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

The two future null directions are proportional to $k_{+}=(1,1)$ and $k_{-}=(1,-1)$. The null energy condition requiresEquivalently,

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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

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A future unit timelike vector is proportional to $(1,v)$ with $|v|<1$. The measured energy density has the sign ofUnder the assumption $c>a$, the weak energy condition first requires $a=q(0)\ge 0$, and hence $c>0$. The convex quadratic has its minimum at $v=-b/c$. Nonnegativity for every $|v|<1$ is then equivalent toIndeed this inequality implies $|b|/c\le \sqrt{a/c}<1$, so the minimum lies inside the allowed interval. Thereforeis necessary and sufficient when $c>a$.

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