Spacetime topology

Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional 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. Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional 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. There are two main types of topology for a spacetime M. As with any manifold, a spacetime possesses a natural manifold topology. Here the open sets are the image of open sets in R 4 {displaystyle mathbb {R} ^{4}} . Definition: The topology ρ {displaystyle ho } in which a subset E ⊂ M {displaystyle Esubset M} is open if for every timelike curve c {displaystyle c} there is a set O {displaystyle O} in the manifold topology such that E ∩ c = O ∩ c {displaystyle Ecap c=Ocap c} . It is the finest topology which induces the same topology as M {displaystyle M} does on timelike curves. Strictly finer than the manifold topology. It is therefore Hausdorff, separable but not locally compact. A base 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 pin M} and some convex normal neighbourhood U ⊂ M {displaystyle Usubset M} . ( Y ± {displaystyle Y^{pm }} denote the chronological past and future). The Alexandrov topology on spacetime, is the coarsest topology such that both Y + ( E ) {displaystyle Y^{+}(E)} and Y − ( E ) {displaystyle Y^{-}(E)} are open for all subsets E ⊂ M {displaystyle Esubset M} .