From racing to relativity

There are three different contexts for 3D time with 1D space (stance), depending on whether stance is continuously increasing and, if so, whether there is a conversion factor between 3D space and 3D time:

(A) Stance is not continuously increasing. This is the situation of a race or sport in which game time has a definite beginning and ending. For example, in many sports the game lasts a specific time. In a race, the length of the course is set and the time for each contestant ends when they cross the finish line. The average pace of a contestant is their race time divided by the course length.

(B) Stance is continuously increasing and there is a general conversion factor between length and time. This is the situation of the special theory of relativity and some transportation settings in which the conversion pace is the minimum pace (and maximum speed).

In this case, there is an increasing stance whether or not a positive time interval is measured. Without a time interval increase the pace is at a minimum (or the speed is a maximum). As the amount of time measured increases, the pace increases (or the speed decreases). Remember that a small amount of time per unit distance is a fast motion, whereas a large amount of time per unit distance is a slow motion.

In this way, the pace increases indefinitely. A pace of infinity would be at rest. A pace of zero is the minimum pace, which in relativity is the speed of light. That is, the speed counts down from the speed of light. This has been misinterpreted as a transformation with superluminal speeds, but because speed decreases as pace increases, object speeds are subluminal.

The dual Lorentz transformation (see here) is

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ur) \; \textup{with}\; \lambda = 1/{\sqrt{1-cu}}

with the understanding that c represents the inverse of the pace of light. The cu in λ is the pace of the object divided by the pace of light, with the stance increasing at the conversion rate. As the time of motion increases, the pace increases (and the speed decreases) from that of light toward the pace or speed of rest. So, the square root never becomes negative here.

(C) Stance is continuously increasing but there is no general conversion factor between length and time. This is the situation of general relativity and transportation in general. Conversion of length and time are local, not global, and the optimal route depends on whether length or time are optimized.