Special relativity
Comparison between flat Euclidean space and Minkowski space

Special relativity uses a "flat" 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.

In 3D space, the differential of distance (line element) ds is defined by d s 2 = d x ⋅ d x = d x 1 2 + d x 2 2 + d x 3 2, {\displaystyle ds^{2}=d\mathbf {x} \cdot d\mathbf {x} =dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2},} where d x = (dx, dx, dx) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X derived from time, such that the distance differential fulfills d s 2 = − d X 0 2 + d X 1 2 + d X 2 2 + d X 3 2, {\displaystyle ds^{2}=-dX_{0}^{2}+dX_{1}^{2}+dX_{2}^{2}+dX_{3}^{2},} where d X = (dX, dX, dX, dX) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10-1). Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski spacetime.

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: X = 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.

Some authors use X = t, with factors of c elsewhere to compensate; for instance, spatial coordinates are divided by c or factors of c are included in the metric tensor. These numerous conventions can be superseded by using natural units where c = 1. Then space and time have equivalent units, and no factors of c appear anywhere.
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