Globally stationary but locally static space-times: A gravitational analog of the Aharonov-Bohm effect

It is well known that gravitational fields may be locally the same but globally distinct due to differences in the topology of their underlying manifolds. Globally stationary but locally static gravitational fields provide an example of gravitational fields which are locally the same but globally distinct in spite of the homeomorphism of their underlying manifolds. Any static metric on a space-time manifold with nonvanishing first Betti number $R_1$ is shown to generate an $R_1$-parameter family of such solutions. These fields are seen to provide a gravitational analog of the electromagnetic Aharonov-Bohm effect. The exterior field of a rotating infinite cylinder of matter is discussed as an exactly soluble example.