ADM energy

Definition

Let $(\Sigma, h, K)$ be an initial data set, as in the initial value formulation, with $\Sigma$ a 3-manifold and $h$ a Riemannian metric. Assume it is asymptotically flat in the following sense. There are a compact set $C \subseteq \Sigma$, a radius $r_0$ and a diffeomorphism $\Sigma \setminus C \to \set{x \in \R^3 : \abs{x} > r_0}$. In these coordinates, for some $q > 1/2$, $$h_{ij} - \delta_{ij} = O(\abs{x}^{-q}), \quad \partial_k h_{ij} = O(\abs{x}^{-q-1}), \quad K_{ij} = O(\abs{x}^{-q-1}),$$ and the energy and momentum densities defined by the constraint equations are integrable. In units with $G = 1$, the ADM energy and the ADM momentum are $$E = \frac{1}{16\pi} \lim_{r \to \infty} \int_{\abs{x} = r} \sum_{i,j} \bigl( \partial_j h_{ij} - \partial_i h_{jj} \bigr) \nu^i \, dA,$$ $$P_i = \frac{1}{8\pi} \lim_{r \to \infty} \int_{\abs{x} = r} \sum_{j} \bigl( K_{ij} - (\operatorname{tr}_h K) \, h_{ij} \bigr) \nu^j \, dA.$$ Here $\nu = x/r$ is the Euclidean unit normal and $dA$ the Euclidean area element of the sphere. When $E \ge \abs{P}$, the ADM mass is $m = \sqrt{E^2 - \abs{P}^2}$.