Theodore Frankel

Energy density and spatial curvature in general relativity

Positive energy density tends to limit the size of space. This effect is studied within several contexts. We obtain sufficient conditions (which involve the energy density in a crucial way) for the compactness of spatial hypersurfaces in space‐time. We then obtain some results concerning static or, more generally, stationary space‐times. The Schwarzchild solution puts an upper bound on the size of a static spherically symmetric fluid with density bounded from below. We derive a result of roughly the same nature which, however, requires no symmetry and allows for rotation. Also, we show that static or rotating universes with Λ = 0 require that the density along some spatial geodesic must fall off rapidly with distance from a point.

Electric currents in multiply connected spaces

It is shown that the topology of 3-space determines whether or not a current can be maintained in a wire loop.