{{Short description|Topological structure of 4D spacetime}} {{Spacetime|cTopic=Mathematics}} '''Spacetime topology''' is the [[Topological space|topological structure]] of [[spacetime]], a topic studied primarily in [[general relativity]]. This [[physical theory]] models [[gravitation]] as the [[curvature]] of a [[four dimensional]] [[Pseudo-Riemannian_manifold#Lorentzian_manifold|Lorentzian manifold]] (a spacetime) and the concepts of [[topology]] thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in [[physical cosmology]].
== Types of topology ==
There are two main types of topology for a spacetime ''M''.
=== Manifold topology ===
As with any manifold, a spacetime possesses a natural [[manifold]] topology. Here the [[open set]]s are the image of open sets in <math>\mathbb{R}^4</math>.
=== Path or Zeeman topology ===
''Definition'':<ref name="Bombelli">[http://www.phy.olemiss.edu/%7Eluca/Topics/t/top_st.html Luca Bombelli website] {{webarchive|url=https://web.archive.org/web/20100616043659/http://www.phy.olemiss.edu/%7Eluca/Topics/t/top_st.html |date=2010-06-16 }}</ref> The topology <math>\rho</math> in which a subset <math>E \subset M</math> is [[open (topology)|open]] if for every [[timelike curve]] <math>c</math> there is a set <math>O</math> in the manifold topology such that <math>E \cap c = O \cap c</math>.
It is the [[comparison of topologies|finest topology]] which induces the same topology as <math>M</math> does on timelike curves.<ref>{{cite journal|last1=Zeeman|first1=E.C.|authorlink=E. C. Zeeman|title=The topology of Minkowski space|journal=[[Topology (journal)|Topology]]|date= 1967|volume=6|issue=2|pages=161–170|doi=10.1016/0040-9383(67)90033-X|doi-access=}}</ref>
==== Properties ====
Strictly [[finer topology|finer]] than the manifold topology. It is therefore [[Hausdorff space|Hausdorff]], [[Separable (topology)|separable]] but not [[Locally compact space|locally compact]].
A [[Base (topology)|base]] for the topology is sets of the form <math>Y^+(p,U) \cup Y^-(p,U) \cup p</math> for some point <math>p \in M</math> and some convex normal neighbourhood <math>U \subset M</math>.
(<math>Y^\pm</math> denote the [[Causal structure#Causal structure|chronological past and future]]).
=== Alexandrov topology === {{more|Alexandrov topology}}
The Alexandrov topology on spacetime, is the [[Comparison of topologies|coarsest topology]] such that both <math>Y^+(E)</math> and <math>Y^-(E)</math> are open for all subsets <math>E \subset M</math>.
Here the [[Base (topology)|base]] of open sets for the topology are sets of the form <math>Y^+(x) \cap Y^-(y)</math> for some points <math>\,x,y \in M</math>.
This topology coincides with the manifold topology if and only if the manifold is [[Causality conditions#Strongly causal|strongly causal]] but it is coarser in general.<ref name="Penrose">{{Citation|last= Penrose |first= Roger|title=Techniques of Differential Topology in Relativity|year=1972|series=CBMS-NSF Regional Conference Series in Applied Mathematics|pages = 34}}</ref>
Note that in mathematics, an [[Alexandrov topology]] on a partial order is usually taken to be the coarsest topology in which only the upper sets <math>Y^+(E)</math> are required to be open. This topology goes back to [[Pavel Alexandrov]].
Nowadays, the correct mathematical term for the Alexandrov topology on spacetime (which goes back to [[Aleksandr Danilovich Aleksandrov|Alexandr D. Alexandrov]]) would be the '''interval topology''', but when Kronheimer and Penrose introduced the term this difference in nomenclature was not as clear{{citation needed|date=September 2017}}, and in physics the term Alexandrov topology remains in use.
==Planar spacetime== [[File:Simple_light_cone_diagram.svg|thumb|right|200px|Spacetime plane with here-now at A, B an event in the future (F), and C in space-right (D)]]
Events connected by light have a [[spacetime interval]] of zero. The plenum of spacetime in the plane is split into four quadrants, each of which has the topology of R<sup>2</sup>. The dividing lines are the trajectory of inbound and outbound photons at (0,0). The planar-cosmology topological segmentation is the future F, the past P, space left L, and space right D. The homeomorphism of F with R<sup>2</sup> amounts to [[polar decomposition#Alternative planar decomposions|polar decomposition]] of [[split-complex number]]s: :<math>z = \exp(a + j b) = e^a (\cosh b + j \sinh b) \to (a, b),</math> so that :<math>z \to (a, b)</math> is the split-complex logarithm and the required [[homeomorphism]] F → R<sup>2</sup>, Note that ''b'' is the [[rapidity]] parameter for relative motion in F.
F is in [[bijection|bijective correspondence]] with each of P, L, and D under the mappings ''z'' → –''z'', ''z'' → j''z'', and z → – j ''z'', so each acquires the same topology. The union U = F ∪ P ∪ L ∪ D then has a topology nearly covering the plane, leaving out only the [[null cone]] on (0,0). [[Hyperbolic rotation]] of the plane does not mingle the quadrants, in fact, each one is an [[invariant set]] under the [[unit hyperbola#Complex plane algebra|unit hyperbola group]].
==See also== * [[4-manifold]] * [[Clifford-Klein form]] * [[Closed timelike curve]] * [[Complex spacetime]] * [[Geometrodynamics]] * [[Gravitational singularity]] * [[Hantzsche-Wendt manifold]] * [[Spacetime curvature]] * [[Wormhole]]
== Notes == <references/>
== References == * {{cite journal|last1=Zeeman|first1=E. C.|authorlink=Christopher Zeeman|title=Causality Implies the Lorentz Group|journal=Journal of Mathematical Physics|date= 1964|volume=5|issue=4|pages=490–493|doi=10.1063/1.1704140|bibcode=1964JMP.....5..490Z}} * {{cite journal|last1=Hawking|first1=S. W.|last2=King|first2=A. R.|last3=McCarthy|first3=P. J.|title=A new topology for curved space–time which incorporates the causal, differential, and conformal structures|journal=Journal of Mathematical Physics|date=1976|volume=17|issue=2|pages=174–181|doi=10.1063/1.522874|bibcode=1976JMP....17..174H|url=https://authors.library.caltech.edu/11027/1/HAWjmp76.pdf}}
[[Category:General relativity]] [[Category:Lorentzian manifolds]]