Geodesics in spacetimes with expanding impulsive gravitational waves

TL;DR

This work analyzes geodesic motion in expanding spherical impulsive gravitational waves propagating in Minkowski space by employing a continuous metric formulation. It identifies a broad class of privileged geodesics with $Z=Z_0$ and derives their refraction and shift across the impulse, linking front and behind data through the generating function $h(Z)$. In the snapping cosmic string case (deficit angle $oldsymbol{\delta}$) it provides explicit matching formulas and reveals deficit-angle induced refraction and focusing, including in the plane perpendicular to the string. The results offer physical interpretation of particle motion in these spacetimes and lay groundwork for a rigorous distributional treatment of impulsive Robinson Trautman solutions, serving as a bridge between distributional and continuous formulations.

Abstract

We study geodesic motion in expanding spherical impulsive gravitational waves propagating in a Minkowski background. Employing the continuous form of the metric we find and examine a large family of geometrically preferred geodesics. For the special class of axially symmetric spacetimes with the spherical impulse generated by a snapping cosmic string we give a detailed physical interpretation of the motion of test particles.

Figures (4)

• Figure 1: The angle $\alpha$ identifies the point where the particle interacts with the impulse (indicated by a circle) generated by a snapped string localized on the $z$-axis. The angle $\beta$ characterizes the inclination of its trajectory. The superscripts "+" and "--" correspond to the respective values in the (local) Minkowskian coordinate system outside ($U>0$) and the (different) Minkowskian system inside ($U<0$) the impulse.
• Figure 2: Typical behavior of geodesic trajectories, which are refracted and shifted by the expanding spherical impulse, for various values of $\alpha^+$, $\beta^+$ and the parameter $\epsilon$. Here $\delta=0.3$.
• Figure 3: Plots of $\beta^+_\parallel$, $\beta^+_\perp$ and $\beta^+_{null}\,$, introduced in the text, as functions of the angle $\alpha^+$ (again $\delta=0.3$). The causal character of the geodesics with specific initial data is indicated by the shading of the respective regions: grey resp. white corresponds to timelike resp. spacelike geodesics.
• Figure 4: In the region $U<0$ behind the expanding spherical impulse with $\epsilon=+1$, the motion of test particles is always radial, i.e., exactly (de)focusing. Incoming trajectories for various $\beta^+$ and $\alpha^+$ in the region $U>0$, with the deficit angle parameter $\delta=0.3$, are indicated by dashed lines. The velocity vectors behind the wave are indicated by arrows of the corresponding length and orientation (tachyons are denoted by double arrows).