Planck length (changes) in nLab

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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

fundamental scales (fundamental/natural physical units)

• speed of light \, cc

• Planck's constant \, ℏ\hbar

• gravitational constant \, G N=κ 2/8πG_N = \kappa^2/8\pi

• Planck scale

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

  • Planck mass \, m p=ℏc/Gm_p = \sqrt{\hbar c / G}

• depending on a given mass mm

  • Compton wavelength \, λ m=ℏ/mc\lambda_m = \hbar / m c

  • Schwarzschild radius \, 2mG/c 22 m G / c^2

  • depending also on a given charge ee

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

• GUT scale

• string scale

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

  • string length scale \, ℓ s=α′\ell_s = \sqrt{\alpha'}

  • string coupling constant \, g s=e λg_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