Regularization of odd-dimensional AdS gravity: Kounterterms

Rodrigo Olea

REVIEW

Regularization of odd-dimensional AdS gravity: Kounterterms

Rodrigo Olea

High Energy Physics - Theory · Submitted 20 Oct 2006 · 56 references

Pith's verdict: A real construction with good checks, but the 'all odd dimensions' action principle rests on an unproved general-n cancellation.

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arXiv IDhep-th/0610230 (version 2)

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CitationRodrigo Olea. "Regularization of odd-dimensional AdS gravity: Kounterterms." arXiv:hep-th/0610230 (2006). https://pith.science/paper/77BSFD5Y

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Editor's note

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Olea's Kounterterms paper is worth your time if you work on AdS gravity regularization, but read §5 with the right expectations. The construction is genuine: a closed-form boundary term B_{2n} (eqs. 22-24, 145-146) built from a transgression/Chern-Simons-like form, a coupling c_{2n} fixed by the asymptotic constant-curvature condition (eqs. 149-150), and a Noether-charge split Q = q + q0 where q gives mass and angular momentum and q0 is identified as the vacuum energy. The general vacuum-energy formula (162) reproduces the Emparan-Johnson-Myers conjecture, and the 5D and 7D checks against Balasubramanian-Kraus, Awad-Johnson, and Papadimitriou-Skenderis are explicit and correct. The Euclidean action finiteness and the entropy = Area/4G checks are done properly. That is a lot of independent confirmation for one paper, and it is the reason to take the construction seriously rather than file it as another counterterm variant. The soft spot is the one the stress-test flags, and it is real. The action principle in general n requires the curl term in the surface variation (147) to vanish under K = (1/ℓ)δ, δK = 0. That cancellation is worked out explicitly for D=5 (eq. 43) and D=7 (eq. 105), then §5 asserts it for arbitrary n. Appendix D gives the general variation (210) but does not carry out the cancellation. The structure is uniform in n, so it is plausible the mechanism is dimension-independent, but plausible is not shown. A referee should ask for either the general proof or an explicit statement that this is an assumption. The abstract's 'compatible with any AAdS spacetime' (s1) is broader than what is demonstrated. Second, smaller: the identification of q0 as the vacuum energy is not unambiguous. The paper itself cites [42] arguing the Casimir energy should be rotation-independent; Olea's q0 for Kerr-AdS depends on the rotation parameters. The argument 'q vanishes for global AdS, so q0 must be the vacuum energy' is suggestive, not conclusive. The paper acknowledges this, which is to its credit. None of this is fatal. The construction is well-motivated, the checks are real, and the paper is honest about what it has not done (general-n charges beyond topological SAdS, equivalence with Dirichlet counterterms in higher dimensions). Send it to review; the general-n cancellation is exactly the question a good referee should press.

Referee report

Major objections (3)

