Spacetime Topology Change| Spacetime Topology Change |
In General
* 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, a Riemannian cobordism 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 topology].
@ General references: Tanaka & Nagami IJGMP(13) [dark-matter production];
Antoniou et al a1812 [and surgery, wormholes].
@ 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.
@ And cosmic censorship:
Joshi & Saraykar PLA(87);
Etesi a1905 [strong cosmic censorship violations].
@ Related topics: Komorowski pr(71) [topology on superspace];
Gibbons CQG(93) [and matter fields, skyrmions];
Maia IJMPCS(12)-a1211 [and the cosmological constant].
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send feedback and suggestions to bombelli at olemiss.edu – modified 10 jun 2019