Suppose a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
contains a
closed timelike curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van St ...
(CTC). No CTC can be continuously deformed as a CTC (is
timelike homotopic) to a point, as that point would not be causally well behaved.
Therefore, any Lorentzian manifold containing a CTC is said to be
timelike multiply connected. A Lorentzian manifold that does not contain a CTC is said to be timelike simply connected.
Any Lorentzian manifold which is timelike multiply connected has a
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two man ...
universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
which is timelike simply connected. For instance, a three-sphere with a Lorentzian metric is timelike multiply connected, (because any compact Lorentzian manifold contains a CTC), but has a diffeomorphic universal covering space which contains no CTC (and is therefore not compact). By contrast, a three-sphere with the standard metric is simply connected, and is therefore its own universal cover.
References
Algebraic topology
Homotopy theory
Lorentzian manifolds
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