The post continues the ones *here*, *here*, and *here*.

There are three dimensions of motion. The extent of motion in each dimension may be measured by either length or time (duration). There are three dimensions of length and three dimensions of time (duration) for a total of six dimensions.

But there is no six-dimensional metric. Why? Because a metric requires all dimensions to have the same units, which requires a ratio to convert one unit into the other unit. The denominator on a ratio is a one-dimensional quantity, which means either the length or time dimensions need to be reduced by two dimensions.

This ratio is a conversion factor that is either a speed, which multiplied by a time equals a length, or a pace, which multiplied by a length equals a time. In general a speed is the ratio Δ*d***r**²/|Δ*d***t**²| = Δ*d***r**²/(*Δt*_{1}² + *Δt*_{2}² + *Δt*_{3}²)^{1/2}, and a pace is the ratio Δ*d***t**²/|Δ*d***r**²| = Δ*d***t**²/(Δ*x*² + Δ*y*² + Δ*z*²)^{1/2}. The denominator is a distance or distime, which is a linear measure of length or time (duration).

The conversion factors required are the speed of light in a vacuum, *c*, or its inverse, the pace of light in a vacuum, *k*. The resulting four-dimensional metric is either *c*²*dt*² − *dx*² − *dy*² − *dz*² (with time reduced to one dimension) or *dt*_{1}² − *dt*_{2}² − *dt*_{3}² − *k*²*dr*² (with space reduced to one dimension).

These metrics are often simplified by taking *c* = 1 and *k* = 1 so that symbolically they are the same. Their units are not the same, however.

Each metric may be further reduced by separating space and time, as in classical physics. Then the space metric is |Δ*d***r**²| = (Δ*x*² + Δ*y*² + Δ*z*²)^{1/2} and the time metric is |Δ*d***t**²| = (*Δt*_{1}² + *Δt*_{2}² + *Δt*_{3}²)^{1/2}. In the classical (3+1) of three space dimensions and one time dimension, time is replaced by its metric, and in the classical (1+3) of one space dimension and three time dimensions, space is replaced by its metric.