The Uses of Natural Units

During my first semester of grad school I worked as a teaching assistant for a freshman physics lab, where I occasionally encountered statements like “the mass of the ball was ten.” I’m paraphrasing here: students would actually write more innocuous-looking formulas like \(m_\text{ball} = 10\). But this is equally problematic, as it is still missing a unit of measurement: ten grams is very different from ten solar masses! Other examples of units of measurement include seconds, kilowatts, and miles per hour — basically, any symbol you can put next to a number to quantify some aspect of the world.

We tend to take units of measurement for granted in daily life, and are most aware of them in frustrating situations: when we are trying to decode a confusing recipe, for instance, or understand why a piece of furniture isn’t going to fit in the new apartment after all. But not only are units of measurement indispensable to the practice of science, they can also be a fascinating subject in their own right. The redefinition of the kilogram in the fall of 2018 received some well-deserved press attention, and the history of the metric system, born of the ideals of the French Revolution, deserves to be more widely known.

In this inaugural blog post, I’ll try to explain why I’m so fond of natural units, a system of measurement which is mainly used by theoretical particle physicists who never have to actually measure anything. We’re all familiar with the fact that we can measure the same physical quantity in different units: I could quote the distance from Boulder to Denver in miles, kilometers, or even megaparsecs if I wanted to be very obnoxious. Natural units invert this logic, using universal physical constants to express as many different quantities as possible in the same units.

This post will start simple, with examples from daily life to establish the necessary formalism and explain everything you need to know about the process of constructing a system of natural units. Then I’ll briefly introduce the universal constants I will use to define natural units: the speed of light, Planck’s constant, Boltzmann’s constant, and the permittivity of free space. If that last sentence meant nothing to you, don’t be alarmed. I won’t assume you know anything about these constants going in, and I won’t attempt to tell you everything there is to know about them: my goal is just to show how they can be used to relate seemingly disparate physical quantities.¹For further slightly more technical reading on the physical basis for natural units, this short article is a good overview.

After defining these constants, we’ll explore what the world looks like through the lens of natural units, and end by using natural units to estimate the surface temperature of the sun. This is a great example of how an appropriate choice of units can greatly simplify back-of-the-envelope calculations.

Units in everyday life

First, some notation. This post will make frequent references to frequency, which in physics denotes the inverse of time and is usually expressed in units of hertz (Hz), or inverse seconds.²Oops, this is not entirely correct. Strictly speaking 1 Hz ought to mean one cycle per second, where a cycle can refer to a full rotation of a wheel or one oscillation of a sine wave. This is not mathematically equivalent to one inverse second basically because \(2\pi\) does not equal one. The distinction will not really matter for the qualitative discussion in this post. I thank Hillel Wayne and Zack Lasner for pointing this out. Movies are shot and displayed at 24 frames per second; equivalently we might say the frame is updated at 24 Hz.

In particular, we will be interested in frequencies of waves. The frequency of any wave is determined by its speed and its wavelength: if we use \(f\) to denote frequency, \(v\) for speed, and the Greek letter \(\lambda\) for wavelength, the relation between these quantities is \(f = v/\lambda\). If you’re wading in the ocean and a wave hits you every ten seconds or so, this could be because the crests of the waves are separated by about one hundred feet and the water is moving at ten feet per second, or because the crests are separated by a mile, and the water is moving at about five hundred feet per second. So much for daily life. As a physicist, I get to use dumb examples like this to illustrate mathematical relationships; fortunately, in the real world, you don’t have to worry about the latter scenario.

I’ll use square brackets throughout this post to mean “units of,” as in
\([\)frequency\(]\) \(=\) \([\)time\(]^{-1}\). For a less trivial example, let’s consider acceleration. The acceleration due to gravity on earth’s surface is (roughly) 9.8 meters per second squared. If you haven’t thought about acceleration in a while, this means that in the absence of air resistance, the speed of a falling object would increase by 9.8 meters per second every second. We will write this as

\[ [\text{acceleration}] = [\text{length}]\cdot[\text{time}]^{-2}.\]

This statement is true regardless of what units we use to measure distance or time.

Now let’s imagine you asked me how far Denver is from Boulder: rather than answer in miles, kilometers, or megaparsecs, I would probably say that Denver is about half an hour away. Answering the question this way amounts to setting \([\)length\(]\) \(=\) \([\)time\(]\); more precisely, I have implicitly asserted 60 miles \(=\) 1 hour, which is of course not always true. If we assume that a light rainstorm doesn’t bring US-36 to a standstill and increase the commute time to two hours (true story), we can divide both sides of this equation by one hour to rephrase it in the more provocative form

\[60\text{ miles/hour} = 1. \]

In this new system of measurement, velocity has no units — pure numbers like this are said to be dimensionless. Velocity being dimensionless implies that

\[ [\text{acceleration}] = [\text{time}]^{-1}.\]

An acceleration of 1 Hz in these units means your velocity changes by 60 miles per hour every second. I wouldn’t recommend maintaining this for very long.

