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 RP2 # R2 –nonorientable– or
T2 # R2–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 
= 
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, (M)
= 0; In odd dimensions, (M1)
= (M2);
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); 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.
References > s.a. Cobordism; models
of topology change; spacetime foam; wormholes [scale-dependent
topology].
@ Intros, reviews: in Sorkin in(90); Gibbons in(93); Callender & Weingard SHPMP(00)
[conceptual]; Dowker gq/02-in.
@ 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]; Borde pr(87); Anderson PLB(88);
Banks NPB(88);
De Ritis et al NCB(88); Visser PRD(90);
Horowitz CQG(91);
Gibbons & Hawking CMP(92),
PRL(92);
del Campo PRD(95);
Konstantinov IJMPD(98)gq/95;
Borowiec et al IJGMP(07)
[Lagrangian formalism].
@ And quantum coherence: Coleman NPB(88); Lavrelashvili et al NPB(88).
@ 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].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jun 2008