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 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.

Significance

The Planck mass is nature’s maximum allowed mass for point-masses (quanta). If two quanta of the Planck mass or greater met, they could spontaneously form a black hole whose Schwarzschild radius equals their Compton wavelength.[citation needed] 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 smaller 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.

$E=\frac{G m_\text{P}^2}{r}=\frac{\hbar c}{r}$

Multiplying through,

$G m_\text{P}^2=\hbar c$

This equation has units of energy times length and equals the value $\hbar c$, 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:

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

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.

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

$\lambda_c = \frac{h}{mc}$

$r_s = \frac{2Gm}{c^2}$

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 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.