All this is to say, you can define a system of units in which “the mass of the ball was ten” is actually a perfectly sensible statement. If you’re an undergrad reading this post, you have my blessing to annoy your TAs by pointing this out. But the fact remains that if you want to make yourself understood, you need to be clear about what units you’re using.

Universal Physical Constants

Ok, on to the physics! As in the example above, we will construct our system of natural units by setting various quantities that normally have units equal to one. Specifically, we will define

\[ c = h = k_B = \epsilon_0 = 1.\]

In this equation, \(c\) represents the speed of light, \(h\) is Planck’s constant, \(k_B\) is Boltzmann’s constant, and \(\epsilon_0\) is the permittivity of free space. Let’s briefly walk through the meaning each of these quantities.

The Speed of Light

Setting \(c=1\) means measuring length and time in the same units, as in the more mundane example presented above. But unlike the speed of traffic, the speed of light is baked into the laws of physics. Consider how relative velocities work in our everyday experience: if you are driving along the highway at the same speed as another car, it will appear stationary from your vantage point, while a car traveling in the other direction passes you faster than it would if you were not moving. But it is an experimentally verifiable fact that the speed of light (about three hundred million meters per second) remains the same for an observer traveling at any velocity.

This observation motivated the development of Einstein’s theory of special relativity, in which space and time get mixed together (very loosely speaking) at velocities approaching the speed of light. For our present purposes, the takeaway point is just that there are deep reasons the speed of light is a less fickle reference point than any other velocity.

One detail we will need for our discussion going forward is the fact that light is an electromagnetic wave (though this is not quite the whole story, as we will see below). The constant speed of light implies a fixed relationship between its wavelength and frequency: specifically, longer-wavelength light must have a lower frequency.

Length and time are not the only units affected by our decision to set \(c=1\). In particular, Einstein’s famous relation \(E=mc^2\) becomes \(E=m\), which doesn’t have quite the same ring to it, but simplifies the underlying physics: mass and energy are measured in the same units, because mass is a form of energy.

Planck’s constant

Setting \(h=1\) means measuring frequency and energy in the same units, or equivalently measuring time in units of inverse energy. Planck’s constant \(h\) arises in the theory of quantum mechanics, the other major development in early 20th-century physics besides the theory of relativity. One of the strange things about quantum mechanics is that quantum objects behave like particles in certain circumstances and like waves in others. Planck’s constant relates the quantities that characterize an object when it is described as a particle (mass, energy, and momentum) to the quantities used to describe it as a wave (wavelength and frequency). In particular, the frequency \(f\) of the wave describing a particle with total energy \(E\) is \(f = E/h\).

The most commonly cited example of this wave-particle duality is light, which can be described either as an electromagnetic wave or as a stream of discrete particles called photons. But wave-particle duality is really a general feature of all objects described by quantum mechanics. If we want to know the equivalent frequency for a particle with known energy (as opposed to the other way around), we need to use the inverse of Planck’s constant, whose numerical value in more standard units is about \(h^{-1} = 1.5\times 10^{33}\text{ Hz/J}\), or about 1.5 billion trillion trillion hertz per joule. This is a very large number! It implies that everyday kinetic and potential energies correspond to absurdly high frequencies, which is one of the reasons (there are other very important reasons!) that we don’t see objects behaving like waves in daily life.

Boltzmann’s constant

In the preceding discussion I have largely ignored the questions of interpretation that arise in the course of defining natural units. What are we really doing when we set \(c=1\)? Are we actually saying that the speed of light is on the same footing as any other factor relating different units, like 2.2 pounds per kilogram? Or is this just notational shorthand for a more cumbersome statement along the lines of “we are measuring time in units of seconds and length in units of the distance that light travels in a second?” The numerical value of \(c\) in the latter interpretation happens to be one, but there is still a meaningful sense in which length and time are measured in different units. Similar questions can be raised about the meaning of setting \(h = 1\), and physicists aren’t all in agreement about how to answer such questions.³See here for a debate on this subject among three theoretical particle physicists. Much of this discussion is way over my head, but Part III in particular is quite fun to skim through.

