# Physics/Essays/Fedosin/Planck mass

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In physics, the Planck mass (mP) is the unit of mass in the system of natural units known as Planck units. It is defined so that

${\displaystyle m_{P}={\sqrt {\frac {\hbar c}{G}}}=1.2209\cdot 10^{19}\,{\text{GeV/c}}^{2}=2.17644\cdot 10^{-8}\,{\text{kg}}\ }$

where c is the speed of light in vacuum, G is the gravitational constant, and h is the reduced Planck constant.

Particle physicists and cosmologists often use the reduced Planck mass, which is

${\displaystyle {\sqrt {\frac {\hbar {}c}{8\pi G}}}}$ = 4.341 × 10-9 kg = 2.435 × 1018 GeV/c2.

The added factor of ${\displaystyle 1/{\sqrt {8\pi }}}$ simplifies a number of equations in general relativity.

The name honors Max Planck, who was the first to propose it. The Planck mass is not so far from Stoney mass ${\displaystyle m_{S}\ }$ and connect with it through fine structure constant ${\displaystyle \alpha ={\frac {e^{2}}{2\varepsilon _{0}hc}}}$ in the way: ${\displaystyle m_{P}={\frac {m_{S}}{\sqrt {\alpha }}}\ .}$

## Derivations

### Dimensional analysis

The formula for the Planck mass can be derived by dimensional analysis. In this approach, one starts with the three physical constants ħ, c, and G, and attempt to combine them to get a quantity with units of mass. The expected formula 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 matching the dimensions of both sides. Using the symbol L for length, T for time, M for mass, and writing "[x]" for the dimensions of some physical quantity x, we have the following:

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

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 dimensions of mass, the following equations must 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:

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

## Significance

Unlike all other Planck units and most Planck derived units, the Planck mass is a macroscopic amount, having a scale more or less conceivable to humans. For example, the body mass of a flea is roughly 4000 to 5000 mP.

The Planck mass has the Schwarzschild radius equals to its Compton wavelength divided by ${\displaystyle \pi \ }$. The Planck mass is also the mass of the Planck particle, a hypothetical tiny black hole whose Schwarzschild radius equals the Planck length.

The Planck mass is an idealized mass thought to have special significance for quantum gravity when general relativity and the fundamentals of quantum physics become mutually important to describe mechanics.