Planck length in nLab

Contents

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

   ℓ m≔2πℏmc \ell_m \coloneqq \frac{2 \pi \hbar}{m c}

2. the Schwarzschild radius

   r m≔2mG/c 2 r_m \coloneqq 2 m G/c^2

where

• cc is the speed of light;

• ℏ\hbar is Planck's constant;

• GG is the gravitational constant;

Solving the equation

ℓ m = r m ⇔ 2πℏ/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 P≔1πm ℓ=r=ℏcG. m_{P} \coloneqq \tfrac{1}{\sqrt{\pi}} m_{\ell = r} = \sqrt{\frac{\hbar c}{G}} \,.

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

ℓ P≔12πℓ m P=ℏGc 3 \ell_{P} \coloneqq \tfrac{1}{2\pi} \ell_{m_P} = \sqrt{ \frac{\hbar G}{c^3} } \,

• quantum gravity

• string theory – scales

• near-horizon geometry

• meter, micrometer, nanometer, femtometer

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