# Planck acceleration

Planck acceleration is the acceleration from zero speed to the speed of light during one Planck time. It is a derived unit in the Planck system of natural units.

## Formula and value

Planck acceleration may be stated as

${\displaystyle a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\frac {299792458\ {\text{m/s}}}{5.39116\times 10^{-44}\ {\text{s}}}}}$
${\displaystyle \approx 5.560815\times 10^{51}\ {\text{m/s}}^{2}}$
${\displaystyle \approx 5.670453\times 10^{50}\ g}$

where aP is the Planck acceleration, c is the speed of light, tP is the Planck time[1] and g is the standard acceleration of gravity.[2]

## Meaning

The Planck acceleration is the highest acceleration conceivable in the Universe, as the speed of light is the highest possible speed and the Planck time is the shortest possible duration of any meaningful physical process. This limitation does assume that relativity has natural units. However, it is not clear whether any object in the Universe actually reaches or can reach the Planck acceleration. One event where the Planck acceleration was possibly reached was the Big Bang, in regard to the acceleration of the expanding Universe during the Planck epoch. Also, within a black hole, such acceleration might be possible beyond its horizon, but that is certainly unknown (and what lies beyond a horizon is beyond the reach of physical observation).

In a classical description, photons emanating from their subluminal source (for example, a particle-antiparticle collision) experience zero acceleration, as they always travel at the speed of light. However, since the Planck time is the least conceivable delay, the "instantaneous" production of a photon pair can not be distinguished from the acceleration of a photon from zero to the speed of light in a Planck time, thereby achieving a Planck acceleration. In other words, nothing higher than the Planck acceleration can be measured, since that would imply measuring a time interval shorter than the Planck time.

As with other quantities at the Planck scale, the physics of the Planck acceleration are not yet fully understood.

If the mass of the body is given, then a second limit, Caianiello's maximal acceleration, also applies:[3]

${\displaystyle a_{\text{Caianiello}}={\frac {2mc^{3}}{\hbar }}}$

If the length L of the body is given, another limit exists on the acceleration that can be supported[4]

${\displaystyle a_{\text{Max}}={\frac {c^{2}}{L}}}$

## References

1. ^ "CODATA Value: Planck time". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2016-07-24.
2. ^ "CODATA Value: standard acceleration of gravity". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2016-07-28.
3. ^ Papini, Giorgio (2006). "Caianiello's Maximal Acceleration. Recent Developments". Imagination and Rigor. p. 119. arXiv:quant-ph/0407115. doi:10.1007/88-470-0472-1_10. ISBN 88-470-0320-2.
4. ^ Taylor, Edwin F.; French, A. P. (1983). "Limitation on proper length in special relativity". American Journal of Physics. 51 (10): 889. Bibcode:1983AmJPh..51..889T. doi:10.1119/1.13372.