The Quantum Stress Tensor in the Three Dimensional Black Hole

Alan R. Steif

REVIEW

The Quantum Stress Tensor in the Three Dimensional Black Hole

Alan R. Steif

General Relativity and Quantum Cosmology · Submitted 27 Aug 1993 · 16 references

Pith's verdict: The rotating BTZ stress tensor in closed form is a real, checkable result; the jump from a measure-zero divergence to 'the inner horizon is unstable' is asserted in one line and should be softened, not rejected.

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arXiv IDgr-qc/9308032 (version 2)

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CitationAlan R. Steif. "The Quantum Stress Tensor in the Three Dimensional Black Hole." arXiv:gr-qc/9308032 (1993). https://pith.science/paper/OHJU72MM

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Editor's note

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The thing worth your time here is Eq. (19): the renormalized stress tensor for a conformally coupled scalar on the rotating BTZ quotient, written as an explicit image sum. The static case (18) is not a first — Steif says so himself in the closing note, citing Shiraishi-Maki and Lifschytz-Ortiz as concurrent — but the rotating series and the identification of the polarized hypersurfaces (20) are new, and the derivation is short enough to check line by line. Green's function from the covering space, image sum, point split, subtract n=0. That is a clean piece of work and it will get cited by anyone doing back reaction in 2+1. I'd cite it myself for the rotating expression alone. The static answer is also pleasingly compact, and the observation that the divergence at r=0 traces back to Λ having fixed points there is the right way to explain it. One more thing in its favour: the paper is honest about a subtlety most readers would miss — is finite at the inner horizon approached from outside, because the horizon is not compactly generated. That is stated, not buried. Where it goes soft is the last step. The denominators d_n vanish on the surfaces (20), which sit inside the inner horizon and accumulate at ρ=0; at ρ=0 itself every d_n is positive and the series converges. So the divergence lives on a countable set of measure-zero surfaces with finite stress tensor between them. The sentence 'these surfaces approach r=0, [so] the stress tensor will diverge there' is an accumulation argument that is asserted, not shown, and the back-reaction estimate that follows — δg ~ s^{-1/2} from 'integrating twice' — has no linearized equation written and is self-undermining in the obvious way: a diverging perturbation can flag breakdown but cannot fix the endpoint. The paper does say 'suggests', which is the right verb, but the abstract and the closing paragraph lean harder than the calculation supports. Worth noting too that the extremal case (21) comes out smooth everywhere, which cuts against a uniform instability story and gets a sentence rather than a discussion. On the reader's second worry, about the λ=0 boundary condition: that one doesn't land. The leading divergence is λ-independent, so the transparent/Dirichlet/Neumann choice doesn't touch it. Send it out. The calculation is sound and the interpretive claim needs a referee to ask for either a real linearized treatment or a weaker conclusion.

Referee report

Significance

This is a clean, parameter-free calculation of a quantity that is normally intractable: the image-sum construction gives closed-form expressions for <T_{μν}> in both the static and rotating 2+1 black holes, with no fitted constants and no approximation beyond the choice of vacuum state. The results are checkable and falsifiable in the ordinary sense — the finiteness on the event horizon, the r^{-3} behaviour at the singularity, and the location of the polarized surfaces are all statements that follow from the geometry of Eq. (20) and could be contradicted by an independent computation. The paper explicitly notes agreement of its Lorentzian Green's function with the Euclidean continuation of Shiraishi and Maki, Ref. [12], and cites the contemporaneous preprints of Shiraishi–Maki and Lifschytz–Ortiz for the static case, which is appropriate disclosure. The rotating case is the genuinely new content, and the identification of the inner horizon as a quantum-mechanically distinguished surface is a useful contribution to the classical-stability literature of Reissner–Nordström and Kerr.

