nLab
Planck length

Contents

Context

Gravity

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


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} } \,

References

  • 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 September 4, 2018 at 07:38:55. See the history of this page for a list of all contributions to it.