 Solution

Use the future-directed null vector $k=-{\partial }_r$, whose dual is $k_a=-(dv{)}_a$. The stress-energy tensor has the null dust formFor every future-directed timelike vector $t$, $k\cdot t<0$. The energy current appearing in the dominant energy condition isIf $\epsilon \ge 0$, this is a future-directed null vector, or zero; also $T_{ab}{t}^a{t}^b=\epsilon (k\cdot t{)}^2\ge 0$. If $\epsilon <0$, the current is past-directed and that energy density is negative. Therefore, given the already stipulated positivity and smoothness of $m$, the necessary and sufficient additional condition is