Colliding Plane Waves

Two plane waves can be individually regular before they meet and nevertheless generate a curvature singularity in their interaction region. Exact collision spacetimes make the mechanism explicit: characteristic data on two null fronts focus both transverse congruences, while the nonlinear Einstein equations couple their polarizations. Exceptional data can replace the singular boundary by a horizon, so singularity formation is generic within important families, not an unconditional theorem for every collision.

Required background. Frames, Coframes, and Anholonomy supplies the aligned null frame; Affine and Variational Geodesics supplies null focusing.

Helpful background. Einstein’s Equation supplies the characteristic field equations; The Riemann Curvature Tensor supplies singularity diagnostics; Exact Solutions of Einstein’s Equations supplies the wave families; The Wave Equation and Characteristics supplies characteristic data and domains of dependence.

Four characteristic regions

Let the wavefronts be $u=0$ and $v=0$. The idealized spacetime divides into:

• region I, $u<0$, $v<0$: the background before either wave;
• region II, $u>0$, $v<0$: the first incoming plane wave;
• region III, $u<0$, $v>0$: the second incoming plane wave; and
• region IV, $u>0$, $v>0$: their interaction.

For aligned linear polarization, a useful double-null form is

$$ds^2=-2e^{-M(u,v)}\,du\,dv +e^{-U(u,v)} \left(e^{V(u,v)}dx^2+e^{-V(u,v)}dy^2\right).$$

The two incoming metrics provide characteristic data for $U,V,M$ on the null boundaries of region IV. Continuity of the induced metric is not the only requirement: derivative matching determines whether the fronts carry intended impulsive pieces, finite shock profiles, or unwanted matter shells. Cross-polarized collisions require an additional off-diagonal polarization function.

Focusing equation

One reduced Einstein equation in the aligned vacuum problem is

$$U_{uv}=U_uU_v.$$

Therefore

$$\big(e^{-U}\big)_{uv}=0,$$

and characteristic data give

$$e^{-U}=f(u)+g(v)$$

after absorbing an additive normalization into $f$ and $g$. The quantity $e^{-U}$ is the transverse area scale. Both waves can decrease it, and it may reach zero at a finite curve in the interaction region even though $f$ and $g$ were regular on the incoming branches.

This area collapse is the geometric core of the answer. The remaining equations drive $V$ and $M$ on the shrinking transverse geometry. For generic vacuum collision data, curvature components and scalar invariants diverge at the focusing boundary. In the Khan–Penrose impulsive collision, for example, the post-collision boundary is a spacelike curvature singularity. The region structure and principal exact families are reviewed in Bičák 2000, §8, pp. 71–76, Open PDF.

Singular and horizon outcomes

Area collapse does not alone prove scalar-curvature blow-up. Special polarization and matter data can arrange cancellations so that the focusing surface is a Killing–Cauchy horizon rather than a curvature singularity. The Bell–Szekeres collision of electromagnetic shock waves and certain Chandrasekhar–Xanthopoulos families illustrate exceptional horizon outcomes. Such horizons are sensitive to perturbations of characteristic data, but stability is a separate dynamical claim and is not inferred from the exact solution.

The correct classification therefore checks, in order: the determinant of the transverse metric; a parallelly propagated curvature frame; scalar invariants where discriminating; extendibility of the candidate boundary; and the causal character of any extension. A coordinate coefficient diverging in $(u,v)$ is not sufficient, just as a finite polynomial scalar need not exclude a parallelly propagated singularity.

Interpretation and limitations

Exact plane waves extend infinitely across their wavefronts and are not localized asymptotically flat beams. Their collision solutions are controlled laboratories for nonlinear characteristic evolution, focusing, and singular boundaries, not direct models of two finite astrophysical wave packets. The characteristic construction also ends at a Cauchy horizon without uniquely specifying what lies beyond unless additional data or an extension principle is supplied.

An independent check of any candidate family is to recover regions II and III by setting one null coordinate negative, verify the intended junction regularity at each front, and then evaluate at least one curvature diagnostic as $e^{-U}\to0$. Griffiths and Podolský develop the characteristic initial data and interaction region in Griffiths and Podolský 2009, §§21.1–21.2, pp. 413–418.

Exercise

Suppose the incoming data are normalized so that $e^{-U(u,0)}=1-u^2$ and $e^{-U(0,v)}=1-v^2$. Find $e^{-U}$ in the interaction region and its first focusing curve.

Solution

Because $(e^{-U})_{uv}=0$, write $e^{-U}=f(u)+g(v)$. Matching both axes and avoiding double-counting their common value gives $e^{-U}=1-u^2-v^2$. The transverse area first vanishes on $u^2+v^2=1$ in the quadrant $u,v>0$. Determining whether this curve is a curvature singularity still requires the remaining functions $V$ and $M$.

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.
• Griffiths, Jerry B., and Jiří Podolský. Exact Space-Times in Einstein’s General Relativity. Cambridge: Cambridge University Press, 2009. DOI.