There seems to be more agreement among physicists that Boltzmann’s constant really is just a conversion factor relating different ways of measuring energy. More precisely, setting \(k_B=1\) means measuring temperature and energy in the same units. From a microscopic perspective, temperature is just a manifestation of the random motions of atoms and molecules comprising any large quantity of matter, whether gas, liquid, or solid; the energy associated with this motion is what we experience as temperature. Boltzmann’s constant appears in equations that relate these microscopic and macroscopic energy scales.

In order to relate temperature to energy using Boltzmann’s constant, we need to use units of absolute temperature, such as the kelvin (K). A temperature change of one kelvin is equal to a change of one degree Celsius, but zero kelvin is set to the minimum physical temperature called “absolute zero” rather than to the freezing point of water. Absolute zero is about \(-\)273 °C, or \(-\)460 °F.

The permittivity of free space

The permittivity of free space \(\epsilon_0\) is the most boring of the quantities we will set equal to one; the units in which it is normally measured are an artifact of the historical development of the theory of electromagnetism. In a nutshell, the role of \(\epsilon_0\) is to convert electrical units to mechanical ones: electric charges exert forces on each other, but the units in which physicists normally measure electric charge, unlike the units of force, cannot be decomposed into powers of mass, length, and time. It’s not really so useful to set \(\epsilon_0 = 1\) unless we also set \(c=1\). Setting \(c = \epsilon_0 = 1\) means measuring electric and magnetic fields in the same units.

Putting it all together

Let’s recap what we’ve learned: setting \(c = 1\) means setting

\[ [\text{length}]^{-1} = [\text{frequency}]\]

and

\[ [\text{mass}] = [\text{energy}].\]

Setting \(h = 1\) implies that

\[ [\text{frequency}] = [\text{energy}],\]

and setting \(k_ B = 1\) implies that

\[ [\text{temperature}] = [\text{energy}].\]

Putting this all together, we can express all of these quantities in energy units:

\[ [\text{length}]^{-1} = [\text{frequency}] = [\text{temperature}] = [\text{mass}] = [\text{energy}].\]

If we also set \(\epsilon_0=1\), we can express electromagnetic quantities in energy units: for example, \([\)voltage\(]\) \(=\) \([\)electric current\(]\) \(=\) \([\)energy\(]\), and
\([\)electric field\(]\) \(=\) \([\)magnetic field\(]\) \(=\) \([\)energy\(]^{2}\).

In fact, in this system of measurement, all physical quantities have units of energy raised to some power. There’s nothing special about energy: we could alternatively choose to measure everything in powers of time or length units, but energy is the more common choice. We are free to choose any units of energy we like, as long as we are consistent. Physicists employing natural units typically measure energy in electron-volts (eV), which are a legacy of the role of particle accelerators in the experimental development of particle physics.

Certain physical quantities which normally have units are dimensionless in natural units.⁴Just as any number raised to the zeroth power equals one, any unit raised to the zeroth power is dimensionless. So in a technical sense I wasn’t lying when I said all physical quantities could be expressed in units of energy raised to some power. This is unsurprising: we forced velocity to be dimensionless by setting \(c = 1\). A few of the other quantities that end up dimensionless are angular momentum (from \(h=1\)), entropy (from \(k_B=1\)), and electric charge (from \(c = h = \epsilon_0 = 1\)).

It can be fun to contemplate the implications of this dimensionlessness, with the caveat that one shouldn’t read too much into it. For example, in quantum mechanics electrons and many other particles have a property called “spin” with units of angular momentum, but all sorts of problems arise when we try to interpret spin as the angular momentum of a particle spinning around its axis; instead, spin manifests as a discrete set of quantum states independent of the particle’s position in the world. No change of units can explain why spin differs in some respects from classical angular momentum, but setting \(h=1\) raises the question of whether this needs explaining. If energy and frequency really are equivalent in the quantum world, then angular momentum is just a number.⁵Age, unfortunately, is not just a number: it has units of inverse energy, which actually kind of makes sense. From this vantage point, the fact that spin works like a label that counts internal states seems less surprising.  

One universal constant that has not come up in our discussion so far is the gravitational constant \(G\), which describes the strength of gravity both in Newton’s formulation of gravity as a force between massive objects and Einstein’s theory of general relativity. In natural units
\([G]\) \(=\) \([\)energy\(]^{-2}\). We can still go one step further and set \(G=1\): this defines a system of measurement called Planck units in which every physical quantity is dimensionless.⁶This last step is less confusing if you interpret setting \(c=1\) to mean choosing units in which \(c\) happens to have the numerical value one: you can use any units you like to measure time, but this choice then determines the units of length. Each universal constant you set to one imposes a similar constraint. In natural units, as noted above, we are still free to use any units to measure energy. Setting \(G=1\) then imposes one final constraint that fixes the energy scale. Planck units are useful for thinking about the intersection of quantum mechanics and gravity, but the fact that everything is dimensionless can make things quite confusing, so they’re not much good for estimating quantities in the real world.

