“You can have time without moving but you can’t move without any time.” Actually, no, that is not correct. I introduced this topic here but let me go into more detail in this post.
The previous post on measurement sets the background: we need to be very careful what it is we’re measuring and how. I’ll use the illustration of two trains again but will analyze it further. We won’t get into relativistic issues like clock synchronization here.
Let’s start with a standard scenario: there are two frames of reference, one attached to train A and the other attached to train A’. Both trains (or frames) are moving with constant speed in the same direction. Train A’ is traveling with a speed V relative to train A, that is, the speed of train A’ = V1 + V where V1 is the speed of train A.
It is important to note that both train A and train A’ are moving relative to the ground. The ground is an important aspect because, like an electrical ground, it provides the context for measurement of train movements. So there is an implicit frame G that represents the external perspective from the ground.
In order to make the parallels between space and time clearer, let’s dispense with ordinary clocks and rods and use two methods that work for measuring both distance and time. For movement relative to the ground we’ll use the click-clack of the train wheels as a measure of time (say, one second) and distance traveled (say, 25 metres). For movement relative to each train we’ll use a measuring wheel traveling down the aisle of each train at a constant speed.
Here’s where the independence of the denominator becomes important. With velocity, the elapsed time is the independent variable but with the dual velocity, the distance traveled is the independent variable. These independent variables are chosen first, then the dependent variable is measured. The independent variable must be chosen to be non-zero in order to avoid zero in the denominator.
The two trains are traveling along side one another and gradually train A’ pulls ahead of train A since it’s traveling faster. Their relative position is measured by the measuring wheel going down the aisle. Now say that when the two trains are side-by-side that train A’ slows down and goes at the same velocity as train A so they are at rest relative to one another.
The click-clack of the trains acts like a clock, showing that time flows ever onward. But it also shows that the distance traveled flows ever onward, too. None of this business about time being unique or unidirectional.
The measuring wheel now shows the two trains with a relative distance traveled of zero: the measuring wheel just begins and it’s done measuring their relative positions. But the measuring wheel also measures a relative elapsed time of zero; it takes no time to go between the beginning and ending of the relative positions of the trains.
Let’s put this together. When we measure velocity, we pick an independent time interval and then measure the distance traveled during this interval. Say we pick 10 click-clacks as the time interval (10 seconds). What distance does the measuring wheel measure during this time? Zero, so the relative velocity is zero.
When we measure the dual velocity, we pick an independent distance interval and then measure the travel time during this interval. Say we pick 10 click-clacks as the distance traveled (250 metres). What elapsed time does the measuring wheel measure during this distance traveled? Zero, for the same reason that the relative velocity is zero. So there is a distance traveled with zero elapsed time.
The key is that the denominator comes from an external movement, whereas the numerator comes from an internal movement. Is this correct? It’s exactly what is done with clocks: they are external to the movement observed, keeping time without regard to the phenomena under observation. In order to switch space and time, we have to completely switch them: so we use travel distances instead of clocks and relative elapsed times instead of distances traveled.
We have again shown that travel time and distance traveled may be interchanged, that is, space and time are symmetric.