Impulsive Gravitational Waves

An impulsive gravitational wave concentrates curvature on a null hypersurface. It can be described by a distributional Brinkmann metric, by a continuous metric whose first derivatives jump, or by cutting two smooth regions and reattaching them with a null shift. These descriptions encode the same geometry only after their coordinate maps, regularity class, and geodesic matching rules have been specified.

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

Helpful background. Einstein’s Equation supplies distributional source interpretation; The Riemann Curvature Tensor fixes curvature signs; Exact Solutions of Einstein’s Equations supplies the Kundt setting; The Wave Equation and Characteristics supplies null wavefronts and weak solutions.

Distributional Brinkmann form

For a nonexpanding impulse in a flat background, write

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

The wavefront is $u=0$. Formally,

$$R_{uiuj}=-\frac12\partial_i\partial_jF\,\delta(u), \qquad R_{uu}=-\frac12\Delta_\perp F\,\delta(u).$$

Thus $\Delta_\perp F=0$ away from transverse singularities gives a vacuum gravitational impulse; a nonzero transverse Laplacian represents a null matter layer. This compact form is useful but its metric coefficient is itself a distribution. Arbitrary nonlinear coordinate changes or multiplication of singular connection terms is not legitimate in ordinary Schwartz distribution theory.

A safe construction starts from smooth sandwich profiles $\rho_\varepsilon(u)$ with unit integral and support shrinking to zero, replaces $\delta$ by $\rho_\varepsilon$, solves the smooth geometry, and then proves the desired weak or generalized-function limit. Steinbauer and Vickers explain the regularity and generalized-function issues in Steinbauer and Vickers 2006, §§4–6, pp. R96–R114.

Continuous coordinates and cut-and-paste geometry

Let

$$u_+=u\,\Theta(u).$$

This function is continuous and locally Lipschitz, with $u_+'=\Theta(u)$ and $u_+''=\delta(u)$ in the weak sense. A profile-dependent transformation built from $u_+$ moves the delta term from the metric coefficients into a jump of their first derivatives. The resulting metric is continuous across $u=0$ but not $C^1$ there; its second weak derivatives contain the impulsive curvature. Because the coefficients are locally Lipschitz, geodesics can be handled in a substantially better-posed coordinate form than the formal delta-metric equation.

Penrose’s cut-and-paste picture gives the same idea without differentiating singular coordinates. Cut Minkowski or constant-curvature spacetime along a null hypersurface, retain the transverse point $(x,y)$, and reglue the two banks after a profile-dependent shift along the null generators. The induced degenerate metric on the wavefront agrees from both sides, while the transverse derivative data do not. Podolský constructs and compares distributional, continuous, and cut-and-paste forms in Podolský 2002, §§3.1–3.3, arXiv manuscript pp. 7–13, Open PDF.

Geodesic matching

Use $u$ itself as affine parameter through a transverse crossing and the displayed $-2\,du\,dv$ normalization. Integrating the regularized transverse equation across the shrinking sandwich gives

$$x^i(0^+)=x^i(0^-), \qquad \dot x^i(0^+)-\dot x^i(0^-) =\frac12\partial^iF\big(x(0)\big).$$

The transverse position is continuous, while the Brinkmann transverse velocity receives a finite refraction kick. The null coordinate $v$ and its derivative have convention-dependent jumps fixed by the profile and by conservation of $g_{ab}\dot x^a\dot x^b$ in the regularized limit. In continuous coordinates the same geodesic is $C^1$; the apparent position or velocity discontinuity has moved into the nonsmooth coordinate transformation. One must therefore state which chart defines a quoted jump.

The Hessian, rather than the gradient, controls relative motion. Two nearby geodesics crossing at transverse separation $\eta^j$ acquire

$$\Delta\dot\eta^i =\frac12\partial^i\partial_jF\,\eta^j.$$

An affine or constant part of $F$ changes coordinate matching but has zero Hessian and no local gravitational tidal content.

Scope and checks

The impulse idealization is conditional on its regularization or gluing construction. It does not justify squaring delta-valued curvature, assigning a conventional pointwise curvature at $u=0$, or treating every discontinuous chart as physically singular. Independent checks are: recover the delta curvature from $u_+''$; derive the transverse kick from a smooth unit-area profile; and verify that the spacetime is smooth on each side of the wavefront.

Exercise

Let $F=A(x^2-y^2)$. Find the velocity kick and show that it preserves the trace-free character of vacuum tidal focusing.

Solution

$\nabla F=(2Ax,-2Ay)$, so $\Delta\dot x=Ax$ and $\Delta\dot y=-Ay$. The relative-velocity map is $\operatorname{diag}(A,-A)$, whose trace is zero. Equivalently $\Delta_\perp F=2A-2A=0$, so the impulsive Ricci component vanishes while the Weyl impulse remains.

References

• Podolský, Jiří. “Exact Impulsive Gravitational Waves in Spacetimes of Constant Curvature.” In Gravitation: Following the Prague Inspiration, edited by Oldřich Semerák, Jiří Podolský, and Martin Žofka, 205–246. Singapore: World Scientific, 2002. DOI; Open PDF (arXiv manuscript).
• Steinbauer, Roland, and James A. Vickers. “The Use of Generalized Functions and Distributions in General Relativity.” Classical and Quantum Gravity 23 (2006): R91–R114. DOI.