Advanced Topics and Applications
Exact solutions, colliding waves, and connections to string theory
Plane Waves: The Homogeneous Case
The most important subclass of PP waves consists of plane waves, for which the profile function \( H \) is a homogeneous quadratic polynomial in the transverse coordinates. A general plane wave metric takes the form:
$$ds^2 = 2\,du\,dv + H_{ij}(u)\,x^i x^j\,du^2 + \delta_{ij}\,dx^i dx^j$$
where \( H_{ij}(u) \) is a symmetric traceless \( 2 \times 2 \) matrix
The tracelessness condition \( H_{ii} = 0 \) is precisely the vacuum Einstein equation\( \nabla^2_\perp H = 0 \). Plane waves enjoy several remarkable properties not shared by general PP waves:
1. Exact solvability: The geodesic equation reduces to a linear ODE with\( u \)-dependent coefficients. This means the geodesic motion can, in principle, be solved exactly for any choice of \( H_{ij}(u) \). The solution is expressed through a\( 2 \times 2 \) matrix \( P^i_{\ j}(u) \) satisfying the Jacobi equation:
$$\ddot{P}^i_{\ j}(u) + H^i_{\ k}(u)\,P^k_{\ j}(u) = 0, \qquad P(u_0) = \mathbb{1}, \quad \dot{P}(u_0) = 0$$
The Jacobi propagator determines the transverse geodesic deviation
2. Maximal symmetry: Plane waves possess a (4+1)-parameter Heisenberg symmetry group. The Killing vectors are:
$$\xi_a = f_a^i(u)\,\partial_i - f_a^i(u)\,\dot{f}_{ai}(u)\,x^i\,\partial_v, \qquad \partial_v$$
Five Killing vectors forming a Heisenberg algebra: \( [\xi_a, \xi_b] = \omega_{ab}\,\partial_v \)
3. Causal structure: Plane waves are causally trivial — they are globally hyperbolic and have the same causal structure as Minkowski space. There are no horizons, no closed timelike curves, and no singularities (unless the profile \( H_{ij} \)itself is singular).
Sandwich Waves
A sandwich wave is a PP wave whose profile function has compact support in \( u \): the spacetime is flat for \( u < u_1 \) and\( u > u_2 \), with a gravitational wave "sandwich" for\( u_1 \leq u \leq u_2 \). This models a localized gravitational wave burst.
$$H(u, x, y) = \begin{cases} 0 & u < u_1 \\ h(u, x, y) & u_1 \leq u \leq u_2 \\ 0 & u > u_2 \end{cases}$$
A sandwich wave: flat → curved → flat, modeling a gravitational wave burst
The power of sandwich waves lies in their clean initial and final states. In the flat regions, we can unambiguously define inertial observers, making physical quantities like the memory effect and velocity kick well-defined. The scattering matrix relating the incoming flat region to the outgoing flat region encodes all observable effects:
$$\begin{pmatrix} x_\text{out} \\ \dot{x}_\text{out} \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\begin{pmatrix} x_\text{in} \\ \dot{x}_\text{in} \end{pmatrix}$$
The ABCD transfer matrix is symplectic: \( AD - BC = \mathbb{1} \)
A particularly important special case is the impulsive wave, where the burst duration shrinks to zero: \( H(u) = h(x,y)\,\delta(u) \). In this limit the transfer matrix simplifies dramatically: the \( A \) block becomes the identity (no displacement) while the \( C \) block gives an instantaneous velocity kick proportional to \( \partial_i h \). The Aichelburg-Sexl solution discussed on the previous page is the canonical example.
Colliding PP Waves: The Khan-Penrose Solution
One of the most fascinating applications of PP waves is the study of head-on collisions of gravitational waves. When two PP waves traveling in opposite directions meet, the interaction region is governed by the full nonlinear Einstein equations — there is no superposition principle.
The setup involves two impulsive plane waves approaching each other. Before the collision, the spacetime consists of four regions: a flat region (before either wave), two single-wave regions, and the interaction region. The metric in the interaction region is obtained by solving the characteristic initial value problem. Khan and Penrose (1971) found the exact solution for two colliding impulsive plane waves:
$$ds^2 = -2\,e^{-M}\,du\,dv + \frac{(1 - u^2)(1 - v^2)\left[(1 - u^2 v^2)\,dx - 2uv\sqrt{(1-u^2)(1-v^2)}\,dy\right]^2}{(1 - u^2 v^2)^2}$$
$$+ \frac{(1 - u^2)(1 - v^2)\left[2uv\sqrt{(1-u^2)(1-v^2)}\,dx + (1 - u^2 v^2)\,dy\right]^2}{(1 - u^2 v^2)^2}$$
The Khan-Penrose metric in the interaction region (schematic form)
The most dramatic feature of this solution is the formation of a curvature singularity in the interaction region. Two perfectly smooth, finite-amplitude gravitational waves, upon collision, produce a spacetime singularity! This singularity forms on a spacelike surface and is a focusing singularity: the mutual gravitational attraction of the wave energy densities causes them to focus each other into a singular state.
