I encountered a mental block in my understanding of natural units. According to natural units, $c = 1$, which would imply that if we wanted to convert velocity from SI units ($m/s$) to natural units (dimensionless), we would use the relation:
$$c = 1 \implies 3 \times 10^8 m/s = 1$$
Thus,
$$1 \frac ms = \frac 13 \times 10^{-8}$$ (in natural units)
Now, if in a given problem, the velocity is $v_{SI} = 220 \times 10^3 m/s$ (same as $220 km/s$), then the corresponding value in natural units should be:
$$v_{nat} = 220 \times 10^{3} \times \frac 13 \times 10^{-8} = 0.000000733 = 7.33 \times 10^{-4}$$
If I am asked to square the velocity ($v_{SI}^{2}$ or $v_{nat}^{2}$), then I seem to arrive at a problem.
$$v_{SI}^2 = 4.84 \times 10^{10} (m/s)^2$$
Converting to natural units,
$$\frac{v_{SI}^2}{c^2} = 5.38 \times 10^{-7}$$
which I also obtain if I square the velocity in natural units,
$$v_{nat}^{2} = 5.38 \times 10^{-7}$$
However, I find this counter intuitive, because while working entirely in conventional units (SI, CGS, MKS, etc.), the square of a velocity ($v_{SI}^{2} = 4.84 \times 10^{10} (m/s)^2$) would yield a larger value of the velocity, whereas in the natural units, it yields a significantly smaller value. Am I doing something wrong, or have I got the intuition wrong? Is this an intended effect of utilizing natural units, and if so, how does this justify effects we observe?
Best Answer
I kind of understand why you find this unintuitive. It may be easier if we visualize it using areas (in SI units for the moment).
We get an analogous - as you call it - unintuitive result if we express the side length of a square in meters, say $10\,\mathrm{m}$. The area of the square is $100\,\mathrm{m}^2$, a number that is larger than the side length. If we express the side in kilometers, we have $0.01\,\mathrm{km}$. Squaring this gives an area of $0.0001\,\mathrm{km}^2$, a number that is smaller than the side length.
But, what this actually means is that the reference area, namely $\mathrm{m}^2$ and $\mathrm{km}^2$ have a much bigger ratio than the reference lengths $\mathrm{m}$ and $\mathrm{km}$. And since all we do is express the areas in multiples of these reference areas and the lengths as multiples as the reference lengths, it is obvious that one area can yield a smaller number than the reference length, and the other a larger number than the reference length.
For your question in particular, one time you express the velocity in multiples of $\mathrm{m}/\mathrm{s}$ and the other in multiples of $c$. For these, the same argument as above holds.