Rational zeta series

Planck force is the derived unit of force resulting from the definition of the base Planck units for time, length, and mass. It is equal to the natural unit of momentum divided by the natural unit of time.

$F_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{G}}=1.21027\times 10^{44}{\mbox{ N.}}$

Other derivations

The Planck force is also associated with the equivalence of gravitational potential energy and electromagnetic energy and in this context it can be understood as the force that confines a self-gravitating mass to half its Schwarzschild radius:

$F_{\text{P}}={\frac {Gm^{2}}{r_{\text{G}}^{2}}}$,

$r_{\text{G}}={\frac {r_{\text{s}}}{2}}={\frac {Gm}{c^{2}}}.$,

where G is the gravitational constant, c is the speed of light, m is any mass and r_G is half the Schwarzschild radius, r_s, of the given mass. Since the dimension of force is also a ratio of energy per length, the Planck force can be calculated as energy divided by half the Schwarzschild radius:

$F_{\text{P}}={\frac {mc^{2}}{\frac {Gm}{c^{2}}}}={\frac {c^{4}}{G}}.$

As mentioned above, Planck force has a unique association with the Planck mass. This unique association also manifests itself when force is calculated as any energy divided by the reduced Compton wavelength (reduced by 2π) of that same energy:

$F={\frac {mc^{2}}{\frac {\hbar }{mc}}}={\frac {m^{2}c^{3}}{\hbar }}.$

Here the force is different for every mass (for the electron, for example, the force is responsible for the Schwinger effect (see page 3 here [1]). It is Planck force only for the Planck mass (approximately 2.18 × 10^-8 kg). This follows from the fact that the Planck length is a reduced Compton wavelength equal to half the Schwarzschild radius of the Planck mass:

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

which in turn follows from another relation of fundamental significance:

$c\hbar =Gm_{\text{P}}^{2}.$

General relativity

Planck force is often useful in scientific calculations as a ratio of electromagnetic energy per gravitational length. Thus for example it appears in the Einstein field equations, describing the properties of a gravitational field surrounding any given mass:

$G_{\mu \nu }=8\pi {\frac {G}{c^{4}}}T_{\mu \nu }$

where $G_{\mu \nu }$ is the Einstein tensor and $T_{\mu \nu }$ is the energy–momentum tensor.

Notes and references

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