The Off-Shell Black Hole

C. Teitelboim; S. Carlip

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The Off-Shell Black Hole

S. Carlip, C. Teitelboim

General Relativity and Quantum Cosmology · Submitted 1 Dec 1993

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Summing over all horizon areas yields the black hole entropy.

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The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wave function on this new degree of freedom is governed by an extended Wheeler-DeWitt equation. Summing over all horizon areas yields the black hole entropy.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black hole topology, the opening angle at the horizon and the horizon area are canonical conjugates.	background 0.97	0.10	0.22	asserted 1.00
s1	It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity.	result 0.96	0.82	0.23	asserted 1.00
s2	The dependence of the wave function on this new degree of freedom is governed by an extended Wheeler-DeWitt equation.	contribution 0.45	0.63	0.12	asserted 1.00
s3	Summing over all horizon areas yields the black hole entropy.	result 0.68	0.70	0.41	asserted 1.00

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

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The paper

Original abstract

The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wave function on this new degree of freedom is governed by an extended Wheeler-DeWitt equation. Summing over all horizon areas yields the black hole entropy.

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How later papers used it

50 later papers cite it

Evidence: substantial. 48 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 24 of the 73 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

2 later papers by other authors are listed below: 1 built on it, 1 used its method or tool and 1 compared its own results with it. 1 of them appears under more than one label.

Builds on it 1 of 2 papers with this label

Said that the Lewkowycz-Maldacena procedure is a direct generalization of the Carlip-Teitelboim approach to black hole entropy to cases that break the U(1) symmetry.

One-loop universality of holographic codes (2019) · 0.83 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 2 papers with this label

Performed, in the spirit of this paper, a Hamilton-Jacobi-like variation with respect to the defect angle boundary condition.

One-loop universality of holographic codes (2019) · 0.93 · paraphrased: the licence does not allow quoting

Compares with it 1 of 3 papers with this label

Found an expression exactly matching this paper's, derived using the metric formalism, with the boundary term as a horizon Hamiltonian times an evolution parameter.

Horizon energy as the boost boundary term in general relativity and loop gravity (2012) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

46 arXiv papers cite this in 71 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 · 55 sentences without a confident label

Builds on it 2 papers

In particular, this confirms that the Lewkowycz-Maldacena procedure is a direct generalization of the Carlip-Teitelboim approach to black hole entropy [ref] to cases that break the $U(1)$ symmetry of [ref] .

One-loop universality of holographic codes · Introduction · 0.83
The latter form in particular gives a standard description of field evolution, and so evolves the matter perturbation together with the perturbative gravitational field corresponding to its dressing in standard field theory fashion.[While Lorentzian evolution for black holes has been considered previously, see e.g. [ref] , the treatment outlined here extends to more general slicings than Schwarzschild, and including the black hole interior.]

Perturbative quantum evolution of the gravitational state and dressing in general backgrounds · Other specific examples · 0.81

Uses its method or tool 1 paper

In the spirit of [ref] , we may then perform a Hamilton-Jacobi-like variation with respect to the defect angle boundary condition.

One-loop universality of holographic codes · Reformulation and higher derivative corrections · 0.93

Compares with it 3 papers

Calling $A$ the area of $M_2$ with the orientation induced by $Σ_0$, [equation] we find [equation] This is exactly the expression derived by Carlip and Teitelboim in [ref] using the metric formalism: the boundary term $I_2$ is the product of a horizon Hamiltonian times an evolution parameter $η$.

Horizon energy as the boost boundary term in general relativity and loop gravity · Introduction · 0.99
Since the local $$ symmetry is a new feature, we briefly comment on the relation between this symmetry and the more familiar boost symmetry of Carlip and Teitelboim [ref] .

Local subsystems in gauge theory and gravity · Algebra of surface symmetries · 0.98
A very similar equation can also be found in (3+1)-dimensions, where the area of a surface equates to the generator of boosts into the orthogonal direction, see [ref] .

Conformal boundary conditions, loop gravity and the continuum · Covariant phase space, Virasoro generators, vanishing central charge · 0.86

Cites it as support 2 papers

Although one would at first sight think that a conical space represents non-equilibrium,[And thus one cannot infinitesimally vary the temperature to nearby equilibrium configurations, as one is instructed to do in thermodynamical calculations.] it can be proven that if one only varies the temperature and hence fixes the transverse horizon area [ref] [ref] , one indeed gets these conical spaces.

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We also argue that this form of the flux is preferred from the perspective of gluing subregions together in the gravitational path integral [ref] .

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Cites it as background 8 papers

We refer to [ref] for the precise definitions of these quantities.

Wave function of the quantum black hole · Introduction · 0.97
(Closely related formulas had been proposed earlier for the entropy of a black hole [ref] .) The can be written in terms of the response of the quantum effective action to a conical singularity at the entangling surface: CWintro S_ent = -lim_→0 ( 2π/ + 1 ) W_E,, where $$ is the deficit angle associated with the conical singularity, and $W_E,$ is the Euclidean quantum effective action in the presence of the conical singularity.

Renormalization of Entanglement Entropy and the Gravitational Effective Action · Introduction · 0.83
Historically, such a conical geometry is called a `squashed cone' [ref] since it has a discrete group of transformations $τ→τ+2πk$ for integer $k$, opposed to spacetimes with conical singularities such that $𝒞$ is a fixed point set of a global $U(1)$ isometry (as in the case of stationary black holes, e.g., [ref] ).[A simple physical example distinguishing a squashed cone from a $U(1)$ symmetric cone is spherical versus planar Rindler space.

Cornering gravitational entropy · Replicated manifolds and squashed cones: a brief history · 0.95

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