[§5, paragraph following Eq. (147)] The cancellation of the curl term ε(δK e − K δe)(R − t²KK + t²/ℓ² ee)^{n−1} in the surface variation (147) is demonstrated explicitly only for D=5 (Eq. 43) and D=7 (Eq. 105), and then asserted for general n by the sentence 'As we have already seen in the five and seven-dimensional cases, no matter the terms that multiply the curl ..., the asymptotic conditions (41) and (42) cancel it identically.' This is an inductive leap, and it is load-bearing for the paper's headline claim of a finite action principle in all odd dimensions. The argument does appear to go through uniformly in n: with K^a=(1/ℓ)e^a, the K δe term is proportional to e^{a1}δe^{a2}, symmetric under a1↔a2 and hence vanishing when contracted with the totally antisymmetric ε; the δK e term vanishes by δK^i_j=0. But the paper should state this explicitly rather than leave it to analogy, and should note that the argument does not depend on the power n−1 of the bracket.
[§5.1, Eqs. (159)-(162)] The general vacuum-energy formula (162) is the paper's headline general result and is claimed to corroborate the Emparan-Johnson-Myers conjecture. It should reduce to the explicit five- and seven-dimensional results (64) and (119) at n=2,3. As typeset, the coefficients do not appear to agree: for n=2, (162) gives Vol ℓ²/(128πG) whereas (64) gives 3Vol ℓ²/(64πG); for n=3, (162) gives Vol ℓ⁴/(768πG) whereas (119) gives 5Vol ℓ⁴/(128πG). There is also an apparent sign discrepancy between (159) and (64) at n=2. Since (64) and (119) match the quoted Balasubramanian-Kraus and Emparan-Johnson-Myers values, the discrepancy is most likely in the typesetting of (162) or (159), but it should be resolved explicitly before publication.
[§6 and abstract] The identification of q0 with the vacuum energy for rotating solutions is in tension with Ref. [42] (Gibbons-Perry-Pope), which argues that the Casimir energy should be rotation-independent and match the SAdS value. The paper acknowledges this in §6 but does not resolve it. The abstract's claim is appropriately hedged ('reproduces the results obtained by the Dirichlet counterterms method in a number of cases'), but the physical interpretation of q0 as the vacuum energy remains open for rotating spacetimes. The paper should state what would settle the question — for example, a direct comparison of the boundary stress tensor, or a dual CFT computation on the rotating boundary geometry — and should be explicit that agreement with Dirichlet counterterms is agreement between two prescriptions, not independent evidence for the physical value.

Minor objections (7)

[§2 vs §3.1] The statement in §2 that 'we do not assume any particular expansion of the boundary metric as in eq.(1),(2)' is in tension with the use of the Fefferman-Graham expansion (40) in §3.1 to derive the boundary condition (41)-(42). The construction of B_d is indeed expansion-independent; only the boundary condition uses the asymptotic form. A sentence distinguishing the two would help the reader.
[§3.1, Eqs. (41)-(42)] The condition K^i_j=(1/ℓ)δ^i_j is Weyl-invariant in mixed-index form (unlike fixing h_{ij}), which is a strength of the construction and should be stated explicitly. The paper should also note that the leading term is unaffected by the log ρ term that appears at order ρ^n in odd dimensions, which the introduction mentions.
[§2.2, Ref. [25]] The universality claim for generic Lovelock theories rests partly on Ref. [25], listed as 'in preparation.' The published Ref. [24] covers Einstein-Gauss-Bonnet; the generic Lovelock case should be flagged as unpublished.
[Eqs. (64), (65), (119), (162)] The numerical coefficients in the vacuum-energy and critical-mass formulas should be checked for internal consistency. In particular, Eq. (65) has ω_D in the numerator, whereas the extremality condition for the metric (56)-(57) gives μ_c ∝ 1/(ω_D G) with the exponent structure ((D−3)/(D−1))^{(D−3)/2}; the placement of ω_D and the exponents should be verified. The claim that the five-dimensional critical mass equals the vacuum energy (64) with opposite sign should also be checked against the stated formula.
[§3.2, Eq. (53)] The reference connection ω̄ introduced in Eq. (17) drops out of the final charge formula (53). The paper should state this explicitly to justify the 'background-independent' characterization, since the construction does introduce an auxiliary connection on the boundary.
[§3.2, after Eq. (47)] The paper notes that a quasilocal stress tensor cannot be defined from the variation of the total action. This is an honest limitation for AdS/CFT applications; the paper should state more explicitly what is lost relative to the Dirichlet approach (for example, the Weyl anomaly and the stress-tensor two-point function).
[Eq. (12) and typographical issues] The expanded form of B_3 in Eq. (12) contains K(K_{ij}K^{ji} − K²/3), which is not obviously the Gauss-Bonnet generalization of the Gibbons-Hawking term; a sentence connecting to the standard Gauss-Bonnet boundary term of Refs. [19,20] would help. 'Gauss-Coddazzi' should be 'Gauss-Codazzi' (Eqs. 33-34 and throughout), and 'Hennigson' should be 'Henningson' in the introduction.

