iSoul In the beginning is reality

# Space and time expanded

Space-time is relativistic 3D space + 1D time. It obscures the 3D nature of time. The opposite is time-space with 3D time + 1D space, which obscures the 3D nature of space. Both of these have their advantages and disadvantages.

To avoid the disadvantage of obscuring 3D space or 3D time use 3D space + 3D time with an invariant interval and without measures such as speed or pace that require combining dimensions.

The invariant interval with coordinates in 3D space (r) and 3D time (t) between two events 1 and 2 (second subscript) is:

(t11 – t12)² + (t21 – t22 )² + (t31 – t32 )² – (r11 – r12)² – (r21 – r22 )² – (r31 – r32 )²,

which can be plus or minus depending on the sign convention. Here c² is a conversion constant that does not favor spatial over temporal coordinates.

The invariance of the interval under linear coordinate transformations between inertial frames follows from the invariance of

c² t11² + c² t21² + c² t31² – r11² – r21² – t31²

for any point event. This quadratic form can be used to define a bilinear form

u · v = t11² t12² + t21² t22² + t31² t32² – r11² r12² – r21² r22² – r31² r32²,

which is often written in matrix form. The signature is then (+ + + – – –).