Block diagonalisation of four-dimensional metrics

Description/Abstract

It is shown that, in 4-dimensions, it is possible to introduce coordinates so that
an analytic metric locally takes block diagonal form. i.e. one can find coordinates such that
g = 0 for (, ) 2 S where S = {(1, 3), (1, 4), (2, 3), (2, 4)}. We call a coordinate system in
which the metric takes this form a ‘doubly biorthogonal coordinate system’. We show that all
such coordinate systems are determined by a pair of coupled second-order partial differential
equations