Major objections (2)

[Paragraph following Eq. (20)] The inference from the divergence of <T_{μν}> to the instability of the inner horizon rests on the sentence 'Integrating twice, one finds that the metric perturbation diverges as δg_{μν} ~ s^{-1/2} on each of the polarized hypersurfaces.' No linearized equation is written, no gauge is specified, and it is not stated what source term is being integrated (the full <T_{μν}>, or only its singular part). Since the abstract's main claim, [abstract:s4], depends on this step, the author should display the linearized equation actually used, state the gauge, and say explicitly that the estimate is a breakdown-of-linearizability argument rather than a controlled solution. As written, the divergence of δg means the linearized treatment cannot by itself determine the endpoint, and the text should say so rather than leave the reader to infer a curvature singularity.
[Eq. (20) and the sentence 'Since these surfaces in the n → ∞ limit approach the inner horizon, r = 0, the stress tensor,] The step from a countable family of measure-zero surfaces to a divergence 'at' the inner horizon is asserted rather than demonstrated. The paper itself states, correctly, that <T_{μν}> is finite at r = 0 approached from the outside, and that d_n > 0 at ρ = 0 for every finite n, so the series of Eq. (19) converges there. What is needed is a precise statement of the sense in which the divergence is attributed to the horizon: for instance, that the polarized surfaces accumulate at ρ = 0, that the curvature scalar of the back-reacted metric diverges along any sequence approaching the horizon, or that the integrated stress-energy flux through a Cauchy surface is infinite. A short paragraph making the accumulation argument explicit, and distinguishing 'divergent on a dense-in-the-limit set of surfaces' from 'divergent at a point', would materially strengthen the central claim.

Minor objections (6)

[Eq. (19)] The individual terms of the sums in Eq. (19) are singular on the surfaces d_n = 0, so inside the inner horizon the series is not absolutely convergent and the displayed expression needs a stated interpretation (principal value, analytic continuation, or order of summation). The text notes only that the sums converge exponentially outside the inner horizon; a sentence on how the expression is defined inside would remove an ambiguity.
[Eq. (10) and the choice λ = 0] The paper introduces three boundary conditions and then restricts to λ = 0 without comment on whether the conclusions depend on this choice. Since the antipodal term in Eq. (10) is regular where |x − Λ^n x| → 0, the leading s^{-5/2} divergence is presumably λ-independent; stating this explicitly would make clear that the instability conclusion is not an artifact of the transparent vacuum.
[Renormalization, paragraph following Eq. (16)] Subtracting the coincident n = 0 term is a choice of state (normal ordering with respect to the anti-de Sitter vacuum), not a removal of a universal divergence, and in odd dimensions a finite local counterterm proportional to g_{μν} is in principle admissible. The paper should say plainly that <T_{μν}> is state-dependent and that the quoted numbers refer to this particular vacuum.
[Eq. (21) and the extremal case] The statement that the extremal stress tensor is 'smooth everywhere but diverges asymptotically' should be spelled out: the growth is <T^t_t> ~ 3Kr^2 at large r, and the reader would benefit from a remark on whether this reflects the choice of state, the slow fall-off of the extremal geometry, or a genuine back-reaction problem. It would also help to say whether the polarized surfaces of Eq. (20) survive in the extremal limit, where the two horizons coincide.
[Abstract] The abstract says the stress tensor 'diverges near the inner horizon', whereas the body of the paper carefully states that it is finite at the inner horizon approached from the outside and diverges on surfaces that accumulate there. The abstract wording should be brought into line with the more precise statement in the text.
[Notation and typography] The name of the first author of Ref. [1] is spelled 'Bañados' in the literature and 'Banados' throughout this manuscript; 'Greens function' should be 'Green's function'; 'anti-deSitter' should be hyphenated consistently as 'anti-de Sitter'. In Eq. (19) the mixed-index components are listed as <T^t_t>, <T^r_r>, <T^φ_φ>, <T^φ_t>; the index placement of the off-diagonal component should be made uniform with the others, and its J → 0 limit checked against Eq. (18).

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The paper

The paper claims

A quantum field's stress tensor in the 2+1 dimensional black hole stays finite at the outer horizon, diverges at the static singularity, and blows up on surfaces that pile up at the rotating solution's inner horizon.

The reading

Steif asks what a quantum field does to the (2+1)-dimensional black hole of Banados, Teitelboim and Zanelli — a spacetime that is anti-de Sitter space with points identified by a boost, so that it has a horizon, a singularity, and, when spinning, both an outer and an inner horizon. The tool is the Green's function, built for the covering anti-de Sitter space and then summed over the identified images; the stress tensor follows by point-splitting and subtracting the coincident term. For the non-rotating hole the answer is well behaved on the event horizon and blows up as $r^{-3}$ at the singularity, with a negative energy density in the manner of the Casimir effect. For the rotating hole the outer horizon is again smooth, but inside the hole the stress tensor diverges on a discrete family of surfaces where a point and its image are joined by a null geodesic, and those surfaces accumulate at the inner horizon. The author reads this as evidence that the inner horizon does not survive quantum effects.

