 Solution

In two dimensions every future causal vector is a nonnegative linear combination of $k_{+}$ and $k_{-}$. The dominant energy condition is therefore equivalent to nonnegativity of $T(k_{+},k_{+})$, $T(k_{-},k_{-})$, and the mixed pairingThus the necessary and sufficient conditions are

Solved by gpt-5.6-sol high.