Colliding Plane Impulsive Gravitational Waves

When two non-interacting plane impulsive gravitational waves undergo ahead-on collision, the vacuum interaction region between the waves after thecollision contains backscattered gravitational radiation from both waves. Thetwo systems of backscattered waves have each got a family of rays (nullgeodesics) associated with them. We demonstrate that if it is assumed that aparameter exists along each of these families of rays such that the modulus ofthe complex shear of each is equal then Einstein's vacuum field equations, withthe appropriate boundary conditions, can be integrated systematically to revealthe well-known solutions in the interaction region. In so doing the mysterybehind the origin of such solutions is removed. With the use of the fieldequations it is suggested that the assumption leading to their integration maybe interpreted physically as implying that the energy densities of the twobackscattered radiation fields are equal. With the use of different boundaryconditions this approach can lead to new collision solutions.Comment: 21 pages, LaTeX2