It may seem an exaggeration at this point to speak of “spacetime” while focusing on examples from everyday life rather than the physics laboratory. Yet after all we live in a physical world so physics should include that, too. But we’ve put off considerations such as the constant speed of light until we have a better handle on what the symmetry of space and time means.
Consider two vehicles on a highway, traveling in the same direction on different lanes. Vehicle S is traveling at speed b and vehicle S’ at speed b’ which is greater than b so at some point vehicle S’ overtakes vehicle S: call this point in spacetime the origin. The relative speed of vehicle S from vehicle S’ is b’ – b. The relative speed of vehicle S’ from vehicle S is b – b’.
Now consider that each vehicle has an odometer and a speedometer; we will ignore any clock or watch since they are not integrated with the frame. From the odometer readings, say r and r’ for vehicle S and S’ respectively, and the speedometer readings, which are b and b’, we can determine the travel times: r/b and r’/b’. The problem now is that we want to use b or b’ as conversion factors between space and time, that is, length and duration. It would be confusing or perhaps contradictory to use both b and b’ for this purpose, so it is natural to use their average:
b* = (b + b’)/2.
Then b* is the spacetime conversion factor in this case. It should not be surprising that an average would be used for the conversion factor; after all, the average or typical travel time between two points is what an isochrone map or travel distance and duration map would show. We’re not saying that the whole fabric of spacetime depends on these two vehicles — that would be ridiculous — but that the spacetime geometry of a transportation system would be represented in this way.
We may then follow the logic of the post on Galileo revised to determine that the transformation from one frame of reference to the other is the same as that given except for time:
t’ = t – vx / (b*)².
The average speed, b*, could be a typical speed rather than a calculated average. There might be reason to take it as the posted speed limit instead. When we open it to the speed of any object, there is good reason to take it as the absolute speed limit, c, the speed of light.