Geodesics and Physical Effects
How test particles respond to PP waves: memory, Penrose limits, and ultrarelativistic boosts
Geodesic Equations in a PP-Wave Background
Understanding the motion of free-falling test particles in a PP-wave spacetime is essential for interpreting gravitational wave observations. We derive the geodesic equations starting from the PP-wave metric in Brinkmann coordinates:
$$ds^2 = 2\,du\,dv + H(u, x, y)\,du^2 + dx^2 + dy^2$$
The geodesic Lagrangian is \( \mathcal{L} = \frac{1}{2}g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu \), where dots denote differentiation with respect to an affine parameter \( \lambda \). Since the metric is independent of \( v \), the conjugate momentum \( p_v \) is conserved:
$$p_v = \frac{\partial \mathcal{L}}{\partial \dot{v}} = \dot{u} = \text{const} \equiv E$$
Conservation of null momentum — \( \partial_v \) is a Killing vector
We can use \( u \) itself as the affine parameter (choosing \( E = 1 \)), so that \( \dot{u} = 1 \). The remaining geodesic equations become remarkably simple. The transverse equations decouple completely:
$$\frac{d^2 x}{du^2} = -\frac{1}{2}\frac{\partial H}{\partial x}, \qquad \frac{d^2 y}{du^2} = -\frac{1}{2}\frac{\partial H}{\partial y}$$
Transverse geodesic equations — Newtonian-like motion in a potential \( -H/2 \)
This is a striking result: the transverse motion of geodesics in a PP-wave background is governed by Newton's second law with a "potential"\( V = -H/2 \). The gravitational wave acts as a time-dependent (actually \( u \)-dependent) transverse force field. The \( v \)-equation then determines the longitudinal motion:
$$\frac{dv}{du} = -\frac{1}{2}\left[H + \left(\frac{dx}{du}\right)^2 + \left(\frac{dy}{du}\right)^2\right] + \text{const}$$
The longitudinal coordinate \( v \) is determined by quadrature once \( x(u), y(u) \) are known
For the plane wave case where \( H = A_+(u)(x^2 - y^2) + 2A_\times(u)\,xy \), the transverse geodesic equations become a system of coupled linear ODEs:
$$\frac{d^2}{du^2}\begin{pmatrix} x \\ y \end{pmatrix} = -\begin{pmatrix} A_+(u) & A_\times(u) \\ A_\times(u) & -A_+(u) \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$$
The geodesic deviation matrix is traceless — reflecting the transverse, traceless nature of gravitational waves
The matrix appearing here is symmetric and traceless, which corresponds to the two independent polarizations. For pure \( + \)-polarization (\( A_\times = 0 \)), the\( x \) and \( y \) equations decouple, producing the characteristic stretching in one direction and compression in the orthogonal direction.
Gravitational Memory Effect
One of the most physically striking phenomena associated with PP waves is the gravitational memory effect. Consider a gravitational wave burst described by a profile function \( H(u, x, y) \) that is nonzero only for\( u_1 < u < u_2 \) (a "sandwich wave"). Before the wave arrives (\( u < u_1 \)), two initially comoving test particles are separated by some displacement \( \Delta x^i_\text{before} \).
After the wave has passed (\( u > u_2 \)), the particles are again in flat spacetime and move on straight lines. However, their final separation \( \Delta x^i_\text{after} \)generically differs from the initial separation:
$$\Delta x^i_\text{after} = B^i_{\ j}\,\Delta x^j_\text{before}$$
where \( B^i_{\ j} \) is the transfer matrix obtained by solving the geodesic deviation equation through the burst
The transfer matrix \( B^i_{\ j} \) is determined by the fundamental solution matrix of the geodesic deviation equation. Crucially, \( B \neq \mathbb{1} \) in general, meaning the wave leaves a permanent displacement between test particles. Furthermore, the particles may also acquire a relative velocity (the velocity memory):
$$\Delta\dot{x}^i_\text{after} = C^i_{\ j}\,\Delta x^j_\text{before}$$
Velocity memory: particles acquire relative velocity even if initially comoving
This memory effect, proposed by Zel'dovich and Polnarev (1974) and further developed by Christodoulou (1991), is a genuine nonlinear effect of general relativity. It has deep connections to the BMS (Bondi-van der Burg-Metzner-Sachs) symmetry group of asymptotically flat spacetimes and to Weinberg's soft graviton theorem via the infrared triangle of Strominger. Detecting this memory effect is a major goal of next-generation gravitational wave detectors.
The Penrose Limit
One of the most remarkable results in general relativity is Penrose's theorem (1976) that any spacetime, when examined in a neighborhood of any null geodesic, looks like a PP wave. This is the gravitational analogue of the fact that any manifold looks flat when zoomed in sufficiently (but now the "zooming in" is along a null direction).
The construction proceeds as follows. Given any spacetime \( (M, g_{\mu\nu}) \) and a null geodesic \( \gamma \), introduce Fermi-like coordinates adapted to \( \gamma \): a null coordinate \( U \) along the geodesic, a complementary null coordinate \( V \), and transverse coordinates \( Y^i \). Then perform the Penrose scaling:
$$U = u, \qquad V = \Omega^2 v, \qquad Y^i = \Omega\, y^i$$
The Penrose scaling with parameter \( \Omega \to 0 \)
In the limit \( \Omega \to 0 \), the rescaled metric converges to a PP-wave metric:
$$\lim_{\Omega \to 0} \Omega^{-2}\,g = 2\,du\,dv + H_{ij}(u)\,y^i y^j\,du^2 + \delta_{ij}\,dy^i\,dy^j$$
The Penrose limit is always a plane wave, with \( H_{ij} = -R_{uiuj}|_\gamma \)
The profile matrix \( H_{ij}(u) \) of the resulting plane wave is determined by the curvature of the original spacetime evaluated along the null geodesic:
$$H_{ij}(u) = -R_{uiuj}\big|_\gamma$$
The plane wave profile is the null sectional curvature of the original spacetime
This theorem has profound consequences. It means that plane wave spacetimes form auniversal class of exact solutions — they capture the local physics near any null ray in any spacetime. This universality is what makes PP waves indispensable in theoretical physics, particularly in string theory where the Penrose limit produces exactly solvable backgrounds.
The Aichelburg-Sexl Ultrarelativistic Boost
A fascinating connection between black holes and PP waves comes from the Aichelburg-Sexl boost (1971). The idea is to take a Schwarzschild black hole of mass \( M \) and boost it to the speed of light while simultaneously taking \( M \to 0 \), keeping the total energy\( E = M/\sqrt{1 - v^2} \) fixed. The result is an impulsive PP wave:
$$ds^2 = 2\,du\,dv - 8GE\,\delta(u)\ln\left(\frac{\rho}{\rho_0}\right)du^2 + d\rho^2 + \rho^2 d\phi^2$$
The Aichelburg-Sexl metric: gravitational field of a massless particle
where \( \rho = \sqrt{x^2 + y^2} \) is the transverse radial coordinate and\( \rho_0 \) is an arbitrary reference scale. The profile function is:
$$H(u, x, y) = -8GE\,\delta(u)\ln\left(\frac{x^2 + y^2}{\rho_0^2}\right)$$
Profile function: logarithmic in transverse distance, impulsive in null time
One can verify that this satisfies the sourced PP-wave equation:
$$\nabla^2_\perp H = -16\pi GE\,\delta(u)\,\delta^{(2)}(\vec{x}_\perp)$$
Point particle source: a massless particle carrying energy E along a null ray
The Aichelburg-Sexl solution describes the exact gravitational field of a massless point particle (or equivalently, a photon at the classical level). It is an impulsive wave: spacetime is flat everywhere except on the null plane \( u = 0 \), where there is a delta-function curvature singularity.
When a test particle crosses this shock wave, it receives an instantaneous velocity kick in the transverse direction and an instantaneous shift in \( v \). This solution has been extensively used to study high-energy gravitational scattering and trans-Planckian collisions, where it provides the leading approximation to the gravitational field of an ultrarelativistic particle. The scattering angle for a test particle passing at impact parameter \( b \) is:
$$\theta = \frac{4GE}{b}$$
Exactly twice the linearized result — an exact non-perturbative deflection angle