ADM Mass

This total energy is called the ADM mass of slice $\Sigma_t$ $$ \displaystyle \fbox{$ M_{\rm ADM} = {\dfrac{1}{16\pi}} \lim\limits_{{{S}}_{t}\rightarrow\infty} \displaystyle\oint_{{{S}}_{t}} \left[ \bar{D}^{j} \gamma_{ij} - \bar{D}_i (f^{kl} \gamma_{kl}) \right] s^i \sqrt{q}\;\hbox{d}^2 y $}\, $$ - Example Let us consider Schwarzschild spacetime and use the standard *Schwarzschild coordinates*, $$ g_{\mu{\it v}} \hbox{d} x^\mu \hbox{d} x^{\it v} = - \left( 1 - {{2m}\over {r}} \right) \;\hbox{d} t^2 + \left( 1 - {{2m}\over {r}} \right)^{-1} \hbox{d} r^2 + r^2 (\hbox{d}\theta^2 + \sin^2\theta \; \hbox{d}\varphi^2) . $$ the induced metric in the coordinates $(r,\theta,\varphi)$ $$ \gamma_{ij} = {\hbox{diag}}\left[ \left( 1 - {{2m}\over {r}} \right) ^{-1}, r^{2}, r^{2} \sin^{2} \theta \right]. $$ On the other side, the components of the flat metric in the same coordinates are $$ f_{ij} = {\hbox{diag}}\left( 1, r^2,r^2 \sin^2 \theta \right) \qquad \hbox{and}\qquad f^{ij} = {\hbox{diag}}\left( 1, r^{-2},r^{-2} \sin^{-2} \theta \right). $$ $$ M_{\rm ADM} = {{1}\over {16\pi}} \lim\limits_{r \rightarrow\infty} \oint_{r={\scriptsize\hbox{const}}} \left[ \bar{D}^j \gamma_{rj} - \bar{D}_{r} (f^{kl} \gamma_{kl}) \right] r^{2} \sin\theta \; \hbox{d}\theta \;\hbox{d}\varphi , $$ $$ M_{ADM}=m $$