Radiation and Asymptotics for Spacetimes with Non-Isotropic Mass

Abstract

We derive new results on radiation, angular momentum at future null infinity and peeling for a general class of spacetimes. For asymptotically-flat solutions of the Einstein vacuum equations with a term homogeneous of degree $-1$ in the initial data metric, that is it may include a non-isotropic mass term, we prove new detailed behavior of the radiation field and curvature components at future null infinity. In particular, the limit along the null hypersurface $C_u$ as $t \to \infty$ of the curvature component $\rho = \frac{1}{4} R_{3434}$ multiplied with $r^3$ tends to a function $P(u, \theta, \phi)$ on $\mathbb{R} \times S^2$. When taking the limit $u \to + \infty$ (which corresponds to the limit at spacelike infinity), this function tends to a function $P^+ (\theta, \phi)$ on $S^2$. We prove that the latter limit does not have any $l=1$ modes. However, it has all the other modes, $l=0, l \geq 2$. Important derivatives of crucial curvature components do not decay in $u$, which is a special feature of these more general spacetimes. We show that peeling of the Weyl curvature components at future null infinity stops at the order $r^{-3}$, that is $r^{-4} |u|^{+1}$, for large data, and at order $r^{- \frac{7}{2}}$ for small data. Despite this fact, we prove that angular momentum at future null infinity is well defined for these spacetimes, due to the good behavior of the $l=1$ modes involved.