The Planck units

This is a little bit beyond IB, but worth a read. The image is of the silicon surface of a microchip.


There is this idea that the Planck units present some kind of fundamental limit on the values of physical quantities. This answers:

  • what the Planck units are and why they are interesting
  • which of them can be interpreted as some kind of “limit”, and how.

What are the Planck units and how are they derived?

Physics is all about establishing the relationships between various physical parameters. In the Système Internationale (SI), the “big 5” physical parameters that form the basis in which everything else can be expressed are mass [𝑀], length [𝐿], time [𝑇], electriccurrent [𝐼], and temperature [Θ]. These are called the dimensions of a quantity (note this is an entirely separate use of the word to the 3 dimensions of space and 1 of time).

The SI units of the base quantities are kilograms, metres, seconds, amperes and kelvins, respectively. These units are entirely arbitrary- there is no fundamental reason why we should use metres for [𝐿]and not inches, furlongs or cubits instead. To take an example, speed has dimensions [𝐿][𝑇]−1and SI units of metres per second.

We know that  there exist certain fundamental constants. Among the most important of these are:

  • the speed of light 𝑐, which has dimensions of a speed [𝐿][𝑇]−1and appears across all of physics,
  • Newton’s gravitational constant 𝐺, which governs the strength of gravity. By considering either Newton’s gravitational force law or general relativity (GR) one can see it has dimensions of [𝑀]1[𝐿]3[𝑇]−2
  • The reduced Planck’s constant ℏ, whose value governs quantum mechanics and which has dimensions [𝑀][𝐿]2[𝑇]−1.

In 1899 Max Planck observed that by multiplying these constants with one another in various combinations, one could create specific-valued quantities with any combination of the dimensions of length, time and mass. For example, we have:

Screenshot 2019-03-02 at 09.18.56

Note that while some of these seem extremely small, there is nothing remotely “limiting” about others— in fact, their values are astonishingly “human-scale” considering they are derived only from fundamental constants. The Planck mass is quite small by everyday standards but not exceptionally so- it is about the mass of a flea and you can easily see Planck-mass objects with simple magnifying equipment. We can derive others. The Planck momentum is similar to that of a served tennis ball. The Planck energy is just enough to boil enough water to make a cup of tea for everyone in the Royal Albert Hall.

By also including Boltzmann’s constant 𝑘 we can extend the Planck system to include temperature and hence the other thermodynamic parameters, and by including the permittivity of free space 𝜀we can include electric current and hence electromagnetism too.

Why are they interesting?

The Planck units are interesting for essentially two reasons: one very pragmatic, the other fundamental.

  1. Since our choice of SI units are entirely arbitrary, in the SI these constants all take very ugly values. ℏ, for example, has a value of  6.63×1034kg.m2s1. By replacing the SI units with the Planck units as the basic units of measurement, the fundamental constants 𝑐, 𝐺, ℏ, 𝑘and 𝜀all take the value of exactly one. This presents a convenient opportunity for physicists to declutter their equations, since we no longer need to keep track of all those wretched constants. In complex computer simulations, it also provides a speed-up by removing the need to multiply every term by the arbitrary SI values of the fundamental constants, which must be specified to high precision.In reality, most real physicists use a “hybrid” system of units because they work on systems that are far away from the Planck scale. I’ll explain what that means below, but suffice it to say that particle physicists don’t often care what value 𝐺 takes in their system of units, because gravitational effects are irrelevant to their line of work. So they will use units such that 𝑐=ℏ=1, but also that the energies in their system, which are always far below the Planck energy, take reasonable values. Similarly, physicists simulating black hole mergers disregard quantum effects and so don’t care about the value of ℏ in their unit system: they will use units where 𝑐=𝐺=1, but where the masses of their black holes are not unimaginably huge multiples of the mass unit.
  2. The more interesting reason why Planck units are important is that in a certain sense, they demarcate the scale where gravitational, quantum, electrodynamic and thermodynamic effects all become comparable. Since all the relevant constants in the laws of physics are simply 1 in the Planck system, if the input values to our equations are similar to the Planck values, then effects from all of these branches of physics are present in equal measure and we cannot neglect anything. For example, if we consider a black hole with the Planck mass, we find that both the radius of its event horizon (governing the scale where gravitational effects become dominant) and its de Broglie wavelength (which tells us when quantum effects move from negligible to dominant) are about the same- around the Planck length, in fact. Additionally, the mass-energy of this thing would be the Planck energy, and the Hawking temperature of its event horizon would be the Planck temperature, and so on. Imbue it with the Planck charge too, and you basically have nearly all of physics playing together in a single system. Of course, such a black hole (a so-called Planck hole) is both far smaller than the black holes we can observe in the sky, where gravity dominates and quantum effects are not discernible, yet also far more energetic than the particle collisions we can make at CERN, where the effects of self-gravity are immeasurably small and quantum effects dominate. This is basically because the Planck length is so small. As such, we are very far away from ever being able to see a Planck hole, have no idea how gravitational and quantum effects interplay for such an object, and can make no predictions for how it would behave. Experimentally, therefore, we are limited to observing physics only in either of the regimes where one of gravitational or quantum mechanics becomes too small to matter, and so have no prospect of figuring out the rules in the general case where both must be considered. This is why, in the 21st century, fundamental physics, as an empirical science, is stuck in such a deep rut. As we wrest the secrets about dark matter and dark energy from their hiding places, more light (pun) might be shed on such matters.

What Planck units represent “limits”, and how?

The only three Planck units that can in any sense be thought of as some kind of limit are the Planck length, the Planck time, and the Planck speed (which is of course simply the speed of light).

To finally answer the question: the reason the Planck length is considered a limit. Anything localised in at least one direction to something close to the Planck length has, by the Heisenberg uncertainty principle, an energy expectation value that approaches the Planck energy, and hence a gravitational radius approaching its spatial extent. In other words, the more localised it becomes, the closer it becomes to a Planck hole. Now, we know nearly nothing about Planck holes, so at this point physicists start talking out of their backsides to some extent. But from classical general relativity, we know that black holes distort the geometry of spacetime. It therefore doesn’t make much sense to talk about a physical system with substructure below the Planck length, since the sub-Planck scale variations in energy density would shroud the whole object in an event horizon and leave it causally disconnected from the rest of spacetime.

The Planck time is “limiting” simply because it is the minimum possible time it would take to traverse the shortest meaningful distance, so there is no physical process characterised as having a timescale faster than the Planck time.

It is important to state that the Planck length and time are not “hard” barriers like the sound barrier. They merely represent the scale where different physical phenomena move from being negligible to being significant. Indeed, one can define many different systems of Planck units, for example by taking the non-reduced Planck’s constant ℎ=2𝜋ℏ=1instead of ℏ=1. A factor of 2𝜋 doesn’t really matter- they should be considered instead a “here be dragons” warning post that when energy is around this localised, quantum phenomena cannot be considered approximately independent of the spacetime they occupy.

To dispute what some others have said: in the current widely accepted fundamental theories of physics (GR and the Standard Model), spacetime itself is continuous, self-similar at all scales, and not in any way quantised or pixelated. It does seem, however, that there are theories of quantum gravity that challenge this assumption, notably loop quantum gravity, but these are still very edgy and far from achieving consensus. The Planck length is only currently considered somehow minimal because of the rules governing energetic phenomena that live in that spacetime, not the spacetime itself.

One final subtlety. There are still certainly occasions in which it makes sense to speak about values of physical parameters on the wrong “side” of the limiting Planck unit. The classic example of this is that if I shoot a laser pointer at the Moon and sweep it across the sky very fast, the point of light can move across the surface of the Moon with a speed not limited by the speed of light (the Planck speed). This is because the point of light is not itself a physical object, cannot carry either energy or information along its path, and is not subject to the laws that constrain the photons themselves.

There are analogous examples where one can still talk about lengths below the Planck length. For example, in spectroscopy, physicists often talk about “linewidth”. No source of light emits a pure electromagnetic wave at a single frequency. If you measure the peak-to-peak distance for each wave of a laser, you will see some small variation in wavelength- this is known as the “linewidth”. There is no physical lower bound to this; the linewidth of the laser at LIGO used to detect gravitational waves is about a billion times smaller than the diameter of a proton. Potentially, this quantity could be smaller than the Planck length.

Finally, if we examine the emission lines of the hydrogen atom, there is no fundamental lower limit to the width of these lines – they may be narrower than the Planck length.

My thanks to a number of creatives for help.