The Belinski–Zakharov (inverse) transform is a nonlinear transformation that generates new exact solutions of the vacuum Einstein's field equation. It was developed by Vladimir Belinski and Vladimir Zakharov in 1978. The Belinski–Zakharov transform is a generalization of the inverse scattering transform. The solutions produced by this transform are called gravitational solitons (gravisolitons). Despite the term 'soliton' being used to describe gravitational solitons, their behavior is very different from other (classical) solitons. In particular, gravitational solitons do not preserve their amplitude and shape in time, and up to June 2012 their general interpretation remains unknown. What is known however, is that most black holes (and particularly the Schwarzschild metric and the Kerr metric) are special cases of gravitational solitons.
Contents
Introduction
The Belinski–Zakharov transform works for spacetime intervals of the form
where we use Einstein's summation convention for
In this case, Einstein's vacuum equation
where
The second set of equations is
Taking the trace of the matrix equation for
The Lax Pair
Consider the linear operators
where
The gist behind the inverse scattering transform is rewriting the nonlinear Einstein equation as an overdetermined linear system of equation for a new matrix function
By operating on the left-hand side of the first equation with
This means that the overdetermined linear Belinski–Zakharov equations are solvable simultaneously exactly when
Thus a solution of the nonlinear