Physics/Essays/Fedosin/Planck mass

Planck mass (m_P) is the unit of mass in the system of natural units known as Planck units. It is defined so that

$m_P=\sqrt{\frac{\hslash c}{G}}=1.2209\cdot 10^{19}{\text{GeV/c}}^2=2.17644\cdot 10^{-8}\text{kg},$

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

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

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

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

The name “Planck mass” honors Max Planck, who was the first to propose the mass. The Planck mass is not so far from Stoney mass $m_S$ and connects with it through fine structure constant $\alpha =\frac{e^2}{2{\epsilon }_{0}hc}$ in the way: $m_P=\frac{m_S}{\sqrt{\alpha }}.$

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

$m_{\text{P}}=c^{n_1}{G}^{n_2}{\hslash }^{n_3},$

where $n_1,n_2,n_3$ 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:

$[c]={𝖫𝖳}^{-1}$

$[G]={𝖬}^{-1}{𝖫}^3{𝖳}^{-2}$

$[\hslash ]={𝖬}^1{𝖫}^2{𝖳}^{-1}$.

Therefore,

$[c^{n_1}{G}^{n_2}{\hslash }^{n_3}]={𝖬}^{-n_2+n_3}{𝖫}^{n_1+3n_2+2n_3}{𝖳}^{-n_1-2n_2-n_3}$

If one wants dimensions of mass, the following equations must hold:

$-n_2+n_3=1$

$n_1+3n_2+2n_3=0$

$-n_1-2n_2-n_3=0$.

The solution of this system is:

$n_1=1/2,n_2=-1/2,n_3=1/2.$

Thus, the Planck mass is:

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

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 m_P.

The Planck mass has the Schwarzschild radius equals to its Compton wavelength divided by $\pi$. 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.

See also

• Planck scale
• Selfconsistent gravitational constants
• Selfconsistent electromagnetic constants
• Maxwell-like gravitational equations
• Gravitational characteristic impedance of free space
• Vacuum constants
• Stoney mass

Sources

1. Sivaram C. WHAT IS SPECIAL ABOUT THE PLANCK MASS? PDF
2. Johnstone Stoney, Phil. Trans. Roy. Soc. 11, (1881)
3. Stephen J. Crothers and Jeremy Dunning-Davies†. Planck Particles and Quantum Gravity. PROGRESS IN PHYSICS, Vol.3, July, 2006

External links

• The NIST Reference on Constants, Units, and Uncertainty
• Planck Units
• Planck Units