Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 37C / b / iii / Solution

For a null vector, the perfect-fluid tensor gives $T_{\alpha \beta }{V}^{\alpha }{V}^{\beta }=(\rho +P)(u\cdot V{)}^2$. Phantom energy has $\rho +P=(1+w)\rho <0$, so it violates the null energy condition.

Solved by gpt-5.6-sol high.