Planck Units - Other Possible Normalizations

Other Possible Normalizations

As already stated above, Planck units are derived by "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible nor necessarily the best. Moreover, the choice of what factors to normalize, among the factors appearing in the fundamental equations of physics, is not evident, and the values of the Planck units are sensitive to this choice.

Possible alternative normalizations include:

• The permittivity of free space ε₀ = 1.

Planck normalized to 1 the Coulomb force constant 1/(4πε₀) (as does the cgs system of units). This sets the Planck impedance, Z_P equal to Z₀/4π, where Z₀ is the characteristic impedance of free space. On the other hand, if ε₀ = 1:

• Sets the permeability of free space µ₀ = 1, (because c = 1).
• Equates the unit impedance, Z_P, to the characteristic impedance of free space Z₀;
• Eliminates 4π from the nondimensionalized form of Maxwell's equations;
• Introduces a factor of (4π)−1 into the nondimensionalized form of Coulomb's law.

• Boltzmann constant k_B = 2. This:
  • Removes the factor of 1/2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom;
  • Introduces a factor of 2 into the nondimensionalized form of Boltzmann's entropy formula;
  • Does not affect the value of any base or derived Planck unit other than the Planck temperature.

The factor 4π is ubiquitous in theoretical physics because the surface area of a sphere is 4πr2. This, along with the concept of flux is the basis for the inverse-square law. For example, gravitational and electrostatic fields produced by point charges have spherical symmetry (Barrow 2002: 214–15). The 4πr2 appearing in the denominator of Coulomb's law, for example, follows from the flux of an electrostatic field being distributed uniformly on the surface of a sphere. If space had more than three spacial dimensions, the factor 4π would have to be changed.

In 1899, Newton's law of universal gravitation was still seen as exact, rather than as a convenient approximation holding for "small" velocities and distances (the non-fundamental nature of Newton's law was shown to be true following the development of general relativity in 1915). Hence Planck normalized to 1 the gravitational constant G in Newton's law. In theories emerging after 1899, G nearly always appears multiplied by 4π or a small integer multiple thereof. Hence a fundamental choice that has to be made when designing a system of natural units is which, if any, instances of 4nπ appearing in the equations of physics are to be eliminated via the normalization:

• 4πG = 1. This would eliminate the factor 4πG appearing in:
  • Gauss's law for gravity, Φ_g = −4πGM;
  • The Bekenstein–Hawking formula for the entropy of a black hole in terms of its mass m_BH and the area of its event horizon A_BH, simplifies to S_BH = πA_BH = (m_BH)2 where A_BH and m_BH are both measured in a slight modification of reduced Planck units, described below;
  • Normalizes the characteristic impedance of gravitational radiation in free space, Z₀ = 1. In any system of units, Z₀ = 4πG/c. General relativity predicts that gravitational radiation propagates at the same speed as electromagnetic radiation;
  • The gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or reasonably flat space-time. These equations have the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetism, with mass density replacing charge density, and with 1/(4πG) replacing ε₀.
• 8πG = 1. This would eliminate 8πG from the Einstein field equations, Einstein–Hilbert action, Friedmann equations, and the Poisson equation for gravitation. Planck units modified so that 8πG = 1 are known as reduced Planck units, because the Planck mass is divided by
  • The Bekenstein–Hawking formula for the entropy of a black hole simplifies to 2(m_BH)2 and 2πA_BH.
• 16πG = 1. This would eliminate the constant c4/(16πG) from the Einstein–Hilbert action. The Einstein field equations with cosmological constant Λ becomes R_μν − Λg_μν = (Rg_μν − T_μν)/2.

Hence a substantial body of physical theory discovered since Planck (1899) suggests normalizing to 1 not G but 4nπG, for one of n = 1, 2, or 4. Doing so would introduce a factor of 1/(4nπ) into the nondimensionalized form of the law of universal gravitation, consistent with the modern formulation of Coulomb's law in terms of the vacuum permittivity. In fact, alternate normalizations frequently preserve the (4π)−1 in the nondimensionalized form of Coulomb's law as well, so that the nondimensionalized Maxwell's equations for electromagnetism and gravitomagnetism both take the same form as those for EM in SI, which is devoid of multiples of 4π.