2 Velocities in n Dimensions

In [1], velocities in 4D Euclidean space-time have been defined from the invariant 4D velocity of magnitude $c$ according to:

$$c^2=(cd\tau/dt)^2+(dx/dt)^2+(dy/dt)^2+(dz/dt)^2$$ (1)

Indexing $x_{1..3}$ for the spatial dimensions, $x_4$ for proper time $\tau$ , and $x_5$ for time $t$ , this leads to definitions for the velocity component in the proper time dimension (time-speed)

$$\chi=cdx_4/dx_5=\sqrt{c^2-v^2}$$ (2)

and for spatial velocity components

$$v_i=dx_i/dx_5\;\;(i=1,2,3)$$ (3)

These definitions for velocity components in 4D Euclidean space-time can be generalized to $n$ -dimensional Euclidean spaces, defined as:

$$X_n =(x_1 ,x_2 ,x_3 ,x_4 ......x_n)$$ (4)

The generalized time-speed in $n$ -dimensional space $X_n$ becomes

$$\chi_n =dx_n/dx_{n+1}$$ (5)

whereas the generalized spatial velocity components become

$$v_a =dx_a/dx_{n+1} \;\;(a=1,2,...n-1)$$ (6)

[depending on the choice of units in $x_a$ or $x_n$ a factor may have to be added to Eqs. (5) and (6).] Together with these expressions the following conditions define the dimensional viewpoint in $X_n$ :

• Observers in $X_n$ have the skill to observe dimensions $1$ to $n-1$ as spatial dimensions.
• The dimension $n$ is the equivalent for 'proper time' for observers in $X_n$ .

This definition means that we are observers in $X_4$ where $x_4$ is our proper time dimension but that for instance in 3D Euclidean space $X_3$ , observers are 'Flatlanders', i.e., they live in a 2D space. See also [3]. They experience the third dimension $x_3$ as their equivalence of proper time, while their basis for speed measurements is $x_4$ .

A conclusion from this section is that time-speed in $X_n$ is to be regarded a spatial speed in $X_{n+1}$ . From the dimensional viewpoint of $X_{n+1}$ , no distinction can be made between spatial and time speed in $X_n$ .