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Register a new userThe global existence, uniqueness and C^1-regularity of geodesics in nonexpanding impulsive gravitational waves
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We study geodesics in the complete family of nonexpanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we prove existence and uniqueness of continuously differentiable geodesics (in the sense of Filippov) and use a C^1-matching procedure to explicitly derive their form.
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We consider the geodesic equation in impulsive pp-wave space-times in Rosen form, where the metric is of Lipschitz regularity. We prove that the geodesics (in the sense of Caratheodory) are actually continuously differentiable, thereby rigorously justifying the $C^1$-matching procedure which has been used in the literature to explicitly derive the geodesics in space-times of this form.
Impulsive gravitational waves in Minkowski space were introduced by Roger Penrose at the end of the 1960s, and have been widely studied over the decades. Here we focus on non-expanding waves which later have been generalised to impulses travelling in all constant-curvature backgrounds, that is also the (anti-)de Sitter universe. While Penroses original construction was based on his vivid geometric `scissors-and-paste approach in a flat background, until now a comparably powerful visualisation and understanding have been missing in the ${Lambda ot=0}$ case. In this work we provide such a picture: The (anti-)de Sitter hyperboloid is cut along the null wave surface, and the `halves are then re-attached with a suitable shift of their null generators across the wave surface. This special family of global null geodesics defines an appropriate comoving coordinate system, leading to the continuous form of the metric. Moreover, it provides a complete understanding of the nature of the Penrose junction conditions and their specific form. These findings shed light on recent discussions of the memory effect in impulsive waves.
We consider a family of globally stationary (horizonless), asymptotically flat solutions of five-dimensional supergravity. We prove that massless linear scalar waves in such soliton spacetimes cannot have a uniform decay rate faster than inverse logarithmically in time. This slow decay can be attributed to the stable trapping of null geodesics. Our proof uses the construction of quasimodes which are time periodic approximate solutions to the wave equation. The proof is based on previous work to prove an analogous result in Kerr-AdS black holes cite{holzegel:2013kna}. We remark that this slow decay is suggestive of an instability at the nonlinear level.
Recently the memory effect has been studied in plane gravitational waves and, in particular, in impulsive plane waves. Based on an analysis of the particle motion (mainly in Baldwin-Jeffery-Rosen coordinates) a velocity memory effect is claimed to be found in [P.-M. Zhang, C. Duval, and P. A. Horvathy. Memory effect for impulsive gravitational waves. Classical Quantum Gravity, 35(6):065011, 20, 2018]. Here we point out a conceptual mistake in this account and employ earlier works to explain how to correctly derive the particle motion and how to correctly deal with the notorious distributional Brinkmann form of the metric and its relation to the continuous Rosen form.
We investigate a class of gravitational pp-waves which represent the exterior vacuum field of spinning particles moving with the speed of light. Such exact spacetimes are described by the original Brinkmann form of the pp-wave metric including the often neglected off-diagonal terms. We put emphasis on a clear physical and geometrical interpretation of these off-diagonal metric components. We explicitly analyze several new properties of these spacetimes associated with the spinning character of the source, such as rotational dragging of frames, geodesic deviation, impulsive limits and the corresponding behavior of geodesics.
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