Geodesics in nonexpanding impulsive gravitational waves with Λ. II

Abstract

Abstract. We investigate all geodesics in the entire class of nonexpanding impulsive gravitational waves propagating in an (anti-)de Sitter universe using the distributional metric. We extend the regularization approach of part I (Geodesics in nonexpanding impulsive gravitational waves with Λ, Part I, Class. Quantum Grav., 33(11):115002, 2016) to a full nonlinear distributional analysis within the geometric theory of generalized functions. We prove global existence and uniqueness of geodesics that cross the impulsive wave and hence geodesic completeness in full generality for this class of low regularity spacetimes. This, in particular, prepares the ground for a mathematically rigorous account on the 'physical equivalence' of the continuous with the distributional 'from' of the metric.

Cut-and-paste for impulsive gravitational waves with Λ : The geometric picture

“…3]). Then the two parts are reattached in the following specific way: The null geodesic generators of the two halves are joined according to (39), or (28) with (27) substituted into (29) in the 5-dim picture, depending on H i (Z 2 , Z 3 , Z 4 ). We will refer to this 5-dim picture in the remainder of this discussion, since it is more directly related to Figure 2.…”

“…Indeed, in this case we have A j = 0 (cf. (27), (29)) and the interaction with such an impulse does not induce any motion in the Z j -directions since by (28) Z 2 (λ) = 0 = Z 3 (λ) for all λ. So the null generators of dS − and dS + depicted in Figure 2 are indeed matched to one another.…”

“…There the key step was a detailed study of the geodesics in the distributional form of the metric. This study has recently been generalized in [31,29] to the Λ = 0 case, and we will employ these results as well as those of [26] to give a clear geometric interpretation of the transformation (5) in the general case.…”

“…Here we draw on recent works [31,29] which are in turn based on the analysis of [26]. The fact that it is essentially the behaviour of the null geodesics that determines the Penrose junction conditions, was first explicitly noted for Λ = 0 in [34] and put to use in [11].…”

“…We have to base our approach on the rigorous results on such geodesics in this representation [26,31,29] rather than on the analysis using the 4-dim approach given in [27], which uses the continuous form of the metric (3). Indeed, in the latter the transformation (5) was explicitly used to derive the form of the geodesics and hence such an approach would lead to a circular argument.…”