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. Unlike some other Planck units, such as Planck length, Planck mass is not a fundamental lower or upper bound; instead, Planck mass is a unit of mass defined using only what Max Planck considered fundamental and universal units. One Planck mass is roughly the mass of a flea egg^[1]. For comparison, this value is of the order of 10¹⁵ (a quadrillion) times larger than the highest energy available to contemporary particle accelerators.^[2]

It is defined as:

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

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

Substituting values for the various components in this definition gives the approximate equivalent value of this unit in terms of other units of mass:

1\ m_{\mathrm {P} }

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

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.

History

The Planck mass was first suggested by Max Planck^[5] in 1899. He suggested that there existed some fundamental natural units for length, mass, time and energy. He derived these units using only dimensional analysis of what he considered the most fundamental universal constants: the speed of light, the Newton gravitational constant and the Planck constant.

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 whose dimension is mass. The formula sought 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 equating the dimensions of both sides. Using the symbols ${\mathsf {M}}$ for mass, ${\mathsf {L}}$ for length and ${\mathsf {T}}$ for time, and writing [ $x$ ] to denote the dimension of some physical quantity $x$ , we have the following:

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

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

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

.

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 this to equal ${\mathsf {M}}$ , the dimension of mass, using ${\mathsf {M}}={\mathsf {M}}^{1}{\mathsf {L}}^{0}{\mathsf {T}}^{0}$ , the following equations need to 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}\hbar ^{1/2}={\sqrt {\frac {c\hbar }{G}}}.

To be sure, 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.

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 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.

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.

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
↑ M. Planck. Naturlische Masseinheiten. Der Koniglich Preussischen Akademie Der Wissenschaften, p. 479, 1899

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

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[1] "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.

[2] Maximum energy of the Large Hadron Collider: 13 TeV (as of 2015).

[physconst-mP-3] "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

[4] CODATA 2016: value in GeV, value in kg

[5] M. Planck. Naturlische Masseinheiten. Der Koniglich Preussischen Akademie Der Wissenschaften, p. 479, 1899

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 Planck angle