Spacetime Topology |

**In General** > s.a. cosmic geometry;
spacetime boundaries [including compactification].

* __Manifold topology__:
The topology \(\cal M\) spacetime inherits from the manifold structure; Its global
structure can be studied using topological invariants, notably the Euler class and
the Pontryagin class.

* __Restrictions__: An even-dimensional
compact manifold without boundary with a Lorentz metric must have *χ*(*M*)
= 0; In 4D, this implies that the manifold is not simply connected.

> __Online resources__:
see Wikipedia page.

**Path or Zeeman Topology**

$ __Def__: The
topology \(\cal P\) in which *E* ⊂ *M* is
open iff for every timelike curve *c* there is an *O* ∈
\(\cal M\) such that *E* ∩ *c* = *O* ∩ *c*;
The finest which induces the same topology as \(\cal M\) does on timelike curves.

* __Properties__: Strictly finer
than the manifold topology, and therefore it is Hausdorff; Separable, but not
locally compact, not Lindelöf, not normal, and not first countable; There are
sequences of points in *M* which converge in the manifold topology (e.g.,
a sequence of distinct points on the light cone of *p* which converges
to *p*) but does not converge in the path topology.

* __Base__: Sets of the form
*I*^{ +}(*p*,* U*) ∪
*I*^{ –}(*p*,* U*)
∪ {*p*}, for convex normal neighborhoods *U*.

* __For Minkowski space__:
The open (closed) sets in \(\cal P\) are the subsets of *M* whose
intersections with all timelike geodesics and spacelike hyperplanes are open
(closed) in the natural topology on those subsets; Any null geodesic is discrete
(its points are isolated); The homeomorphism group is generated by the Poincaré
group and dilatations; The space is path connected, but not simply connected.

* __For curved spacetimes__:
The homeomorphism group is the group of all homothetic transformations.

* __Variation__: Fullwood's
topology, defined only in terms of causal structure, equivalent to \(\cal P\) iff
the distinguishing condition holds.

@ __For Minkowski space__: Zeeman JMP(64),
Top(67);
Whiston IJTP(72);
Dossena a0704-laurea,
JMP(07) [properties];
Sainz a0803-wd,
a1003-wd [criticism];
Dossena a1103 [constructive response to criticism].

@ __For curved spacetimes__: Göbel CMP(76), JMP(76);
Hawking et al JMP(76);
Fullwood JMP(92);
Struchiner & Rosa mp/05 [for
Kaluza-Klein and gauge theories]; Kim JMP(06);
Papadopoulos & Papadopoulos a1706 [three distinct Zeeman topologies], a1712 [more Zeeman topologies, and the Limit Curve Theorem]; Papadopoulos et al IJGMP(18)-a1710; > s.a. spacetime boundaries.

**Alexandrov (or Interval) Topology** > s.a. causality conditions [global hyperbolicity].

$ __Def__: The coarsest topology on *M* in which *I*^{ +}(*E*)
is open for all *E* ⊂ *M*.

* __Base__: In a full chronological
space, one is given by the Alexandrov neighborhoods {[*x*,* y*]} [@ Lerner
in(72)].

* __Special cases__: It coincides
with the manifold topology iff (*M*,* g*)
is strongly causal (in which case it is Hausdorff), but in general it is
coarser; In the discrete case it is often trivial, in the sense that it gives the discrete topology.

@ __And strong causality__: McWilliams IJTP(81);
Martin & Panangaden gq/04 [globally
hyperbolic case]; > s.a. causality.

**Topology of Space** > s.a. initial-value
formulation of general relativity; topology
at cosmological scales; topology
change.

* __Restrictions__: There are none on the spatial topology
for an asymptotically flat vacuum spacetime, although in most cases singularities will develop.

@ __References__: Isenberg et al AHP(03)gq/02 [vacuum].

**References** > s.a. boundaries
in field theory; cosmological models
in general relativity;
initial-value formulation; particle models.

@ __General__: Alonso & Ynduráin CMP(67);
Cel'nik SMD(68); Whiston IJTP(73),
IJTP(74),
IJTP(75);
Briginshaw IJTP(80);
Lee GRG(83);
Heathcote BJPS(88);
in Naber 88 (pr ch1);
Lester JMP(89),
Kirillov gq/94 [phenomenological description]; Acherjee et al a1710 [Euclidean topology, order topology from horismos, and global topological properties of spacetime manifolds];
Papadopoulos & Scardigli a1804-in [critical review].

@ __Dimension__: Barrow PTRS(83);
Mirman LNC(84);
Zeilinger & Svozil PRL(85);
Svozil & Zeilinger IJMPA(86);
Müller & Schäfer PRL(86);
Mirman IJTP(88);
Hochberg & Wheeler PRD(91);
NCA(91)469 [from wormholes];
Tegmark CQG(97)gq [from strings, anthropic];
Callender SHPMP(05) ["proofs" of 3-dimensionality];
Gersten FP(05) [proposed test of 4-dimensionality];
in Petkov 09;
> s.a. fractals in physics.

@ __Fundamental group__: Smith AJM(60),
PNAS(60);
Lee GRG(75).

@ __Special cases__: Lee CMB(75),
Yurtsever JMP(90) [compact];
Chamblin gq/95-conf
[singular, and causality]; > types of spacetimes.

@ __Phenomenology__: Cassa PAMS(93) [and geodesics];
Parfionov & Zapatrin gq/97,
Breslav et al HJ(99)qp [measurement, histories appproach];
> s.a. Detectors in Quantum Theory.

@ __Related topics__: in Steenrod 51, p207 [restrictions];
in Hawking & Ellis 73, 181-182 [orientability];
Kovár & Chernikava a1311
[causal sites, weakly causal topologies and their de Groot duals];
> s.a. stiefel-whitney classes.

> __Related topics__: see causal structures
[recovery of spacetime structure]; diffeomorphisms; lines [topology
on the space of causal lines/geodesics].

**Quantum Aspects** > s.a. observables in gravity;
quantum cosmology [sum over topologies];
quantum spacetime.

@ __Topology at Planck length__: Yetter ed-94;
Madore & Saeger CQG(98)gq/97.

@ __Scale-dependent topology__: Seriu PLB(93),
ViA(93);
> s.a. Coarse Structures in Geometry.

@ __Related topics__: Friedman in(91) [and quantum gravity];
Jonsson PLB(98)ht [2D, handle width];
Raptis et al IJTP(06)gq/05,
IJTP(06)gq/05 [tomographic histories approach];
Atyabi IJGMP(15)-a1412 [topology fluctuations and non-commutative spectral geometry, effect of matter].

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