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The paper

The paper claims

A boundary term built from the extrinsic curvature makes the Einstein–Hilbert action with negative cosmological constant finite in every odd dimension, and splits the resulting conserved charges into an ordinary mass/angular-momentum piece,

The reading

The paper asks whether the standard way of taming infinities in anti-de Sitter gravity — adding a series of boundary counterterms built only from the boundary metric and its intrinsic curvature — can be replaced by a single, closed-form boundary term that depends explicitly on how the boundary is embedded, i.e. on the extrinsic curvature $K_{ij}$. Olea calls these terms Kounterterms and gives their tensorial form in all odd dimensions $D = 2n+1$, where no Euler-class topological invariant exists and the even-dimensional trick therefore fails. The claim is that with a boundary condition fixing the extrinsic curvature to $K^i{}_j = (1/ℓ)δ^i{}_j$ rather than fixing the induced metric, the surface term from an arbitrary variation cancels, the coupling constant is fixed by demanding constant curvature asymptotically, and the action becomes stationary and finite. If that holds, the payoff is a background-independent definition of conserved charges and of the AdS vacuum energy that never requires expanding the metric in a Fefferman–Graham series or inverting that series to recover covariant boundary functionals.

What carries the argument

The load-bearing object is the Kounterterms series $B_{2n}$, written as a double parametric integral over $t$ and $s$ of antisymmetrized products of $K$, the boundary curvature $R^{ab}$ and the vielbein $e^a$ (eqs. 22–24). Its structural origin is the transgression form: the difference between two invariant polynomials of the curvature equals an exterior derivative of a form built from the difference of two connections, the second fundamental form $θ^{AB} = ω^{AB} - øverline{ω}^{AB}$. The Cartan homotopy operator supplies the relative binomial coefficients automatically, which is why a single closed expression covers every dimension instead of an ever-growing list of counterterms. In even dimensions the same object is the $n$-th Chern form tied to the Euler theorem; in odd dimensions the author assumes the form carries over and shows it still does the regularizing job for Einstein–Hilbert, not just for Chern–Simons or Gauss–Bonnet theory.

Extended reading notes

Core claim

Olea proposes that the regularizing boundary term for Einstein–Hilbert–AdS gravity in $D = 2n+1$ dimensions is the transgression-form expression $B_{2n}$ of eqs. (22)–(24), a polynomial in the extrinsic curvature $K_{ij}$, the boundary Riemann tensor and the boundary vielbein, obtained as the odd-dimensional analogue of the Chern-form boundary terms that work in even dimensions. The action principle is not Dirichlet: instead of holding the induced metric fixed, one imposes $K^i{}_j = (1/ℓ)δ^i{}_j$ together with $δK^i{}_j = 0$ on the conformal boundary, a condition the author derives from the asymptotic Fefferman–Graham expansion of the extrinsic curvature. With that condition the residual surface term in the variation collapses to a single piece whose coefficient fixes the coupling $c_{2n} = -(-ℓ^2)^{n-1}/(4^{n+1}πG n[(n-1)!]^2)$. The Noether charge then splits naturally into $q(ξ)$, which reproduces the mass and angular momentum of asymptotically AdS black holes and vanishes identically for globally AdS spacetimes because it factorizes into the AdS-curvature combination $ÂR_{ij}{}^{kl} + ℓ^{-2}δ_{[ij]}{}^{[kl]}$, and $q_0(ξ)$, which the author identifies as a covariant formula for the vacuum energy. The general vacuum-energy result $E_0 = (-k)^n [Vol(Σ_k^{D-2})/8πG] ℓ^{2n-2} (2n-1)!!/(2(2n)!)$ matches the value previously conjectured from extrapolating counterterm calculations, and the paper checks the whole package — finite charges, finite Euclidean action, entropy equal to one quarter the horizon area — against Schwarzschild–AdS, five- and seven-dimensional Kerr–AdS, and the Clarkson–Mann AdS soliton.

