The strange beauty of Planck units

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What does it mean to say that the speed of light is 1?

We know the speed of light in the vacuum of space to be 299,792,458 m/s – or about 300,000 km/s. It’s a quantity of speed that’s very hard to visualise with the human brain. In fact, it’s so fast as to practically be instantaneous for the human experience. In some contexts it might be reassuring to remember the 300,000 km/s figure, such as when you’re a theoretical physicist working on quantum physics problems and you need to remember that reality is often (but not always) local, meaning that when a force appears to to transmit its effects on its surroundings really rapidly, the transmission is still limited by the speed of light. (‘Not always’ because quantum entanglement appears to break this rule.)

Another way to understand the speed of light is as an expression of proportionality. If another entity, which we’ll call X, can move at best at 150,000 km/s in the vacuum of space, we can say the speed of light is 2x the speed of X in this medium. Let’s say that instead of km/s we adopt a unit of speed called kb/s, where b stands for bloop: 1 bloop = 79 km. So the speed of light in vacuum becomes 3,797 kb/s and the speed of X in vacuum becomes 1,898.5 kb/s. The proportionality between the two entities – the speeds of light and X in vacuum – you’ll notice is still 2x.

Let’s change things up a bit more, to expressing the speed of light as the nth power of 2. n = 18 comes closest for light and n = 17 for X. (The exact answer in each case would be log s/log 2, where s is the speed of each entity.) The constant of proportionality is not even close to 2 in this case. The reason is that we switched from linear units to logarithmic units.

This example shows how even our SI units – which allow us to make sense of how much a mile is relative to a kilometre and how much a solar year is in terms of seconds, and thus standardise our sense of various dimensions – aren’t universally standard. The SI units have been defined keeping the human experience of reality in mind – as opposed to, say, those of tardigrades or blue whales.

As it happens, when you’re a theoretical physicist, the human experience isn’t very helpful as you’re trying to understand the vast scales on which gravity operates and the infinitesimal realm of quantum phenomena. Instead, physicists set aside their physical experiences and turned to the universal physical constants: numbers whose values are constant in space and time, and which together control the physical properties of our universe.

By combining only four universal physical constants, the German physicist Max Planck found in 1899 that he could express certain values of length, mass, time and temperature in units related to the human experience. Put another way, these are the smallest distance, mass, duration and temperature values that we can express using the constants of our universe. These are:

  • G, the gravitational constant (roughly speaking, defines the strength of the gravitational force between two massive bodies)
  • c, the speed of light in vacuum
  • h, the Planck constant (the constant of proportionality between a photon’s energy and frequency)
  • kB, the Boltzmann constant (the constant of proportionality between the average kinetic energy of a group of particles and the temperature of the group)

Based on Planck’s idea and calculations, physicists have been able to determine the following:

Credit: Planck units/Wikipedia

(Note here that the Planck constant, h, has been replaced with the reduced Planck constant ħ, which is h divided by 2π.)

When the speed of light is expressed in these Planck units, it comes out to a value of 1 (i.e. 1 times 1.616255×10−35 m per 5.391247×10−44 s). The same goes for the values of the gravitational constant, the Boltzmann constant and the reduced Planck constant.

Remember that units are expressions of proportionality. Because the Planck units are all expressed in terms of universal physical constants, they give us a better sense of what is and isn’t proportionate. To borrow Frank Wilczek’s example, we know that the binding energy due to gravity contributes only ~0.000000000000000000000000000000000000003% of a proton’s mass; the rest comes from its constituent particles and their energy fields. Why this enormous disparity? We don’t know. More importantly, which entity has the onus of providing an explanation for why it’s so out of proportion: gravity or the proton’s mass?

The answer is in the Planck units, in which the value of the gravitational constant G is the desired 1, whereas the proton’s mass is the one out of proportion – a ridiculously small 10-19 (approx.). So the onus is on the proton to explain why it’s so light, rather than on gravity to explain why it acts so feebly on the proton.

More broadly, the Planck units define our universe’s “truly fundamental” units. All other units – of length, mass, time, temperature, etc. – ought to be expressible in terms of the Planck units. If they can’t be, physicists will take that as a sign that their calculations are incomplete, wrong or that there’s a part of physics that they haven’t discovered yet. The use of Planck units can reveal such sources of tension.

For example, since our current theories of physics are founded on the universal physical constants, the theories can’t describe reality beyond the scale described by the Planck units. This is why we don’t really know what happened in the first 10-43 seconds after the Big Bang (and for that matter any events that happen for a duration shorter than this), how matter behaves beyond the Planck temperature or what gravity feels like at distances shorter than 10-35 m.

In fact, just like how gravity dominates the human experience of reality while quantum physics dominates the microscopic experience, physicists expect that theories of quantum gravity (like string theory) will dominate the experience of reality at the Planck length. What will this reality look like? We don’t know, but we know that it’s a good question.

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