Characteristic initial value problems for integrable hyperbolic reductions of Einstein's equations

TL;DR

This paper develops a unified method for solving characteristic initial value problems for integrable hyperbolic reductions of Einstein's equations, using linear systems and integral evolution equations to construct solutions from initial data.

Contribution

It introduces a general approach utilizing linear systems and scattering matrices to solve initial value problems for Einstein's equations with two symmetries, advancing solution construction techniques.

Findings

Solutions can be expressed in quadratures from integral evolution equations.

The method applies to space-times with two commuting isometries.

Initial data fully determine the solution via scalar kernels and integral equations.

Abstract

A unified general approach is presented for construction of solutions of the characteristic initial value problems for various integrable hyperbolic reductions of Einstein's equations for space-times with two commuting isometries in General Relativity and in some string theory induced gravity models. In all cases the associated linear systems of similar structures are used, and their fundamental solutions admit an alternative representations by two ``scattering'' matrices of a simple analytical structures on the spectral plane. The condition of equivalence of these representations leads to the linear ``integral evolution equations'' whose scalar kernels and right hand sides are determined completely by the initial data for the fields specified on the two initial characteristics. If the initial data for the fields are given, all field components of the corresponding solution can be expressed in quadratures in terms of a unique solution of these quasi - Fredholm integral evolution equations.