nLab Planck length





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The fundamental physical unit of length.

In comparison to macorscopic physical units such as the meter, the approximate value of the Planck length is 1.610 35\sim 1.6 \;10^{-35} meter.


Two important physical units of length induced by a mass mm are

  1. the Compton wavelength

    m2πmc \ell_m \coloneqq \frac{2 \pi \hbar}{m c}
  2. the Schwarzschild radius

    r m2mG/c 2 r_m \coloneqq 2 m G/c^2


Solving the equation

m = r m 2π/mc = 2mG/c 2 \array{ & \ell_m &=& r_m \\ \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2 }

for mm yields the Planck mass

m P1πm =r=cG. m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}} \,.

The corresponding Compton wavelength m P\ell_{m_{P}} is given by the Planck length P\ell_P

P12π m P=Gc 3 \ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} } \,

fundamental scales (fundamental/natural physical units)


The notion was introduced in:

  • Max Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. 5: 440–480. pp. 478–80, 1899, (10.1002/andp.19003060105)

See also

Last revised on October 27, 2020 at 15:13:19. See the history of this page for a list of all contributions to it.