|Spacetime Topology Change|
* Motivation: (1) In a path-integral formulation of quantum gravity, we would like to sum over all metrics, but also over all different topologies, interpolating between two given manifolds; (2) Possibility of creating monopole-antimonopole pairs; (3) Validity of the spin-stats theorem; (4) Possibility of getting fermions and internal symm multiplets in pure general relativity; (5) Allow second quantization of geons (consistency); (6) Quantum topology change at small scales would cost little action.
* Early ideas: Jordan (different spaces may unite–astronomically motivated).
* Criteria: We want to exclude models with infinite particle production, and possibly also those with double light cones; It might be possible to ensure this by requiring continuity of time as volume of past light cone.
* Mechanism: It would presumably be a quantum phenomenon, occurring only at microscopic scales, since there are no classical topology-changing solutions in general relativity; In quantum gravity one could get changing amplitudes for various topologies; One possibility however is through some modification of the Einstein equation.
* Controversy: DeWitt & Anderson, Castagnino, Dray & Manogue (not ok, infinite particle production in trousers); Sorkin et al (ok, but in higher dimensions some elementary cobordisms might not be equally suppressed).
Kinematics: Cobordism > s.a. models of topology change.
* Idea: We require the existence of an interpolating manifold between two given spatial geometries (topological cobordism), on which we can then put a metric (Riemannian, Lorentzian or causal cobordism).
* Topological: It always exists for pair creation (e.g., in 3-dimensions, creating two geons of the kind \(\mathbb R\)P2 # \(\mathbb R\)2 –non-orientable– or T2 # \(\mathbb R\)2–orientable); More generally it exists if the initial and final manifolds are cobordant, which happens iff their Stiefel-Whitney numbers are equal; One can use surgery to obtain the desired cobordism (allows Δχ = ±2 for n > 3), a cobordism is like a sequence of localized surgeries.
* Riemannian: Given a topological one, it is always possible.
* Lorentzian: If the manifold is time-orientable, it is possible only if we allow the metric to have closed timelike curves (likely to be very small, for dynamical reasons); Conditions: if ∂M = M1 ∪ M2, in even dimensions, χ(M) = 0; In odd dimensions, χ(M1) = χ(M2); There is no possible Lorentzian topology change in 0+1, 1+1 and 2+1 dimensions.
* Causal: We require no causality violations, but allow the metric to be singular (= 0) at isolated points:
- Pair creation: In even (> 1+1) dimensions it can always be obtained; In 4+1 Kaluza-Klein monopole-antimonopole pairs can be created (with non time-orientable metrics).
- Local causality structure: In 1+1 dimensions both future and past light cones of singular points are double; In 2+1 only one of them need split; In 3+1 neither.
@ General references: Treder AdP(62); Kreisel et al AdP(63) [degenerate]; Crampin PCPS(68); Antonelli & Williams IJTP(79) [and kink field theories]; Borde gq/94.
@ Degenerate metrics, causality: Horowitz CQG(91); Louko & Sorkin CQG(97)gq/95 [complex action]; Matschull CQG(96)gq/95; Borde et al CQG(99)gq.
Phenomenology > s.a. models of topology change; wormholes [scale-dependent
@ General references: Tanaka & Nagami IJGMP(13) [dark-matter production].
@ And quantum coherence: Coleman NPB(88); Lavrelashvili et al NPB(88).
@ And black-hole information, unitarity: Barbón & Rabinovici ht/05-conf; Hsu PLB(07)ht/06 [baby universes].
References > s.a. Cobordism; models
of topology change; spacetime foam.
@ Intros, reviews: in Sorkin in(90); Gibbons in(92)-a1110, in(93); Callender & Weingard SHPMP(00) [conceptual]; Dowker gq/02-proc; Asorey et al a1211.
@ General: Misner & Wheeler AP(57); Geroch JMP(67); Brill in(72); Yodzis CMP(72), GRG(73); Tipler PRL(76), AP(77); Lee PRS(78); Strominger PRL(84); Konstantinov & Melnikov CQG(86); Sorkin PRD(86) [conditions, and monople creation]; Anderson PLB(88); Banks NPB(88); De Ritis et al NCB(88); Visser PRD(90); Horowitz CQG(91); Gibbons & Hawking CMP(92), PRL(92); Borde gq/94; del Campo PRD(95); Konstantinov IJMPD(98)gq/95; Borowiec et al IJGMP(07) [Lagrangian formalism].
@ And causal continuity: Dowker & Surya PRD(98)gq/97; Dowker et al CQG(00)gq/99.
@ Related topics: Komorowski pr(71) [topology on superspace]; Joshi & Saraykar PLA(87) [and cosmic censorship]; Gibbons CQG(93) [and matter fields, skyrmions]; Maia IJMPCS(12)-a1211 [and the cosmological constant].
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