Cosmological Multi-Black Hole Solutions

David Kastor; Jennie Traschen

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Cosmological Multi-Black Hole Solutions

David Kastor, Jennie Traschen

High Energy Physics - Theory · Submitted 4 Dec 1992

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

The authors' main claim We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant $\Lambda$.

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What the abstract itself states, as probabilities: claims first 0.88 · states limitation 0.73 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.94 · 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.

We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant $\Lambda$. In the limit $\Lambda=0$, these solutions reduce to the well known Majumdar-Papapetrou (MP) solutions. Like the MP solutions, each black hole in a $\Lambda >0$ solution has charge $Q$ equal to its mass $M$, up to a possible overall sign. Unlike the $\Lambda = 0$ limit, however, solutions with $\Lambda >0$ are highly dynamical. The black holes move with respect to one another, following natural trajectories in the background deSitter spacetime. Black holes moving apart eventually go out of causal contact. Black holes on approaching trajectories ultimately merge. To our knowledge, these solutions give the first analytic description of coalescing black holes. Likewise, the thermodynamics of the $\Lambda >0$ solutions is quite interesting. Taken individually, a $|Q|=M$ black hole is in thermal equilibrium with the background deSitter Hawking radiation. With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined. In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established.

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	We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant $\Lambda$.	contribution 1.00	0.96	0.58	asserted 1.00
s1	In the limit $\Lambda=0$, these solutions reduce to the well known Majumdar-Papapetrou (MP) solutions.	result 0.70	0.32	0.06	asserted 1.00
s2	Like the MP solutions, each black hole in a $\Lambda >0$ solution has charge $Q$ equal to its mass $M$, up to a possible overall sign.	result 0.97	0.68	0.11	asserted 0.87
s3	Unlike the $\Lambda = 0$ limit, however, solutions with $\Lambda >0$ are highly dynamical.	result 0.90	0.75	0.18	asserted 1.00
s4	The black holes move with respect to one another, following natural trajectories in the background deSitter spacetime.	result 0.93	0.81	0.12	asserted 1.00
s5	Black holes moving apart eventually go out of causal contact.	result 0.95	0.77	0.06	asserted 1.00
s6	Black holes on approaching trajectories ultimately merge.	result 0.99	0.79	0.11	asserted 1.00
s7	To our knowledge, these solutions give the first analytic description of coalescing black holes.	contribution 0.53	0.79	0.34	asserted 0.92
s8	Likewise, the thermodynamics of the $\Lambda >0$ solutions is quite interesting.	interpretation 0.43	0.47	0.05	asserted 0.95
s9	Taken individually, a $\|Q\|=M$ black hole is in thermal equilibrium with the background deSitter Hawking radiation.	result 0.81	0.41	0.05	asserted 1.00
s10	With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined.	result 0.69	0.74	0.06	asserted 1.00
s11	In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established.	result 0.72	0.77	0.07	asserted 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · 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

The paper

Original abstract

We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant $\Lambda$. In the limit $\Lambda=0$, these solutions reduce to the well known Majumdar-Papapetrou (MP) solutions. Like the MP solutions, each black hole in a $\Lambda >0$ solution has charge $Q$ equal to its mass $M$, up to a possible overall sign. Unlike the $\Lambda = 0$ limit, however, solutions with $\Lambda >0$ are highly dynamical. The black holes move with respect to one another, following natural trajectories in the background deSitter spacetime. Black holes moving apart eventually go out of causal contact. Black holes on approaching trajectories ultimately merge. To our knowledge, these solutions give the first analytic description of coalescing black holes. Likewise, the thermodynamics of the $\Lambda >0$ solutions is quite interesting. Taken individually, a $|Q|=M$ black hole is in thermal equilibrium with the background deSitter Hawking radiation. With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined. In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established.

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References

Forward citations

Cited by 6 Pith papers

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

Thermodynamics of horizons: de Sitter black holes and reentrant phase transitions
hep-th 2015-07 conditional novelty 5.0 of 10

A single Gibbs free energy built from the three horizons of a de Sitter black hole reveals a small/large/small reentrant phase transition for six-dimensional rotating de Sitter black holes.
Shadows of Multi-Black Holes: Analytic Exploration
gr-qc 2012-08 conditional novelty 5.0 of 10

Analytic estimates show that the eyebrow-like fringes outside a multi-black-hole shadow scale inversely with the hole separation, and that each shadow is distorted into an ellipse by its neighbor.
On black hole formation in higher dimensions
gr-qc 2020-09 conditional novelty 4.0 of 10

In higher-dimensional general relativity, rotating black holes cannot form via gravitational collapse or accretion because no stable circular orbits exist, whereas pure Lovelock gravity avoids this obstruction.
Supersymmetry of the static Reissner-Nordstr\"om black hole in Bertotti-Robinson ($\mathrm{AdS}_2 \times \mathbb{S}^2$)
hep-th 2026-06 unverdicted novelty 3.0 of 10

The static Reissner-Nordström black hole embedded in Bertotti-Robinson (AdS₂ × S²) is supersymmetric in N=2 D=4 supergravity, saturates the BPS bound, and yields mass and thermodynamic relations with a cosmological-co...
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...

