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

Einstein's equations are nonlinear, therefore, when gravitational waves meet, they must interact. This interaction process has been studied in detail for some cases, particularly those involving plane waves. To understand the structure of the space-time resulting from collisions of this type, many solutions have been generated. However, these have been obtained by first taking a "candidate" resulting space-time, and then extending it back to give rise to the originating waves. While these techniques are not too complex, it is not an easy task to obtain physically acceptable initial waves, and this is the greatest disadvantage of this indirect method. The main aim of this thesis is to consider a direct approach, to find a method that can overcome the difficulties indicated above, giving rise to solutions from arbitrary colliding plane waves. A well posed initial value problem is formulated for the collinear case. This is achieved by making use of generalised Abel transforms. This method is successfully tested for some particularly well-known cases. However, when it is applied to more general cases, a number of problems arise. Along the direct and inverse transformation process, there are several successive integrations involved, and it is in these integrations that the main difficulties appear, as the integrands themselves contain elliptic integrals. Nevertheless, a final way is found to obtain a final solution, which gives the solution as a series expansion involving hypergeometric functions. Consequently, assuming we can obtain the spectral functions generated by the Abel transforms, the problem would be theoretically solved, although the calculations tend to become extremely complicated when arbitrary colliding waves are taken.