What carries the argument

The load-bearing device is the method of images applied to the anti-de Sitter Green's function. In the covering space, the conformally coupled scalar's Green's function is the flat three-dimensional form $1/(4π|x-x'|)$ in the four-dimensional embedding, with $|x-x'|$ the chordal distance; the black hole is that space quotiented by the boost $Λ$ of Eq. (8), so the Green's function becomes $G(x,x') = (1/4π)∑_n |x - Λ^n x'|^{-1}$. Renormalization is then a subtraction: drop the $n=0$ term. The stress tensor is obtained by point-splitting the classical expression and pulling the resulting embedding-space tensor back to the three-dimensional spacetime, Eq. (17). The polarized hypersurface — a surface whose points are joined to their own images by null geodesics — is the object that converts a formal divergence in the image sum into a physical statement about where the stress tensor becomes singular.

Extended reading notes

Core claim

The paper's central claim is that the renormalized quantum stress tensor of a conformally coupled massless scalar in the rotating 2+1 dimensional black hole is finite at the outer horizon but divergent near the inner horizon, and that this divergence signals a quantum-mechanical instability of the inner horizon against the formation of a curvature singularity. The mechanism is geometric: the image sum for the Green's function has denominators $d_n = |x-Λ^n x|^2$, and $d_n$ vanishes precisely on the surfaces $ρ = ρ_n = -l^2(��cosh nα_- - 1)/(��cosh nα_+ - ��cosh nα_-)$, where a spacetime point and its $n$-th image are connected by a null geodesic. On each such polarized hypersurface the stress tensor diverges as $s^{-5/2}$ in geodesic distance $s$, and integrating the field equation twice gives a metric perturbation growing as $s^{-1/2}$. Because the surfaces approach $r=0$ as $n o ∞$, the inner horizon is where the effect concentrates. The author is careful to note that the stress tensor is finite at the inner horizon when approached from outside, since that surface is not compactly generated; the divergence lives on the polarized surfaces inside.

Load-bearing premise

The step from a divergent stress tensor to a destroyed inner horizon assumes that substituting the fixed-background stress tensor into the linearized field equation and integrating twice gives a metric perturbation that stays small enough for the linear approximation to remain valid as the divergence is approached.

Editorial extensions

If this is right

For the non-rotating hole, the stress tensor is smooth on the event horizon, so the horizon itself is not where quantum effects break down; the breakdown is at $r=0$, where it scales as $r^{-3}$.
The energy density outside the non-rotating hole is negative, in analogy with the Casimir effect, and the stress tensor is traceless and conserved.
For the rotating hole, the outer horizon is regular, but the inner horizon is approached by a sequence of surfaces on which the stress tensor diverges, so the interior geometry is not the classical one.
The back-reacted metric perturbation grows as $s^{-1/2}$ near each polarized hypersurface, which the author reads as the inner horizon being replaced by a curvature singularity.
In the extremal case $M = J/l$ the stress tensor is smooth everywhere but diverges asymptotically, so extremal and non-extremal rotating holes behave differently.
As $M o ∞$ the identification becomes trivial and the stress tensor vanishes, matching the expectation that the anti-de Sitter covering space is recovered.

Reading between the lines

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

The divergence is a property of the image sum and the boost's fixed-point structure, so the same polarized-hypersurface mechanism should appear for any quotient of anti-de Sitter space by a boost with two unequal parameters, not only for this black hole.
Because the effect is largest where a point and its image are null-separated, it is a self-intersection of null geodesics under the identification — the same geometric ingredient that drives classical mass inflation at inner horizons, which suggests the classical and quantum stories here share a cause.
The transparent boundary condition ($λ=0$) was chosen for convenience; the Dirichlet and Neumann Green's functions differ by an antipodal term, so a reader could check whether the $s^{-5/2}$ divergence and its location survive those choices.
A full calculation would need the stress tensor evaluated on the back-reacted metric rather than the fixed classical one; until then the instability claim is an estimate rather than a solution.

