where they enjoy similar properties to those we have discussed. C. M. Hull has shown that such higher-dimensional pp-waves are essential building blocks for eleven-dimensional supergravity. Geometric and physical properties PP-waves enjoy numerous striking properties. Some of their more abstract mathematical properties have already been mentioned. In this section a few additional properties are presented. Consider an inertial observer in Minkowski spacetime who encounters a sandwich plane wave. Such an observer will experience some interesting optical effects. If he looks into the oncoming wavefronts at distant galaxies which have already encountered the wave, he will see their images undistorted. This must be the case, since he cannot know the wave is coming until it reaches his location, for it is traveling at the speed of light. However, this can be confirmed by direct computation of the optical scalars of the null congruence . Now suppose that after the wave passes, our observer turns about face and looks through the departing wavefronts at distant galaxies which the wave has not yet reached. Now he sees their optical images sheared and magnified (or demagnified) in a time-dependent manner. If the wave happens to be a polarized gravitational plane wave, he will see circular images alternately squeezed horizontally while expanded vertically, and squeezed vertically while expanded horizontally. This directly exhibits the characteristic effect of a gravitational wave in general relativity on light. The effect of a passing polarized gravitational plane wave on the relative positions of a cloud of (initially static) test particles will be qualitatively very similar. We might mention here that in general, the motion of test particles in pp-wave spacetimes can exhibit chaos. The fact that Einstein's field equation is nonlinear is well known. This implies that if you have two exact solutions, there is almost never any way to linearly superimpose them. PP waves provide a rare exception to this rule: if you have two PP waves sharing the same covariantly constant null vector (the same geodesic null congruence, i.e. the same wave vector field), with metric functions respectively, then gives a third exact solution. Roger Penrose has observed that near a null geodesic, every Lorentzian spacetime looks like a plane wave. To show this, he used techniques imported from algebraic geometry to "blow up" the spacetime so that the given null geodesic becomes the covariantly constant null geodesic congruence of a plane wave. This construction is called a