In some ways transportation is more general than physics, which is surprising. In terms of extent, from the microscopic to the astronomical, from extremes of temperature, etc., physics is the more general subject. But because transportation includes people, there are some additional possibilities. Let’s look at one transportation situation in which this is the case. (Note: we are not talking about transport theory here.)
Consider transportation in terms of positions in space and time, directions and speeds plus the expectations people have for a trip — in particular, what they see as their typical or expected travel speeds. The point is that people use a particular speed for trip planning and forecasting purposes, which may reflect general travel conditions or their personal travel experience, or simply their driving style. Call this the reference speed to distinguish it from their actual speed(s).
Let there be observer-travelers going in the same direction but in different vehicles (or trains, boats, etc.). Distinguish them by their frame of reference, unprimed or primed. Call their frames S and S’, their positions in space r and r’, in time t and t’, the actual speed of the second frame relative to the first v, and their reference travel speeds b and c respectively. Allowing different reference speeds is more general than the Lorentz transformation.
To make it more general we could say they may begin at different positions or their units of measure are different, but we’ll leave these as an exercise for the reader. The actual speeds could also vary over time but we’ll consider them constant.
Consider only the path/trajectory followed, i.e., one dimension of space and time each. Then we have: r = bt and r’ = ct’ as time-space conversions for each frame. We will follow the derivation of the Lorentz transformation (wavefront approach). A general linear transformation between (r, t) and (r’, t’) can be written as: r’ = ex + ft and t’ = gr + ht where the constants e, f, g, and h depend only on b, c, and v. The derivation is an exercise in algebraic manipulation with the following result:
e = 1 / sqrt(1 – v2/ b2) = γb,
f = -v e = -v / sqrt(1 – v2/ b2) = – v γb,
g = – (v / (bc)) γc,
h = (b/c) / sqrt(1 – v2/ c2) = (b/c) γc,
where γc = 1 / sqrt(1 – v2/ c2).
So the general Lorentz transformation is:
r’ = γb (x – vt),
ct’ = γc (b t – vx / b).
If b = c, there is only one reference speed for both traveler-observers, which is the requirement of the Lorentz transformation.
r’ = γ (x – vt),
t’ = γ (t – vx / c2).
This is the case with the speed of light, which acts as a reference speed to which all speeds can be compared.