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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics