PP-Wave Spacetimes — Fundamentals
Exact gravitational wave solutions with planar wavefronts and parallel rays
Introduction to PP Waves
PP waves — standing for plane-fronted gravitational waves with parallel rays — are among the most important exact solutions to Einstein's field equations. Unlike the linearized gravitational wave solutions commonly discussed in introductory treatments, PP waves are exact solutions to the full nonlinear Einstein equations. This remarkable property makes them invaluable laboratories for studying gravitational radiation in the strong-field regime.
The history of PP waves traces back to the work of Brinkmann (1925), who first discovered the general class of metrics admitting a covariantly constant null vector. The physical interpretation as gravitational waves was later clarified by Bondi, Pirani, and Robinson in the 1950s. The name "PP wave" encodes two geometric properties: the wavefronts are planar (plane-fronted), and the null rays generating these wavefronts are parallel — meaning they have zero shear, twist, and expansion.
Formally, a PP-wave spacetime is defined by the existence of a covariantly constant null Killing vector field \( k^\mu \):
$$\nabla_\mu k_\nu = 0, \qquad k^\mu k_\mu = 0$$
A covariantly constant null vector: the defining property of PP-wave spacetimes
The condition \( \nabla_\mu k_\nu = 0 \) is extremely restrictive. It implies that\( k^\mu \) is simultaneously geodesic, shear-free, expansion-free, and twist-free. These are precisely the conditions that characterize a congruence of parallel null rays, giving PP waves their name.
Derivation of the PP-Wave Metric
We begin with a general ansatz adapted to null coordinates. Introduce coordinates\( (u, v, x, y) \) where \( u \) and \( v \) are null coordinates related to the usual time and spatial coordinates by \( u = t - z \) and \( v = t + z \)(the retarded and advanced time, respectively), and \( x, y \) are transverse coordinates.
The most general metric admitting a covariantly constant null vector\( k^\mu = \delta^\mu_v \) (so that \( k = \partial_v \)) takes the form:
$$ds^2 = 2\,du\,dv + H(u, x, y)\,du^2 + dx^2 + dy^2$$
The PP-wave metric in Brinkmann coordinates
To see why this is the unique form, note that \( \nabla_\mu k_\nu = 0 \) implies\( k_\mu \) is a closed 1-form (\( dk = 0 \)), so locally\( k_\mu = \partial_\mu u \) for some scalar \( u \). The null condition\( k^\mu k_\mu = 0 \) requires \( g^{uu} = 0 \). The covariant constancy then constrains the metric to depend on \( u \) only through the single profile function\( H(u, x, y) \).
The key simplification is that this metric has only one free function \( H(u, x, y) \)that encodes all the gravitational wave information. The \( u \)-dependence of \( H \)determines the time profile of the wave, while the \( (x, y) \)-dependence determines the transverse structure.
Brinkmann vs. Rosen Coordinates
The metric form above uses Brinkmann coordinates, which are the most natural and widely used. However, there exists an alternative coordinate system known as Rosen coordinates that makes the plane-wave nature more manifest. In Rosen coordinates, the metric takes the form:
$$ds^2 = 2\,dU\,dV + g_{ij}(U)\,dX^i\,dX^j, \qquad i,j \in \{1,2\}$$
PP-wave metric in Rosen coordinates — transverse metric depends only on U
Here \( g_{ij}(U) \) is a 2x2 positive-definite matrix that depends only on the null coordinate \( U \). The relationship between Brinkmann and Rosen coordinates involves a\( U \)-dependent transformation of the transverse coordinates:
$$x^i = e^i_{\ a}(U)\,X^a, \qquad v = V + \frac{1}{2}\dot{e}^i_{\ a}\,e_{i b}\,X^a X^b$$
Coordinate transformation via the vielbein \( e^i_{\ a}(U) \)
The advantage of Brinkmann coordinates is that they are globally well-defined (no coordinate singularities), while Rosen coordinates can develop coordinate singularities when the transverse metric \( g_{ij}(U) \) becomes degenerate. Rosen coordinates, however, are more intuitive: they make the notion of "transverse distance between geodesics" more transparent.
The Vacuum Einstein Equation
A remarkable feature of the PP-wave metric is the dramatic simplification of the Einstein equations. Computing the Ricci tensor for the Brinkmann metric, one finds that all components vanish except\( R_{uu} \):
$$R_{\mu\nu} = -\frac{1}{2}\nabla^2_\perp H\;\delta_\mu^u\,\delta_\nu^u$$
where \( \nabla^2_\perp = \partial_x^2 + \partial_y^2 \) is the transverse Laplacian
Therefore, the vacuum Einstein equation \( R_{\mu\nu} = 0 \) reduces to the single condition:
$$\nabla^2_\perp H = \frac{\partial^2 H}{\partial x^2} + \frac{\partial^2 H}{\partial y^2} = 0$$
The profile function H must be harmonic in the transverse coordinates
This is simply Laplace's equation in two dimensions! The full nonlinear Einstein equations have collapsed to a linear PDE. This linearization is not an approximation — it is exact, a consequence of the special algebraic structure of PP waves (they are type N in the Petrov classification).
The most general solution to the 2D Laplace equation can be written using complex coordinates\( w = x + iy \):
$$H(u, x, y) = \text{Re}\left[f(u, w)\right]$$
where \( f(u, w) \) is an arbitrary holomorphic function of \( w \) for each \( u \)
For a gravitational wave with matter sources (e.g., a null dust or electromagnetic field), the right-hand side of the Einstein equation is modified:
$$\nabla^2_\perp H = -16\pi G\,T_{uu}$$
The sourced PP-wave equation — analogous to Poisson's equation
Physical Interpretation
PP waves represent gravitational radiation propagating at the speed of light along the\( z \)-direction. The null coordinate \( u = t - z \) plays the role of retarded time: surfaces of constant \( u \) are the wavefronts, and they are flat planes perpendicular to the direction of propagation.
The profile function \( H(u, x, y) \) encodes the amplitude and polarization of the wave. For the most physically relevant case of plane waves(waves with homogeneous wavefronts), \( H \) is at most quadratic in the transverse coordinates:
$$H_{\text{plane}} = A_+(u)(x^2 - y^2) + 2A_\times(u)\,xy$$
\( A_+(u) \) and \( A_\times(u) \) are the two polarization amplitudes
This quadratic form automatically satisfies \( \nabla^2_\perp H = 0 \) because the traces of \( x^2 - y^2 \) and \( xy \) under the transverse Laplacian cancel. The two independent functions \( A_+(u) \) and \( A_\times(u) \)correspond to the two polarization states of gravitational waves, the "plus" and "cross" polarizations familiar from linearized theory. In the exact PP-wave setting, these polarizations are encoded without any approximation.
An important property of PP waves is their vanishing scalar curvature invariants. All polynomial scalar invariants constructed from the Riemann tensor (such as the Kretschner scalar \( R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} \)) vanish identically for PP waves, despite the spacetime being curved. This makes them examples ofVSI (vanishing scalar invariant) spacetimes and means they cannot be detected by local scalar measurements alone.
The Weyl tensor of a PP wave is of Petrov type N, meaning it has a single principal null direction (aligned with \( k^\mu \)) with multiplicity four. This is the algebraic type associated with pure gravitational radiation — confirming the wave interpretation.