Planck mass: Definition from Answers.com

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

m_P = √ħc/G ≈ 1.2209×10¹⁹ GeV/c² = 2.17644(11)×10⁻⁸ kg,

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

$\sqrt\frac{\hbar{}c}{8\pi G}$ ≈ 4.340×10⁻⁶ g = 2.43 × 10¹⁸ GeV/c².

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

The name honors Max Planck, who was the first to propose it.

1 Derivations
- 1.1 Dimensional analysis
- 1.2 Compton wavelength and Schwartzchild radius
2 Significance
3 See also
4 References
5 Sources
6 External links

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 $\hbar$ , c, and G, and attempt to combine them to get a quantity with units of mass. The expected formula is of the form

$m_P = c^{n_1} G^{n_2} \hbar^{n_3},$

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] = LT^{-1} \$

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

$[\hbar] = M^1L^2T^{-1} \$ .

Therefore,

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

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

$-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:

$m_P = c^{1/2}G^{-1/2}\hbar^{1/2} = \sqrt{\frac{c\hbar}{G}}.$

Compton wavelength and Schwartzchild radius

The Planck mass can be derived approximately by setting it as the mass whose Compton wavelength and Schwartzchild radius are equal.^[1] 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 Schwartzchild radius is the radius in which a mass, if confined, would become a black hole; the heavier the particle, the larger the Schwartzchild radius. If a particle were massive enough that its Compton wavelength and Schwartzchild 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 Schwartzchild 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 is a heuristic derivation, only intended to get the right order of magnitude.

Significance

Unlike all other Planck base 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 is the mass of the Planck particle, a hypothetical minuscule 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.

References

^ 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)
Stephen J. Crothers and Jeremy Dunning-Davies†. Planck Particles and Quantum Gravity. PROGRESS IN PHYSICS, Vol.3, July, 2006

External links

v • d • e Planck's natural 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

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