Propagation and interaction of gravitational waves in some expanding backgrounds

Gravitational Waves Propagating into Friedmann–Robertson–Walker Universes

“…However, complete metrics were not given and the structure of the wavefront was not analysed. A method for obtaining complete solutions was later described in [9].…”

“…We will describe the waves by considering the most general form of the solution of the main field equation for a distinct wavefront. Using the method outlined in [9], we will complete the integration of the subsidiary equations, thus determining the whole metric. We will also analyse the structure of the wavefront in each case.…”

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

“…We may, however, recall that there exists a family of self-similar solutions of (24) that have been described in [14]- [16] which represent waves with distinct wavefronts. These were originally presented in the context of different backgrounds, and in a form in which the wavefront u = 0 is given by f = 0.…”

“…In order to consider the wave with the wavefront u = 0, f = 1 2 , we replace the coordinates f and g in [14]- [16] by f − 1 2 and g + 1 2 respectively. We then introduce new coordinates τ, ζ in the interaction region defined by…”

“…On the other hand, it may well not be possible to explicitly evaluate the quadratures for the remaining metric function M . Nevertheless, many terms can be evaluated using the methods described in [14]- [16] and some qualitative properties may be deduced. The solution described above may be considered to be valid throughout the interaction region f < 1 2 , g < 1 2 and f + g > 0.…”