Load-bearing premise

The whole construction rests on the boundary condition $K^i{}_j = (1/ℓ)δ^i{}_j$ with $δK^i{}_j = 0$ being compatible with every asymptotically AdS spacetime, which the paper argues from the leading terms of the Fefferman–Graham expansion but verifies only on the specific black holes and solitons it examines.

Editorial extensions

If this is right

The regularized action and the conserved charges are finite in every odd dimension without solving the asymptotic Einstein equations order by order, removing the practical blow-up in the number of Dirichlet counterterms as dimension grows.
The vacuum energy of asymptotically AdS spacetimes acquires a closed covariant formula rather than a case-by-case byproduct of holographic renormalization, reproducing the Balasubramanian–Kraus value in five dimensions and the Emparan–Johnson–Myers extrapolation in general odd dimensions.
Black-hole thermodynamics closes consistently: the thermodynamic energy defined from the Euclidean action equals the Noether charge $q + q_0$, and the entropy comes out as one quarter of the horizon area for the static and rotating cases checked.
Because $q(ξ)$ vanishes identically for globally AdS spacetime, the split gives a clean operational definition of what counts as vacuum energy versus dynamical charge in a background-independent setting.
The same boundary-term form is argued to be universal across Lovelock-type theories, changing only its coupling constant when the bulk theory changes.

Reading between the lines

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

If the Kounterterms prescription really coincides with Dirichlet counterterms up to a boundary topological invariant in every odd dimension — proven so far only in three dimensions — then the vacuum energy and the Weyl anomaly would have to come from the same corner of the boundary stress tensor, which the author flags as unexplained.
The construction suggests a practical calculational shortcut: any question that only needs finite charges for a new odd-dimensional AAdS solution can be answered from eqs. (154) and (155) without setting up a counterterm series at all.
The disagreement noted with later work on Kerr–AdS vacuum energy — whether the Casimir energy should depend on rotation parameters or be a pure constant fixed by the Einstein static universe boundary metric — is a concrete place where the two regularization schemes could be told apart by an explicit higher-dimensional computation.
Because the boundary condition fixes $K^i{}_j$ rather than the conformal metric, the scheme may extend to boundary geometries where the Fefferman–Graham expansion is irregular, which is exactly where the Dirichlet prescription is hardest to run.

Original abstract

As an alternative to the Dirichlet counterterms prescription, I introduce the concept of Kounterterms as the boundary terms with explicit dependence on the extrinsic curvature K_{ij} that regularize the AdS gravity action. Instead of a Dirichlet boundary condition on the metric, a suitable choice of the boundary conditions --compatible with any asymptotically AdS (AAdS) spacetime-- ensures a finite action principle for all odd dimensions. Background-independent conserved quantities are obtained as Noether charges associated to asymptotic symmetries and their general expression appears naturally split in two parts. The first one gives the correct mass and angular momentum for AAdS black holes and vanishes identically for globally AdS spacetimes. Thus, the second part is a covariant formula for the vacuum energy in AAdS spacetimes and reproduces the results obtained by the Dirichlet counterterms method in a number of cases. It is also shown that this Kounterterms series regularizes the Euclidean action and recovers the correct black hole thermodynamics in odd dimensions.

