pp-Wave Spacetimes

Wave terminology is not completely uniform. On this page a parallel wave admits a nonzero covariantly constant null vector field, while a pp-wave means the classical plane-fronted subclass whose screen-space connection is flat. In the standard local form used below, those conditions reduce the curvature and field equations to a transverse Poisson problem, producing exact vacuum waves, pure-radiation solutions, and plane-wave subclasses. The construction is local; global identifications or completeness require separate analysis.

Required background. Frames, Coframes, and Anholonomy supplies null frames; Affine and Variational Geodesics supplies null congruences and deviation.

Helpful background. Einstein’s Equation supplies the vacuum and pure-radiation equations; The Riemann Curvature Tensor fixes signs; Exact Solutions of Einstein’s Equations supplies the invariant-classification context; The Wave Equation and Characteristics supplies wavefront terminology.

Covariantly constant null geometry

Let $k^a$ satisfy

$$k^ak_a=0, \qquad \nabla_a k^b=0.$$

Parallelism is much stronger than the geodesic condition $k^a\nabla_ak^b=0$. It implies that $k^a$ is Killing, affinely geodesic, and has zero expansion, shear, and twist. By itself, however, it defines the broader parallel-wave class; it does not force a flat two-dimensional wavefront. Here the additional pp-wave condition is that the curvature induced on the screen bundle $k^\perp/k$ vanishes. In the standard local subclass with removable transverse cross terms, coordinates $(u,v,x,y)$ can then be chosen so that $k=\partial_v$ and

$$ds^2=-2\,du\,dv+dx^2+dy^2+H(u,x,y)\,du^2.$$

The null hypersurfaces $u=\text{constant}$ are wavefronts, while $v$ is an affine coordinate along the parallel rays. Terms in $H$ that are constant or linear in $(x,y)$ can be removed locally by a $u$-dependent transverse translation and a compensating change of $v$; their presence is not tidal curvature.

Some references use “pp-wave” for every parallel wave and do not require a planar screen. The convention adopted here follows the explicit parallel-wave/flat-wavefront distinction in Roche, Aazami, and Cederbaum 2023, Definitions 1 and 3 and §§2–3, Open PDF. Stating the curvature condition avoids silently deriving the flat transverse metric from parallelism alone.

Curvature and field equations

With the site’s curvature convention, the independent nonzero coordinate components are

$$R_{uiuj}=-\frac12\partial_i\partial_jH, \qquad R_{uu}=-\frac12(\partial_x^2+\partial_y^2)H,$$

where $i,j\in\{x,y\}$. The vacuum equation is therefore

$$\Delta_\perp H=0.$$

This provides a direct independent check: for the displayed metric, the nonzero Christoffel families are $\Gamma^i_{uu}=-\tfrac12\partial_iH$, $\Gamma^v_{ui}=-\tfrac12\partial_iH$, and $\Gamma^v_{uu}=-\tfrac12\partial_uH$. They recover the Hessian curvature and its trace. A source $T_{ab}=\rho\,k_a k_b$ changes the equation to $R_{uu}=8\pi G T_{uu}$; stress-energy conservation constrains the allowed profile but does not change the parallel null direction.

The plane-wave subclass has a profile exactly quadratic in transverse position,

$$H=A_{ij}(u)x^ix^j,$$

with $A_{ij}=A_{ji}$. Vacuum requires $\delta^{ij}A_{ij}=0$, leaving the two polarization functions. A general harmonic $H$ may contain higher transverse multipoles and is a pp-wave without being a plane wave. Bičák locates these families and their invariant interpretation in Bičák 2000, §8, pp. 66–71, Open PDF.

Algebraic and scalar diagnostics

For a nonflat vacuum pp-wave, the Weyl tensor is Petrov type N with repeated principal null direction $k^a$; conformally flat pure-radiation cases are type O. More surprisingly, every polynomial scalar formed from the Riemann tensor and its covariant derivatives vanishes for the standard VSI pp-wave class. Minkowski space has the same scalar list, although $R_{uiuj}$ can be nonzero.

Thus vanishing scalar invariants do not establish flatness. A parallel-propagated frame detects the tidal matrix $-R^i{}_{uju}$ through geodesic deviation. Cartan invariants or the explicit Hessian of $H$ distinguish a nonflat wave from Minkowski space. This is the canonical counterexample to identifying a spacetime by scalar curvature invariants alone.

Physical and coordinate limits

If $\partial_i\partial_jH=0$, the curvature vanishes and the metric is locally Minkowski even when $H$ itself is nonzero. If $A_{ij}(u)$ has compact support, the result is a sandwich plane wave; outside the support, each region is flat in Brinkmann coordinates, although freely falling probes can emerge with changed separation data. A delta-function $A_{ij}$ defines an impulsive idealization only in a distributionally controlled formulation.

Exactness does not make every pp-wave globally regular. Profiles can grow at transverse infinity, quotients can alter causality, and some plane waves are geodesically incomplete. Those questions depend on the chosen $H$, domain, and global construction.

Exercise

For $H=A(u)(x^2-y^2)+2B(u)xy$, verify the vacuum equation and identify the transverse tidal matrix.

Solution

$\partial_x^2H=2A$ and $\partial_y^2H=-2A$, so $\Delta_\perp H=0$. The Hessian is $2\begin{pmatrix}A&B\\B&-A\end{pmatrix}$, hence $R_{uiuj}=-\begin{pmatrix}A&B\\B&-A\end{pmatrix}$. Its trace vanishes, and rotations in the $(x,y)$ plane mix the two displayed polarization functions.

References

• Bičák, Jiří. “Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, 1–126. Lecture Notes in Physics 540. Berlin: Springer, 2000. DOI; Open PDF.
• Roche, Cian, Amir Babak Aazami, and Carla Cederbaum. “Exact Parallel Waves in General Relativity.” General Relativity and Gravitation 55 (2023): 40. doi:10.1007/s10714-023-03083-x; arXiv:2207.03591, Open PDF.
• Stephani, Hans, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt. Exact Solutions of Einstein’s Field Equations. 2nd ed. Cambridge: Cambridge University Press, 2003, §§24.5–24.6. DOI.