Geometrized Units

When reading about theoretical physics, like relativity or quantum mechanics, sooner or later you come across statements like "... with $c=1$ ..." meaning the speed of light is set to be $1$. And similarly, for example with the gravitational constant: $G=1$.

This always made me grind my teeth, because: "One? One what?" And also I take some confidence from checking units in a formula for better understanding. If $c$ and $G$ just disappear, this can be quite confusing. Or, as Alan L Myers puts it:

However, for a person educated in the SI system of units, the lingo of “natural units” used by cosmologists is confusing and seems (incorrectly) to display a casual disregard of the importance of units in calculations.

This PDF by Myers is quite formal, which is nice, but still it left me scratching my head a bit. So I toyed around with the presented ideas to come up with an explanation I personally like. Whether it is 100% correct, I can't say, but ... well.

What follows directly from setting $c=1$ ?"

$$\def\um{\text{m}} \def\us{\text{s}} \def\ukg{\text{kg}} \def\uN{\text{N}}$$

Myers writes two things in different formulas, which, together amount to:

$$1 = c = 2.9979\cdot 10^8\, \text{m/s}\strut$$

And to not carry around the unwieldy number, lets call it $c_0 := 2.9979\cdot 10^8$, so $c_0$ is really just shortcut for a number. Similarly, lets have $G_0$ as the numerical value of the gravitional constant such that we get:

\begin{align*} 1 = c &= c_0 \frac{\um}{\us}\\ 1 = G &= G_0 \frac{\um^3}{\ukg\,\us^2} \end{align*}

Taking this formally seriously, we see that

$$1\,\us = c_0\,\um\,$$

which basically tells us that meter and second are now "the same" unit connected by a mere dimensionless conversion factor. Distance and time are the same thing and can be converted into each other with the speed of light as the numerical factor. Which, for everyone having heard the term space-time, should not be a surprise. Fine.

What about the $G$? Rearranging the formula we get

\begin{align*} 1 &= G_0 \frac{\um^3}{\ukg\,\us^2} \\ \Leftrightarrow\quad 1\,\ukg &= G_0 \frac{\um^3}{\us^2} \\ \Leftrightarrow\quad 1\,\ukg &= \frac{G_0}{c_0^2}\,\um \end{align*}

telling us that the units of mass (or energy) and length are the same too. Weird.

Lets try some more.

Force in Newton

Force is measured in Newton, defined as $1\,\uN = \frac{\ukg\,\um}{\us^2}$, so we get

\begin{align*} 1\,\uN &= \frac{G_0}{c_0^2}\frac{\um^2}{\us^2}\\ &= \frac{G_0}{c_0^4} \end{align*}

which makes force dimensionless.

Schwarzschild Radius

The Scharzschild Radius conventionally is $r_s = 2GM/c^2$ which reduces to $r_s = 2M$ and luckily, mass is measured in $\um$ now, so this fits.

Planck's Constant

If, similar to the above, we chose the number $h_0$ such that in SI units we have $h=h_0\, \ukg\,\um^2/\us$, we get, by replacing the mass and one velocity term:

\begin{align*} h &= h_0 \frac{G_0}{c_0^2}\,\um \frac{1}{c_0}\um \\ &= \frac{h_0 G_0}{c_0^3}\um^2\,. \end{align*}

Personal Comment

Time has told that the above system of units makes sense and helps to make equations simpler. I think I understand it now a bit better. The PDF cited above formalises how to replace combinations of $\ukg^\alpha \um^\beta \us^\gamma$ easily with combinations of $G$ and $c$ to see the resulting unit, but it helped me to just do it step by step for a few examples.

Yet:

• It feels like cheating, but that's probably because I am not used to this.
• I am somewhat surprised that this works. While diluting the difference between time and length seems well founded by what we know from General Relativity, having mass and length "the same" is weird.
• If we would, somewhat arbitrarily, equate $\ukg$ and $1/\us$, would we still have working physics math, if for the price of over-complex equations? Or would this lead to contradictions, meaning the above selection is not as arbitrary as it looks — apart from it making formulars slicker?

And finally, the above gymnastics raise the question: what are physical units anyway, in particular if we can dissolve them into each other so easily?