ADM formalism

The ADM mass ( by Richard Arnowitt, Stanley Deser, and Charles W. Misner 1961) assigns solutions of the field equations of general relativity to a mass that can be read on its gravitational effect at a great distance. The ADM mass is defined for asymptotically flat spacetimes.

Definition

Be an asymptotically flat Riemannian manifold (ie a space whose curvature tensor vanishes at infinity ) with metric. Then the ADM mass is given by

It is a sphere with radius and surface is the outwardly facing surface normal.

The ADM mass can thus be determined from metric sizes at a great distance from the matter. After the Schoen -Yau theorem, the ADM mass is positive if the weak energy condition is satisfied.

Example

For the Schwarzschild metric, the ADM mass is equal to the mass of the black hole, which is read off at the Schwarzschild radius. It is everywhere, except at the origin, vacuum: the energy -momentum tensor vanishes.