Because of the directionality, symmetry, and convertibility of space and time, there could be three dimensions of both (3S+3T). However, the formation of rates, notably velocity and celerity, effectively reduces the dimensionality to either three dimensions of space and one of time (3S+1T) or one dimension of space and three of time (1S+3T). Furthermore, the on-board or proper measures of space and time form one dimension each of space and time (1S+1T). So the six dimensions of *potential* space-time are commonly reduced to four or less dimensions.

If there are six dimensions, a point-event of space-time would be indicated by its coordinates: (*ct _{1}, ct_{2}, ct_{3}, r_{1}, r_{2}, r_{3}*) = (

*c*

**t**

*, r*) = (

_{1}, r_{2}, r_{3}*ct*

_{1}, ct_{2}, ct_{3},**r**) = (

*c*

**t**,

**r**), where

*c*is the speed of light in a vacuum and lowercase bold indicates a three-vector. If either or both three-vectors is replaced with its distance (or duration) from the origin, the dimensionality is reduced to four, (

*ct, r*) or (

_{1}, r_{2}, r_{3}*ct*), or two, (

_{1}, ct_{2}, ct_{3}, r*ct, r*), in which

*t*= |

**t**| and

*r*= |

**r**|.

The spacetime interval (or invariant interval) between two point-events would be defined as

*s² = Δ***r***² – c²Δ***t***² = Δr _{1}² + Δr_{2}² + Δr_{3}² – c²Δ*

**t**

*² = Δ*

**r**

*² – c²Δt*.

_{1}² – c²Δt_{2}² – c²Δt_{3}² = Δr_{1}² + Δr_{2}² + Δr_{3}² – c²Δt_{1}² – c²Δt_{2}² – c²Δt_{3}²This is commonly given in its differential form, which in six dimensions would be

*d***s***² = d***r***² – c²d***t***² = dr _{1}² + dr_{2}² + dr_{3}² – c²d*

**t**

*² = d*

**r**

*² – c²dt*.

_{1}² – c²dt_{2}² – c²dt_{3}² = dr_{1}² + dr_{2}² + dr_{3}² – c²dt_{1}² – c²dt_{2}² – c²dt_{3}²The proper time, *dτ²*, would be defined similarly:

*dτ² = d***s***²/c² = d***r***²/c² – d***t***² *= (*dr _{1}² + dr_{2}² + dr_{3}²*)

*/c² – d*

**t**

*² = d*

**r**

*²/c² – dt*= (

_{1}² – dt_{2}² – dt_{3}²*dr*.

_{1}² + dr_{2}² + dr_{3}²)/c² – dt_{1}² – dt_{2}² – dt_{3}²Note that (*dτ/dt)²* = 1 – (*dr/dt*)²/*c²* = 1 – *v²/c²*, with *v* the speed of the object. And so *dt/dτ* = *γ* and *dτ* = *dt/γ*, where *γ* is the factor from the Lorentz transformation.

The six dimensions of potential space-time are reduced to four in order to represent rates of time (speeds) or rates of distance (paces). For this purpose time (duration) or space (distance) are converted into a scalar:

*r²* := *r _{1}² + r_{2}² + r_{3}²* or

*t²*:=

*t*.

_{1}² + t_{2}² + t_{3}²The four-vector for velocity is thus:

**V** = *ds/dτ* = *γ ds/dt* = γ (*v _{1}, v_{2}, v_{3}*, 1), where the uppercase bold indicates a four-vector.

This shows “*four-theory as a special case*” of the potential six-dimensions of space-time.