Planck mass

Planck mass

In physics, the Planck mass, denoted by mP, is the unit of mass in the system of natural units known as Planck units. It is defined so that

m_\text{P}=\sqrt{\frac{\hbar c}{G}}1.2209×1019 GeV/c2 = 2.17651(13)×10−8 kg = 21.7651 µg = 1.3107×1019 amu,[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 an alternative normalization with the reduced Planck mass, which is

\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.

Contents

• Significance 1
• Derivations 2
• Dimensional analysis 2.1
• Elimination of a coupling constant 2.2
• Compton wavelength and Schwarzschild radius 2.3
• Notes and references 4
• Bibliography 5

Significance

The Planck mass is nature’s maximum allowed mass for point-masses (quanta) – in other words, a mass capable of holding a single elementary charge. If two quanta of the Planck mass or greater met, they could spontaneously form a black hole whose Schwarzschild radius equals their Compton wavelength. Once such a hole formed, other particles would fall in, and the black hole would experience runaway, explosive growth (assuming it did not evaporate via Hawking radiation). Nature’s stable point-mass particles, such as electrons and quarks, are many, many orders of magnitude lighter than the Planck mass and cannot form black holes in this manner. On the other hand, extended objects (as opposed to point-masses) can have any mass.

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.

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 attempt 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] = \mathsf{LT}^{-1} \
[G] = \mathsf{M}^{-1}\mathsf{L}^3\mathsf{T}^{-2} \
[\hbar] = \mathsf{M}^1\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 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_\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 mP 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{G m_\text{P}^2}{r}=\frac{\hbar c}{r}.

Isolating mP, 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.

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 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}

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.