2 The Time Dimension

Minkowski interpretations of special relativity treat time differently from spatial dimensions, showing from the Minkowski metric where the time component is given the opposite sign. Some alternative interpretations (e.g. [1-4]) seek positive definite Euclidean metrics for space-time. Also in this article, the time dimension will be treated as a regular fourth dimension in Euclidean space-time.

If time is considered a fourth spatial dimension, then it must show properties similar to those found in the other three. In there we encounter properties like length, speed, acceleration, curvature etc., expressed respectively as $s$ , $ds/dt$ , $d^2s$ $/dt^2$ , $R_{bcd}^a$ etc. Of those properties, a single one can be measured relatively easily in the time dimension: the 'length' or timeduration $\Delta t$ . That raises the question of how a hypothetical speed in time, let us call it $\chi$ , should be expressed mathematically. In [6], Greene has given a derivation of an expression that can be used as the velocity component in the Euclidean time dimension. Rewriting the usual Minkowski invariant

$$c^2=(dct/d\tau)^2-(dx/d\tau)^2-(dy/d\tau)^2-(dz/d\tau)^2$$ (1)

into Euclidean form:

$$c^2=(cd\tau/dt)^2+(dx/dt)^2+(dy/dt)^2+(dz/dt)^2$$ (2)

one arrives at the temporal velocity component

$$\chi=cd\tau/dt$$ (3)

This clearly defines $\tau$ as the coordinate for the fourth Euclidean dimension, and it says that the velocity components in all four dimensions involve derivatives with respect to $t$ , which then can no longer represent the fourth dimension. It can only be an extra, fifth dimension, $x_5$ (provided we index the other four $x_1$ , $x_2$ , $x_3$ , and $x_4$ respectively, with $\tau=x_4$ ). This fifth dimension is sometimes treated as a parameter in Euclidean approaches similar to special relativity, e.g. in [1,2], but here it will be treated as a genuine extra Euclidean dimension. A general expression for speed in the time dimension (henceforth refereed to as time-speed) is now:

$$\chi=cdx_4/dx_5$$ (4)

while the scalar value of time-speed $\chi$ is

$$\chi=\sqrt {\vphantom{/}c^2-v^2}$$ (5)

The general expression for spatial velocity components in 4D Euclidean space-time is

$$v_i=dx_i/dx_5$$ (6)