Belinski–Zakharov transform

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 than other (classical) solitons. In particular, gravitational solitons do not preserve their amplitude and shape in time, and up to the present moment, their general interpretation is 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.

Introduction

The Belinski-Zakharov transform works for spacetime intervals of the form

$ds^{2}=f(-d(x^{0})^{2}+d(x^{1})^{2})+g_{ab}dx^{a}dx^{b}$

where we use Einstein's summation convention for $a,b=2,3$. It is assumed that both the function $f$ and the matrix $g=g_{ab}$ depend on the coordinates $x^{0}$ and $x^{1}$ only. Despite being a specific form of the spacetime interval that depends only on two variables, it includes a great number of interesting solutions a special cases, such as the Schwarzschild metric, the Kerr metric, Einstein-Rosen metric, and many others.

In this case, Einstein's vacuum equation $R_{\mu \nu }=0$ decomposes into two sets of equations for the matrix $g=g_{ab}$ and the function $f$. Using light-cone coordinates $\zeta =x^{0}+x^{1},\eta =x^{0}-x^{1}$, the first equation for the matrix $g$ is

$(\alpha g_{,\zeta }g^{-1})_{,\eta }+(\alpha g_{,\eta }g^{-1})_{,\zeta }=0$

where $\alpha$ is the square root of the determinant of $g$, namely

$\det g=\alpha ^{2}$

The second set of equations is

$(\ln f)_{,\zeta }={\frac {(\ln \alpha )_{,\zeta \zeta }}{(\ln \alpha )_{,\zeta }}}+{\frac {\alpha }{4\alpha _{,\zeta }}}tr(g_{,\zeta }g^{-1}g_{,\zeta }g^{-1})$

$(\ln f)_{,\eta }={\frac {(\ln \alpha )_{,\eta \eta }}{(\ln \alpha )_{,\eta }}}+{\frac {\alpha }{4\alpha _{,\eta }}}tr(g_{,\eta }g^{-1}g_{,\eta }g^{-1})$