Plane Waves in Brinkmann and Rosen Coordinates

The same plane wave can appear either as a quadratic potential on a flat transverse plane or as a time-dependent transverse metric. Brinkmann coordinates keep curvature regular and polarization transparent; Rosen coordinates move with a congruence of geodesics and can become singular when that congruence focuses. The singularity of a Rosen chart is therefore not, by itself, a spacetime singularity.

Required background. Frames, Coframes, and Anholonomy supplies transverse frame changes; Affine and Variational Geodesics supplies geodesic congruences and Jacobi matrices.

Helpful background. Einstein’s Equation supplies the trace-free vacuum condition; The Riemann Curvature Tensor fixes the tidal sign; Exact Solutions of Einstein’s Equations supplies the pp-wave family; The Wave Equation and Characteristics supplies wavefront language.

Brinkmann profile

Write a plane wave as

$$ds^2=-2\,du\,dv+\delta_{ij}dx^idx^j +A_{ij}(u)x^ix^j\,du^2,$$

where $A_{ij}$ is symmetric. In vacuum $\operatorname{tr}A=0$. The curvature is

$$R_{uiuj}=-A_{ij}(u),$$

so $A$ is directly the negative transverse tidal matrix in this frame. A constant rotation of $(x,y)$ changes the polarization basis by $A\mapsto O^TAO$ without changing its eigenvalues. More general $u$-dependent rotations introduce cross terms and must not be treated as the same Brinkmann gauge without accounting for them.

Transverse geodesics parametrized by $u$ obey

$$\frac{d^2x}{du^2}=A(u)x.$$

This linear system is also the Jacobi equation for the separation of neighboring reference rays.

Constructing Rosen coordinates

Let $E(u)$ be an invertible $2\times2$ matrix solving

$$E''=AE, \qquad E^TE'=E'^TE.$$

The second condition can be imposed on initial data and is preserved by the equation when $A$ is symmetric. Define

$$x=E(u)X, \qquad v=V+\frac12X^TE^TE'X.$$

Substitution cancels the $du\,dX$ terms and the $du^2$ profile, yielding Rosen form

$$ds^2=-2\,du\,dV+\gamma_{ab}(u)dX^adX^b, \qquad \gamma=E^TE.$$

The transformation is valid only where $\det E\neq0$. Different fundamental matrices $E$ can generate different Rosen charts for the same Brinkmann wave, so the individual entries of $\gamma$ are not invariant polarization amplitudes. The relation between these forms and the focusing interpretation is reviewed in Bičák 2000, §8, pp. 66–71, Open PDF.

Why the Rosen chart can fail

Columns of $E$ are transverse Jacobi fields. When $\det E=0$, a nontrivial combination vanishes: the chosen geodesic congruence develops a caustic. Then $\det\gamma=(\det E)^2$ vanishes and the Rosen metric degenerates, although $A(u)$ and the Brinkmann curvature can remain smooth. This need not be a point conjugate to a chosen initial event: pairwise conjugacy requires a Jacobi field that vanishes at both events, whereas a Rosen congruence can start with nonzero separations.

For example, take

$$A=\begin{pmatrix}\kappa^2&0\\0&-\kappa^2\end{pmatrix}, \qquad E=\begin{pmatrix}\cosh(\kappa u)&0\\0&\cos(\kappa u)\end{pmatrix}.$$

Then $E''=AE$ and the vacuum trace condition holds. Here $E(0)=I$ and $E'(0)=0$: the reference rays start separated and initially parallel. Brinkmann curvature is constant, but the Rosen chart fails at $\kappa u=\pi/2$ because the second congruence direction focuses. No polynomial curvature scalar diverges there. A new Rosen patch, or the original Brinkmann chart, continues the geometry.

For comparison, rays emitted from one event at $u=0$ have Jacobi matrix $B(u)=\operatorname{diag}(\sinh\kappa u,\sin\kappa u)/\kappa$, with $B(0)=0$ and $B'(0)=I$. Its first positive conjugate value is $u=\pi/\kappa$, not the initially parallel congruence’s caustic at $\pi/(2\kappa)$; assume $\kappa\gt0$ in this comparison.

Polarization and invariant content

In a chosen parallel transverse frame, decompose the trace-free matrix as

$$A=\begin{pmatrix}A_+&A_\times\\A_\times&-A_+\end{pmatrix}.$$

A constant rotation through angle $\alpha$ rotates the pair $(A_+,A_\times)$ through $2\alpha$. Thus “plus” and “cross” depend on the screen basis, while the eigenvalues of $A$ and the tidal map between specified parallel frames are invariantly comparable. Rosen scale factors mix that tidal information with the initial position and velocity chosen for the reference congruence.

The practical rule is to compute curvature in Brinkmann form, use Rosen form when a comoving congruence is pedagogically useful, and test every Rosen singularity by returning to $A_{ij}$ or a regular frame. Blau, Figueroa-O’Farrill, Hull, and Papadopoulos place this construction in the broader plane-wave setting in Blau et al. 2002, pp. L87–L95.

Exercise

Verify that the displayed constant-profile example satisfies the Rosen vacuum equation

$$\operatorname{tr}\!\left(\gamma^{-1}\gamma'' -\frac12\gamma^{-1}\gamma'\gamma^{-1}\gamma'\right)=0.$$

Solution

For diagonal $E=\operatorname{diag}(e_1,e_2)$, $\gamma=\operatorname{diag}(e_1^2,e_2^2)$. The displayed trace equals $2(e_1''/e_1+e_2''/e_2)$. Here those ratios are $\kappa^2$ and $-\kappa^2$, so the sum vanishes. The equation holds wherever both scale factors are nonzero; its coordinate form ceases to be valid at the Rosen caustic.

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.
• Blau, Matthias, José Figueroa-O’Farrill, Christopher Hull, and George Papadopoulos. “Penrose Limits and Maximal Supersymmetry.” Classical and Quantum Gravity 19 (2002): L87–L95. DOI.