Asymptotically Anti-de Sitter Space-times: Conserved Quantities

Abhay Ashtekar; Penn State University); Saurya Das (CGPG

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Asymptotically Anti-de Sitter Space-times: Conserved Quantities

Abhay Ashtekar, Saurya Das (CGPG, Penn State University)

High Energy Physics - Theory · Submitted 29 Nov 1999 · 24 references

Pith's verdict: The derivation and the d=4/d=5 comparison are solid and worth citing; the covariance no-go is a good argument carrying one unstated premise, and the paper never actually checks its own charges against the residual conformal freedom.

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CitationAbhay Ashtekar, Penn State University), Saurya Das (CGPG. "Asymptotically Anti-de Sitter Space-times: Conserved Quantities." arXiv:hep-th/9911230 (1999). https://pith.science/paper/GCUYAKME

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Significance

The paper addresses a live and consequential question — whether the classical gravitational charges in asymptotically AdS space-times agree with those extracted from the counter-term subtraction procedure used in the AdS/CFT literature — and it does so with a method that is both general in d and free of adjustable parameters. The derivation of the balance equation is clean and the comparison formula (4.8) is explicit and checkable: it makes a sharp, falsifiable statement (agreement at d = 4, disagreement at d = 5) rather than a qualitative claim. The covariance argument in Section III B is a genuinely useful observation that constrains any proposed charge definition. The paper is also careful to state its conclusions as a tension with the conjectured duality rather than as a refutation of it, which is the appropriate register.

Major objections (2)

[Section III B, Eq. (3.2)] The covariance argument that forces Q_ξ = 0 in pure AdS requires Q_ξ to be a functional of the physical metric alone, i.e., independent of the conformal completion used to define it. Section II A asserts that physical quantities at I must be conformally invariant, but the text never verifies that (3.2) is invariant under the residual freedom Ω → ωΩ with n^a∇_aω = 0 on I. The invariance does in fact hold — under this rescaling E_ab → ω^{3-d}E_ab and dS^b → ω^{d-3}dS^b, so the integrand has weight zero — but since the argument in Section III B is the paper's headline claim, the weight counting should be written out explicitly rather than left to the reader.
[Section IV, Eq. (4.8)] The inference that the two charge definitions agree when d = 4 requires Δ_ab to vanish on I in four dimensions, not merely to be trace-free. As written, (4.8) gives Q_ξ − Q̃_ξ = (1/8πG)∮ Ω^{d−5}Δ_ab ξ^a dS^b, which for d = 4 is ∮ Ω^{−1}Δ_ab ξ^a dS^b and vanishes only if Δ_ab = O(Ω). The trace argument given in the paragraph following (4.10) establishes only that Δ_ab is trace-free when d = 4. Please add the identity, or the constraint equation on I_Ω, that makes Δ_ab vanish at I in d = 4, or state explicitly that this is the step on which the d = 4 agreement rests.

Minor objections (9)

[Section III B] The balance equation is cited as '(2.20)', but the relevant equation is (2.16); please correct the cross-reference.
[Section IV] The reference to 'Section 3.3' does not correspond to any section of the paper; the intended reference appears to be Section III B, remark 3.
[Eqs. (3.2) and (4.2)] The spacetime dimension is written as 'n' in these two equations but as 'd' everywhere else; please use a single symbol throughout.
[Eq. (4.7)] The last term of the displayed expression contains 'h_ac', which should presumably read 'h_ab'.
[Eq. (2.9)] The symbol '1l2' should be '1/ℓ^2'.
[Eq. (2.2)] Please check the (d−3) factor in the expression for M_(d); the standard ADM mass for Schwarzschild–AdS_d is (d−2)A_(d−2)r_0^{d−3}/(16πG_d), and the extra factor would affect the identification Q_ξ = M stated in Section III B for d > 4.
[Section III A] For d = 4 the boundary I is three-dimensional, and the conformal isometry group of a conformally flat three-metric is infinite-dimensional; a sentence explaining how the reflective condition B_ab = 0 reduces G to the ten-dimensional AdS group (or a pointer to the specific argument in [6]) would help the reader.
[Sections IV and V] The covariance argument establishes that Q̃_ξ requires additional boundary structure; it does not by itself show that Q̃_ξ is physically meaningless. In the AdS/CFT reading the boundary metric is a specified source, so frame dependence is expected rather than pathological. The paper would be sharper if it distinguished 'not covariant in the sense of Section III B' from 'not a legitimate Hamiltonian for a theory with a fixed boundary metric'.
[Throughout] Several typographical errors should be corrected: 'Maldecena' for 'Maldacena' (throughout), 'confor field theory' (Section IV), 'a a clear example' and 'in in the treatment' (Introduction), 'than than' (Section V), and 'to bridging' (Introduction).

