Block diagonalisation of four-dimensional metrics

Details Block diagonalisation of four-dimensional metrics Block diagonalisation of four-dimensional metrics

2008-09-19

arxiv.org/abs/0809.3327v2

Class. Quantum Grav. 26 (2009) 235014

Mathematics

Differential Geometry

20 pages, Latex, no figures. Typos corrected and some brief comments added in Sections 1 and 3. Version published in Classical

Scientific paper

10.1088/0264-9381/26/23/235014

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.

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