nLab Planck length

Redirected from "Planck lengths".
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Context

Gravity

Physics

physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

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.

Definition

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

where

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)

References

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.