Colliding Plane Impulsive Gravitational Waves

TL;DR

This paper demonstrates a systematic method to derive solutions for the interaction region of colliding plane impulsive gravitational waves, revealing the physical significance of certain assumptions and enabling the discovery of new solutions.

Contribution

It introduces a new approach to integrate Einstein's vacuum equations for colliding gravitational waves using shear assumptions, clarifying the origin of known solutions and allowing for new ones.

Findings

Systematic integration of Einstein's equations for wave collision

Physical interpretation of shear assumptions as equal energy densities

Potential to generate new collision solutions with different boundary conditions

Abstract

When two non-interacting plane impulsive gravitational waves undergo a head-on collision, the vacuum interaction region between the waves after the collision contains backscattered gravitational radiation from both waves. The two systems of backscattered waves have each got a family of rays (null geodesics) associated with them. We demonstrate that if it is assumed that a parameter exists along each of these families of rays such that the modulus of the complex shear of each is equal then Einstein's vacuum field equations, with the appropriate boundary conditions, can be integrated systematically to reveal the well-known solutions in the interaction region. In so doing the mystery behind the origin of such solutions is removed. With the use of the field equations it is suggested that the assumption leading to their integration may be interpreted physically as implying that the energy densities of the two backscattered radiation fields are equal. With the use of different boundary conditions this approach can lead to new collision solutions.