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P6.5.1Planck units of measurement

In physics, the Planck units of measurement are those defined such that the five universal physical constants \(c\) (the speed of light), \(G\) (the gravitational constant), \(\hbar\) (the reduced Planck constant), \((4\pi\epsilon_0)^{-1}\) (the Coulomb constant) and \(k_\mathrm{B}\) (the Boltzmann constant) are set to unity. The dimensions of these quantities in terms of length (L), mass (M), time (T), charge (Q) and thermodynamic temperature (\(\mathrm{\Theta}\)) are given in the table below, along with their values in SI units.

ConstantValueDimensions
\(c\)Speed of light\(2.99792458 \times 10^8 \;\mathrm{m\,s^{-1}}\)\(\mathrm{L\,T^{-1}}\)
\(G\)Gravitational constant\(6.6743\times 10^{-11} \;\mathrm{m^3\,kg^{-1}\,s^{-2} }\)\(\mathrm{L^3\,M^{-1}\,T^{-2}}\)
\(\hbar\)Reduced Planck constant\(1.0545718176461565\times 10^{-34} \;\mathrm{J\,s}\)\(\mathrm{L^2\,M\,T^{-1}}\)
\((4\pi\epsilon_0)^{-1}\)Coulomb constant\(8.987551786170798 \times 10^9 \;\mathrm{N\,m^2\,C^{-2}}\)\(\mathrm{L^3\,M\,T^{-2}\,Q^{-2}}\)
\(k_\mathrm{B}\)Boltzmann constant\(1.380649 \times 10^{-23} \;\mathrm{J\,K^{-1}}\)\(\mathrm{L^2\,M\,T^{-2}\,\Theta^{-1}}\)

This suggests the following matrix relationship between the constants and their dimensions:

\[\begin{array}{crrrrr} {}&\mathrm{L} & \mathrm{M} & \mathrm{T} & \mathrm{Q} & \mathrm{\Theta}\\ c & 1 & 0 & -1 & 0 & 0\\ G & 3 & -1 & -2 & 0 & 0\\ \hbar & 2 & 1 & -1 & 0 & 0\\ (4\pi\epsilon_0)^{-1} & 3 & 1 & -2 & -2 & 0\\ k_\mathrm{B} & 2 & 1 & -2 & 0 & -1 \end{array}\]

Using the inverse of this matrix, determine the SI values of length, mass, time, charge and temperature in the base Planck units; that is, the combination of these physical constants yielding the dimensions L, M, T, Q and \(\mathrm{\Theta}\). For example, the Planck length is found to be \(l_\mathrm{P} = \sqrt{\hbar G / c^3} = 1.616199 \times 10^{-35}\;\mathrm{m}\).

Show solution

Here is one possible solution:

planckunits
import numpy as np

# The physical constants set to unity in the Planck unit system:
# c, G, hbar, 1/(4.pi.eps0), kB
constants = np.array(
[
    299792458,
    6.6743e-11,
    1.0545718176461565e-34,
    8.987551786170798e9,
    1.380649e-23,
]
)
dimensions = ("length", "mass", "time", "charge", "temperature")
si_units = ("m", "kg", "s", "C", "K")

M = np.array(
[
    [1, 0, -1, 0, 0],
    [3, -1, -2, 0, 0],
    [2, 1, -1, 0, 0],
    [3, 1, -2, -2, 0],
    [2, 1, -2, 0, -1],
]
)

# Invert the matrix to find the Planck quantities in SI units
Minv = np.linalg.inv(M)
planck_set = np.prod(constants**Minv, axis=1)
for dimension, planck_quant, si_unit in zip(dimensions, planck_set, si_units):
print(f"Planck {dimension} = {planck_quant} {si_unit}")

Which gives:

Planck length = 1.6162550244237098e-35 m
Planck mass = 2.1764343427178775e-08 kg
Planck time = 5.391246448313618e-44 s
Planck charge = 1.8755460384151295e-18 C
Planck temperature = 1.416784162157341e+32 K

(Note that not all of these digits are significant).