A continuous Riemann-Hilbert problem for colliding plane gravitational waves

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Written by Stefan Palenta

I will present the foundations of a new solution technique for the characteristic initial value problem (IVP) of colliding plane gravitational waves. The corresponding spacetime features a two-dimensional orthogonally transitive group of isometries essentially reducing the Einstein equations to the hyperbolic Ernst equation. Its solution can be constructed via the so-called inverse scattering method using a linear system of partial differential equations and a Riemann-Hilbert problem (RHP). Inevitable nonanalytic behaviour of the initial data at the wavefronts leads to singularities in the integral equation determining the RHP solution. Therefore, a transformation to a continuous RHP with a solution given in terms of non-singular integral equations is introduced. Ambiguities in this procedure lead to the construction of a family of spacetimes containing the solution to the IVP. Hence the described technique may also serve as an interesting solution generating method.