(Note: I am using some Greek letters and math symbols in this post and HTML can be dicey in how they appear. They look ok to me using the Safari browser on my Mac but if it looks weird to you, let me know in the comments and I will try and tweak the HTML to make it look better.)

In our lives we need to have some system of units for quantities like mass (M), length (L), and time (T). The most common system, used almost everywhere in the world with the US being a notable exception, is the metric system where length is in meters, time is in seconds, and mass is in kilograms. But any such system of units is purely conventional and if we were to make contact with any extra-terrestrials, it is almost certain that they would have a different set of units. But could there be units that are universal?

Max Planck introduced the constant h in 1900 that now bears his name as part of his explanation for the blackbody radiation spectrum, and it became realized that it was a universal constant of nature, like the gravitational constant G. In 1905, Einstein’s special theory of relativity suggested that the speed of light c was another universal constant and so we had three universal constants of nature.

ℏ=h/2π = 1.055×10^-34 Js with dimensions ML²/T

G = 6.670×10^-11 Nm²/kg² with dimensions L³/(MT²)

c = 3.000×10⁸ m/s with dimensions L/T

In 1913 Planck noted that these three constants could be used to obtain new units of length, time, and mass that would be the same everywhere, and thus could serve as a universal set of units that are now associated with his name (Cosmology by Edward Harrison, p. 486).

Planck length = √(Gℏ/c³) =1.614×10^-35 meters

Planck time = √(Gℏ/c⁵) = 5.381×10^-44 seconds

Planck mass = √(ℏc/G) = 2.718×10^-8 kg

The first thing to note is while all three quantities are very small, only the first two are smaller than anything that we could possibly measure. The Planck mass is of the order of micrograms, not extreme on an everyday scale. But on a particle physics scale, the Planck mass in very much larger than (say) the mass of the proton, which is 1.672×10^-27 kg, and about 15 orders of magnitude bigger than any mass that we could possibly produce at the highest energy particle accelerators we now have.

Are the Planck units just a universal standard or do they have some physical meaning as well? This question remained unanswered for some time. It used to be believed that both space and time were infinitely divisible, like points on the real number line. But over time, folklore grew that the Planck length represented the minimal distance below which speaking of a ‘point in space’ ceased to be meaningful and space becomes some kind of foam that cannot be specified more precisely. The Planck time is the amount of time that light takes to traverse the Planck length. Both those quantities are thought to represent the values at which our present theories become inapplicable and quantum gravity becomes important.

But as I said, this seemed to be largely folklore. What prompted this post was that just recently I came across a 1964 paper Possible Connection Between Gravitation and Fundamental Length by C. Alden Mead (Physical Review, vol. 135, no. 3B, B849-B862, 10 August 1964) that I believe deserves to be much more widely known. (Sabine Hossenfelder has a nice post discussing Mead’s paper.) In his paper, Mead finds that there is a fundamental meaning to the Planck length and time. Since Mead expanded upon the basic idea that Heisenberg used to arrive at his Uncertainty Principle, for those not familiar with that argument I will first sketch the HUP before getting to the meaning of the Planck units.

To measure the position of any particle, we basically bounce a wave off it and detect the deflected wave through a microscope. The direction of the trajectory of the deflected photon from the particle into the microscope is not known exactly because the width of the microscope lens allows a range of trajectories to enter. Assume that it is spread over an angle θ. Knowledge of the particle’s actual position is limited to a small range Δx given by the resolving power of the microscope. If the wavelength of the wave used for the measurement is given by λ, then this minimal value is given by Δx~λ/sinθ. In theory, we could measure position as precisely as we wish by making the wavelength as small as possible.

But a wave bundle also has momentum given by p=h/λ and energy given by E=hc/λ and so when the target particle gets hit by the wave, its measured momentum is uncertain by an amount given by Δp~psinθ=hsinθ/λ. To measure the momentum more precisely, we have to increase the wavelength of the wave, thus reducing its momentum and energy, so that its collision with the target particle is gentler.

So if we reduce the wavelength of the wave, we can measure the position of the target particle more precisely but its momentum less precisely. Increasing the wavelength does the opposite. There seems to be an unavoidable tradeoff. If we combine the two uncertainties, we get ΔxΔp~h, a constant. This is Heisenberg’s Uncertainty Principle and gives the lower limit on the simultaneous measurement of position and momentum.

But it still leaves open the possibility that if we are willing to abandon the desire for any knowledge of the momentum of the particle, then we could measure the position of the particle with arbitrary level of precision by simply making the wavelength of the measuring wave as small as possible.

