This is a topic I have addressed before, such as here. In this post I want to show some similarities and differences between the Galilean (Galileo) transformation and what I’m now calling the dual-Galilean transformation.
There are many expositions of the Galilean transformation, such as here. It is standard to describe them in terms of two reference frames and their coordinate systems in uniform relative motion along the x-axis. Here we take the spatial axis to be the r-axis, which represents the spatial direction of motion, whatever that is.
Similarly, the temporal axis is taken to be the t-axis, which represents the temporal direction of motion, whatever that is.
One aspect of the exposition here is that the notation is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames.
The two frames are differentiated by primed and unprimed letters. Their relative speed is v. Their relative pace is ℓ = 1/v. The key difference between speed and pace is their independent unit of measure: speed is per unit of time (duration), whereas pace is per unit of space (length).
In the Galilean transformation, the measurement of time is the same for all observers, which is often called absolute time, whereas the measurement of space is relative to the motion of each observer. So the relationship between the coordinates is as follows:
speed: r´ = r – vt,
pace: r´ = r – t/ℓ,
and for all other coordinates the primed and unprimed values are equal. Note that the reason the other coordinates are equal may be either because there is no motion in their direction or other directions are not known to exist.
In the dual-Galilean transformation, the measurement of space is the same for all observers, which is called absolute space, whereas the measurement of time is relative to the motion of each observer. So the relationship between the coordinates is as follows:
speed: t´ = t – r/v,
pace: t´ = t – ℓr,
and for all other coordinates the primed and unprimed values are equal. Again, the reason the other coordinates are equal may be either because there is no motion in their direction or other directions are not known to exist.