Spacetime topology

Spacetime topology, the topological structure of spacetime, is a subject studied primarily in general relativity. This physical theory models gravitation as a 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 sets are the image of open sets in $mathbb\{R\}^n$.

Path or Zeeman topology

"Definition": [http://www.phy.olemiss.edu/%7Eluca/Topics/t/top_st.html Luca Bombelli website] ] The topology $ho$ in which a subset $E\; subset\; M$ is open if for every timelike curve $c$ there is a subset $O\; subset\; M$ such that $E\; cap\; c\; =\; O\; cap\; c$.

It is the finest topology which induces the same topology as $M$ does on timelike curves.

Properties

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 $I^+(p,U)\; cup\; I^-(p,U)\; cup\; p$ for some point $p\; in\; M$ and some convex normal neighbourhood $U\; subset\; M$.

($I^pm$ denote the chronological past and future).

Alexandrov topology

The Alexandrov topology, also called the interval topology, is defined in terms of the causality relations in the spacetime.

It is the coarsest topology such that $I^+(E)$ is open for all subsets $E\; subset\; M$.

Here the base of open sets for the topology are sets of the form $I^+(x)\; cap\; I^-(y)$ for some points $,x,y\; in\; M$.

This topology coincides with the manifold topology iff the manifold is strongly causal but in general it is coarser.

ee also

*Spacetime
*Causal structure
*Gravitational singularity

References

* E. C. Zeeman [http://dx.doi.org/10.1063/1.1704140 Causality Implies the Lorentz Group] J. Math. Phys. April 1964 Volume 5, Issue 4, pp. 490-493

* S. W. Hawking, A. R. King, P. J. McCarthy [http://dx.doi.org/10.1063/1.522874 A new topology for curved space–time which incorporates the causal, differential, and conformal structures] J. Math. Phys. February 1976 Volume 17, Issue 2, pp. 174-181