Planck Units - Uncertainties in Values

Uncertainties in Values

Table 2 clearly defines Planck units in terms of the fundamental constants. Yet relative to other units of measurement such as SI, the values of the Planck units are only known approximately. This is mostly due to uncertainty in the value of the gravitational constant G.

Today the value of the speed of light c in SI units is not subject to measurement error, because the SI base unit of length, the metre, is now defined as the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. Hence the value of c is now exact by definition, and contributes no uncertainty to the SI equivalents of the Planck units. The same is true of the value of the vacuum permittivity ε₀, due to the definition of ampere which sets the vacuum permeability μ₀ to 4π × 10−7 H/m and the fact that μ₀ε₀ = 1/c2. The numerical value of the reduced Planck constant ℏ has been determined experimentally to 44 parts per billion, while that of G has been determined experimentally to no better than 1 part in 8300 (or 120000 parts per billion). G appears in the definition of almost every Planck unit in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3 SI equivalents of the Planck units derives almost entirely from uncertainty in the value of G. (The propagation of the error in G is a function of the exponent of G in the algebraic expression for a unit. Since that exponent is ±1⁄₂ for every base unit other than Planck charge, the relative uncertainty of each base unit is about one half that of G. This is indeed the case; according to CODATA, the experimental values of the SI equivalents of the base Planck units are known to about 1 part in 16600, or 60000 parts per billion.)