Planck mass

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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 approximately 0.02 milligrams (roughly the mass of a flea egg[1]). For comparison, this value is of the order of 1015 (one million billion) times larger than the highest energy available to contemporary particle accelerators.[2]

It is defined so that

1.220910×1019 GeV/c2 = 2.176470(51)×10−8 kg[3] = 21.76470 µg = 1.3107×1019 u,[4]

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

4.341×10−9 kg = 2.435×1018 GeV/c2.

The factor of simplifies a number of equations in general relativity.

Derivations[edit]

Dimensional analysis[edit]

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 attempts to combine them to get a quantity with units of mass. The expected formula is of the form

where 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:

.

Therefore,

If one wants dimensions of mass, the following equations must hold:

.

The solution of this system is:

Thus, the Planck mass is:

Elimination of a coupling constant[edit]

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.

Isolating mP, we get that

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.

See also[edit]

Notes and references[edit]

  1. ^ "average weight of a flea egg (3.42×10−2 mg)" M. W. Dryden, Blood consumption and feeding behavior of the cat flea (1990) p. 51.
  2. ^ Maximum energy of the Large Hadron Collider: 13 TeV (as of 2015).
  3. ^ "CODATA Value: Planck mass". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2017-06-22. 2014 CODATA recommended values 
  4. ^ CODATA 2016: value in GeV, value in kg

Bibliography[edit]

External links[edit]