ADM ground

The ADM mass (according to Richard A rnowitt , Stanley D eser and Charles W. M isner 1961) assigns solutions to the field equations of the general theory of relativity to a mass that can be read from its gravitational effect at a large distance. The ADM mass is defined for asymptotically flat spacetime .

definition

Let be an asymptotically flat Riemannian manifold (i.e. a space whose curvature tensor vanishes at infinity) with metric . Then the ADM mass is given by ${\displaystyle M}$ ${\displaystyle g}$

${\displaystyle m_{ADM}(M,g):=\lim _{R\to \infty }\,{\frac {1}{16\,\pi }}\sum _{\mu ,\nu =1,2,3}\,\,\int _{\partial K_{R}}\left({\frac {\partial }{\partial x_{\mu }}}\,g_{\nu \nu }-{\frac {\partial }{\partial x_{\nu }}}\,g_{\nu \mu }\right)\,\mathrm {d} n^{\mu }}$,

Here is a sphere with a radius and the surface is the surface normal pointing outwards. ${\displaystyle K_{R}}$ ${\displaystyle R}$ ${\displaystyle \partial K_{R}\,,}$ ${\displaystyle n}$

The ADM mass can therefore be determined from metric quantities at a great distance from matter. According to the Schoen-Yau theorem, the ADM mass is positive when the weak energy condition is met. ${\displaystyle m_{ADM}>0\,,}$

example

For the Schwarzschild metric , the ADM mass is equal to the mass of the black hole , which can be read from the Schwarzschild radius. There is a vacuum everywhere, except in the origin: the energy-momentum tensor disappears. ${\displaystyle m_{ADM}}$ ${\displaystyle T_{\mu \nu }=0}$

literature

• R. Arnowitt, S. Deser, C. Misner: Coordinate Invariance and Energy Expressions in General Relativity , Phys. Rev. 122 (1961) 997-1006.
• RM Wald: General Relativity . Chicago: University of Chicago Press, 1984.

See also

• general theory of relativity
• Mass (physics)