The fact that we can choose units in which everything is dimensionless may not seem particularly meaningful. I could construct a system of measurement in which this is true by setting one foot, one year, one kilowatt-hour, one degree Fahrenheit, and one volt all equal to one. The curious thing is that nature has seen fit to supply us with exactly the right number of universal constants to make this possible. This would no longer be the case if we were to discover a new fundamental law of physics in which a velocity called \(b\) completely unrelated to \(c\) played a special role: we’d have to choose whether to set \(b\) or \(c\) to one.

Estimation with natural units

In practice, natural units are most often used to simplify expressions in quantum field theory, where the calculations are already convoluted enough without extra powers of \(c\) and \(h\) floating around. But natural units can still be useful to those of us who aren’t likely to be doing such calculations anytime soon. The virtue of using natural units to estimate answers to questions about physics is that it primes you to ignore the messy details  of the physical processes at work and focus on the energy scales involved. You’ll certainly get a more accurate answer if you actually do the relevant calculations, but this will take more time, and it’s possible for the essential physical insights to get buried in the details.

To estimate anything in practice, you’ll need an expression that translates the correspondences that define natural units into whatever units are actually used to measure the quantities you care about. For example, we can use the numerical values of \(c\), \(h\), and \(k_B\) to rewrite the expression

\[ [\text{length}]^{-1} = [\text{frequency}] = [\text{energy}] = [\text{temperature}] \]

as

\[\frac{1}{1\text{ μm}} =  300\text{ THz} = 1.25\text{ eV} = 14{,}000\text{ K}.\]

In this last expression, the prefixes μ and T stand for “micro” and “Tera;” 1 μm (officially called a micrometer, but more commonly referred to as a micron) is a millionth of a meter, and 1 THz is a trillion hertz.

With this formalism out of the way, let’s say we want to estimate the temperature of the surface of the sun. Go ahead and venture a guess before we begin the exercise — no calculations, just go with your gut.

Whether or not you’ve made your guess, you’ve clearly already started reading this sentence, so let’s get started. One thing we know about the sun is that we can see the light it emits. The sun is emitting light over a range of different wavelengths from the infrared into the ultraviolet, but let’s assume it’s emitting mostly at 500 nanometers, a nice round number in the middle of the visible spectrum.⁷I chose 500 nm because I wanted an easy number to work with; it turned out to be a surprisingly accurate guess. Without thinking at all about the details, we might suspect the photons emitted by the sun should have some connection to the energy scale set by the sun’s surface temperature.

500 nm is half of 1 μm. A shorter wavelength means higher frequency, energy, and temperature, so dividing the length scale in the above expression by two means doubling the corresponding temperature scale. So our best guess is 28,000 K.

The real answer turns out to be about 6,000 K. Where did we go wrong? The limitation of this kind of estimation is that it entirely ignores the existence of pure numbers other than one. We shouldn’t have expected a photon at the peak-emission wavelength to have an energy precisely equal to \(k_BT\), and indeed, in the exact expression from the theory of thermal radiation there is a numerical factor pretty close to 5.

Our estimate may not seem very accurate, but considering how little calculation went into it, a factor-of-five discrepancy is entirely reasonable. Even when you think you might already know the answer, and even when you’re about to start a more detailed calculation, it can be useful to do a quick estimate along these lines as a sanity check. Generally speaking, we should expect the true answer to be within a factor of 10 in either direction from your estimate.

This kind of order-of-magnitude estimation ultimately has nothing to do with natural units per se. Natural units are a convenient choice for questions involving light or other forms of electromagnetic radiation; more mundane units are more suitable in most other situations. Fermi problems are a fun way to practice the art of estimation, and I can enthusiastically recommend Randall Munroe’s blog/book What If?, which tackles extreme hypothetical scenarios with back-of-the-envelope estimation and a great sense of humor.

At the end of the day, choosing units is always a matter of convention. You cannot change the laws of physics by changing the units you use to describe them. But our choice of units can prime us to approach certain questions from a different perspective than we might otherwise. Natural units are ultimately just a tool for facilitating calculations, but they have prompted us to think about the essential interconnectedness of space, time, and energy in all its different forms. You don’t need to start measuring your commute in inverse electron-volts to think that’s cool.