Planck mass

In physics, the Planck mass, denoted by m_P, is the unit of mass in the system of natural units known as Planck units. It is approximately 0.02 milligrams (roughly the mass of a flea egg^[1]). For comparison, this value is of the order of 10¹⁵ (one million billion) times larger than the highest energy available to contemporary particle accelerators.^[2]

It is defined so that

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

≈ 1.220910×10¹⁹ GeV/c² = 2.176470(51)×10⁻⁸ kg^[3] = 21.76470 µg = 1.3107×10¹⁹ u,^[4]

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

Particle physicists and cosmologists often use an alternative normalization with the reduced Planck mass, which is

M_{\text{P}}={\sqrt {\frac {\hbar c}{8\pi G}}}

≈ 4.341×10⁻⁹ kg = 2.435×10¹⁸ GeV/c².

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

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 attempts 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}}\hbar ^{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=L/T\

G=L^{3}/MT^{2}

\hbar =ML^{2}/T\

Therefore,

[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:

n_{3}-n_{2}=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}\hbar ^{1/2}={\sqrt {\frac {c\hbar }{G}}}.

Elimination of a coupling constant

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 graviton) of angular wavelength r (see the Planck relation), or that their ratio equals one.

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

Isolating m_P, we get that

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

Note that if, instead of Planck masses, the electron mass were used, the equation would require a gravitational coupling constant, analogous to how the equation of the fine-structure constant relates the elementary charge and the Planck charge. Thus, the Planck mass can be viewed as resulting from absorbing the gravitational coupling constant into the unit of mass (and those of distance/time as well), like the Planck charge does for the fine-structure constant.

Compton wavelength and Schwarzschild radius

The Planck mass can be derived approximately by setting it as the mass whose Compton wavelength and Schwarzschild radius are equal.^[5] The Compton wavelength is, loosely speaking, the length-scale where quantum effects start to become important for a particle; the heavier the particle, the smaller the Compton wavelength. The Schwarzschild radius is the radius in which a mass, if it were a black hole, would have its event horizon located; the heavier the particle, the larger the Schwarzschild radius. If a particle were massive enough that its Compton wavelength and Schwarzschild radius were approximately equal, its dynamics would be strongly affected by quantum gravity. This mass is (approximately) the Planck mass.

The Compton wavelength is

\lambda _{c}={\frac {h}{mc}}

and the Schwarzschild radius is

r_{s}={\frac {2Gm}{c^{2}}}

Setting them equal:

m={\sqrt {\frac {hc}{2G}}}={\sqrt {\frac {\pi c\hbar }{G}}}

This is not quite the Planck mass: It is a factor of ${\sqrt {\pi }}$ larger. However, this heuristic derivation gives the right order of magnitude.

Notes and references

↑ "average weight of a flea egg (6992342000000000000♠3.42×10⁻² mg)" M. W. Dryden, Blood consumption and feeding behavior of the cat flea (1990) p. 51.
↑ Maximum energy of the Large Hadron Collider: 13 TeV (as of 2015).
↑ "CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values
↑ CODATA 2016: value in GeV, value in kg
↑ The riddle of gravitation by Peter Gabriel Bergmann, page x

Bibliography

Sivaram, C. (2007). "What is Special About the Planck Mass?". arXiv:0707.0058v1  [gr-qc].

External links

The NIST Reference on Constants, Units, and Uncertainty

Planck's natural units
Planck constant Planck units
Base Planck units	Planck time Planck length Planck mass Planck charge Planck temperature
Derived Planck units	Planck energy Planck force Planck power Planck density Planck angular frequency Planck pressure Planck current Planck voltage Planck impedance Planck momentum Planck area Planck volume Planck acceleration

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