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The paper

The paper claims

Using Penrose's conformal completion in any dimension $d$ from 4 up, the paper defines a charge for each generator of the asymptotic anti-de Sitter symmetry group, shows those charges balance against matter flux across infinity, and argues—

The reading

The paper asks a narrow but consequential question: in a space-time that approaches anti-de Sitter space at infinity, what are the conserved energy, momentum and angular momentum, and what do they equal when the space-time is exactly anti-de Sitter? Working with Penrose's conformal completion rather than coordinate comparisons against a reference metric, the authors show that the boundary conditions force the asymptotic symmetry group to be the anti-de Sitter group $O(d-1,2)$ in every dimension $d$ (with an extra 'reflective' condition needed when $d=4$). The asymptotic field equations then yield a charge $Q_ξ$ for each symmetry generator $ξ$, defined as an integral over a $(d-2)$-sphere cross-section of the electric part of the Weyl tensor contracted with $ξ$. These charges satisfy a balance equation: they change from one cross-section to another only by the flux of physical matter across the piece of infinity between them, and are exactly conserved when no such matter flux exists. The striking consequence is that in pure anti-de Sitter space every one of these charges vanishes, in all dimensions, and the authors argue by covariance that any charge definition not tied to extra boundary structure must do the same. That puts the result in direct tension with charges built by counterterm subtraction, which assign a nonzero 'vacuum energy' in five dimensions.

What carries the argument

The load-bearing device is the conformal completion: a rescaled metric $g_{ab} = Ω^2 ĝ_{ab}$ that brings infinity to a finite surface $I$ where $Ω=0$ and $∇_aΩ$ is nonvanishing, with a conformal frame chosen so that $∇_a n_b = 0$ on $I$. This turns delicate limits into ordinary local geometry at $I$. On that surface the intrinsic metric $h_{ab}$ is conformally flat, which is what fixes the asymptotic symmetry group to be the conformal isometry group of $I$ — the anti-de Sitter group. The charge itself is the electric part of the leading Weyl fall-off, $E_{ab} = ℓ^2 K_{ambn}n^m n^n$, integrated against a conformal Killing field; its trace-free property is exactly what makes the charges conserved when no matter flows through infinity.

Extended reading notes

Core claim

The central claim is that for asymptotically anti-de Sitter space-times in any dimension $d ≥ 4$, the asymptotic symmetry group is the anti-de Sitter group, and each of its generators carries a conserved quantity built from the leading fall-off of the Weyl tensor at conformal infinity. The derivation runs through a key identity, $D_d E^{md} = -8πG^{(d)}(d-3)T_{ab}n^a h^{bm}$, relating the divergence of the electric Weyl field $E_{ab}$ on the boundary to the pull-back of the matter stress-energy; integrating it gives the balance equation and, in the absence of matter flux, absolute conservation. Because the Weyl tensor vanishes identically in anti-de Sitter space, all these charges vanish there, and the authors add a covariance argument: if a charge assignment is defined purely from the boundary conditions, then acting with an asymptotic symmetry must rotate the charges among themselves, and since the anti-de Sitter metric is invariant under its own isometry group, only the zero assignment is compatible. The paper then compares its charges with those of the counterterm-subtraction method, finding agreement in $d=4$ for all asymptotically anti-de Sitter space-times and disagreement in $d=5$, where the difference is a constant shift equal to $3A^{(3)}ℓ/(64πG)$ in the pure anti-de Sitter case.

