nLab Planck length

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 $\sim 1.6 \;10^{-35}$ meter.

Definition

Two important physical units of length induced by a mass $m$ are

1. the Compton wavelength

   $$\ell_m \coloneqq \frac{2 \pi \hbar}{m c}$$

2. the Schwarzschild radius

   $$r_m \coloneqq 2 m G/c^2$$

where

• $c$ is the speed of light;

• $\hbar$ is Planck's constant;

• $G$ is the gravitational constant;

Solving the equation

$$\array{ & \ell_m &=& r_m \\ \Leftrightarrow & 2\pi\hbar / m c &=& 2 m G / c^2 }$$

for $m$ yields the Planck mass

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

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

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

Related entries

• quantum gravity

• string theory – scales

• near-horizon geometry

• meter, micrometer, nanometer, femtometer

fundamental scales (fundamental/natural physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

  • Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

  • Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

  • Compton wavelength$\,$ $\lambda_m = \hbar / m c$

  • Schwarzschild radius$\,$ $2 m G / c^2$

  • depending also on a given charge $e$

    • Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$

• GUT scale

• string scale

  • string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

  • string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

  • string coupling constant$\,$ $g_s = e^\lambda$

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

• Wikipedia, Planck length