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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.

FP=mPctP=c4G=1.210295×1044 N.{displaystyle F_{text{P}}={frac {m_{text{P}}c}{t_{text{P}}}}={frac {c^{4}}{G}}=1.210295times 10^{44}{mbox{ N.}}}

Other derivations

The Planck force is also associated with the equivalence of gravitational potential energy and electromagnetic energy^[1] and in this context it can be understood as the force that confines a self-gravitating mass to half its Schwarzschild radius:

FP=Gm2rG2{displaystyle F_{text{P}}={frac {Gm^{2}}{r_{text{G}}^{2}}}},

rG=rs2=Gmc2.{displaystyle 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:

FP=mc2Gmc2=c4G.{displaystyle 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. The gravitational attractive force of two bodies of 1 Planck mass each, set apart by 1 Planck length is 1 Planck force. 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=mc2ℏmc=m2c3ℏ.{displaystyle 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:

ℏmPc=GmPc2{displaystyle {frac {hbar }{m_{text{P}}c}}={frac {Gm_{text{P}}}{c^{2}}}}

which in turn follows from another relation of fundamental significance:

cℏ=GmP2.{displaystyle chbar =Gm_{text{P}}^{2}.}

General relativity

The Planck force appears in the Einstein field equations, describing the properties of a gravitational field surrounding any given mass:

Gμν=8πGc4Tμν{displaystyle G_{mu nu }=8pi {frac {G}{c^{4}}}T_{mu nu }}

where Gμν{displaystyle G_{mu nu }} is the Einstein tensor and Tμν{displaystyle T_{mu nu }} is the energy–momentum tensor. The Planck force thus describes how much or how easily space-time is curved by a given amount of mass-energy.

Since 1993, various authors (De Sabbata & Sivaram, Massa, Kostro & Lange, Gibbons, Schiller) have argued that the Planck force is the maximum force value that can be observed in nature. This limit property is valid both for gravitational force and for any other type of force.

Planck force as a tension constant of the space time fabric

According to new research ^[2] Planck force might be a tension constant of the space time fabric:

Aμν=1Gμν{displaystyle A^{mu nu }={frac {1}{G_{mu nu }}}} where Aμν{displaystyle A^{mu nu }} is the Estakhr tensor which is inverse of Einstein tensor. so then in a different representation of Einstein field equations : 8πTμνAμν=Fp{displaystyle 8pi T_{mu nu }A^{mu nu }=F_{p}} Planck force turn out to be actually "a tension constant of the space time fabric" Fp=T={displaystyle F_{p}=T=}
constant.

Notes and references