There are four perspectives on space and time depending on whether the observer is internal or external to space or time. The four perspectives are internal space with internal time, external space with internal time, internal space with external time, and external space with external time.

The internal perspective is that of an observer traveling along a route or trajectory. It does not track direction because direction refers to the external perspective. So the internal perspective on movement is one-dimensional, a curve or world line, which has an arc length in space or time. The internal perspective makes space or time absolute because they travel with the observer.

The external perspective is that of one who observes an object move through a space or time with all its directional possibilities. So the external perspective on movement is three-dimensional, an abstract space or manifold, which contains a curve or world line that has direction at every point. Note that an internal perspective is external to the movement of another observer. The external perspective makes space or time relative because the observer and observed move relative to each other.

Each of these perspectives may be “classical” or “relativistic” depending on whether a Lorentz-type transformation is applied to them. The internal-internal perspective is absolute so its space and time are Lorentz invariant. Time and distance are measured along the world line; they are the proper time and proper distance (not to be confused with the comoving distance).

Otherwise the ratio of movement should be one that is relative and three-dimensional in the numerator and absolute and one-dimensional in the denominator so that a vector is divided by a scalar. Thus the velocity should be used with 3D space and 1D time and the pace should be used with 1D space and 3D time.

The external-external perspective requires a different approach. Velocity and acceleration are defined relative to one-dimensional time so they cannot be used if time is three-dimensional. An inverse quantity (e.g., pace) is no better: that would be relative to one-dimensional space, which cannot be used if space is three-dimensional. One way would be to go back and forth between 3D space with 1D time and 1D space with 3D time.

A better way is via Minkowski’s approach to space-time: a new geometry. One advantage of this is that the invariant interval is defined without reference to velocity (though it includes the speed of light, a scalar). The Lorentz transformation can be represented as a hyperbolic rotation in a Minkowski space-time.