More precisely, the Kretschner scalar diverges on the surface where \( u^2 + v^2 = 1 \)within the interaction region:
$$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \to \infty \quad \text{as} \quad u^2 + v^2 \to 1^-$$
A curvature singularity forms from the collision of two smooth impulsive waves
This result, while initially surprising, is now understood as a consequence of the focusing theorem (Raychaudhuri equation) applied to the null generators of the wave fronts. Each wave acts as a gravitational lens for the other, and the cumulative focusing leads to caustic formation and eventually a singularity. The study of colliding plane waves remains an active area of research, with connections to the BKL (Belinski-Khalatnikov-Lifshitz) approach to cosmological singularities.
PP Waves in String Theory
PP waves play a pivotal role in string theory, primarily through two interconnected developments: the BMN (Berenstein-Maldacena-Nastase) limit and the exact solvability of string theory on PP-wave backgrounds.
The starting point is the AdS/CFT correspondence, which relates string theory on\( \text{AdS}_5 \times S^5 \) to \( \mathcal{N} = 4 \) super Yang-Mills theory. Taking the Penrose limit of \( \text{AdS}_5 \times S^5 \) along a null geodesic that wraps the \( S^5 \) produces a maximally supersymmetric PP wave:
$$ds^2 = 2\,du\,dv - \mu^2 \sum_{i=1}^{8} (x^i)^2\,du^2 + \sum_{i=1}^{8} (dx^i)^2$$
The maximally supersymmetric PP wave in 10 dimensions: 24 supercharges preserved
This background preserves 24 of the 32 supercharges — making it the maximally supersymmetric plane wave. The crucial breakthrough of Metsaev (2002) and Metsaev-Tseytlin was showing that the worldsheet string theory on this background is exactly solvable: the Green-Schwarz action in light-cone gauge reduces to a massive free field theory:
$$S = \frac{1}{4\pi\alpha'}\int d\tau\,d\sigma\left[\dot{x}^i\dot{x}^i - x'^i x'^i - \mu^2 x^i x^i + \text{fermions}\right]$$
The string worldsheet action becomes a free massive theory — exactly quantizable
The string spectrum on this background is:
$$p^- = \frac{1}{p^+}\left[\mu^2(p^+)^2 + \sum_{n=-\infty}^{\infty} N_n\sqrt{\mu^2 + \frac{n^2}{(\alpha' p^+)^2}}\right]$$
Exact string spectrum: \( N_n \) is the occupation number of the \( n \)-th mode
On the gauge theory side (via AdS/CFT), the Penrose limit corresponds to focusing on operators with large R-charge \( J \sim \sqrt{N} \). The string oscillator excitations\( N_n \) map to insertions of "impurity" fields into the BMN operator\( \text{Tr}(Z^J) \), where \( Z \) is a complex scalar. This provides the first quantitative verification of string theory on a curved Ramond-Ramond background and opened the door to precision tests of the AdS/CFT correspondence through integrability.
Key Concepts Summary
- PP-wave metric: \( ds^2 = 2\,du\,dv + H(u,x,y)\,du^2 + dx^2 + dy^2 \) with a single profile function \( H \) encoding the wave
- Vacuum condition: The profile function satisfies the 2D Laplace equation \( \nabla^2_\perp H = 0 \), an exact simplification of the full nonlinear Einstein equations
- Brinkmann vs. Rosen coordinates: Brinkmann coordinates are globally well-defined; Rosen coordinates may develop singularities but make the transverse geometry more intuitive
- Geodesic motion: Transverse geodesics obey Newtonian-like equations\( \ddot{x}^i = -\frac{1}{2}\partial_i H \), making them exactly solvable for plane waves
- Memory effect: Sandwich waves produce permanent displacement and velocity changes in test particles, connected to BMS symmetries and soft graviton theorems
- Penrose limit: Any spacetime looks like a plane wave near any null geodesic, making PP waves universally relevant
- Aichelburg-Sexl metric: The ultrarelativistic limit of Schwarzschild produces an impulsive PP wave describing a massless particle's gravitational field
- Colliding waves: The Khan-Penrose solution shows that colliding plane waves produce curvature singularities — a purely nonlinear gravitational effect
- String theory: PP waves are exactly solvable string backgrounds; the BMN limit connects them to precision tests of AdS/CFT via integrability