Bondi mass

Definition

Let $(M, g)$ be a vacuum spacetime that is asymptotically flat at future null infinity. Assume that near null infinity it has Bondi–Sachs coordinates $(u, r, x^A)$, with $x^A$ coordinates on the unit sphere $S^2$, in which $$g = -\frac{V}{r} e^{2\beta} \, du^2 - 2 e^{2\beta} \, du \, dr + r^2 h_{AB} \bigl( dx^A - U^A du \bigr) \bigl( dx^B - U^B du \bigr).$$ Here $\det(h_{AB})$ equals the determinant of the round metric of $S^2$, and $V, \beta, U^A, h_{AB}$ are functions of all coordinates. Assume the asymptotic expansion $$\frac{V}{r} = 1 - \frac{2 \mathcal{M}(u, x^A)}{r} + O(r^{-2}) \quad \text{as } r \to \infty.$$ The function $\mathcal{M}$ is the mass aspect. In units with $G = 1$, the Bondi mass at retarded time $u$ is $$m_B(u) = \frac{1}{4\pi} \int_{S^2} \mathcal{M}(u, x^A) \, d\Omega,$$ where $d\Omega$ is the area element of the unit sphere.