# Planck mass

In physics, the **Planck mass** (*m*_{P}) is the unit of mass in the system of natural units known as Planck units. It is defined so that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m_P=\sqrt{\frac{\hbar c}{G}}}**≈ ×10^{19}GeV/c^{2}= 1.220951(13)×10^{−8}kg, (or 2.176), 21.7651 mug^{[1]}

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 the **reduced Planck mass**, which is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \sqrt\frac{\hbar{}c}{8\pi G}}**≈ ×10^{−9}kg = 2.435 × 10 4.341^{18}GeV/c^{2}.

The added factor of 1/√8π simplifies a number of equations in general relativity.

The name honors Max Planck because the unit measures the approximate scale at which quantum effects, here in the case of gravity, become important. Quantum effects are typified by the magnitude of Planck's constant, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle h = 2\pi\hbar}**
.

## Contents

## Significance

The Planck mass is approximately the mass of the Planck particle, a hypothetical minuscule black hole whose Schwarzschild radius equals the Planck length.

Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg.

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.

## Derivations

### Dimensional analysis

The formula for the Planck mass can be derived by dimensional analysis. In this approach, you start 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m_P = c^{n_1} G^{n_2} \hbar^{n_3},}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle [c] = LT^{-1} \ }****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle [G] = M^{-1}L^3T^{-2} \ }****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle [\hbar] = M^1L^2T^{-1} \ }**.

Therefore,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle [c^{n_1} G^{n_2} \hbar^{n_3}] = M^{-n_2+n_3} L^{n_1+3n_2+2n_3} T^{-n_1-2n_2-n_3}}**

If we want dimensions of mass, the following equations must hold:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle -n_2 + n_3 = 1 \ }****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_1 + 3n_2 + 2n_3 = 0 \ }****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle -n_1 - 2n_2 - n_3 = 0 \ }**.

The solution of this system is:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle n_1 = 1/2, n_2 = -1/2, n_3 = 1/2. \ }**

Thus, the Planck mass is:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m_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.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle E=\frac{G m_p^2}{r}=\frac{\hbar c}{r}}**

Multiplying through,

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle G m_p^2=\hbar c}**

This equation has units of energy times length and equals the value **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \hbar c}**
, a ubiquitous quantity when deriving the Planck units. Since the two quantities are equal their ratio equals one. From here, it is easy to isolate the mass that would satisfy this equation in our system of units:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m_p=\sqrt{\frac{\hbar c}{G}}}**

Note in the second equation that if instead of planck masses the electron mass were used, the equation would no longer be unitary and instead equal a gravitational coupling constant, analogous to how the equation of the fine-structure constant operates with respect to the elementary charge and the Planck charge. Thus, the planck mass is an attempt to absorb the gravitational coupling constant into the unit of mass (and those of distance/time as well), as the planck charge does for the fine-structure constant; naturally it is impossible to truly set either of these dimensionless numbers to zero.

### 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.^{[2]} 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 confined, would become a black hole; 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda_c = \frac{h}{mc}}**

and the Schwarzschild radius is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r_s = \frac{2Gm}{c^2}}**

Setting them equal:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m=\sqrt{\frac{hc}{2G}}=\sqrt{\frac{\pi c \hbar}{G}}}**

This is not quite the Planck mass: It is a factor of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \sqrt{\pi}}**
larger. However, this is a heuristic derivation, only intended to get the right order of magnitude. On the other hand, the previous "derivation" of the Planck mass should have had a proportional sign in the initial expression rather than an equal sign. Therefore, the extra factor might be the correct one.

## See also

## References

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (December 2010) (Learn how and when to remove this template message) |

- ↑ CODATA 2010: value in GeV, value in kg
- ↑ The riddle of gravitation by Peter Gabriel Bergmann, page x

## Sources

- Sivaram C. WHAT IS SPECIAL ABOUT THE PLANCK MASS? PDF
- Johnstone Stoney, Phil. Trans. Roy. Soc. 11, (1881)