iSoul In the beginning is reality

6D invariant interval

Since one may associate either the arclength (travel length) or the arctime (travel time) with direction, one might think that the full coordinates for every event are of the form (s, t, ê), with arclength s, arctime t, and unit vector ê. Since the direction is a function of either the arclength or the arctime, the coordinates would be either (s, t, ê(s)) or (s, t, ê(t)).

However, since s = ∫ || r′(τ) || , where the integral is from 0 to t, and t = ∫ || w′(σ) || , where the integral is from 0 to s (see here), this reduces to either (t, r) or (s, w).

But science seeks unification and so must combine these forms into one. In that case, both s and t are redundant, and the full coordinates for every event are of the form [r, w]. That is, there are three dimensions of space and three dimensions of time. The arclength and arctime are implicit, and may be made explicit through integration.

The standard exposition of special relativity looks at one dimension of space and one dimension of time. This is convenient and makes Δs = Δx and Δt = Δw1. But in general Δs and Δt will either be measured directly or found through integration.

What is the distance-like invariant interval then between two events? The interval in length units (proper length) is (dσ)² = (cdw)² – (dr)²,  where c is the speed of light. The interval in time units (proper time) is (dτ)² = (dw)² – (dr/c)².

This appears different from special relativity because it substitutes the vector dw for the scalar dt. However, the scalar (dt)² = (dw1)² + (dw2)² + (dw3)² so there is no discrepancy.

In order to demonstrate that this interval is invariant for two observers traveling at different rates, one must either convert dw to dt or convert dr to ds, which reduces the six dimensions to four.

The intervals above may be generalized for general relativity with the relation L = cP √(–gμν dxμ dxν), where P is the path, gμν is the metric tensor, and there are six coordinates xμ and xν.

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