Base Planck Units
All systems of measurement feature base units: in the International System of Units (SI), for example, the base unit of length is the meter. In the system of Planck units, the Planck base unit of length is known simply as the Planck length, the base unit of time is the Planck time, and so on. These units are derived from the five dimensional universal physical constants of Table 1, in such a manner that these constants are eliminated from fundamental equations of physical law when physical quantities are expressed in terms of Planck units. For example, Newton's law of gravitation:
is equivalent to
Both equations are dimensionally consistent and equally valid in any system of units, but the second equation, with G missing, is relating only dimensionless quantities since any ratio of two like-dimensioned quantities is a dimensionless quantity. If, by a shorthand convention, it is axiomatically understood that all physical quantities are expressed in terms of Planck units, the ratios above may be expressed simply with the symbols of physical quantity, without being scaled by their corresponding unit:
In order for this last equation to be valid (without G present), F, m1, m2, and r are understood to be the dimensionless numerical values of these quantities measured in terms of Planck units. This is why Planck units or any other use of natural units should be employed with care. Referring to G = c = 1, Paul S. Wesson wrote that, "Mathematically it is an acceptable trick which saves labour. Physically it represents a loss of information and can lead to confusion."
| Constant | Symbol | Dimension | Value in SI units with uncertainties |
|---|---|---|---|
| Speed of light in vacuum | c | L T −1 | 2.99792458×108 m s−1 (exact by definition of meter) |
| Gravitational constant | G | L3 M−1 T −2 | 6.67384(80)×10−11 m3kg−1s−2 |
| Reduced Planck constant | ħ = h/2π where h is Planck constant |
L2 M T −1 | 1.054571726(47)×10−34 J s |
| Coulomb constant | (4πε0)−1 where ε0 is the permittivity of free space |
L3 M T −2 Q−2 | 8.9875517873681764×10^9 kg m3 s−2 C−2 (exact by definition of ampere and meter) |
| Boltzmann constant | kB | L2 M T −2 Θ−1 | 1.3806488(13)×10−23 J/K |
Key: L = length, M = mass, T = time, Q = electric charge, Θ = temperature.
As can be seen above, the gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 Planck force. Likewise, the distance traveled by light during 1 Planck time is 1 Planck length. To determine, in terms of SI or another existing system of units, the quantitative values of the five base Planck units, those two equations and three others must be satisfied to determine the five unknown quantities that define the base Planck units:
Solving the five equations above for the five unknowns results in a unique set of values for the five base Planck units:
| Name | Dimension | Expression | Value (SI units) |
|---|---|---|---|
| Planck length | Length (L) | 1.616 199(97) × 10−35 m | |
| Planck mass | Mass (M) | 2.176 51(13) × 10−8 kg | |
| Planck time | Time (T) | 5.391 06(32) × 10−44 s | |
| Planck charge | Electric charge (Q) | 1.875 545 956(41) × 10−18 C | |
| Planck temperature | Temperature (Θ) | 1.416 833(85) × 1032 K |
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