Expansion of bundles of light rays in the Lema\^itre -- Tolman models

Abstract

The locus of $\theta \equiv {k^{\mu}};_{\mu} = 0$ for bundles of light rays emitted at noncentral points is investigated for Lema\^{\i}tre -- Tolman (L--T) models. The three loci that coincide for a central emission point: (1) maxima of $R$ along the rays, (2) $\theta = 0$, (3) $R = 2M$ are all different for a noncentral emitter. If an extremum of $R$ along a nonradial ray exists, then it must lie in the region $R > 2M$. In $2M < R \leq 3M$ it can only be a maximum; in $R > 3M$ both minima and maxima can exist. The intersection of (1) with the equatorial hypersurface (EHS) $\vartheta = \pi/2$ 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, $\theta$ has no zeros or two zeros along a ray, and becomes $+ \infty$ at the Big Crunch (BC). The only rays on which $\theta \to - \infty$ at the BC are the radial ones. Along rays on the boundaries between the no-zeros and the two-zeros regions $\theta$ has one zero, but still tends to $+ \infty$ at the BC. When the emitter is sufficiently close to the center, $\theta$ 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, $R = 2M$ is still the ultimate barrier behind which events become invisible from outside; loci (1) and (2) are not such barriers.