Asymmetric collision of gravitational plane waves: A new class of exact solutions

A new three-parameter class of solutions to the Einstein vacuum equations is presented which represents the collision of a pair of gravitational plane waves. Depending on the choice of the parameters, one of the colliding waves has a smooth or unbounded wavefront, or it is a shock, or impulsive, or shock accompanied by an impulsive wave, while the second is any of the above types. A subfamily of the solutions develops no curvature singularity in the interaction region formed by the colliding waves.Expanded version of a talk presented at the Third National Workshop on Recent Developments in Gravitation, September 12–16, 1988, Ioannina, Greece.     

Riemann–Hilbert formulation for the KdV equation on a finite interval

The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I. Hitzazis, D. Tsoubelis, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Inertial frames and tidal forces along the symmetry axis of the Kerr spacetime

The gravitational field along the symmetry axis of the Kerr spacetime is examined. The equations of parallel transport along this axis are solved for the timelike geodesics case, and the corresponding tidal tensor is constructed.     

Impulsive shells of null dust colliding with gravitational plane waves

A two-parameter family of solutions of Einstein's equations, corresponding to distribution valued stress-energy tensors with support on a (pair of intersecting) null hypersurface(s), is presented. They describe the collision of infinitely thin shells of null dust colliding with shells of the same kind and/or gravitational plane waves. For a subclass of this new family of solutions, the typical spacelike singularity that develops after the collision and forms the future boundary of the interaction region gives its place to a nonsingular Killing-Cauchy horizon.     

The Inverse Spectral Method for Colliding Gravitational Waves

The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 x < y 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possesses a Lax pair formulation. Using the simultaneous spectral analysis of this Lax pair we study the above Goursat problem as well as its linearized version. It is shown that the linear problem reduces to a scalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problem reduces to a 2 × 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.     

Polar orbits in the Kerr space-time

The motion of test particles in polar orbit about the source of the Kerr field of gravity is studied, using Carter's first integrals for timelike geodesies in the Kerr space-time. Expressions giving the angular coordinates of such particles as functions of the radial one are derived, both for the case of a rotating black hole as well as for that of a naked singularity.     

On the systematic approach to the classification of differential equations by group theoretical methods

Complete symmetry groups enable one to characterise fully a given differential equation. By considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of Bateman, Monge–Ampère and Born–Infeld as special cases. We develop a symbolic algorithm to decrease the complexity of the calculations involved.