The hyperbolic Ernst equation in a triangular domain

The collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann–Hilbert problem. The formulation of the Riemann–Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle’s corners—this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann–Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.

Icosahedral Symmetry and Quantum Gravity

In this note we will show that Theta functions are a solution of the icosahedron equation and also a solution of the Ernst equation for the stationary axisymmetric case of Einstein’s gravitational equation.

GRAVITY-ASSISTED SOLUTION OF THE MASS GAP PROBLEM FOR PURE YANG–MILLS FIELDS

In 1979, Louis Witten demonstrated that stationary axially symmetric Einstein field equations and those for static axially symmetric self-dual SU (2) gauge fields can both be reduced to the same (Ernst) equation. In this paper, we use this result as point of departure to prove the existence of the mass gap for quantum source-free Yang–Mills (Y–M) fields. The proof is facilitated by results of our recently published paper, J. Geom. Phys.59 (2009) 600–619. Since both pure gravity, the Einstein–Maxwell and pure Y–M fields are described for axially symmetric configurations by the Ernst equation classically, their quantum descriptions are likely to be interrelated. Correctness of this conjecture is successfully checked by reproducing (by different methods) results of Korotkin and Nicolai, Nucl. Phys. B475 (1996) 397–439, on dimensionally reduced quantum gravity. Consequently, numerous new results supporting the Faddeev–Skyrme (F–S)-type models are obtained. We found that the F–S-like model is best suited for description of electroweak interactions while strong interactions require extension of Witten's results to the SU(3) gauge group. Such an extension is nontrivial. It is linked with the symmetry group SU (3) × SU (2) × U (1) of the standard model. This result is quite rigid and should be taken into account in development of all grand unified theories. Also, the alternative (to the F–S-like) model emerges as by-product of such an extension. Both models are related to each other via known symmetry transformation. Both models possess gap in their excitation spectrum and are capable of producing knotted/linked configurations of gauge/gravity fields. In addition, the paper discusses relevance of the obtained results to heterotic strings and to scattering processes involving topology change. It ends with discussion about usefulness of this information for searches of Higgs boson.