imaginary Wick rotation Misner, Thorne and Wheeler Minkowski metric Euclidean metric general relativity
The actual form of ds above depends on the metric and on the choices for the X coordinate.
To make the time coordinate look like the space coordinates, it can be treated as imaginary: X0 = ict (this is called a Wick rotation).
According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X = ct, rather than a "disguised" Euclidean metric using ict as the time coordinate.
