Note: this is Euclidean relativity, not Minkowski.
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3 Using Time-Speed in Special Relativity

It will be shown that the Lorentz transformation equations for length and time can be reproduced from the Euclidean context.

Maintaining orthogonality for all Euclidean dimensions, Eqs. (2) and (5) imply that the axes for the proper time dimension and the spatial dimension in the direction of the initial motion must have rotated for the moving object, as seen from the rest frame of the observer, in fact defining Lorentz transformations as rotations in SO(4). See also [1], where this is referred to as a Relative Euclidean Space-Time. In the approach that follows now, these axes will therefor (unlike in the Minkowski diagram) both rotate in the same direction, clockwise or counter clockwise, depending on the direction of the motion. The diagrams in Fig. 1 and Fig. 2 should illustrate this.

**Figure 1:** 4D representation of an observer at O and an object A, both at rest.
$\begin{figure}\centerline{\includegraphics[width=0.5\textwidth]{figure1.eps}}\end{figure}$

Figure 1 depicts an object A at rest together with an observer at O, also at rest. The horizontal axis shows both the spatial dimensions , , for the object A as well as the spatial dimensions for the observer. The vertical axis shows both time dimensions with notation conform Eq. (2), so $x_4=c\tau$ . Due to object A being at rest, relative to the observer, the axes overlap. The circle is just a tool to better show the rotation that will be depicted in Fig. 2.

**Figure 2:** Object A in motion relative to observer. The dimensional axes of object A have rotated relative to the observer.
$\begin{figure}\centerline{\includegraphics[width=0.5\textwidth]{figure2.eps}}\end{figure}$

Definitions are as follows:

Vector $\mathbf{C}$ indicates the 4D velocity, having magnitude , of object A.
Vector $\mathbf{V}$ , of magnitude , and $\mathbf{X}$ , of magnitude $\chi$ , are the projections of this velocity $\mathbf{C}$ on, respectively, the spatial dimensions and the proper time dimension of the observer.
indicates the proper length of object A in the spatial direction in the rest frame of object A (in this example is also set to ).
and are, respectively, the projections of this proper length on the spatial dimensions and the proper time dimension of the observer.

In Fig. 2, object A moves with speed relative to the observer. This leads to a relative rotation of dimensions and such that $\mathbf{V}$ is the projection of the original 4D velocity $\mathbf{C}$ of object A on the axis of the observer at rest. The situation is examined at the instant where .

The Lorentz transformation equation for is

$\displaystyle x'=\gamma(x-vt)$

(7)

where

$\displaystyle \gamma =1/\sqrt {1-v^2/c^2}$

(8)

but this factor can also be written as

$\displaystyle \gamma=c/\sqrt {c^2-v^2}=c/\chi$

(9)

leading to

$\displaystyle x'=c(x-vt)/\chi$

(10)

, the length of object A will be contracted, as measured by the observer, according to

$\displaystyle x=x'\chi/c$

(11)

so the contraction of length

can be written as

$\displaystyle l=l'\chi/c$

(12)

which shows that

, as measured by the observer at rest, is indeed the goniometric projection of the proper length

on the

axis.

Arrow is the projected 'length' component of the moving object A on the proper time axis of the observer as a result of the rotation of the dimension . This length is the manifestation of the difference in proper time (the non-simultaneity) between the endpoints of object A in motion according to the Lorentz transformation equation for time: