REVIEW
Cosmological production of charged black hole pairs
General Relativity and Quantum Cosmology · Submitted 10 Apr 1995 · 12 references
Pith's verdict: Narrow but honest: the real new content is the C-metric-with-Λ regularity result and the identification of the Type I instanton as Euclidean Reissner-Nordström; the instanton taxonomy is Romans' and the authors say so.
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Referee report
The paper
The paper claims
The paper argues that a positive cosmological constant can pair-create charged black holes, and that the two instantons which describe this process yield pair-creation rates fixed entirely by the entropy of the horizons involved.
The reading
What carries the argument
The engine is the charged C metric with a cosmological constant, written in the standard accelerating coordinates with two structure functions $G(x)$ and $H(y)$, and the demand that the conical singularities at the two poles of the $(x,φ)$ section cancel. That cancellation requires $G'(x_3) = -G'(x_4)$, which the authors show can only be satisfied by taking $x_3 = x_4$, i.e. by sending the acceleration parameter to $A = √(Λ/3)$; a coordinate rescaling $x = ǫ cos θ$, $φ = φ/ǫ$ then turns the metric into Reissner–Nordström de Sitter with $V(r) = (1 - r_+/r)(1 - r_-/r) - Λr^2/3$. The same collapse happens in the second coordinate patch, at a critical acceleration $A_c$ given explicitly in the text. Having reduced the problem to Reissner–Nordström de Sitter, the argument is carried by the horizon structure: matching the surface gravities $κ_3 = κ_4$ of the black hole and cosmological horizons is what selects the lukewarm ($Q = M$) and Nariai ($r_3 = r_4$) cases, while a double root of $V(r)$ selects the cold case. The actions, computed from the Einstein–Maxwell action with the Gibbons–Hawking boundary term and, in the electric case, an extra $F^{μν}n_μ A_ν$ surface term needed to hold the charge fixed, are then compared with the de Sitter action to read off suppression factors and with the horizon areas to check the entropy relation.
Extended reading notes
Core claim
The paper's central claim is that the only non-singular versions of the charged C metric with a positive cosmological constant are the Reissner–Nordström de Sitter metric, and that this metric nonetheless supports two distinct instantons that can be read as black hole pair creation. The first is the lukewarm solution, in which the black hole horizon and the cosmological horizon share a common surface gravity (which forces charge equal to mass, $Q = M$); its Euclidean section has topology $S^2$ times $S^2$ and its action is $I_L = -3π/(2Λ) + πM/√(3Λ)$. The second is the cold solution, in which the inner and outer black hole horizons coincide so that the horizon sits at infinite proper distance; its Euclidean section has topology $S^2$ times $R^2$ and its action is $I_C = -πb^2/2$. In both cases the pair-creation rate $Γ ≈ e^{-2I}$ equals $e^{S}$ with $S = A/4$, the quarter-area law, and since de Sitter space itself has the lowest action ($-3π/2Λ$), every black hole pair-creation channel is suppressed relative to the creation of empty de Sitter space. The authors also note two further solutions, the charged Nariai and the ultracold cases, and observe that the charged Nariai instanton does not describe black hole pair creation because its Lorentzian section is simply $dS_2$ times a two-sphere.
Load-bearing premise
The whole rate calculation assumes that the Euclidean path integral is dominated by these classical instantons, and that for a closed cosmology the partition function $Z = Ψ^2$ is a density of states, so that the entropy is simply $S = -2I$.
Editorial extensions
If this is right
- Charged black hole pairs can be created by a positive cosmological constant alone, with a rate $Γ ≈ e^{S}$ set by one quarter of the total horizon area of the instanton.
- Because de Sitter space has the lowest action of all these solutions, every such pair-creation channel is suppressed relative to the spontaneous creation of empty de Sitter space.
- Pair creation of extreme charged black holes is suppressed relative to non-extreme pairs by a factor $e^{S_{bh}}$, where $S_{bh}$ is the entropy of the black hole horizon, supporting the reading of $e^{S_{bh}}$ as a count of internal black hole states.
- Extreme black hole horizons contribute nothing to the gravitational entropy even when they have non-zero area, since only horizons that appear on the Euclidean section enter the quarter-area count.
- The Type I instanton of the ordinary (zero cosmological constant) charged C metric is not a new solution at all: it is the Reissner–Nordström metric with the azimuthal coordinate playing the role of Euclidean time, which points to an interpretation as the decay of a three-dimensional Kaluza–Klein vacuum.
Reading between the lines
- If the quarter-area counting survives beyond the saddle point, then the pair-creation rate is a direct thermodynamic statement about horizon areas rather than a dynamical detail of the creation process, which would make the result robust against the details of how the pairs form.
- The parallel the authors draw between these instantons and the Ernst instantons of pair creation in an electromagnetic field suggests the qualitative structure — one extreme, one non-extreme, one degenerate 'Type I-like' solution — is a generic feature of pair creation by any agent that supplies negative potential energy, and could be tested in dilaton gravity with an effective cosmological consta
- The reidentification of the Type I instanton as Reissner–Nordström with a compactified time direction implies that the physical content of that solution lies in Kaluza–Klein vacuum decay rather than black hole pair creation, which would remove it from the catalogue of black hole creation channels and leave the cold and lukewarm instantons as the only genuine ones.
