Refraction of geodesics by impulsive spherical gravitational waves in constant-curvature spacetimes with a cosmological constant

TL;DR

The study analyzes test-particle geodesics crossing expanding spherical impulsive gravitational waves in backgrounds with constant curvature, encompassing Minkowski, de Sitter, and anti-de Sitter spacetimes via $\Lambda$. It develops a continuous metric framework with a Penrose-type impulse defined by a holomorphic map $h(Z)$ and employs both a conformally flat 4D description and a 5D embedding for $\Lambda\neq0$, delivering explicit junction conditions and refraction formulae for all causal geodesics. A detailed application to a snapped cosmic string yields explicit position/velocity matching and reveals strong dragging effects along the moving strings, with near-light speeds for nearby particles and marked deformation of a ring of test particles. These results extend prior Minkowski analyses, provide practical tools for analyzing impulsive waves in $\Lambda$-backgrounds, and illuminate the dynamical impact of cosmic strings on particle motion in curved spacetimes.

Abstract

We investigate motion of test particles in exact spacetimes with an expanding impulsive gravitational wave which propagates in Minkowski, de Sitter or anti-de Sitter universe. Using the continuous form of these metrics we derive explicit junction conditions and simple refraction formulae for null, timelike and spacelike geodesics crossing a general impulse of this type. In particular, we present a detailed geometrical description of the motion of test particles in a special class of axially symmetric spacetimes in which the impulse is generated by a snapped cosmic string.

Figures (16)

• Figure 1: An expanding spherical impulse can be visualized as a section ${{\rm Z}_4=a}$ of the four-dimensional hyperboloids representing de Sitter (left) and anti-de Sitter (right) spaces. The bold lines are trajectories of opposite poles of an expanding spherical wave surface given by ${{\rm Z}_2=0={\rm Z}_3}$. The time-reversed situation in the region ${{\rm Z}_0<0}$, indicated by dashed lines, corresponds to contracting impulsive waves.
• Figure 2: Mapping in the complex plane $Z \leftrightarrow h(Z)$ is equivalent to identifying the points $P^-$ inside the impulsive spherical surface with the corresponding points $P^+$ outside through the stereographic projection.
• Figure 3: Definition of the angles ${\alpha, \gamma}$ characterizing position of the particle and inclination ${\beta,\delta}$ of its velocity in the ${(x,z)}$ section (top) and ${(y,z)}$ section (bottom), respectively. Here the superscript "$+$" denotes quantities outside the spherical impulse (left), while "$-$" labels analogous quantities inside the impulse (right). The points of interaction ${P^+ = (x_i^+,y_i^+,z_i^+)}$ and ${P^- = (x_i^-,y_i^-,z_i^-)}$ correspond to those in Fig. \ref{['figure2']}. The impulsive gravitation wave is an expanding sphere indicated in each section by the bold outer circle.
• Figure 4: Geometry of a spherical impulse expanding with the speed of light. It is generated by a snapped cosmic string, whose remnants are two semi-infinite strings located along the $z$ axis outside the impulsive wave. Any point $P$ on the impulse is described by two angles $\alpha$ and $\gamma$ which characterize its projections to the $(x,z)$ and $(y,z)$ planes, respectively (cf. Fig. \ref{['figure3']}).
• Figure 5: The function ${\alpha^-(\alpha^+)}$ which determines the displacement of the position of a particle when it crosses the impulse generated by a snapped cosmic string. The curves correspond to different values of the deficit angle parameter ${\delta=0, 0.1,0.2,\ldots,0.8}$.