The characteristic initial value problem for colliding plane waves: the linear case

Abstract

The physical situation of the collision and subsequent interaction of plane gravitational waves in a Minkowski background gives rise to a well-posed characteristic initial value problem in which initial data are specified on the two null characteristics that define the wavefronts. In this paper, we analyse how the Abel transform method can be used in practice to solve this problem for the linear case in which the polarization of the two gravitational waves is constant and aligned. We show how the method works for some known solutions, where problems arise in other cases, and how the problem can always be solved in terms of an infinite series if the spectral functions for the initial data can be evaluated explicitly.

Travelling waves in expanding spatially homogeneous space–times

“…These solutions certainly can be used in farther investigation of propagation of waves with arbitrary amplitudes in curved space-times as well as their collision and nonlinear interaction in this background. Besides that, it is also important for us here, that the classes of solutions (12) and (13) in the above considerations arose from the ansatz of invariance of the solution under the Kramer-Neugebauer transformation of the solution space. This suggests the opportunity to use the similar ansatz for construction of classes of travelling wave solutions for integrable reductions of some other gravity models in four and higher dimensions.…”

“…For different choices of α as a solution of the equation α uv = 0 for which α = const are time-like or space-like surface, the appropriate transformations u → h(u), v → g(v) allow to choose t = α or x = α respectively. The case α = t is the most interesting because (12) with ψ + (u) = 0 as well as (13) with ψ − (u) = 0 represent a cosmological Kasner solution with a special set of exponents (p x , p y , p z ) = (− 3 13 , 4 13 , 12 13 ) ‡ and therefore, for ψ + (u) = 0 or ψ − (v) = 0 these solutions describe the plane-fronted travelling gravitational waves which profiles are determined by the arbitary functions ψ + (u) in (12) and ψ − (v) in (13) and which propagate on this specific Kasner background in the positive and negative directions of the x-axis respectively. ‡ The usual form of the family of vacuum cosmological Kasner solutions is ds 2 = −dτ 2 + τ 2p (x) dx 2 + τ 2p (y) dy 2 + τ 2p (z) dz 2 , where the exponents should satisfy p (x) + p (y) + p (z) = 1, p 2 (x) + p 2 (y) + p 2 (z) = 1.…”

“…It is interesting to note here that this Bonnor transformation changes also the parameters of the background Kasner solution so that from the waves ( 12) and ( 13) on the Kasner background with the exponents (p x , p y , p z ) = (− 3 13 , 4 13 , 12 13 ) we obtain pure electromagnetic waves propagating through the symmetric Kasner background with the exponents (p x , p y , p z ) = (− 1 3 , 2 3 , 2 3 ) along its axis of symmetry. However, we do not follow this way, but instead we use some ansatz (obviously inspired by applicqation of Bonnor transformation to (12) or (13)) which leads to a class of solutions depending on two arbitrary functions which determine the amplitudes of electromagnetic waves of both polarizations.…”

“…To construct the solutions for the travelling "vacuum" (A = 0) bosonic waves, we use the same way as it was used in one of the previous sections of this paper for construction of pure vacuum travelling gravitational waves in General Relativity -the solutions (12) and (13). Namely, we consider the solutions of matrix Ernst equations (32) which are fixed points of the transformation (36).…”

Asymptotics to all orders of the Euler–Darboux equation in a triangle

“…e.g. [11,14,15]) and the main purpose of Theorem 1 is to provide a formulation that is suitable for our needs. We discuss regularity, the behavior at the boundary of D, and uniqueness of the solution under reasonable assumptions on the boundary data.…”