REVIEW
Dynamic wormholes, anti-trapped surfaces, and energy conditions
General Relativity and Quantum Cosmology · Submitted 17 Feb 1998 · 28 references
Pith's verdict: The definitional move is the whole paper, and it's a good one; the NEC theorems follow cleanly once you accept it.
Why Pith says this
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Referee report
The paper
The paper claims
A wormhole throat is where light stops converging, and that forces matter to break the null energy condition.
The reading
What carries the argument
The Raychaudhuri equation for null geodesic congruences is the engine: it relates the rate of change of the expansion $theta$ along the congruence to the shear squared, the twist squared, the Ricci tensor contracted with the null tangent, and torsion terms. Because a throat is defined by $theta = 0$ and $dtheta/du ge 0$, and because $sigma_{ab}sigma^{ab}$ and the twist-squared term are non-negative, the sign of $T_{ab}l^a l^b$ is pinned down on the throat. The companion twist equation is used to show that an initially hypersurface-orthogonal congruence stays twist-free, which is what licenses dropping the twist at the throat and, separately, what lets torsion act as a source of twist away from it.
Extended reading notes
Core claim
The paper's central claim is that the null energy condition is generically violated at and near wormhole throats, with no assumption of staticity, symmetry, asymptotic flatness, or nontrivial topology. The authors define a throat as a marginally anti-trapped surface: a closed spatial two-surface on which the expansion $theta$ of one orthogonal future-directed null congruence vanishes and satisfies $dtheta/du ge 0$, the generalized flare-out condition. Substituting $theta = 0$ into the Raychaudhuri equation gives $dtheta/du + sigma_{ab}sigma^{ab} = -8pi T_{ab}l^a l^b$ in the torsion-free case, so a simply flared throat requires $T_{ab}l^a l^b le 0$, a strongly flared throat requires strict violation somewhere on the surface, and a strongly anti-trapped throat requires it everywhere. Averaged versions of flare-out give sign-averaged violations over the surface, and $N$-fold degenerate throats push the same conclusion onto higher odd derivatives of the expansion. Two further results ride on the same machinery: with totally antisymmetric torsion the torsion contribution to the relevant combination is a positive semidefinite square, so torsion aggravates rather than absorbs the violation at the throat; and in a conformally expanding Morris-Thorne example, choosing the conformal factor to suspend the violation forces $dtheta/du le 0$, which removes the flare-out and hence the throat.
Load-bearing premise
The whole theorem rests on defining a wormhole throat as a marginally anti-trapped surface, and that identification is a definitional choice argued for by Page's suggestion, the static-limit coalescence, and worked examples, rather than derived from something more basic.
Editorial extensions
If this is right
- Any traversable wormhole, however it evolves in time, must have matter that violates the null energy condition at or near its throat; the violation is not a feature of static models.
- A dynamic wormhole generally has two distinct throats, one for each orthogonal null congruence, and these merge into a single throat only in the static limit.
- Adding a Kalb-Ramond-type torsion field does not let you move the energy-condition violation out of the ordinary stress-energy sector, because the torsion contribution to the null-null projection is positive semidefinite.
- Schemes that suspend the energy-condition violation temporarily in a time-dependent wormhole also destroy the flare-out of the extremal surface, so the throat ceases to be a throat.
- The center of a wormhole, defined on a fixed time slice, is not the same place as the throat; only in the static limit do they coincide.
Reading between the lines
- If the definition is accepted, the energy-condition requirement becomes a theorem about light-bundle kinematics rather than about any particular matter model, which suggests searching for wormhole solutions should start from the flare-out condition rather than from an assumed stress-energy ansatz.
- The result is local and pointwise, so it should compose with global no-go results such as topological censorship; a useful check would be to see whether every known dynamic wormhole solution satisfies the inequality at its numerically located extremal surface.
- Because the torsion contribution enters as a square, one could ask whether any non-Riemannian extension whose extra field enters the Raychaudhuri equation with indefinite sign could genuinely evade the conclusion, which would identify the precise structural feature that makes torsion fail to help.
Original abstract
Adapting and extending a suggestion due to Page, we define a wormhole throat to be a marginally anti-trapped surface, that is, a closed two-dimensional spatial hypersurface such that one of the two future-directed null geodesic congruences orthogonal to it is just beginning to diverge. Typically a dynamic wormhole will possess two such throats, corresponding to the two orthogonal null geodesic congruences, and these two throats will not coincide, (though they do coalesce into a single throat in the static limit). The divergence property of the null geodesics at the marginally anti-trapped surface generalizes the ``flare-out'' condition for an arbitrary wormhole. We derive theorems regarding violations of the null energy condition (NEC) at and near these throats and find that, even for wormholes with arbitrary time-dependence, the violation of the NEC is a generic property of wormhole throats. We also discuss wormhole throats in the presence of fully antisymmetric torsion and find that the energy condition violations cannot be dumped into the torsion degrees of freedom. Finally by means of a concrete example we demonstrate that even temporary suspension of energy-condition violations is incompatible with the flare-out property of dynamic throats.
