Planck particle

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A Planck particle, named after physicist Max Planck, is a hypothetical particle defined as a tiny black hole whose Compton wavelength is equal to its Schwarzschild radius.[1] Its mass is thus approximately the Planck mass, and its Compton wavelength and Schwarzschild radius are about the Planck length.[2] Planck particles are sometimes used as an exercise to define the Planck mass and Planck length.[3] They play a role in some models of the evolution of the universe during the Planck epoch.[4]

Compared to a proton, for example, the Planck particle would be extremely small (its radius being equal to the Planck length, which is about 10−20 times the proton's radius) and massive (the Planck mass being 1019 times the proton's mass).[5] The Planck particle would also have a very fleeting existence, evaporating due to Hawking radiation after approximately 5×10−39 seconds.

Derivation

While opinions vary as to its proper definition, the most common definition of a Planck particle is a particle whose Compton wavelength is equal to its Schwarzschild radius. This sets the relationship:

${\displaystyle \lambda ={\frac {h}{mc}}={\frac {2Gm}{c^{2}}}}$

Thus making the mass of such a particle:

${\displaystyle m={\sqrt {\frac {hc}{2G}}}}$

This mass will be ${\displaystyle {\sqrt {\pi }}}$ times larger than the Planck mass, making a Planck particle 1.772 times more massive than the Planck unit mass.

Its radius will be the Compton wavelength:

${\displaystyle r={\frac {h}{mc}}={\sqrt {\frac {2Gh}{c^{3}}}}}$

The Planck length P is defined as

${\displaystyle \ell _{\mathrm {P} }={\frac {r}{2{\sqrt {\pi }}}}={\sqrt {\frac {\hbar G}{c^{3}}}}}$

Dimensions

Using the above derivations we can substitute the universal constants h, G, and c, and determine physical values for the particle's mass and radius. Assuming this radius represents a sphere of uniform density, we can further determine the particle's volume and density.

Table 1: Physical dimensions of a Planck particle
Parameter Dimension Value in SI units
Mass M 3.85763×10−8 kg
Radius L 5.72947×10−35 m
Volume L3 7.87827×10−103 m3
Density M L−3 4.89655×1094 kg m−3
Lifetime T 4.826512×10−39 s

References

1. ^ Michel M. Deza; Elena Deza. Encyclopedia of Distances. Springer; 1 June 2009. ISBN 978-3-642-00233-5. p. 433.
2. ^ "Light element synthesis in Planck fireballs" – SpringerLink
3. ^ B. Roy Frieden; Robert A. Gatenby. Exploratory data analysis using Fisher information. Springer; 2007. ISBN 978-1-84628-506-6. p. 163.
4. ^ Harrison, Edward Robert (2000), Cosmology: the science of the universe, Cambridge University Press, ISBN 978-0-521-66148-5 p. 424
5. ^ Harrison 2000, p. 478.