Six-dimensional space-time

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 lenticity, 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). Also, the linear (e.g., on-board) 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: (ct1, ct2, ct3, r1, r2, r3) = (ct, r1, r2, r3) = (ct1, ct2, ct3, r) = (ct, 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, r1, r2, r3) or (ct1, ct2, ct3, r), or two, (ct, r), in which t = |t| and r = |r|.

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

s² = Δr² – c²Δt² = Δr1² + Δr2² + Δr3² – c²Δt² = Δr² – c²Δt1² – c²Δt2² – c²Δt3² = Δr1² + Δr2² + Δr3² – c²Δt1² – c²Δt2² – c²Δt3².

This is commonly given in its differential form, which in six dimensions would be

ds² = dr² – c²dt² = dr1² + dr2² + dr3² – c²dt² = dr² – c²dt1² – c²dt2² – c²dt3² = dr1² + dr2² + dr3² – c²dt1² – c²dt2² – c²dt3².

The proper time, dτ², would be defined similarly:

dτ² = ds²/c² = dr²/c² – dt² = (dr1² + dr2² + dr3²)/c² – dt² = dr²/c² – dt1² – dt2² – dt3² = (dr1² + dr2² + dr3²)/c² – dt1² – dt2² – dt3².

Note that (dτ/dt)² = 1 – (dr/dt)²/ = 1 – v²/c² = 1/γ², with v the speed of the object. And so dt/dτ = γ and = 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 (length) are converted into a scalar:

:= r1² + r2² + r3² or := t1² + t2² + t3².

The four-vector for velocity is thus:

V = ds/dτ = γ ds/dt = γ (v1, v2, v3, 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.