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1 Displacement

If the Minkowski 4-vector components for displacement in 4D space-time

$$ds^2=d(ct)^2-dx^2-dy^2-dz^2$$ (1)

are alternatively written in Euclidean form (see also [1]) they read:

$$d(ct)^2=ds^2+dx^2+dy^2+dz^2$$ (2)

Equations 1 and 2 both contain the same information but the essential difference is that the roles of the variables have switched. In Eq. 1, $ds^2$ is the Lorentz invariant. In Eq. 2, $dct^2$ is the Lorentz invariant. The mathematics for such a switch remain consistent with special relativity if it is assumed that the vectorsum of velocities in all 4 Euclidean dimensions has constant magnitude $c$ . That means that the speed in the Euclidean time dimension, which is now actually formed by the proper time $\tau$ must be:

$$v_{\tau}^2=c^2-v_{space}^2$$ (3)

The displacement in the proper time dimension for a moving object in an interval $dt$ (according to an observer at rest) then equals:

$$v_{\tau}^2dt^2 = c^2dt^2-v_x^2dt^2-v_y^2dt^2-v_z^2dt^2$$

$$= c^2dt^2-dx^2-dy^2-dz^2$$

$$= ds^2$$ (4)

So as a result of the assumption that was made about the universal speed $c$ in 4D Euclidean space-time, $ds$ is now no longer the invariant Minkowski displacement but the displacement in the proper time dimension. The factor $dct$ that played an equivalent role in the Minkowski 4-vector has become the invariant 4D displacement in Euclidean space-time.

Figure 1: Geometric visualization of Minkowski and Euclidean displacement 4-vectors. Spatial displacement is represented as $dA$ .

This is visualized in Fig. 1. Here the spatial displacement $\sqrt{dx^2+dy^2+dz^2}$ is written as a single variable $dA$ . The 'Minkowski triangle' (left) then is:

$$ds^2=dct^2-dA^2$$ (5)

and the 'Euclidean triangle' (right) is:

$$dct^2=ds^2+dA^2$$ (6)

The dotted lines in Fig. 1 represent the values for $dA$ and $dct$ that would result from a Lorentz transformation.

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