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
Adapting and extending a suggestion due to Page, we define a wormhole throat to be a marginally anti-trapped surface, that is, a closed two-dimensional spatial hypersurface such that one of the two future-directed null geodesic congruences orthogonal to it is just beginning to diverge. Typically a dynamic wormhole will possess two such throats, corresponding to the two orthogonal null geodesic congruences, and these two throats will not coincide, (though they do coalesce into a single throat in the static limit). The divergence property of the null geodesics at the marginally anti-trapped surface generalizes the ``flare-out'' condition for an arbitrary wormhole. We derive theorems regarding violations of the null energy condition (NEC) at and near these throats and find that, even for wormholes with arbitrary time-dependence, the violation of the NEC is a generic property of wormhole throats. We also discuss wormhole throats in the presence of fully antisymmetric torsion and find that the energy condition violations cannot be dumped into the torsion degrees of freedom. Finally by means of a concrete example we demonstrate that even temporary suspension of energy-condition violations is incompatible with the flare-out property of dynamic throats.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Adapting and extending a suggestion due to Page, we define a wormhole throat to be a marginally anti-trapped surface, that is, a closed two-dimensional spatial hypersurface such that one of the two future-directed null geodesic congruences orthogonal to it is just beginning to diverge. | contribution 0.91 | 0.73 | 0.05 | asserted 1.00 |
| s1 | Typically a dynamic wormhole will possess two such throats, corresponding to the two orthogonal null geodesic congruences, and these two throats will not coincide, (though they do coalesce into a single throat in the static limit). | result 0.71 | 0.46 | 0.05 | asserted 0.68 |
| s2 | The divergence property of the null geodesics at the marginally anti-trapped surface generalizes the ``flare-out'' condition for an arbitrary wormhole. | interpretation 0.79 | 0.55 | 0.07 | asserted 1.00 |
| s3 | We derive theorems regarding violations of the null energy condition (NEC) at and near these throats and find that, even for wormholes with arbitrary time-dependence, the violation of the NEC is a generic property of wormhole throats. | result 0.99 | 0.97 | 0.70 | asserted 1.00 |
| s4 | We also discuss wormhole throats in the presence of fully antisymmetric torsion and find that the energy condition violations cannot be dumped into the torsion degrees of freedom. | result 0.79 | 0.91 | 0.06 | asserted 1.00 |
| s5 | Finally by means of a concrete example we demonstrate that even temporary suspension of energy-condition violations is incompatible with the flare-out property of dynamic throats. | result 0.93 | 0.95 | 0.19 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
37 arXiv papers cite this in 72 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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We discuss our recent work proving that the violation of the null energy condition (NEC) is a generic feature of all wormholes, whether they be time-dependent or static, and demonstrate that time-dependent wormholes have two throats, one for each direction through the wormhole, which coalesce only in the static limit.
General Dynamic Wormholes and Violation of the Null Energy Condition · 0.93
Papers citing it · 46 sentences without a confident label
Contests it 1 paper
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In so doing, we are going to pinpoint a key fact in Riemann-Cartan spacetimes that has been missed in Ref. [ref] and that led the authors to such an incomplete conclusion about the NEC for wormholes in Riemann-Cartan spacetimes.
Revisiting wormhole energy conditions in Riemann-Cartan spacetimes and under Weyl transformations · Introduction · contest 0.87
Builds on it 1 paper
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The no- statement for can be further generalized in a few respects [ref] : (i) our inferences remain valid in the presence of any other ghost-free (respecting the NEC) matter; (ii) instead of asymptotic flatness, we can simply require the existence of two spatial infinities (defined in terms of infinitely growing areas of closed 2-surfaces in the spatial sections of and ); (iii) the no- statement holds not only in STT but in any metric theory of gravity (e.g., $F(R)$ or $F(R, Φ)$ [ref] ) admitting a conformal mapping of the physical manifold to where GR holds with ghost-free matter, provided that the conformal factor is everywhere smooth and positive; (iv) under certain restrictions, our statement can even be extended to dynamic based on the no- results for obtained by Hochberg and Visser in [ref] .
Alexei Starobinsky and wormhole physics · Throats and asympotic regions · 0.92
Uses its method or tool 1 paper
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We identify the throat following the definition introduced by Hochberg and Visser [ref] for the case of dynamical wormholes.
A new solution for a generalized cosmological wormhole · Specific model for the generalized cosmological wormhole · 0.98
Compares with it 2 papers
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Suffice it to mention here that after computing explicitly the expansions $θ_±$ using the Morris-Thorne metric in the original frame and then substituting in our conditions ([ref]) one gets exactly the same conditions obtained in Ref. [ref] by starting right away with the conformal frame.
Revisiting black holes and wormholes under Weyl transformations · Hochberg-Visser wormhole · 0.99
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However, it does not describe a dynamical wormhole throat of the kind described in e.g. [ref] .
Black bounces to traversable wormholes from pure gravity in four and higher dimensions · Dynamic wormholes and black bounces · 0.93
Cites it as background 13 papers
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However, the discovery that, to be sustained, wormholes require the violation of the null energy condition (NEC), and hence the presence of the so-called “exotic” matter [ref] , pushed these objects further into the realm of pure theoretical curiosities.
Revisiting wormhole energy conditions in Riemann-Cartan spacetimes and under Weyl transformations · Introduction · 0.84
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As Hochberg and Visser [ref] have remarked that to define a wormhole, one identifies a local property that is operative near its throat through the expansion of the null geodesic congruence propagating orthogonally to a closed two - dimensional spatial hypersurface, generalizing in this way the Morris - Thorne analysis [ref] .
Light Dragging Phenomenon and Expanding Wormholes · INTRODUCTION · 0.81
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However, for dynamical wormholes there are several throat definitions that arise from imposing conditions on the expansion of ingoing or outgoing null geodesics and their derivatives [ref] .
Geometric surfaces: An invariant characterization of spherically symmetric black hole horizons and wormhole throats · Introduction · 0.88
Discussion
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