2 Velocity

We will henceforth use, conform [1], the notation $\chi=ds/dt=cd\tau/dt$ for the speed in the proper time dimension and $v$ for spatial speed. The Euclidean components of the 4-vector for velocity are the derivatives of the displacement 4-vector components with respect to $t$ :

$$\left(\frac{d(ct)}{dt}\right)^2=\left(\frac{ds}{dt}\right)^2+\left(\frac{dA}{dt}\right)^2$$ (7)

or

$$c^2=\chi^2+v^2$$ (8)

Figure 2: Geometric visualization of Minkowski and Euclidean velocity 4-vectors.

In Fig. 2 this is again illustrated.

The corresponding Minkowski 4-vector components for velocity are derived by multiplying the time derivative of the Minkowski displacement 4-vector with the factor $\gamma=1/\sqrt{1-v^2/c^2}$ ,

$$c^2=\gamma^2(c^2-v^2)$$ (9)

which is the same as taking the derivative with respect to the proper time $\tau=t/\gamma$ instead of $t$ . Without this factor the 4-vector does not yield a Lorentz invariant value.

Figure 3: Geometry of Minkowski and Euclidean velocity 4-vectors.

Figure 3 shows the background of this multiplication and here it shows that, from an Euclidean perspective, the Minkowski 4-vector components seem to have a pure mathematical function only. The components of the Euclidean 4-vector on the other hand have an intuitive physical meaning: they represent the actual speeds in Euclidean space and proper time. The vectorsum of these has universal magnitude $c$ and rotates in 4D if the spatial velocity accelerates in 3D.