nLab
Schwarzschild radius

Contents

Contents

Idea

The radius of the horizon of a Schwarzschild spacetime black hole of mass mm is called the Schwarzschild radius,

r m=2mG/c 2 r_m = 2 m G/c^2

where GG denotes the gravitational constant and cc denotes the speed of light.

Properties

Relation to Compton wavelength

Another physical unit of length parameterized by a mass mm is the Compton wavelength m=2πmc\ell_m = \frac{2 \pi \hbar}{m c}. 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} } \,

Created on November 9, 2017 at 05:17:49. See the history of this page for a list of all contributions to it.