 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/37c/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

T_{α β} V^{α} V^{β} = (ρ + P) (u_{α} V^{α})^{2},

because

g_{α β} V^{α} V^{β} = 0

ρ + P = (1 + w) ρ < 0

. A nonzero

u_{α} V^{α}

therefore makes the contraction negative, so phantom energy violates the null energy condition.

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