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. Examples There are many noteworthy explicit examples of pp-waves. ("Explicit" means that the metric functions can be written down in terms of elementary functions or perhaps well-known special functions such as Mathieu functions.) Explicit examples of axisymmetric pp-waves include The Aichelburg–Sexl ultraboost is an impulsive plane wave which models the physical experience of an observer who whizzes by a spherically symmetric gravitating object at nearly the speed of light, The Bonnor beam is an axisymmetric plane wave which models the gravitational field of an infinitely long beam of incoherent electromagnetic radiation. Explicit examples of plane wave spacetimes include exact monochromatic gravitational plane wave and monochromatic electromagnetic plane wave solutions, which generalize solutions which are well-known from weak-field approximation, exact solutions of the gravitational field of a Weyl fermion, the Schwarzschild generating plane wave, a gravitational plane wave which, should it collide head-on with a twin, will produce in the interaction zone of the resulting colliding plane wave solution a region which is locally isometric to part