qc/9403028]

View on Pith Pith review arXiv 1994

[11] [8]

T. Jacobson, G. Kang and R. C. Myers, arXiv:gr-qc/950200 9

View on Pith

[12] [10]

P. Kraus and F. Larsen, arXiv:hep-th/0508218

View on Pith Pith review arXiv

[13] [11]

J. M. Maldacena and A. Strominger, JHEP 9812, 005 (1998) [arXiv:hep-th/9804085]

View on Pith Pith review arXiv 1998

[14] [12]

S. N. Solodukhin, arXiv:hep-th/0509148

View on Pith Pith review arXiv

[15] [13]

A. Achucarro and M. E. Ortiz, Phys. Rev. D 48, 3600 (1993) [arXiv:hep-th/9304068]

View on Pith Pith review arXiv 1993

[16] [16]

A. Sen, arXiv:hep-th/0508042. 11

View on Pith Pith review arXiv