Null geodesics and embedding diagrams of the interior Schwarzschild-de Sitter spacetimes with uniform density - art. no. 044004

Null geodesics and embedding diagrams of central planes in the ordinary spa ce geometry and the optical reference geometry of the interior Schwarzschil d-de Sitter spacetimes with uniform density are studied. For completeness, both positive and negative values of the cosmological constant are consider ed. The null geodesics are restricted to the central planes of these spacet imes, and their properties can be reflected by an "effective potential." If the interior spacetime is extremely compact, the effective potential has a local maximum corresponding to a stable circular null geodesic around whic h bound null geodesics are concentrated. The upper limit on the size of the interior spacetimes containing bound null geodesics is R = 3M, independent ly of the value of the cosmological constant. The embedding diagrams of the central planes of the ordinary geometry into three-dimensional Euclidean s pace are well defined for the complete interior of all spacetimes with a re pulsive cosmological constant, but the planes cannot be embedded into the E uclidean space in the case of spacetimes with subcritical values of an attr active cosmological constant. On the other hand, the embedding diagrams of the optical geometry are well defined for all of the spacetimes, and the tu rning points of these diagrams correspond to the radii of the circular null geodesics. All the embedding diagrams, for both the ordinary and optical g eometry, are smoothly matched to the corresponding embedding diagrams of th e external vacuum Schwarzschild-de Sitter spacetimes.