Planck mass

The formula for the Planck mass can be derived by dimensional analysis. The basic idea is to find a quantity based on "universals" like the "speed of light in vacuum", that are both objective and invariant or unchangeable as we shift from one experimental or observational situation to another. This is a somewhat idealized perspective, as we must think in a framework that is not experimentally accessible. For example, even if the speed of light is observed and estimated and is measured under careful conditions closely approximating "vacuum" (insofar as we can do so), we do not necessarily know if the results (under repeated identical conditions) which may well show some variability, indicate a flaw in our assumption of universal speed, for light, or is correlated with factors such as possible biases or mistakes we ourselves are making, or certain ambiguities inherent in the experimental set up. But the goal is to remove traces of the particular experiments or observations and concomitant specific measured quantities, like length or time that bear on a speed measurement. In this context, too, we must ask ourselves what "universal" quantities are relevant for mass estimations in experiments. The acceleration due to gravity, g, on the surface of the Earth is not quite an objective constant, but close enough for that environment under numerous typical conditions. Likewise, when we consider the general astrophysical context of stars, planets, etc. Newton's gravitation constant G seems to represent an adequate "universal" under numerous ordinary experimental or observational conditions. In this approach, one starts with the three physical constants ħ, c, and G, and attempts to combine them to get a quantity whose dimension is mass. The formula sought is of the form

${\displaystyle m_{\text{P}}=c^{n_{1}}G^{n_{2}}\hbar ^{n_{3}},}$

where ${\displaystyle n_{1},n_{2},n_{3}}$ are constants to be determined by equating the dimensions of both sides. Using the symbols ${\displaystyle {\mathsf {M}}}$ for mass, ${\displaystyle {\mathsf {L}}}$ for length and ${\displaystyle {\mathsf {T}}}$ for time, and writing [x] to denote the dimension of some physical quantity x, we have the following:

${\displaystyle \,[c]\,={\mathsf {L}}{\mathsf {T}}^{-1}\ }$

${\displaystyle [G]={\mathsf {M}}^{-1}{\mathsf {L}}^{3}{\mathsf {T}}^{-2}}$

${\displaystyle \,[\hbar ]\,={\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}\ }$.

Therefore,

${\displaystyle [c^{n_{1}}G^{n_{2}}\hbar ^{n_{3}}]={\mathsf {M}}^{-n_{2}+n_{3}}{\mathsf {L}}^{n_{1}+3n_{2}+2n_{3}}{\mathsf {T}}^{-n_{1}-2n_{2}-n_{3}}.}$

If one wants this to equal ${\displaystyle {\mathsf {M}}}$, the dimension of mass, using ${\displaystyle {\mathsf {M}}={\mathsf {M}}^{1}{\mathsf {L}}^{0}{\mathsf {T}}^{0}}$, the following equations need to hold:

${\displaystyle ~~~~~~~~~-n_{2}+~~n_{3}=1\ }$

${\displaystyle ~~~n_{1}+3n_{2}+2n_{3}=0\ }$

${\displaystyle -n_{1}-2n_{2}-~~n_{3}=0\ }$.

The solution of this system is:

${\displaystyle n_{1}=1/2,n_{2}=-1/2,n_{3}=1/2.\ }$

Thus, the Planck mass is:

${\displaystyle m_{\text{P}}=c^{1/2}G^{-1/2}\hbar ^{1/2}={\sqrt {\frac {c\hbar }{G}}}\ }$.

Dimensional analysis can only determine a formula up to a dimensionless multiplicative factor. There is no a priori reason for starting with the reduced Planck constant ħ instead of the original Planck constant h, which differs from it by a factor of 2π.

Equivalently, the Planck mass is defined such that the gravitational potential energy between two masses m_P of separation r is equal to the energy of a photon (or the mass–energy of a graviton, if such a particle exists) of angular wavelength r (see the Planck relation), or that their ratio equals one.

${\displaystyle E={\frac {Gm_{\text{P}}^{2}}{r}}={\frac {\hbar c}{r}}\ }$.

Isolating m_P, we get that

${\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}}$