Expansion of bundles of light rays in the lemaître-tolman models.

The locus of θ = def k μ ; μ = 0 for bundles of light rays emitted at noncentral points is investigated for Lemaître-Tolman (L-T) models. The three loci that coincide for a central emission point: (1) maxima of R along the rays, (2) θ = 0, (3) R = 2 M are all different for a noncentral emitter. If an extremum of R along a nonradial ray exists, then it must lie in the region R > 2 M. In 2 M < R ≤ 3 M it can only be a maximum; in R > 3 M both minima and maxima can exist. The intersection of (1) with the equatorial hypersurface (EHS) ϑ = π/2 is numerically determined for an exemplary toy model (ETM), for two typical emitter locations. The equation of (2) is derived for a general L-T model, and its intersection with the EHS in the ETM is numerically determined for the same two emitter locations. Typically, θ has no zeros or two zeros along a ray, and becomes +∞ at the Big Crunch (BC). The only rays on which θ → -∞ at the BC are the radial ones. Along rays on the boundaries between the no-zeros and the two-zeros regions θ has one zero, but still tends to +∞ at the BC. When the emitter is sufficiently close to the center, θ has 4 or 6 zeros along some rays (resp. 3 or 5 on the boundary rays). For noncentral emitters in a collapsing L-T model, R = 2 M is still the ultimate barrier behind which events become invisible from outside; loci (1) and (2) are not such barriers. [ABSTRACT FROM AUTHOR]