What Mead did was to note that even this was not possible because since the wave used in the position measurement has energy, it thus exerts a gravitational force on the target particle that causes the latter to move. This force prevents us from knowing even in principle the exact position of the target particle, even if we are willing to forego any knowledge of its momentum. This force gets bigger as the energy (and momentum) of the wave increases, which corresponds to its wavelength decreasing. Mead showed that the uncertainty in target particle position due to this gravitational attraction is given by Δx~Gℏsinθ/(c³λ). By combining this with the resolving power uncertainty Δx~λ/sin θ, he obtained Δx~√(Gℏ/c³). i.e., the intrinsic uncertainty in the space location of the target particle due to the gravitational attraction between the target and probe is of the order of the Planck length.

But if there is an intrinsic uncertainty in space location, then it means that the process of clock synchronization used in relativity also becomes uncertain. The synchronization process depends upon correcting for the time of travel of light from the event location clock to the observer and is given by d/c, where d is the distance to the clock. Since this distance is now uncertain by the Planck length, the time taken by the light is also uncertain by an amount Δt=Δx/c=√(Gℏ/c⁵), which is the Planck time.

Thus the Planck length sets a fundamental lower limit on the precision of ruler readings and hence of location measurements. At the Planck length, space becomes less point-like and more foam-like. Similarly, the Planck time sets a corresponding limit on the precision of clock readings and hence of time measurements.

So we have meanings for the Planck length and time and find that they play fundamental roles in physics. What about the Planck mass? Mead did not address that question but we now have some insight into what it might mean. From general relativity, we know that associated with any non-rotating mass M is a distance called the Schwarzschild radius given by R=2GM/c². If the object is compressed so that its radius becomes less than the Schwarzschild radius, then nothing, not even light, can escape from its surface. i.e., it becomes a black hole. This is true for any mass but for most everyday objects, the Schwarzschild radius is far smaller than the size of the object so there is no danger of it becoming a black hole. For the Earth, for instance, the Schwarzschild radius is about one centimeter. The Earth would have to be compressed to a value smaller than that to become a black hole.

But what about elementary particles, say the electron? If we assume that they are truly point-like, then even though its Schwarzschild radius is really tiny (for an electron it is 1.4×10^-57 m), it will still be bigger than zero. Why isn’t an electron a black hole? We know it isn’t because we can scatter light off it. If it were a black hole, then as soon as a photon went inside the electron’s Schwarzschild radius, it would get trapped and be unable to escape

One possible answer is that no elementary particle is really point-like. We know that at very short distances, an electron is surrounded by a ‘cloud’ of photons and virtual particle-antiparticle pairs. What we measure as the electron’s charge and mass are not the ‘bare’ values of the charge and mass that we would find if we were able to penetrate through the cloud right to the center but is instead the net charge and mass as a result of the shielding of the bare values by the cloud.

Seen this way, it becomes difficult to define what we mean by the size of an elementary particle since the size of the surrounding cloud is not well-defined. Louis de Broglie postulated in 1925 that all particles had wave-like properties with an associated wavelength given by λ=h/p, where p is its momentum. Suppose we assume as a rough estimate of an elementary particle’s size that it is its Compton wavelength given by ℏ/Mc where M is its mass. So for an elementary particle to become a black hole, its mass M has to be such that its Compton wavelength becomes less than its Schwarzschild radius. i.e. ℏ/Mc<(2GM/c²). This gives M>√(ℏc/2G), which, apart from a factor of 2, is the Planck mass. In this model, the Planck mass sets the scale at which an elementary particle becomes a black hole and explains why all the elementary particles we are familiar with are not black holes. They are just not massive enough.

We thus see a connection between all three Plank units, as expressed by Edward Harrison in his book Cosmology (p. 478). (Note that an energy of 10¹⁹ GeV corresponds to a mass of about 10^-8 kg.)

The creation of a quantum black hole requires an enormous energy (10¹⁹ GeV) and these particles have not yet been observed. Particles of Planck mass are possibly the basic constituents of spacetime. If spacetime could be examined on the scale of 10^-33 centimeters, we might see a dense foam of virtual quantum black holes, popping into and out of existence on a time scale of 10^-43 seconds. Conceivably, in the beginning, the universe consisted of an extremely dense sea of real quantum black holes.. On this matter we cannot be certain because we lack a quantum theory of gravity.

Isn’t physics fun?