'Newtonian' time in general relativity

Abstract

A result recently given by Eisenhart suggests an interesting application in general relativity. According to it, we can choose co-ordinates in which a line element showing spherical symmetry would take the form: ds² = - r²(dθ² + sin²θdφ²) + γdt² + 2adrdt, γ = γ(r,t), a = a(r,t) and a radial null vector w^μ will have w² = w³ = w⁴ = 0, so that the velocity of light along radial directions (given by w¹/w⁴) is infinite. Hence we may call the co-ordinates (r,t) the 'Newtonian' co-ordinates.