Load-bearing premise

The covariance argument assumes that under any asymptotic symmetry a candidate charge assignment must transform in the natural way, with the new charges being the old ones evaluated on the transformed generator; a definition allowed to depend on a preferred boundary frame would escape the conclusion.

Editorial extensions

If this is right

Every asymptotically anti-de Sitter space-time in $d ≥ 4$ admits $d(d+1)/2$ conserved charges, one per asymptotic symmetry generator, with a balance law that accounts for matter flux across infinity.
In pure anti-de Sitter space all of these charges vanish in every dimension, so there is no distinguished 'vacuum energy' in the classical gravitational description.
In four dimensions, the conformal-completion charges and the counterterm-subtraction charges agree for all asymptotically anti-de Sitter space-times, not only for Kerr-anti-de Sitter metrics.
In five dimensions the two definitions differ by a term that is a constant shift in the Schwarzschild-anti-de Sitter family but appears to vary from one space-time to another in general.
Because the counterterm integrand is not trace-free, its charges need not be independent of the choice of cross-section, which undermines their use as Hamiltonians generating asymptotic symmetries.

Reading between the lines

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

If the covariance argument holds, the disagreement is not a calculational discrepancy to be fixed by a better counterterm but a structural constraint: any dictionary matching bulk charges to boundary symmetry generators must map the anti-de Sitter ground state to zero charge, and the boundary side must explain where a nonzero Casimir energy sits instead.
The paper's distinction between charges tied to a chosen conformal frame and charges defined from the equivalence class of boundary metrics suggests a clean diagnostic: test candidate holographic stress tensors for cross-section independence in a non-stationary bulk, where the trace-free condition is easiest to violate.
The absence of gravitational radiation carrying charge through infinity in this setting, contrasted with null infinity in asymptotically flat space-times, suggests that the anti-de Sitter boundary behaves partly like spatial infinity; charge definitions that work at spatial infinity may transfer more directly than Bondi-style ones.
A useful extension would be to recompute the difference term for a rotating or collapsing bulk configuration, since the paper only evaluates it explicitly for pure anti-de Sitter and states that the general case was not settled.

Original abstract

Asymptotically anti-de Sitter space-times are considered in a general dimension $d\ge 4$. As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d=4). Asymptotic field equations imply that, associated with each generator $\xi$ of this group, there is a quantity $Q_\xi$ which satisfies the expected `balance equation' if there is flux of physical matter fields across the boundary $\I$ at infinity and is absolutely conserved in absence of this flux. Irrespective of the dimension d, all these quantities vanish if the space-time under considerations is (globally) anti-de Sitter. Furthermore, this result is required by a general covariance argument. However, it contradicts some of the recent findings based on the conjectured ADS/CFT duality. This and other features of our analysis suggest that, if a consistent dictionary between gravity and conformal field theories does exist in fully non-perturbative regimes, it would have to be more subtle than the one used currently.

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.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.78 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.93 · 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.

Asymptotically anti-de Sitter space-times are considered in a general dimension $d\ge 4$. As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d=4). Asymptotic field equations imply that, associated with each generator $\xi$ of this group, there is a quantity $Q_\xi$ which satisfies the expected `balance equation' if there is flux of physical matter fields across the boundary $\I$ at infinity and is absolutely conserved in absence of this flux. Irrespective of the dimension d, all these quantities vanish if the space-time under considerations is (globally) anti-de Sitter. Furthermore, this result is required by a general covariance argument. However, it contradicts some of the recent findings based on the conjectured ADS/CFT duality. This and other features of our analysis suggest that, if a consistent dictionary between gravity and conformal field theories does exist in fully non-perturbative regimes, it would have to be more subtle than the one used currently.

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	Asymptotically anti-de Sitter space-times are considered in a general dimension $d\ge 4$.	background 0.76	0.38	0.02	asserted 0.95
s1	As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d=4).	result 0.72	0.28	0.05	asserted 0.81
s2	Asymptotic field equations imply that, associated with each generator $\xi$ of this group, there is a quantity $Q_\xi$ which satisfies the expected `balance equation' if there is flux of physical matter fields across the boundary $\I$ at infinity and is absolutely conserved in absence of this flux.	result 0.94	0.74	0.30	asserted 1.00
s3	Irrespective of the dimension d, all these quantities vanish if the space-time under considerations is (globally) anti-de Sitter.	result 1.00	0.83	0.26	asserted 1.00
s4	Furthermore, this result is required by a general covariance argument.	interpretation 0.77	0.41	0.07	asserted 0.99
s5	However, it contradicts some of the recent findings based on the conjectured ADS/CFT duality.	interpretation 0.50	0.34	0.13	asserted 0.99
s6	This and other features of our analysis suggest that, if a consistent dictionary between gravity and conformal field theories does exist in fully non-perturbative regimes, it would have to be more subtle than the one used currently.	interpretation 1.00	0.41	0.46	hedged possibility 0.97

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.94.

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sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
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levels run

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References

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hep-th 2005-05 conditional novelty 6.0 of 10

Holographic charges equal Noether charges for asymptotically locally AdS spacetimes, so the first law of black hole mechanics holds for these black holes with the holographic mass, angular momentum, and charge as the ...
Rotating Black Holes in Gauged Supergravities; Thermodynamics, Supersymmetric Limits, Topological Solitons and Time Machines
hep-th 2005-04 accept novelty 6.0 of 10

Rotating charged black holes in gauged supergravity are shown to satisfy the first law of thermodynamics, and their supersymmetric limits yield new regular black holes in seven and four dimensions plus non-singular ro...
The First Law of Thermodynamics for Kerr-Anti-de Sitter Black Holes
hep-th 2004-08 accept novelty 6.0 of 10

The correct thermodynamic mass and angular momenta of Kerr-AdS black holes in all dimensions are the ones that make the first law hold with entropy equal to one quarter of the horizon area.
Dirichlet gravitons in AdS$_4$ and $\mathcal{L}_{\Lambda}w_{1+\infty}$ wedge algebra
hep-th 2026-09 conditional novelty 5.0 of 10

A bulk derivation of the L_Λ w_{1+∞} wedge algebra from linearized AdS4 gravitons with Dirichlet boundary conditions, fixing the helicity pairing through AdS covariance and the Jacobi identities.
The Role of the Volume in Black Hole Thermodynamics
gr-qc 2026-06 unverdicted novelty 5.0 of 10

Conserved charges built from a Kerr-Schild background show the first law requires the Killing vector and AdS background to be held fixed, explaining why the rotating-frame energy F fails while E works, and why the geo...
Holographic CFT Phase Transitions and Criticality for Charged AdS Black Holes
hep-th 2021-12 conditional novelty 5.0 of 10

Enumerating all eight thermodynamic ensembles of the CFT dual to a charged AdS black hole, the authors find criticality only in the charge-potential and central-charge planes, a new zeroth-order transition at fixed ch...
Thermodynamics of Lorentzian Taub-NUT spacetimes
hep-th 2019-03 conditional novelty 5.0 of 10

Keeping the Misner string and adding a charge conjugate to the NUT parameter gives Lorentzian Taub-NUT black holes a consistent first law with positive thermodynamic volume.
Solution Phase Space and Conserved Charges: A General Formulation for Charges Associated with Exact Symmetries
hep-th 2015-12 accept novelty 5.0 of 10

A reformulation of covariant phase space methods shows that black hole entropy and other conserved charges can be computed on any closed surface around the hole, including in theories with gauge fields.
Super-Entropic Black Holes
hep-th 2014-11 accept novelty 5.0 of 10

