Figure 29: Past light cone of an
event in the
spacetime (156)
of a plane gravitational wave. The
picture was produced with profile functions and . Then there is focusing
in the -direction and defocusing in the -direction. In the
(2+1)-dimensional picture, with the -coordinate not shown, the past
light cone is completely refocused into a single point , with the exception of one generator . 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 when they pass through the
first caustic. Taking the -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).
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