Six-dimensional spacetime

First consider the dual to Minkowski spacetime. Recall that the invariant interval of Minkowski spacetime has one dimension of time with three dimensions of space:

(ds)² = (c dt)² – (dx1)² – (dx2)² – (dx3)² = (c dt)² – (dr

where t is the time coordinate and x1, x2, and x3 are space coordinates of r.

This could be called temporal spacetime since speeds and other ratios are referenced to duration, i.e., they have units of time in the denominator. The dual could be called spatial spacetime since measures of movement are referenced to distance, i.e., they have units of length in the denominator. In that case the dual invariant interval is:

(ds)² = (dr)² – (c dt1)² – (c dt2)² – (c dt3)² = (dr)² –  (c dt

where r is the space coordinate and t1, t2, and t3 are time coordinates of t.

Here’s how one might put together the two four-dimension spacetimes into one six-dimension spacetime invariant interval:

(ds)² = (c dt1)² + (c dt2)² + (c dt3)² – (dx1)² – (dx2)² – (dx3)² = (c dt)² – (dr

= (c dt1)² – (dx1)² + (c dt2)² – (dx2)² + (c dt3)² – (dx3

where the three dimensions of direction are the same for space and time. The six dimensions are in two groups of three dimensions, i.e., there are 2 × 3 dimensions or three complex dimensions.

Minkowski and dual Minkowski spacetime have 10 symmetries each. Six-dimension spacetime has 6 translations (one for each dimension), 6 rotations (along the x1-x2, x2-x3, x3-x1, t1-t2, t2-t3, and t3-t1 planes), and 3 Lorentz boosts (about the t1-x1, t2-x2, and t3-x3 planes) for a total of 15 symmetries.