Planck particle

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

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). 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:

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

Thus making the mass of such a particle:

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

This mass will be ${\sqrt {\pi }}$ times as large as the Planck mass, making a Planck particle 1.772 times as massive as the Planck unit mass.

Its radius will be the Compton wavelength:

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

The Planck length ℓ_P is defined as

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

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	Expression	Value in SI units	Value in Planck units
Mass	M	${\sqrt {\frac {hc}{2G}}}$	3.85763×10−8 kg	1.7724 $m_{\text{P}}$
Radius	L	${\sqrt {\frac {2hG}{c^{3}}}}$	5.72947×10−35 m	3.5449 $l_{\text{P}}$
Maximum charge	Q	${\sqrt {\frac {hc}{2k_{\text{e}}}}}$	2.86474×10−18 C	1.7724 $q_{\text{P}}$
Volume	L3	${\frac {8}{3}}\pi {\sqrt {\frac {2h^{3}G^{3}}{c^{9}}}}$	7.87827×10−103 m3	186.6137 $l_{\text{P}}^{3}$
Lifetime	T	$5120\pi ^{2}{\sqrt {\frac {hG}{2c^{5}}}}$	4.826512×10−39 s	89524.9652 $t_{\text{P}}$

See also

• Micro black hole
• Planck units
• Max Planck
• Black hole electron

External links

• Sachs, R.; Narlikar, J. V.; Hoyle, F. (1996). "The quasi-steady state cosmology: Analytical solutions of field equations and their relationship to observations". Astronomy and Astrophysics. 313: 703. Bibcode:1996A&A...313..703S.
• "Mach's principle: from Newton's bucket to quantum gravity" – Google Books
• "Mysteries of Mass: Some Contrarian Views From an Experimenter"
• "The Gauge Hierarchy Problem and Planck Oscillators" – CERN Document Server
• "The First Turbulence and First Fossil Turbulence"
• "Lecture on Nuclear Physics for Plasma Engineers"
• The Planck Length