A new ultraspinning limit of Kerr-AdS black holes yields horizons that are non-compact but finite in area, and their entropy violates the conjectured Reverse Isoperimetric Inequality.
AdS Dyonic Black Hole and its Thermodynamics
hep-th 2013-07 conditional novelty 5.0 of 10

A new asymptotically AdS dyonic black hole is constructed, and its thermodynamics requires an extra conjugate pair beyond the usual temperature, entropy, and electric and magnetic potentials.
Thermodynamics of Black Holes in Horava-Lifshitz Gravity
hep-th 2009-05 conditional novelty 5.0 of 10

For general Horava-Lifshitz coupling λ, black hole entropy is the horizon area times a λ-dependent factor that equals one quarter only when λ = 1.
Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions
hep-th 2007-08 conditional novelty 5.0 of 10

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.
Constraints on extra charges in dyonic Kerr-Newman-Kasuya-Taub-NUT black hole from the observations of quasi-periodic oscillations
astro-ph.HE 2025-07 conditional novelty 4.0 of 10

Using QPO data from five X-ray binaries, the authors place upper limits on electric, magnetic, and NUT charges of a dyonic Kerr black hole, with a tentative nonzero NUT parameter in GRS 1915+105.
Construction of Regular Black Holes in General Relativity
gr-qc 2016-10 conditional novelty 4.0 of 10

A reverse-engineering recipe constructs two-parameter families of regular black holes in general relativity with nonlinear electrodynamics, recovering Bardeen and Hayward as special cases.
Notes on the BMS group in three dimensions: I. Induced representations
hep-th 2014-03 conditional novelty 4.0 of 10

Induced representations of the three-dimensional BMS group are shown to be labelled by Virasoro coadjoint orbits, giving one-particle states with continuous spin and a lower energy bound set by the vacuum orbit.
Notes on solution phase space and BTZ black hole
gr-qc 2024-11 accept novelty 3.0 of 10

A detailed worked example showing that the solution phase space method reproduces the known mass, angular momentum, entropy, first law, and Smarr relation for the BTZ black hole and three-dimensional Kerr-dS spacetime.
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...
Asymptotically Anti-de Sitter spacetimes and their stress energy tensor
hep-th 2000-10 accept novelty 3.0 of 10

The gravitational stress tensor of an asymptotically anti-de Sitter spacetime is a fixed constant times the coefficient g(d) in the near-boundary metric expansion, plus a universal polynomial in the lower coefficients.
Black Holes in Higher Dimensions
hep-th 2008-01 accept novelty 2.0 of 10

Higher-dimensional vacuum gravity admits rotating black holes with non-spherical horizons (black rings) and multi-component horizons, so four-dimensional black-hole uniqueness does not extend beyond d = 4.

How later papers used it

119 later papers cite it

Evidence: substantial. 119 of the citing papers share no author with it, and 54 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 163 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool.

Uses its method or tool 4 of 41 papers with this label

Used the Ashtekar-Das formalism to compute the energy of the hairy black hole in AdS.

A scalar field condensation instability of rotating anti-de Sitter black holes (2010) · 1.00 · paraphrased: the licence does not allow quoting
Computed the black hole mass using the conformal method.

Thermodynamics of charged and accelerating black holes (2025) · 0.98 · paraphrased: the licence does not allow quoting
Computed M also using a 5-dimensional definition involving the intrinsic 5-dimensional Weyl tensor due to Ashtekar and collaborators.

Half BPS states in AdS_5 x Y^{p,q} (2007) · 0.94 · paraphrased: the licence does not allow quoting
In this subsection, we calculate the mass of the hairy black hole using the AMD prescription [ref] .

Analytic topological hairy dyonic black holes and thermodynamics (2021) · Mass from AMD prescription · 1.00

How the literature uses it

106 arXiv papers cite this in 142 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 · 73 sentences without a confident label

Builds on it 2 papers

In nondegenerate Lovelock theories, this polynomial can be truncated so that it becomes linear in the Weyl tensor [ref] [equation] This expression corresponds to Conformal Mass for nondegenerate Lovelock AdS gravity [ref] , as an extension of the concept developed by Ashtekar, Magnon and Das [ref] .

