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
\(\mathbb R\)P^{2} # \(\mathbb R\)^{2}
–non-orientable– or T^{2}
# \(\mathbb R\)^{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 Δ*χ* = ±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* =
*M*_{1} ∪ *M*_{2},
in even dimensions, *χ*(*M*)
= 0; In odd dimensions, *χ*(*M*_{1})
= *χ*(*M*_{2});
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].

@ __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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feb 2016