Time Machine Tales by Paul J. Nahin

Author:Paul J. Nahin
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Writing x 1 = ϕ and x 2 = θ yields the more general form

or, using the symmetry condition g 12 = g 21, we have

From this we immediately have g 11 = a 2 sin 2(θ), g 22 = a 2, and g 12 = g 21 = 0.

Now, suppose we ask whether it is possible to find some new coordinate system (with variables and ) in which the invariant (ds)2 is given by the flat Euclidean metric . In such a coordinate system (if it exists) we would have the ‘flatness conditions’ of and . With such a change of coordinates, each of our original ϕ, θ coordinates would generally be a function of both of the new coordinates—that is, and . Thus, writing the total differential of a function of two variables (see note 96 again), we have

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