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 pairing
$$
T(k_+,k_-)=a-c.
$$
Thus the necessary and sufficient conditions are
$$
\boxed{a+c\geq2|b|,
\qquad a\geq c}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 3 / b / ii / Solution

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The dominant energy condition requires the energy flux measured by every future-directed timelike observer to be future-directed and causal. Equivalently, $T_{ab}u^av^b\geq0$ for every pair of future-directed causal vectors $u$ and $v$ under the $(-,+,+,+)$ convention.


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