3 Acceleration

Figure 4 shows the geometry for the components of, respectively, the Minkowski 4-vector for acceleration $\mathbf{\alpha_M}$ and the Euclidean one, $\mathbf{\alpha_E}$ . Here, a different geometric representation is chosen for the factors $\gamma c$ and $\gamma v$ to remain consistent with the direction of spatial velocity vector $\mathbf{v}$ (note that the scalar values of these factors have been used in the previous examples).

Figure 4: Geometry of Minkowski and Euclidean acceleration 4-vectors.

The components of the Euclidean acceleration 4-vector are

$$\alpha_{E_\mu}=\frac{dU_{E_\mu}}{dt}=(d\chi/dt,d\mathbf{v}/dt)$$ (10)

Any acceleration in 3D space corresponds to a rotation in SO(4) of the Euclidean 4D velocity vector $\mathbf{U_E}$ with invariant magnitude $c$ , implying that the acceleration 4-vector must always be orthogonal to it, or $\mathbf{\alpha_E}\cdot \mathbf{U_E}=0$ . This is in agreement with the Minkowski acceleration 4-vector $\mathbf{\alpha_M}$ with components

$$\alpha_{M_\nu} = dU_{M_\nu}/d\tau$$

$$= [\gamma d(\gamma c)/dt,\gamma d(\gamma \mathbf{v})/dt]$$

$$= \gamma( cd\gamma/dt, \mathbf{v}d\gamma/dt + \gamma d\mathbf{v}/dt)$$ (11)

that is also orthogonal to the Minkowski velocity 4-vector. The components for the Minkowski acceleration 4-vector have their origin in the change of the line elements $\gamma c$ and $\gamma v$ as is shown in the Figure. As these line elements were shown to have a mathematical function only in Euclidean space-time, the derived acceleration components will either. In the Euclidean 4-vector on the other hand, $dv/dt$ and $d\chi/dt$ form the orthogonal vector components and these maintain their intuitive physical interpretation, while the magnitude of $\mathbf{\alpha_E}$ is invariant under rotations in SO(4) (see also the next Section for the physical significance of this invariance). Note that, although $\mathbf{\alpha_M}$ and $\mathbf{\alpha_E}$ in Fig. 4 are parallel, in general their magnitudes are not equal.