Original abstract

The quantum stress tensor $<T_{\mu\nu}>$ is calculated in the 2+1 dimensional black hole found by Banados, Teitelboim, and Zanelli. The Greens function, from which $<T_{\mu\nu}> $ is derived, is obtained by the method of images. For the non-rotating black hole, it is shown that $<T_{\mu\nu}>$ is finite on the event horizon, but diverges at the singularity. For the rotating solution, the stress tensor is finite at the outer horizon, but diverges near the inner horizon. This suggests that the inner horizon is quantum mechanically unstable against the formation of a singularity.

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The abstract, sentence by sentence

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The quantum stress tensor $<T_{\mu\nu}>$ is calculated in the 2+1 dimensional black hole found by Banados, Teitelboim, and Zanelli. The Greens function, from which $<T_{\mu\nu}> $ is derived, is obtained by the method of images. For the non-rotating black hole, it is shown that $<T_{\mu\nu}>$ is finite on the event horizon, but diverges at the singularity. For the rotating solution, the stress tensor is finite at the outer horizon, but diverges near the inner horizon. This suggests that the inner horizon is quantum mechanically unstable against the formation of a singularity.

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 quantum stress tensor $<T_{\mu\nu}>$ is calculated in the 2+1 dimensional black hole found by Banados, Teitelboim, and Zanelli.	contribution 0.98	0.84	0.04	asserted 1.00
s1	The Greens function, from which $<T_{\mu\nu}> $ is derived, is obtained by the method of images.	method 1.00	0.83	0.02	asserted 1.00
s2	For the non-rotating black hole, it is shown that $<T_{\mu\nu}>$ is finite on the event horizon, but diverges at the singularity.	result 1.00	0.89	0.14	asserted 1.00
s3	For the rotating solution, the stress tensor is finite at the outer horizon, but diverges near the inner horizon.	result 1.00	0.88	0.12	asserted 1.00
s4	This suggests that the inner horizon is quantum mechanically unstable against the formation of a singularity.	interpretation 1.00	0.60	0.81	hedged possibility 1.00

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

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References

Works cited (16)

[12]

In K. Shiraishi and T. Maki, “Vacuum Polarization aroun d a Three-Dimensional Black Hole” Akita Junior College preprint AJC-HEP-18, the Greens function for the Eu- clidean continuation of the non-rotating black hole was cal culated using a diﬀerent technique. Its Lorentzian continuation agrees with the Gre ens function derived here

View on Pith
[1]

M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849; M. Ba- nados, M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. D 48 (1993) 1506

View on Pith 1992
[2]

S. F. Ross and R. B. Mann, Phys. Rev. D 47 (1993) 3319; G.T. Horowitz and D.L. Welch, Phys. Rev. Lett. 71 (1993) 328; N. Kaloper, Phys. Rev. D 48 (1993) 2598; A. Achucarro and M. Ortiz, Phys. Rev. D 48 (1993) 3600; D. Cangemi, M. Leblanc, and R. B. Mann, Phys. Rev. D 48 (1993) 3606; C. Farina, J. Gamboa, and A.J. Segui-Santonja, preprint IPNO-TH 93/6

View on Pith 1993
[3]

Black holes in 3 + 1 anti-deSitter space have been discuss ed in S. W. Hawking and D. Page, Comm. Math. Phys. 87 (1983) 577

View on Pith 1983
[4]

For a review, see J. Harvey and A. Strominger, “Quantum As pects of Black Holes”, lectures given at the 1992 TASI Summer School

View on Pith 1992
[5]

M. Simpson and R. Penrose, Int. J. Theor. Phys. 7 (1973) 183; J. McNamara, Proc. Roy. Soc. Lond. A358 (1978) 499; R. Matzner, N. Zamorano, and V. Sandberg, Phys. Rev. D 19 (1979) 2821; N. Zamorano, Phys. Rev. D 26 (1982) 2564; S. Chandrasekhar and J. Hartle, Proc. Roy. Soc. Lond. A384 (1982) 301

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W. Hiscock, Phys. Lett. A 83 (1981) 110; E. Poisson and W. Israel, Phys. Rev. D 41 (1990) 1796; A. Ori, Phys. Rev. Lett. 67 (1991) 789; 68 (1991) 2117

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W. Hiscock, Phys. Rev. D 15 (1977) 3054; N. Birrell and P. Davies, Nature 272 (1978) 35

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R. Balbinot and E. Poisson, Phys. Rev. Lett. 70 (1993) 13; W. Anderson, P. Brady, W. Israel, and S. Morsink, Phys. Rev. Lett. 70 (1993) 1041; W. Israel, in Directions in General Relativity, vol. I, eds. B. Hu, M. Ryan Jr., and C. Vishveshwara, (Cambrid ge Press, 1993)

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S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, 1973)

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W. Hiscock and D. Konkowski, Phys. Rev. D 26 (1982) 1225; J. Grant, Phys. Rev. D 47 (1993) 2388

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B. O’Neill, Semi-Riemannian Geometry (Academic Press, New York, 1983). 10

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Forward citations

Cited by 12 Pith papers

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

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hep-th 2020-07 conditional novelty 7.0 of 10

An exact braneworld construction of a rotating quantum BTZ black hole yields a generalized entropy that satisfies the first law of thermodynamics to all orders in backreaction and curvature corrections.
Quantum Information Bound on the Energy
hep-th 2019-09 conditional novelty 7.0 of 10

A Quantum Penrose Inequality is conjectured, replacing trapped-surface area with lightsheet generalized entropy, and is tested against a semiclassical counterexample.
Quantum Black Holes as Holograms in AdS Braneworlds
hep-th 2002-06 conditional novelty 7.0 of 10

Classical brane-localized black holes in AdS braneworlds are conjectured to be dual to quantum-corrected black holes of a cutoff conformal field theory coupled to gravity, supported by parametric matches in 2+1 dimensions.
Eternal Black Holes in AdS
hep-th 2001-06 conditional novelty 7.0 of 10

The maximally extended AdS black hole is dual to two copies of the boundary CFT in the thermofield double entangled state, and summing over geometries restores unitarity.

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Quasinormal modes and quantum black hole interiors
hep-th 2026-09 conditional novelty 6.0 of 10

Asymptotic quasinormal mode spectra of quantum BTZ black holes reveal the singularity scaling exponent and detect transitions between interior Kasner eons.
Quantum Kerr-de Sitter black holes in three dimensions
hep-th 2023-03 accept novelty 6.0 of 10

Braneworld holography yields an exact three-dimensional quantum Kerr-de Sitter black hole with inner and outer horizons, a ring singularity, and a semi-classical first law.
Black holes in dS$_3$
hep-th 2022-07 accept novelty 6.0 of 10

Quantum backreaction of conformal fields generates a black hole horizon in three-dimensional de Sitter space, with a Nariai limit and thermodynamics governed by generalized entropy.
Holography of 3d Flat Cosmological Horizons
hep-th 2012-08 conditional novelty 6.0 of 10

A Cardy-like state count in a 2d Galilean conformal field theory reproduces the Bekenstein-Hawking entropy of a 3d flat cosmological horizon and matches the bulk first law of thermodynamics.
Strong cosmic censorship in the BTZ black hole
hep-th 2020-02 conditional novelty 5.0 of 10

The BTZ black hole's inner horizon becomes singular once next-order quantum backreaction is included, so strong cosmic censorship is not violated.
CFT descriptions of bulk local states in the AdS black holes
hep-th 2017-03 conditional novelty 5.0 of 10

A symmetry-based map from bulk points to boundary points, plus a dressing by Virasoro generators, reconstructs CFT states dual to local excitations anywhere inside a 3D AdS black hole.
Hawking radiation from AdS black holes
hep-th 2009-11 conditional novelty 5.0 of 10

A double Wick rotation maps Hartle-Hawking states on large AdS black holes to low-temperature states on the AdS soliton, explaining small stress tensors and yielding a phase diagram for M2-brane theory on BTZ.

How the literature uses it

32 arXiv papers cite this in 58 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 · 38 sentences without a confident label

Contests it 1 paper

These statements are based on a calculation of $⟨0|T_ab | 0 ⟩$ by Steif [ref] .

The BTZ black hole violates strong cosmic censorship · Introduction · contest 0.71

Uses its method or tool 4 papers

We can now obtain the renormalized quantum stress tensor $⟨T_μν⟩$ from $G(x,x')$ using the point-splitting method [ref] .

Black holes in dS$_3$ · Quantum stress tensor for a conical defect in $dS_3$ · 1.00
The renormalized quantum stress tensor $⟨T_μν⟩$ is obtain from $G(x,x')$ using the point-splitting method [ref] .

