Жалпы салыстырмалылықтағы PP толқындары

In general relativity, the pp wave spacetimes, or pp waves for short, are an important family of exact solutions of Einstein's field equation. The term pp stands for plane fronted waves with parallel propagation, and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt.

Since they constitute a very simple and natural class of Lorentzian manifolds, defined in terms of a null congruence, it is not very surprising that they are also important in other relativistic classical field theories of gravitation. In particular, pp waves are exact solutions in the Brans–Dicke theory,
various higher curvature theories and Kaluza–Klein theories, and certain gravitation theories of J. W. Moffat. Indeed, B. O. J. Tupper has shown that the common vacuum solutions in general relativity and in the Brans/Dicke theory are precisely the vacuum pp waves (but the Brans/Dicke theory admits further wavelike solutions). Hans Jürgen Schmidt has reformulated the theory of (four dimensional) pp waves in terms of a two dimensional metric dilaton theory of gravity. Pp waves also play an important role in the search for quantum gravity, because as Gary Gibbons has pointed out, all loop term quantum corrections vanish identically for any pp wave spacetime. This means that studying tree level quantizations of pp wave spacetimes offers a glimpse into the yet unknown world of quantum gravity. It is natural to generalize pp waves to higher dimensions, 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.

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 Penrose limit. Penrose also pointed out that in a pp wave spacetime, all the polynomial scalar invariants of the Riemann tensor vanish identically, yet the curvature is almost never zero. This is because in four dimension all pp waves belong to the class of VSI spacetimes. Such statement does not hold in higher dimensions since there are higher dimensional pp waves of algebraic type II with non vanishing polynomial scalar invariants. If you view the Riemann tensor as a second rank tensor acting on bivectors, the vanishing of invariants is analogous to the fact that a nonzero null vector has vanishing squared length. Penrose was also the first to understand the strange nature of causality in pp sandwich wave spacetimes. He showed that some or all of the null geodesics emitted at a given event will be refocused at a later event (or string of events). The details depend upon whether the wave is purely gravitational, purely electromagnetic, or neither. Every pp wave admits many different Brinkmann charts. These are related by coordinate transformations, which in this context may be considered to be gauge transformations. In the case of plane waves, these gauge transformations allow us to always regard two colliding plane waves to have parallel wavefronts, and thus the waves can be said to collide head on. This is an exact result in fully nonlinear general relativity which is analogous to a similar result concerning electromagnetic plane waves as treated in special relativity.