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{{short description |Unit of mass in the system of Planck units}}
In [[Physics:Physics|physics]], the '''Planck mass''', denoted by ''m''<sub>P</sub>, is the unit of [[Physics:Mass|mass]] in the system of [[Natural units|natural units]] known as [[Planck units]]. It is approximately 21&nbsp;[[Kilogram#SI multiples|micrograms]],<ref>{{cite web |url=https://www.wolframalpha.com/input/?i=planck+mass |title=Planck mass |website=wolframalpha.com}}</ref> or roughly the mass of a flea egg.<ref>{{cite journal |quote=average weight of a flea egg ({{val|3.42|e=-2|u=mg}}) |first=M.W. |last=Dryden |title=Blood consumption and feeding behavior of the cat flea |journal=[journal name missing] |year=1990 |page=51}}{{full citation|date=November 2019}}</ref>

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 [[Biography:Max Planck|Max Planck]] considered fundamental and universal units. For comparison, this value is of the order of 10<sup>15</sup> (a quadrillion) times larger than the highest energy available to [[Physics:Particle accelerator|particle accelerator]]s as of 2015.{{efn|Maximum energy of the [[Physics:Large Hadron Collider|Large Hadron Collider]]: 13&nbsp;TeV (as of 2015).{{citation needed|date=November 2019}} }}

It is defined as:
:<math>m_\text{P}=\sqrt{\frac{\hbar c}{G}},</math>

where ''c'' is the [[Physics:Speed of light|speed of light]] in a vacuum, ''G'' is the [[Physics:Gravitational constant|gravitational constant]], and ''ħ'' is the [[Physics:Reduced Planck constant|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:

:<math> 1 \ m_\mathrm{P} \approx </math> {{val|1.220910|e=19|ul=GeV/c2}}
: = {{physconst|mP}}
: = {{val|21.76470|ul=mcg}}
: = {{val|1.3107|e=19|ul=u}}&nbsp;.<ref>{{cite web |series=CODATA |year=2016 |url=http://physics.nist.gov/cgi-bin/cuu/Value?plkmc2gev |title=Planck mass value in GeV}}</ref><ref>{{cite web |series=CODATA |year=2016 |url=http://physics.nist.gov/cgi-bin/cuu/Value?plkm |title=Planck mass value in kg}}</ref>

For the [[Physics:Planck mass|Planck mass]] <math>m_{\rm P} = \sqrt{\hbar c/G}</math>, the [[Astronomy:Schwarzschild radius|Schwarzschild radius]] (<math>r_{\rm S} = 2 l_{\rm P}</math>) and the [[Physics:Compton wavelength|Compton wavelength]] (<math>\lambda_{\rm C} = 2 \pi l_{\rm P}</math>) are of the same order as the [[Planck length]] <math>l_{\rm P} = \sqrt{\hbar G/c^3}</math>.

[[Physics:Particle physics|Particle physicists]] and [[Astronomy:Physical cosmology|cosmologists]] often use an [[Planck units#Gravity|alternative normalization]] with the '''reduced Planck mass''', which is
:<math>M_\text{P} = \sqrt\frac{\hbar c}{8\pi G} \approx\ </math> {{val|4.341|e=-9|u=kg}} = {{val|2.435|e=18|ul=GeV/c2}}&nbsp;.{{efn| The convention of including the factor <math>\frac{1}{\sqrt{8\pi}}</math> simplifies several other equations in [[Physics:General relativity|general relativity]].}}

==History ==

The Planck mass was first suggested by [[Biography:Max Planck|Max Planck]]<ref>{{cite journal |first=M. |last=Planck |title=Naturlische Masseinheiten |publisher=Der Koniglich Preussischen Akademie der Wissenschaften <!-- "The Royal Prussian Academy of Science" --> |language=DE |journal=[journal name missing] <!-- Presumably "The Proceedings of ..." or "Annals of ..." or "Journal of ..." --> |page=479 |year=1899}}{{full citation|date=November 2019}}</ref> 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|dimensional analysis]]. In this approach, one starts with the three [[Physics:Physical constant|physical constant]]s ''ħ'', ''c'', and ''G'', and attempts to combine them to get a quantity whose dimension is mass. The formula sought is of the form
:<math>m_\text{P} = c^{n_1} G^{n_2} \hbar^{n_3},</math>

where <math>n_1,n_2,n_3</math> are constants to be determined by equating the dimensions of both sides. Using the symbols <math>\mathsf{M}</math> for mass, <math>\mathsf{L}</math> for length and <math>\mathsf{T}</math> for time, and writing [{{mvar|x}}] to denote the dimension of some physical quantity {{mvar|x}}, we have the following:
:<math>\,[c]\,=\mathsf{L}\mathsf{T}^{-1} \ </math>
:<math>[G]=\mathsf{M}^{-1}\mathsf{L}^3\mathsf{T}^{-2} </math>
:<math>\,[\hbar]\,=\mathsf{M}\mathsf{L}^2\mathsf{T}^{-1} \ </math>.

Therefore,
:<math>[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}.</math>

If one wants this to equal <math>\mathsf{M}</math>, the dimension of mass, using <math>\mathsf{M} = \mathsf{M}^1\mathsf{L}^0\mathsf{T}^0</math>, the following equations need to hold:
:<math>~~~~~~~~~-n_2 + ~~n_3 = 1 \ </math>
:<math>~~~n_1 + 3n_2 + 2n_3 = 0 \ </math>
:<math>-n_1 - 2n_2 - ~~n_3 = 0 \ </math>.

The solution of this system is:
:<math>n_1 = 1/2, n_2 = -1/2, n_3 = 1/2. \ </math>

Thus, the Planck mass is:
:<math>m_\text{P} = c^{1/2}G^{-1/2}\hbar^{1/2} = \sqrt{\frac{c\hbar}{G}}\ </math>.

Dimensional analysis can only determine a formula [[Up to|up to]] a [[Physics:Dimensionless|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''<sub>P</sub> of separation ''r'' is equal to the energy of a photon (or the mass-energy of a [[Physics:Graviton|graviton]], if such a particle exists) of angular wavelength ''r'' (see the [[Physics:Planck relation|Planck relation]]), or that their ratio equals one.
:<math>E=\frac{G m_\text{P}^2}{r}=\frac{\hbar c}{r}\ </math>.

Isolating ''m''<sub>P</sub>, we get that
:<math>m_\text{P}=\sqrt{\frac{\hbar c}{G}}</math>

Note that if, instead of Planck masses, the electron mass were used, the equation would require a [[Physics:Gravitational coupling constant|gravitational coupling constant]], analogous to how the equation of the [[Physics:Fine-structure constant|fine-structure constant]] relates the [[Physics:Elementary charge|elementary charge]] and the [[Physics:Planck charge|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.

==See also==
* [[Astronomy:Micro black hole|Micro black hole]]
* [[Orders of magnitude (mass)]]
* [[Planck length]]
* [[Astronomy:Planck particle|Planck particle]]
* [[Stoney units]]

== Footnotes ==
{{notelist|1}}

== References==
{{Reflist|25em}}

==Bibliography==
{{Refbegin}}
*{{cite arXiv |last=Sivaram |first=C. |eprint=0707.0058v1 |title=What is special about the Planck mass? |class=gr-qc |year=2007}}
{{Refend}}

==External links==
* {{cite web |url=http://physics.nist.gov/cuu/Constants/index.html |publisher=NIST |title=Reference on Constants, Units, and Uncertainty}}


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[[Category:Physical constants]]
[[Category:Planck units]]
[[Category:Units of mass]]

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