Quantum Kerr-de Sitter black holes in three dimensions · Renormalized quantum stress-tensor · 0.99
We then use the method of images [ref] to construct the BTZ Green function.

Horizon Instability of the Extremal BTZ Black Hole · Introduction · 1.00

Compares with it 3 papers

The rotating and charged holographic quantum black holes, meanwhile, do not match to the non-holographic perturbatively-corrected counterparts (cf. [ref] ).

Three-dimensional quantum black holes: a primer · Holographic quantum black holes: a conjecture · 0.92
Note that the schematic form of the quantum stress tensor agrees qualitatively with the result of [ref] .

Hawking radiation from AdS black holes · Rotating BTZ black hole · 1.00
This form is very similar to the one of a free conformal scalar field [ref] and for a holographic CFT dual to a four-dimensional bulk [ref] , [equation] On the other hand, we find that $fH_rr$ grows toward the AdS boundary and $fH_rr≃H_vv$ near $u=0$ by eq.

Semiclassical Einstein equations from holography and boundary dynamics · Static perturbations · 0.89

Cites it as background 9 papers

Semiclassical analyses employ quantum field theory on curved backgrounds to understand the stress tensor sourced by a point particle as it approaches the inner horizon [ref] .

Holographic Probes of Inner Horizons · Introduction · 0.91
A suitable coordinate for is [ref] ds^2=-(^2-ℓ^2) dt^2+ℓ^2 d^2/^2-ℓ^2+^2dϕ^2 where $t,ϕ∈(-∞,+∞)$.

O-BTZ: Orientifolded BTZ Black Hole · Introduction · 0.81
A quantized free scalar field has been extensively analyzed in various charged and rotating black hole backgrounds [ref] .

Semiclassical Rotating AdS Black Holes with Quantum Hair in Holography · Introduction · 0.97

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] [12]

In K. Shiraishi and T. Maki, “Vacuum Polarization aroun d a Three-Dimensional Black Hole” Akita Junior College preprint AJC-HEP-18, the Greens function for the Eu- clidean continuation of the non-rotating black hole was cal culated using a diﬀerent technique. Its Lorentzian continuation agrees with the Gre ens function derived here

View on Pith

[2] [1]

M. Banados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849; M. Ba- nados, M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. D 48 (1993) 1506

View on Pith 1992

[3] [2]

S. F. Ross and R. B. Mann, Phys. Rev. D 47 (1993) 3319; G.T. Horowitz and D.L. Welch, Phys. Rev. Lett. 71 (1993) 328; N. Kaloper, Phys. Rev. D 48 (1993) 2598; A. Achucarro and M. Ortiz, Phys. Rev. D 48 (1993) 3600; D. Cangemi, M. Leblanc, and R. B. Mann, Phys. Rev. D 48 (1993) 3606; C. Farina, J. Gamboa, and A.J. Segui-Santonja, preprint IPNO-TH 93/6

View on Pith 1993

[4] [3]

Black holes in 3 + 1 anti-deSitter space have been discuss ed in S. W. Hawking and D. Page, Comm. Math. Phys. 87 (1983) 577

View on Pith 1983

[5] [4]

For a review, see J. Harvey and A. Strominger, “Quantum As pects of Black Holes”, lectures given at the 1992 TASI Summer School

View on Pith 1992

[6] [5]

M. Simpson and R. Penrose, Int. J. Theor. Phys. 7 (1973) 183; J. McNamara, Proc. Roy. Soc. Lond. A358 (1978) 499; R. Matzner, N. Zamorano, and V. Sandberg, Phys. Rev. D 19 (1979) 2821; N. Zamorano, Phys. Rev. D 26 (1982) 2564; S. Chandrasekhar and J. Hartle, Proc. Roy. Soc. Lond. A384 (1982) 301

View on Pith 1973

[7] [6]

W. Hiscock, Phys. Lett. A 83 (1981) 110; E. Poisson and W. Israel, Phys. Rev. D 41 (1990) 1796; A. Ori, Phys. Rev. Lett. 67 (1991) 789; 68 (1991) 2117

View on Pith 1981

[8] [7]

W. Hiscock, Phys. Rev. D 15 (1977) 3054; N. Birrell and P. Davies, Nature 272 (1978) 35

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