# Spacetime topology

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Topological structure of 4D spacetime

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**Spacetime topology** is the [topological structure](/source/Topological_space) of [spacetime](/source/Spacetime), a topic studied primarily in [general relativity](/source/General_relativity). This [physical theory](/source/Physical_theory) models [gravitation](/source/Gravitation) as the [curvature](/source/Curvature) of a [four dimensional](/source/Four_dimensional) [Lorentzian manifold](/source/Pseudo-Riemannian_manifold#Lorentzian_manifold) (a spacetime) and the concepts of [topology](/source/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](/source/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](/source/Manifold) topology. Here the [open sets](/source/Open_set) are the image of open sets in R 4 {\displaystyle \mathbb {R} ^{4}} .

### Path or Zeeman topology

*Definition*:[1] The topology ρ {\displaystyle \rho } in which a subset E ⊂ M {\displaystyle E\subset M} is [open](/source/Open_(topology)) if for every [timelike curve](/source/Timelike_curve) c {\displaystyle c} there is a set O {\displaystyle O} in the manifold topology such that E ∩ c = O ∩ c {\displaystyle E\cap c=O\cap c} .

It is the [finest topology](/source/Comparison_of_topologies) which induces the same topology as M {\displaystyle M} does on timelike curves.[2]

#### Properties

Strictly [finer](/source/Finer_topology) than the manifold topology. It is therefore [Hausdorff](/source/Hausdorff_space), [separable](/source/Separable_(topology)) but not [locally compact](/source/Locally_compact_space).

A [base](/source/Base_(topology)) for the topology is sets of the form Y + ( p , U ) ∪ Y − ( p , U ) ∪ p {\displaystyle Y^{+}(p,U)\cup Y^{-}(p,U)\cup p} for some point p ∈ M {\displaystyle p\in M} and some convex normal neighbourhood U ⊂ M {\displaystyle U\subset M} .

( Y ± {\displaystyle Y^{\pm }} denote the [chronological past and future](/source/Causal_structure#Causal_structure)).

### Alexandrov topology

Further information: [Alexandrov topology](/source/Alexandrov_topology)

The Alexandrov topology on spacetime, is the [coarsest topology](/source/Comparison_of_topologies) such that both Y + ( E ) {\displaystyle Y^{+}(E)} and Y − ( E ) {\displaystyle Y^{-}(E)} are open for all subsets E ⊂ M {\displaystyle E\subset M} .

Here the [base](/source/Base_(topology)) of open sets for the topology are sets of the form Y + ( x ) ∩ Y − ( y ) {\displaystyle Y^{+}(x)\cap Y^{-}(y)} for some points x , y ∈ M {\displaystyle \,x,y\in M} .

This topology coincides with the manifold topology if and only if the manifold is [strongly causal](/source/Causality_conditions#Strongly_causal) but it is coarser in general.[3]

Note that in mathematics, an [Alexandrov topology](/source/Alexandrov_topology) on a partial order is usually taken to be the coarsest topology in which only the upper sets Y + ( E ) {\displaystyle Y^{+}(E)} are required to be open. This topology goes back to [Pavel Alexandrov](/source/Pavel_Alexandrov).

Nowadays, the correct mathematical term for the Alexandrov topology on spacetime (which goes back to [Alexandr D. Alexandrov](/source/Aleksandr_Danilovich_Aleksandrov)) would be the **interval topology**, but when Kronheimer and Penrose introduced the term this difference in nomenclature was not as clear[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*], and in physics the term Alexandrov topology remains in use.

## Planar spacetime

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](/source/Spacetime_interval) of zero. The plenum of spacetime in the plane is split into four quadrants, each of which has the topology of R2. 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 R2 amounts to [polar decomposition](/source/Polar_decomposition#Alternative_planar_decomposions) of [split-complex numbers](/source/Split-complex_number):

- z = exp ⁡ ( a + j b ) = e a ( cosh ⁡ b + j sinh ⁡ b ) → ( a , b ) , {\displaystyle z=\exp(a+jb)=e^{a}(\cosh b+j\sinh b)\to (a,b),} so that

- z → ( a , b ) {\displaystyle z\to (a,b)} is the split-complex logarithm and the required [homeomorphism](/source/Homeomorphism) F → R2, Note that *b* is the [rapidity](/source/Rapidity) parameter for relative motion in F.

F is in [bijective correspondence](/source/Bijection) 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](/source/Null_cone) on (0,0). [Hyperbolic rotation](/source/Hyperbolic_rotation) of the plane does not mingle the quadrants, in fact, each one is an [invariant set](/source/Invariant_set) under the [unit hyperbola group](/source/Unit_hyperbola#Complex_plane_algebra).

## See also

- [4-manifold](/source/4-manifold)

- [Clifford-Klein form](/source/Clifford-Klein_form)

- [Closed timelike curve](/source/Closed_timelike_curve)

- [Complex spacetime](/source/Complex_spacetime)

- [Geometrodynamics](/source/Geometrodynamics)

- [Gravitational singularity](/source/Gravitational_singularity)

- [Hantzsche-Wendt manifold](/source/Hantzsche-Wendt_manifold)

- [Spacetime curvature](/source/Spacetime_curvature)

- [Wormhole](/source/Wormhole)

## Notes

1. **[^](#cite_ref-Bombelli_1-0)** [Luca Bombelli website](http://www.phy.olemiss.edu/%7Eluca/Topics/t/top_st.html) [Archived](https://web.archive.org/web/20100616043659/http://www.phy.olemiss.edu/%7Eluca/Topics/t/top_st.html) 2010-06-16 at the [Wayback Machine](/source/Wayback_Machine)

1. **[^](#cite_ref-2)** [Zeeman, E.C.](/source/E._C._Zeeman) (1967). "The topology of Minkowski space". *[Topology](/source/Topology_(journal))*. **6** (2): 161–170. [doi](/source/Doi_(identifier)):[10.1016/0040-9383(67)90033-X](https://doi.org/10.1016%2F0040-9383%2867%2990033-X).

1. **[^](#cite_ref-Penrose_3-0)** Penrose, Roger (1972), *Techniques of Differential Topology in Relativity*, CBMS-NSF Regional Conference Series in Applied Mathematics, p. 34

## References

- [Zeeman, E. C.](/source/Christopher_Zeeman) (1964). "Causality Implies the Lorentz Group". *Journal of Mathematical Physics*. **5** (4): 490–493. [Bibcode](/source/Bibcode_(identifier)):[1964JMP.....5..490Z](https://ui.adsabs.harvard.edu/abs/1964JMP.....5..490Z). [doi](/source/Doi_(identifier)):[10.1063/1.1704140](https://doi.org/10.1063%2F1.1704140).

- Hawking, S. W.; King, A. R.; McCarthy, P. J. (1976). ["A new topology for curved space–time which incorporates the causal, differential, and conformal structures"](https://authors.library.caltech.edu/11027/1/HAWjmp76.pdf) (PDF). *Journal of Mathematical Physics*. **17** (2): 174–181. [Bibcode](/source/Bibcode_(identifier)):[1976JMP....17..174H](https://ui.adsabs.harvard.edu/abs/1976JMP....17..174H). [doi](/source/Doi_(identifier)):[10.1063/1.522874](https://doi.org/10.1063%2F1.522874).

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Adapted from the Wikipedia article [Spacetime topology](https://en.wikipedia.org/wiki/Spacetime_topology) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Spacetime_topology?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.