Spacetime topology
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Spacetime topology is the
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 sets are the image of open sets in .
Path or Zeeman topology
Definition:[1] The topology in which a subset is
It is the finest topology which induces the same topology as does on timelike curves.[2]
Properties
Strictly
A base for the topology is sets of the form for some point and some convex normal neighbourhood .
( denote the chronological past and future).
Alexandrov topology
The Alexandrov topology on spacetime, is the coarsest topology such that both and are open for all subsets .
Here the base of open sets for the topology are sets of the form for some points .
This topology coincides with the manifold topology if and only if the manifold is strongly causal but it is coarser in general.[3]
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 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
Planar spacetime
Events connected by light have zero separation. 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 of split-complex numbers:
- so that
- is the split-complex logarithm and the required homeomorphism F → R2, Note that b is the rapidity parameter for relative motion in F.
F is in
See also
- 4-manifold
- Clifford-Klein form
- Closed timelike curve
- Complex spacetime
- Geometrodynamics
- Gravitational singularity
- Hantzsche–Wendt_manifold
- Wormhole
Notes
- ^ Luca Bombelli website Archived 2010-06-16 at the Wayback Machine
- .
- ^ Penrose, Roger (1972), Techniques of Differential Topology in Relativity, CBMS-NSF Regional Conference Series in Applied Mathematics, p. 34
References
- .
- 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" (PDF). Journal of Mathematical Physics. 17 (2): 174–181. doi:10.1063/1.522874.