Expressing the conversion from units of time to units of space in special relativity rigorously

Question

In their book Spacetime Physics, Taylor and Wheeler stress the analogy of the speed of light $c$ to a conversion factor $k$ which converts between miles and meters and subsequently convert freely between units of length and units of time . In analogy to something like $$ \Delta x = 2 \,mi = 2 \,mi \cdot 1600 \,\frac{m}{mi} = 3800\,m $$ we get something like $$ \Delta t = 100 \,s = 100 \,s \cdot 3 \cdot 10^{8} \,\frac{m}{s} = 3 \cdot 10^{10} \,m \tag{1} $$

Calculating like this is very convenient and powerful (especially if the calculations involve spacetime intervals) but the symbols can't have their usual meaning because ordinary seconds can't be measured in ordinary meters.

My questions are:

1. How does the meaning of the symbols $\Delta t$, $m$, $s$ in equation $(1)$ differ from their ordinary meaning?
2. How would I rewrite equation $(1)$ into a rigorous expression which uses the symbols in their ordinary meaning?

Or is there something wrong with equation $(1)$? As written above, it works well in calculations.

I have read the answers to some related questions (1, 2). They suggest that $\Delta t$ may be a redefinition of time as the length $c \Delta t$. This makes sense but it doesn't explain why units of both length and time occur in equation $(1)$.

Answer 1

A more rigorous form of the equation $(1)$ would be

$$ \Delta t = 100 \,s = 100 \,s \cdot 3 \cdot 10^{8} \,\frac{m}{ls} = 3 \cdot 10^{10}\,m \cdot\dfrac{s}{ls}$$

Where $ls$ is light-second. Then for convenience one can drop $\dfrac{s}{ls}$, but it remains implied for the dimensional analysis.

Answer 2

There is no difference in the symbols $\Delta t$, $m$, and $s$ from their ordinary meaning.

What is different is the assertion that $3\:10^8 m/s=1$ or equivalently that $1s=3\:10^8 m$. The second still retains its original SI definition as does the meter. All that is changed is that a second is a certain number of meters. This means that you can measure distances in seconds or times in meters, where seconds and meters retain their SI definitions.

What does change from ordinary meaning is that length and duration have the same dimensionality. The dimensionality of a quantity is a matter of convention, so that convention can be altered as desired, but the ordinary meaning is to consider them to have different dimensionality.

Answer 3

It is indeed true that there is more human convention involved in the definitions of units and physical dimension than one would suspect before looking into it more closely. In the present example, the physical quantity "1 mile" is by definition equal to a certain number of metres in the SI system. However the quantity "1 second" is not normally set equal to a distance, even though we could agree to do this if we wanted (and that is the point Taylor and Wheeler are making). Looking at spacetime as an abstraction, it is reasonable to say that time intervals and spatial intervals are different examples of essentially the same sort of thing: displacement in spacetime. On the other hand, looking at the physical world more generally, it is not true to say that time and space are just different versions of each other. This is shown by the notion of causality, for example: the interior of light cones is timelike, and worldlines are timelike. Therefore it makes a lot of sense to preserve the distinction between distance and time in our system of units. Having said that, it is often convenient to adopt units in which the value of the speed of light is one distance unit per time unit. This would not in itself make the speed of light dimensionless. However it would also be possible to introduce a system of units in which the speed of light is dimensionless (again, that's what Taylor and Wheeler are pointing out). I personally would not recommend such a system of units, because I think it adds more confusion than it removes.

Answer 4

Rigorous, you asked? Ok. A mile is, by legal definition, 1609.344 meters. A meter is 1/299792458 of a light-second. A second is the duration of 9192631770 cycles at the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom. Therefore, two miles is 2×1609.344×9192631770/299792458 c/ν = 52836095654496/535343675 c/ν, where $c$ is the speed of light in a vacuum and $ν$ is the above-mentioned transition frequency. By similar conversions, 100 seconds = 100×299792458/c meters = 29979245800/c meters. Both c and ν are (currently) the fundamental units in SI, as are e: the negative of the electron charge and h: Planck's constant.