A continuous Riemann-Hilbert problem for colliding plane gravitational waves

Abstract

We present the foundations of a new solution technique for the characteristic initial value problem (IVP) of colliding plane gravitational waves. It has extensive similarities to the approach of Alekseev and Griffiths in 2001, but we use an inverse scattering method with a Riemann-Hilbert problem (RHP), which allows for a transformation to a continuous RHP with a solution given in terms of integral equations for non-singular functions. Ambiguities in this procedure lead to the construction of a family of spacetimes containing the solution to the IVP. Therefore the described technique also serves as an interesting solution generating method. The procedure is exemplified by extending the Szekeres class of colliding wave spacetimes with 2 additional real parameters. The obtained solution seems to feature a limiting case of a new type of impulsive waves, which are circularly polarised.

34 pages, 9 figures, section 7 and appendix added

References in corpus (5)

• Progress in relativistic gravitational theory using the inverse scattering method
• Solving the characteristic initial value problem for colliding plane gravitational and electromagnetic waves
• Collision of plane gravitational and electromagnetic waves in a Minkowski background: solution of the characteristic initial value problem
• The stability of Killing-Cauchy horizons in colliding plane wave space-times
• A continuous Riemann-Hilbert problem for colliding plane gravitational waves

Cited by in corpus (2)

• Asymptotics to all orders of the Euler--Darboux equation in a triangle
• A continuous Riemann-Hilbert problem for colliding plane gravitational waves