REVIEW
Regular Black Holes and Topology Change
General Relativity and Quantum Cosmology · Submitted 20 Dec 1996 · 32 references
Pith's verdict: The structural claim is right and worth having on record, but the compactness step in the proof is a sketch, not a proof.
T0 · AI review
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Referee report
The paper
The paper claims
Borde proves that a singularity-free black hole obeying the weak energy condition must force spatial topology to change from open to closed, so such a spacetime cannot be globally hyperbolic.
The reading
What carries the argument
The engine is the focusing inequality for the expansion $θ$ of the null geodesic generators of the light cone $E^+(T)$: $θ̇ ≤ -½θ^2 - R_{ab}N^aN^b ≤ -½θ^2$. The second inequality is where the null convergence condition enters. Integrating it forces $θ → -∞$ at a finite affine parameter along every generator, so each generator must leave $E^+(T)$ within finite affine distance, which makes $E^+(T)$ compact. The second ingredient is assumption D, future causal simplicity, which identifies the light cone with the full boundary of the future of $T$. A boundary of a future has no edge, so the compact light cone is already the advertised compact slice. The pair of steps — focusing closes the light cone, causal simplicity promotes it to a slice — is what carries the whole argument.
Extended reading notes
Core claim
The claim is that singularity avoidance in black holes is not a local patch but a global restructuring. Borde isolates the mechanism in the Bardeen black hole family: inside the horizon both families of future-directed null geodesics from a trapped surface converge, but they converge to two different $r=0$ origins placed antipodally, so the light cone closes up as a 3-sphere instead of terminating on a singularity. That closure is what lets the spacetime escape Penrose's 1965 theorem, which needs the trapped surface's light cone to have an edge. Borde then shows the escape is not optional. Assume a spacetime (A) contains an eventually future-trapped surface $T$, meaning the divergence of the orthogonal null geodesics becomes negative somewhere to the future along each one; (B) obeys $R_{ab}N^aN^b$ for all null $N^a$, which follows from the weak energy condition plus Einstein's equation; (C) is future null-geodesically complete; and (D) is future causally simple, so $E^+(X) = J^+(X)$-boundary for every set $X$. Then there is a compact slice to the causal future of $T$. Applied to a black hole that starts from a noncompact asymptotic region, the theorem says the spatial slices must evolve from open to closed, and therefore that no global Cauchy hypersurface exists.
Load-bearing premise
The load-bearing premise is assumption D, future causal simplicity: that a point's future light cone is the entire boundary of its future, so that a compact light cone counts as an edgeless slice. The paper concedes that without some causality assumption, Gödel-based scenarios avoid both singularities and topology change.
Editorial extensions
If this is right
- A regular black hole obeying the weak energy condition cannot be globally hyperbolic, so any initial-value formulation built on a global Cauchy surface must break down somewhere inside the horizon.
- The topology change is located concretely: it happens on a compact slice to the causal future of the trapped surface, not somewhere unspecified.
- Known prohibitions on topology change are not violated, because they assume causal compactness or stronger conditions that an open-to-closed transition fails.
- A proposal that removes the singularity while keeping the weak energy condition and causal simplicity is committed to predicting a topology change, not merely a modified metric function near the origin.
- Models built by gluing a de Sitter core into a Schwarzschild interior inherit the same global structure as the Bardeen black holes and fall under the same conclusion.
Reading between the lines
- The theorem converts singularity avoidance into a global consistency check for model builders: anyone patching a regular core into a black hole interior should verify causal simplicity, since the argument needs it to turn a closed light cone into a slice.
- Read as a trade-off, the result suggests a black hole can be singularity-free or admit a well-posed global initial value problem, but not both under these energy and causality assumptions.
- The topology change lies behind the horizon, so no exterior observation distinguishes these models from Schwarzschild; the difference would surface only in the interior or in global quantum state considerations.
- Because the convergence premise fails wherever the weak energy condition is violated, semiclassical negative energy offers a possible route to regular black holes that avoids topology change altogether.
Original abstract
The conditions are clarified under which regular (i.e., singularity-free) black holes can exist. It is shown that in a large class of spacetimes that satisfy the weak energy condition the existence of a regular black hole requires topology change.
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
The conditions are clarified under which regular (i.e., singularity-free) black holes can exist. It is shown that in a large class of spacetimes that satisfy the weak energy condition the existence of a regular black hole requires topology change.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The conditions are clarified under which regular (i.e., singularity-free) black holes can exist. | contribution 0.66 | 0.75 | 0.11 | asserted 0.90 |
| s1 | It is shown that in a large class of spacetimes that satisfy the weak energy condition the existence of a regular black hole requires topology change. | result 1.00 | 0.89 | 0.84 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
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- R cross-paper relations
Figures
Figures reproduced from arXiv:gr-qc/9612057 by the authors.
References
How the literature uses it
65 arXiv papers cite this in 74 sentences
Papers citing it · 36 sentences without a confident label
Cites it as support 2 papers
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Lastly, the QNFs of Kerr and Taub-NUT BHs never cross the barrier established by the QNFs of Schwarzschild BHs owing to the topological protection [ref] .
Phase diagrams of quasinormal frequencies for Schwarzschild, Kerr, and Taub-NUT black holes · Conclusions · 0.96
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Although the monopole sources are unusual in that the associated Lagrangians are somewhat ad-hoc, they indeed produce reasonable stress energy tensors satisfying the weak energy condition (WEC) [ref] .
The classical double copy of non-singular black holes · Introduction · 0.99
Cites it as background 35 papers
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There are many publications discussing properties of different models of non-singular evaporating black holes (see e.g. [ref] ).
Quantum radiation from an evaporating non-singular black hole · Introduction · 0.99
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Therefore, since the seminal work of Bardeen [ref] , investigations of regular black holes, that is, black holes free of curvature singularities [ref] , have attracted considerable attention in both classical and quantum gravity frameworks [ref] .
Finite Curvature Construction of Regular Black Holes and Quasinormal Mode Analysis · Introduction · 0.88
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There are many well-known models of regular black holes [ref] [ref] .
Constructing a counterexample to the black hole complementarity · Frolov, Markov, and Mukhanov's model · 0.99
Discussion
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