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, 5,094 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

As an alternative to the Dirichlet counterterms prescription, I introduce the concept of Kounterterms as the boundary terms with explicit dependence on the extrinsic curvature K_{ij} that regularize the AdS gravity action. Instead of a Dirichlet boundary condition on the metric, a suitable choice of the boundary conditions --compatible with any asymptotically AdS (AAdS) spacetime-- ensures a finite action principle for all odd dimensions. Background-independent conserved quantities are obtained as Noether charges associated to asymptotic symmetries and their general expression appears naturally split in two parts. The first one gives the correct mass and angular momentum for AAdS black holes and vanishes identically for globally AdS spacetimes. Thus, the second part is a covariant formula for the vacuum energy in AAdS spacetimes and reproduces the results obtained by the Dirichlet counterterms method in a number of cases. It is also shown that this Kounterterms series regularizes the Euclidean action and recovers the correct black hole thermodynamics in odd dimensions.

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	As an alternative to the Dirichlet counterterms prescription, I introduce the concept of Kounterterms as the boundary terms with explicit dependence on the extrinsic curvature K_{ij} that regularize the AdS gravity action.	contribution 0.97	0.92	0.26	asserted 1.00
s1	Instead of a Dirichlet boundary condition on the metric, a suitable choice of the boundary conditions --compatible with any asymptotically AdS (AAdS) spacetime-- ensures a finite action principle for all odd dimensions.	method 0.46	0.72	0.24	asserted 0.94
s2	Background-independent conserved quantities are obtained as Noether charges associated to asymptotic symmetries and their general expression appears naturally split in two parts.	result 0.78	0.90	0.14	asserted 1.00
s3	The first one gives the correct mass and angular momentum for AAdS black holes and vanishes identically for globally AdS spacetimes.	result 0.98	0.84	0.11	asserted 1.00
s4	Thus, the second part is a covariant formula for the vacuum energy in AAdS spacetimes and reproduces the results obtained by the Dirichlet counterterms method in a number of cases.	result 0.86	0.84	0.21	asserted 1.00
s5	It is also shown that this Kounterterms series regularizes the Euclidean action and recovers the correct black hole thermodynamics in odd dimensions.	result 0.98	0.93	0.28	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.95 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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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

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Forward citations

Cited by 14 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Large rotating AdS black holes from fluid mechanics
hep-th 2007-08 accept novelty 7.0 of 10

Large rotating AdS black holes are dual to stationary rigidly rotating solutions of the Navier-Stokes equations on a sphere, and their thermodynamics is the static equation of state multiplied by a universal factor of...
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.
Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs
hep-th 2019-08 conditional novelty 6.0 of 10

Kounterterm renormalization is extended to even-dimensional holographic CFTs, yielding a universal entanglement entropy formula whose logarithmic coefficient reproduces the type A central charge.
Holographic renormalization and supersymmetry
hep-th 2016-12 conditional novelty 6.0 of 10

Standard holographic renormalization breaks supersymmetry in five-dimensional minimal gauged supergravity, and a new non-standard finite boundary term restores it and reproduces the field theory Casimir energy.
Holographic renormalization as a canonical transformation
hep-th 2010-07 accept novelty 6.0 of 10

Holographic renormalization is shown to be a canonical transformation that repairs ill-defined variational problems at infinity, and the same construction is carried out for point particles, flat-space scalars, gravit...

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Semiclassical instabilities of Kerr-AdS black holes
gr-qc 2009-05 conditional novelty 6.0 of 10

Solving the coupled metric-perturbation equations for Kerr-AdS numerically, the authors find a single negative mode of the Euclidean action precisely in the region where the grand-canonical specific heat is negative.
Thermodynamic instability of rotating black holes
gr-qc 2009-03 accept novelty 6.0 of 10

