A note on the null surface formulation of GR (Q5947445)

This paper is devoted to General Relativity (GR). The interesting nature along with the structure of the singularities of wavefronts in a flat space-time is investigated. The null surface formulation of GR reformulates the classical theory in terms of two real functions \(u=Z(x^a,\zeta ,\overline\zeta)\) and \(\Omega (x^a,\zeta ,\overline\zeta)\) on the bundle of null directions over the space-time manifold (locally \(\mathcal M\)\(\times S^2\)). The singularities of the arbitrary null surfaces are denoted as \(Z(x^a,\zeta ,\overline\zeta)=\) const. The vacuum Einstein equations in terms of the quantities \(\Lambda (x^a,\zeta ,\overline\zeta)\equiv \partial^2Z\) and \(\overline\Lambda (x^a,\zeta ,\overline\zeta)\equiv \overline\partial^2Z\) and also \(\Omega (x^a,\zeta ,\overline\zeta)\), \(q\equiv 1-\Lambda_{,1}\overline\Lambda_{,1}\), \(Q\), \(W\) have the form: \(D^2\Omega =Q\Omega \), \(\partial \Omega =(1/2)W\Omega \), \(\partial\Lambda_{,1}-2\Lambda_{,-}=[W+\partial (\ln q)]\Lambda_{,1}\), where \(Q\), \(W\) are functions of \(q, \Lambda_{,1}, \overline\Lambda_{,1},\Lambda_{,-}, \overline\Lambda_{,-}\) and their derivatives \(D\equiv \theta_1^a\partial_a=\partial_1=\partial /\partial R\), \(\partial_i\Phi =\theta_{,i}^a\partial_a\Phi =\Phi_{,i}\) (\(i=0,+,-,1\)). Another interesting result in this paper is that the Minkowski space-time with non-trivial null surfaces is a solution of the null surface approach to GR.