For those of us who work with them every day, equations become almost tactile. Some are as sharp as flints, others as edgeless as a beach pebble worn by waves and sand and time. In particular, the deepest relationships in nature are often expressed by equations which, at least typographically, are disarmingly smooth.

Theoretical physicists polish these simple relationships to a shine by working in natural units. These are designed to ensure that many of the “constants of nature” you met in school become precisely and exactly equal to one, and thus neatly vanish from the algebra. 

For example, in everyday units the speed of light – the “c” of “Ee equals em-cee-squared” – is 299,792,458 metres per second. But in what physicists call natural units, light travels one measure of length in one beat of time, so its speed is exactly one.

Natural units clarify deep ideas but they do not neatly match the scales of human experience – it is not convenient to specify lengths in metres while marking time in steps of a billionth of a second.¹ Consequently, we stick with familiar measures for daily use, whereas natural units help turn already simple equations into polished gems.

To give an example of this in practice: black holes slowly leak energy, leaving them with a temperature that is ever so slightly warmer than absolute zero. Using natural units,² the temperature of a black hole with a mass M can be written thus:

This result was one of Stephen Hawking’s crowning achievements, helping to earn him the rare distinction of burial in Westminster Abbey. In testimony to this, the equation itself is now carved into the floor of the abbey. But look closely – the carver has gone for the more complex version (with plenty of space still left).

In this case, it seems that natural units provided an equation that was perhaps deemed too simple for public display. The rarest diamonds are made remarkable by their impurities, and the same can be true of equations. Restoring everyday units gives the more visually complex version engraved on the memorial:

Written out this way, each symbol serves as a reminder of the fundamental tools used by Hawking to arrive at this result. We have Planck’s constant, ℏ, from quantum mechanics; Newton’s constant, G, from gravitational physics; and Boltzman’s constant, k, from thermodynamics – along with our old friend c, the speed of light.

Fundamental concepts in physics – such as general relativity – are often admired for their elegance. The elegance of deep ideas can be akin to a Zen garden: their simplicity is instantly accessible to any viewer, and a full understanding can unfold with time and contemplation.

As you study a Zen garden from different perspectives, its appearance changes; likewise, it invites you to consider the individual rocks, pieces of gravel and grains of sand that make up the whole. Similarly with physics, when you delve into the details of a given theory, the maths becomes complex, spiky and nuanced.

By way of an example, part of my research involves thinking about pairs of black holes, which are governed by Einstein’s general theory of relativity. The fundamental equations of general relativity, which are indeed simple, tell us that these “cosmic eggbeaters” generate gravitational waves as they stir the fabric of space. Doing so costs energy for the black holes involved, and they pay it for by moving closer together – slowly to begin with, and then all at once.

The equation below is the mathematical version of the statement above, and it adamantly refuses to be boiled down to a simple form.

In fact, even this expression is only an approximation – the full result runs to many of pages of algebra and finally becomes so complex that it can only be expressed as a sheaf of numbers rather than in symbolic form.³ However, as you work through the details, you see that everything is there for a reason: the 73/24 has to be 73/24 and not 77/24 or 71/24.

It is this specificity that makes these complex equations just as beautiful as the simpler ideas from which we derive them. But the elegance of these detailed equations is a different kind of beauty from that of their parent theories. Theirs is the sharp, jangly beauty of the multifaceted crystals that grow inside a geode.

For me, discovering their complex charm always rewards the effort, even if investigating them thoroughly is more akin to herding hedgehogs than raking the gravel of a Zen garden.

Banner image: Tom Magliery (CC-BY) – via Flickr

1. Strictly speaking, light travels 29.979 centimetres in a nanosecond. And in practice the metre is actually defined relative to the speed of light, but in such a way as to preserve its original definition. ↩︎
2. An extra definitional tweak will get rid of the 8π, but it can then crop up in equations from which it would be otherwise absent – you have to make a choice and stick with it. ↩︎
3. For the curious, m₁ and m₂ are the masses of the two black holes, and e is the eccentricity (that is, a measure of non-circularity) of the orbit, and a is the size of their mutual orbit. ↩︎