Block diagonalisation of four-dimensional metrics

TL;DR

This paper demonstrates that in four-dimensional spaces, it is possible to choose coordinates that make the metric block diagonal, and characterizes all such coordinate systems through coupled PDEs.

Contribution

It introduces the concept of doubly biorthogonal coordinate systems and derives the governing PDEs for their existence in four-dimensional metrics.

Findings

Existence of coordinate systems with block diagonal metrics in 4D

Characterization of these coordinates via coupled second-order PDEs

Framework for constructing such coordinate systems

Abstract

It is shown that, in four 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 $g_{\alpha\beta} = 0$ for $(\alpha, \beta) \in S$ 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.