- Because the suppression is largest exactly where the black hole entropy is largest, a universe with a small cosmological constant should produce charged pairs at a rate that is exponentially sensitive to the horizon areas, which is in principle a distinguishable signature if the cosmological constant is treated as a fixed background parameter.
Original abstract
We investigate the pair creation of charged black holes in a background with a positive cosmological constant. We consider $C$ metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner-Nordstr\"om de Sitter metric. We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of non-extreme black holes from the Reissner-Nordstr\"om de Sitter metric, and calculate their actions. There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. We also observe that the type I instanton in the ordinary $C$ metric with zero cosmological constant is actually the Reissner-Nordstr\"om solution.
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.
The abstract, sentence by sentence
We investigate the pair creation of charged black holes in a background with a positive cosmological constant. We consider $C$ metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner-Nordstr\"om de Sitter metric. We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of non-extreme black holes from the Reissner-Nordstr\"om de Sitter metric, and calculate their actions. There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. We also observe that the type I instanton in the ordinary $C$ metric with zero cosmological constant is actually the Reissner-Nordstr\"om solution.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the pair creation of charged black holes in a background with a positive cosmological constant. | contribution 0.88 | 0.91 | 0.07 | asserted 0.73 |
| s1 | We consider $C$ metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner-Nordstr\"om de Sitter metric. | result 0.80 | 0.92 | 0.24 | asserted 1.00 |
| s2 | We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of non-extreme black holes from the Reissner-Nordstr\"om de Sitter metric, and calculate their actions. | contribution 0.93 | 0.96 | 0.23 | asserted 1.00 |
| s3 | There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. | interpretation 0.60 | 0.80 | 0.08 | asserted 0.97 |
| s4 | We also observe that the type I instanton in the ordinary $C$ metric with zero cosmological constant is actually the Reissner-Nordstr\"om solution. | result 0.97 | 0.80 | 0.08 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:gr-qc/9504015 by the authors.
References
How the literature uses it
37 arXiv papers cite this in 76 sentences
Papers citing it · 45 sentences without a confident label
Builds on it 1 paper
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Thus our work generalizes the scenario studied in [ref] to include leading quantum corrections.
Quantum Corrections to Pair Production of Charged Black Holes in de Sitter Space · Introduction · 1.00
Uses its method or tool 5 papers
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In order to estimate the PBH nuclear rate, we follow the gravitational instanton approach proposed by Mann & Ross [ref] and Bousso & Hawking [ref] .
Primordial black holes in a dimensionally reduced universe · Introduction · 1.00
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To evaluate the boundary action, we choose the gauge for $A_(1)$ [ref] A_(1)=i√(π)gℓ_0/2√(G)(6_*-π)Ψ_ esinχψdχ .
Quantum Corrections to Pair Production of Charged Black Holes in de Sitter Space · Purely electric case · 0.93
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A suitable gauge choice that is regular everywhere on the Euclidean geometry is [ref] [equation] Further, we choose coordinates on the boundary $Σ$ such that it is located at the surface $τ=0,β/2$ and the coordinate $r$ runs from the black hole horizon to the cosmological horizon.
Action, entropy and pair creation rate of charged black holes in de Sitter space · On-shell Euclidean action · 0.96
Compares with it 3 papers
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Our results are in qualitative agreement with the classical computation [ref] .[Notably, the results of [ref] differ from those found by Mann and Ross [ref] .
Nucleation of charged quantum de-Sitter$_{3}$ black holes · Nucleation of charged quantum-dS$_3$ black holes · 0.99
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We wish to compare the relative rate of production of the above bubbles of nothing to another well known instability in de Sitter space–the nucleation of charged pairs of de Sitter black holes [ref] .
New instabilities of de Sitter spacetimes · Comparison with black hole production · 1.00
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Below we specialize to these true instanton cases and compare our results for the on-shell action to those in the literature, especially to the results in the comprehensive paper by Mann and Ross [ref] .
Action, entropy and pair creation rate of charged black holes in de Sitter space · Extreme cases: cold, lukewarm, Nariai and ultracold instantons · 0.98
Cites it as background 16 papers
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In the standard treatment, instantons describing the pair production of spherically symmetric charged black holes in de Sitter space correspond to smooth configurations of Euclidean charged de Sitter black holes, including the lukewarm, cold and charged Nariai solutions [ref] .
Quantum Corrections to Pair Production of Charged Black Holes in de Sitter Space · Introduction · 0.86
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First, for RNdS$_4$ there are three different extremal limits [ref] , dubbed cold (inner and outer horizon coincide), Nariai (outer and cosmological horizon coincide) and ultracold (confluence of three horizons) configurations.
Near-Extremal Limits of de Sitter Black Holes · Introduction · 0.82
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The Bousso-Hawking normalization requires a careful re-examination of the thermodynamical laws for black holes in de Sitter space[The thermodynamics of de Sitter black holes has been previously studied in [ref] .
Limits on the Statistical Description of Charged de Sitter Black Holes · Introduction · 0.98
Discussion
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