Show 1 more citing papers

De Sitter Musings
hep-th 2012-05 accept novelty 3.0 of 10

A survey of de Sitter physics arguing that the static-patch observer and the far-future-boundary metaobserver are genuinely distinct problems, and that most open questions in the field come from conflating them.

How later papers used it

51 later papers cite it

Evidence: substantial. 49 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 74 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 and 1 used its method or tool.

Builds on it 1 of 2 papers with this label

Built on this paper's result describing Majumdar-Papapetrou spacetime embedded in a universe with a positive cosmological constant to derive an analogous solution for the ModMax case.

Exact multiblack hole spacetimes in Einstein-ModMax theory (2025) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 3 papers with this label

Used the generation technique proposed in this paper to find an exact solution for κ=1/8.

An instability of the Reissner-Nordstrom solution and new hairy black holes in d=5 dimensions (2011) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

41 arXiv papers cite this in 60 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 · 45 sentences without a confident label

Builds on it 1 paper

This configuration describes a non-Abelian deformation of the extremal RN-de Sitter black holes in [ref] .

Asymptotically flat, stable black hole solutions in Einstein--Yang-Mills--Chern-Simons theory · Introduction · 0.84

Uses its method or tool 3 papers

We close the discussion here by noticing that for $κ=1/8$, by using the generation technique proposed in [ref] , one finds the following exact solution [equation] where $F(r,t)^-1= 1+ e^-2t/L(Q-2/r^2+2(r^2+J)/(r^2+J/2)^2)$ and $Λ=6/L^2$.

An instability of the Reissner-Nordstrom solution and new hairy black holes in d=5 dimensions · $Λ>0$: de Sitter solutions · 0.97
For $α=0$, this form of the solution is used to construct cosmological configurations by using the prescription in [ref] .], any progress in this direction appears to require a separate numerical study of these solutions.

Einstein-Gauss-Bonnet black holes in de Sitter spacetime and the quasilocal formalism · Reissner–Nordström–dS-GB black hole · 0.84
By using the methods in [ref] , [ref] , one finds that ([ref]) shares the basic features of the $J=0$ Abelian configuration.

Asymptotically flat, stable black hole solutions in Einstein--Yang-Mills--Chern-Simons theory · Introduction · 0.91

Compares with it 3 papers

As in the case of the Kastor-Traschen solution [ref] , this spacetime is expected to describe a collision of black holes provided the background universe is contracting ($t_0<0$).

Black Hole in the Expanding Universe from Intersecting Branes · Multi black holes in the time-dependent universe · 0.96
Yet, one faces the same situation in the Kastor-Traschen solution [ref] , where the black hole collision can be described by multicenter harmonics.

Black Hole in the Expanding Universe with Arbitrary Power-Law Expansion · Preliminaries · 0.93
This is reminiscent of the Kastor-Traschen solution [ref] in the Einstein-Maxwell-$Λ(>0)$ system.

Static spacetimes haunted by a phantom scalar field III: asymptotically (A)dS solutions · Gibbons solution in dS · 0.97

Cites it as background 7 papers

Kastor and Traschen have found a family of exact solutions to the Einstein-Maxwell equations with a cosmological constant representing an arbitrary number of charged $Q=M$ black holes in an otherwise closed universe [ref] .

Geometric Horizons in the Kastor-Traschen Multi Black Hole Solutions · Introduction · 0.97
One of such solutions in five-dimensional Einstein-Maxwell theory with a positive cosmological constant is the Kastor-Traschen solution(5DKT) [ref] .

Topology Changing Process of Coalescing Black Holes on Eguchi-Hanson Space · Coalescing Black Holes in Five Dimensions · 0.94
The singularities on spacetime that appear due to the existence of time dependence in Einstein-Maxwell theories were studied from several points of view in [ref] .

Violation of cosmic censorship in dynamical $p$-brane systems · Introduction · 0.98

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