Suppose that I have a two-by-two symmetric metric (no time) which has off-diagonal terms with the elements to be some function of the coordinates. My question is how to find a coordinate system in which the metric is diagonal and what constraints should the functions satisfy to do so. I know that I have to do a coordinate transformation to go to that frame but I haven't attempt to do it before, so what I am looking for is a reference that I can see how this is implemented.
2026-03-31 17:23:41.1774977821
Diagonalize an off-diagonal metric
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The two-by-two case can be solved by studying the existence of isothermal coordinates; these are systems in which the metric tensor is not only diagonalized, but in fact equal to multiples of the identity matrix. Their existence requires solving the Beltrami equation: this is nontrivial and in general does not have a simple algorithm.
Your best chance is to somehow identify a non-trivial harmonic (relative to the Laplace-Beltrami operator of your metric) function, and from their compute its harmonic conjugate. The two together will form a system of orthogonal coordinate systems. Computing harmonic conjugates are relatively straightforward: the harder part is to identify one non-trivial harmonic function in the first place.