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 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. Introduction. The Belinski–Zakharov transform works for spacetime intervals of the form where we use Einstein's summation convention for formula_2. It is assumed that both the function formula_3 and the matrix formula_4 depend on the coordinates formula_5 and formula_6 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 formula_7 decomposes into two sets of equations for the matrix formula_4 and the function formula_3. Using light-cone coordinates formula_10, the first equation for the matrix formula_11 is where formula_13 is the square root of the determinant of formula_11, namely The second set of equations is Taking the trace of the matrix equation for formula_11 reveals that in fact formula_13 satisfies the wave equation The Lax pair. Consider the linear operators formula_21 defined by where formula_24 is an auxiliary complex spectral parameter. A simple computation shows that since  formula_13 satisfies the wave equation, formula_26. This pair of operators commute, this is the Lax pair. 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 formula_27. Consider the Belinski–Zakharov equations: By operating on the left-hand side of the first equation with formula_30 and on the left-hand side of the second equation with formula_31 and subtracting the results, the left-hand side vanishes as a result of the commutativity of formula_31 and formula_30. As for the right-hand side, a short computation  shows that indeed it vanishes as well precisely when formula_11 satisfies the nonlinear matrix Einstein equation. This means that the overdetermined linear Belinski–Zakharov equations are solvable simultaneously exactly when formula_11 solves the nonlinear matrix equation . Actually, one can easily restore formula_11 from the matrix-valued function formula_37 by a simple limiting process. Taking the limit formula_38 in the Belinski-Zakharov equations and multiplying by formula_39 from the right gives Thus a solution of the nonlinear formula_11 equation is obtained from a solution of the linear Belinski–Zakharov equation by a simple evaluation