Rotating black hole instantons in four, five, and six dimensions carry at least one non-conformal negative mode precisely when grand-canonical thermodynamic stability fails.
Black objects in the Einstein-Gauss-Bonnet theory with negative cosmological constant and the boundary counterterm method
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Boundary counterterms regularize the action and charges of Einstein-Gauss-Bonnet anti-de Sitter black objects up to nine dimensions, with explicit results for black holes and new numerical black string solutions.
Universal regularization prescription for Lovelock AdS gravity
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A boundary term built from intrinsic and extrinsic boundary curvature, with a dimension-dependent form and a theory-dependent coefficient, regularizes both the Noether charges and the Euclidean action of any Lovelock-...
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Charged anti-de Sitter black holes with an exponentially coupled scalar field undergo a reentrant phase transition, passing from the plain charged phase into a scalarized phase and back as temperature rises.
Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions
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A cohomological surface-charge formalism yields the first law of black-hole mechanics for Einstein gravity coupled to matter and classifies asymptotic charges in three-dimensional gravity.
Counterterms in Dimensionally Continued AdS Gravity
hep-th 2007-06 conditional novelty 5.0 of 10

In Dimensionally Continued AdS gravity, the Dirichlet counterterms that regularize the action equal the difference between the Kounterterm series and the generalized Gibbons-Hawking boundary term, up to a finite piece.
A note on the total action of $4D$ Gauss-Bonnet theory
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The D to 4 limit used to define four-dimensional Gauss-Bonnet gravity produces divergent on-shell and surface terms in the action, because some terms do not carry the (D-4) factor that the rescaled coupling is meant t...
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How the literature uses it

49 arXiv papers cite this in 102 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 · 69 sentences without a confident label

Uses its method or tool 8 papers

We obtain this by making use of the Kounterterm renormalization procedure [ref] .

Renormalized holographic entanglement entropy in Lovelock gravity · Introduction · 1.00
The scheme considers the renormalized Einstein-AdS action obtained through the Kounterterms procedure [ref] , evaluated on a conically-singular manifold [ref] defined via the Replica Trick [ref] .

Topological terms, AdS_2n gravity and renormalized Entanglement Entropy of holographic CFTs · Introduction · 0.95
In more detail, we foliate the solution in boundary spheres, compute the extrinsic curvature $Θ^μ_ν$ of these foliations near the boundary, subtract off the appropriate counter terms contributions [ref] , and finally multiply the answer by the $r^D-1$ to obtain the stress tensor on a unit sphere.

Large rotating AdS black holes from fluid mechanics · Stress tensor from rotating black holes in AdS$_D$ · 0.99

Compares with it 2 papers

Note that for Einstein-Hilbert-AdS gravity, $F_p=0$ and the above expression for $q(ξ)$ recovers the corresponding charge in [ref] .

Universal regularization prescription for Lovelock AdS gravity · Conserved quantities and vacuum energy. · 0.92
The structure of ([ref]) is similar to the Kounterterm action of the metric [ref] , where there is dependence on both the intrinsic and extrinsic curvatures.

Extrinsic Holographic Renormalization for a Scalar Field · Hybrid version of the extrinsic counterterms · 1.00

Cites it as background 19 papers

As for odd spacetime dimensions, the Kounterterms adopt a slightly different form, which is a consequence of a double-parametric integral [ref]

Energy of anti-de Sitter black holes in odd-dimensional Quadratic Curvature Gravity · Kounterterms in AdS gravity · 0.84
First proposed for even bulk dimensions ($d$ odd) in [ref] and later generalized to odd bulk dimensions in [ref] , the Kounterterms constitute a closed form expression for a boundary term applicable to any dimension.

Counterterms, Kounterterms, and the variational problem in AdS gravity · Kounterterms in AdS gravity · 0.87
Let us review the five dimensional Kerr-$AdS_5$ black hole metric as presented in [ref] [equation] where [equation] with $M, a_1,a_2$ real parameters, related to the ADM mass and angular momenta by [ref] [equation] When $M>0$, $a^2_1,a^2_2<1$ all these quantities are physically acceptable, and we have that $r_-,r_+$, the real roots of $Δ_r$, correspond to the inner and outer horizons of the black hole [ref] , whereas $r_0$ is purely imaginary: [equation]

Scalar quasinormal modes of Kerr-AdS$\mathbf{_5}$ · Scalar Fields in Kerr-$𝐀𝐝𝐒_5$ · 0.88

Discussion

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