Figure 25

Figure 25: Past light cone of an event $p O$ in the spacetime of a transparent string of finite radius $ρ∗$ with $k = 0.8$ and $a = 0$ . The metric (133 ) is matched at $ρ = ρ ∗$ to an interior metric, and light rays are allowed to pass through the interior region. The perspective is analogous to Figure 24. The light rays which were blocked by the string in the non-transparent case now form a caustic. In the (2 + 1)-dimensional picture the caustic consists of two lightlike curves that meet in a swallow-tail point (see Figure 26 for a close-up). Taking the $z$ -dimension into account, the caustic actually consists of two lightlike 2-manifolds (fold surfaces) that meet in a spacelike curve (cusp ridge). The third picture in Figure 2 shows the situation projected to 3-space. Each of the past-oriented lightlike geodesics that form the caustic first passes through the cut locus (transverse self-intersection), then smoothly slips over one of the fold surfaces. The fold surfaces are inside the chronological past $− I (pO)$ , the cusp ridge is on its boundary.