Figure 29

Figure 29: Past light cone of an event pO in the spacetime (156View Equation) of a plane gravitational wave. The picture was produced with profile functions f(u) > 0 and g(u) = 0. Then there is focusing in the x-direction and defocusing in the y-direction. In the (2+1)-dimensional picture, with the y-coordinate not shown, the past light cone is completely refocused into a single point q, with the exception of one generator c. It depends on the profile functions whether there is a second, third, and so on, caustic. In any case, the generators leave the boundary of the chronological past I-(pO) when they pass through the first caustic. Taking the y-coordinate into account, the first caustic is not a point but a parabola (“astigmatic focusing”) (see Figure 30). An electromagnetic plane wave (vanishing Weyl tensor rather than vanishing Ricci tensor) can refocus a light cone, with the exception of one generator, even into a point in 3+1 dimensions (“anastigmatic focusing”) (cf. Penrose [259] where a hand-drawing similar to the picture above can be found).