Chapter 23: PP Waves
PP waves (plane-fronted parallel rays) are exact gravitational wave solutions representing waves with planar wave fronts. They're important in theoretical physics and provide exact solutions to Einstein's equations.
PP Wave Metric
\( ds^2 = 2du\,dv + H(u, x, y)du^2 + dx^2 + dy^2 \)
u, v are null coordinates, H is the profile function
Vacuum Condition
\( \nabla^2_{\perp} H = \frac{\partial^2 H}{\partial x^2} + \frac{\partial^2 H}{\partial y^2} = 0 \)
H is harmonic in transverse coordinates
Key Properties
Exact Solutions
Unlike linearized waves, pp waves are exact solutions to the full nonlinear Einstein equations.
Impulsive Waves
H(u) = δ(u)f(x,y) describes an impulsive gravitational wave — a sudden "shock" in spacetime.
Penrose Limit
Any spacetime, zoomed in near a null geodesic, becomes a pp wave. Universally important!