Mass in Lovelock Unique Vacuum gravity theories · Kounterterms and conserved charges · 0.94
As a matter of fact, the above formula corresponds to the generalization of the Ashtekar-Magnon-Das mass [ref] for nondegenerate Lovelock-AdS gravity (see <ref>).

Black Hole Energy in Lovelock Unique Vacuum theories · Even dimensions $D=2n$ · 0.93

Uses its method or tool 37 papers

The first argument uses the Ashtekar–Das definition of conformal mass [ref] , which extracts the mass via conformal regularisation of the AdS C-metric near the boundary.

Holographic Thermodynamics of Accelerating Black Holes · Introduction · 1.00
Note that similar to what was done in [ref] , the thermodynamic quantities [ref] (in the limit $a→ℓ$) can be directly obtained from the Superentropic metric [ref] by using the conformal method [ref] .

Are Superentropic black holes superentropic? · Kerr-AdS with conical deficits · 0.80
To obtain the expression for the mass, we use a prescription due to Ashtekar, Magnon and Das (AMD) [ref] .

Thermodynamics and Entanglement Entropy with Weyl Corrections · First law of Weyl corrected black hole thermodynamics · 1.00

Compares with it 9 papers

The formula obtained for the mass satisfies the first law of thermodynamics and coincide with the value obtained with the Astekar-Magnon-Das (AMD) formalism [ref] .

First order flow for non-extremal AdS black holes and mass from holographic renormalization · Introduction · 0.98
In this paper, we relate Kounterterms charges to another definition of conserved quantities in AAdS spacetimes, which was given for Einstein gravity by Ashtekar-Magnon-Das (AMD) [ref] .

Vacuum Degeneracy and Conformal Mass in Lovelock AdS Gravity · Introduction · 0.83
We also mention that Ref. [ref] gives the same result as ours for the neutral black hole based on the surface integral using the 'effective stress-energy tensor'.

Conserved charges in general relativity · The Komar integral · 0.98

Cites it as support 1 paper

This choice of setting the vacuum energy to zero is also in agreement with other definitions of the asymptotic energy in asymptotically AdS spacetimes [ref] , which are not based on the counterterm subtraction method.

Holographic CFT Phase Transitions and Criticality for Charged AdS Black Holes · CFT thermodynamics with a chemical potential for the central charge · 0.99

Cites it as background 12 papers

It is known that the Riemann curvature of a maximally symmetric spacetime can be written explicitly as [equation] Note that this condition can be written in terms of the Riemann curvature 2-form $ℛ^ab$ as [equation] The curvature 2-form $ℛ̅^ab$ also satisfies the differential Bianchi equation: [equation] Inspired by equation [ref], we define alAdS spacetimes [ref] (see also [ref] ) to be a solution of ([ref]) such that its metric obeys the fall-off condition [equation] with $ℐ$ the conformal boundary of $ℳ$.[In contrast, we define an asymptotically AdS spacetime to be alAdS spacetime together with a Dirichlet boundary condition that fixes the induced boundary $ℝ×S^2$.] It was proven by Fefferman-Graham [ref] that one can always find a set of coordinates $(z,x^a)$ at least near the conformal boundary $ℐ$ such that ${z=0}⊂ℐ$ and the metric is of the form [equation] where we have an expansi

Dual Charges for AdS Spacetimes and the First Law of Black Hole Mechanics · Definition, action and topological renormalization · 0.89
Conformal Mass is the expression for energy in AAdS spacetimes [ref] .

Black Hole Energy in Lovelock Unique Vacuum theories · Odd dimensions · 0.83
In this section, we will study the dynamics of the “physical boundary” in the vacuum asymptotically AdS$_3$ $(_3)$ spacetime [ref] .

On boundary degrees of freedom in three dimensional Anti-de Sitter spacetime and thermofield-double · The Gibbons-Hawking-York action close to the boundary of AdS$_3$ · 0.89

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.

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