Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 3 / c / i

Along the affine null geodesic write

Φ^{'} = U^{a} \nabla_{a} Φ

. Contracting the scalar stress tensor with

U^{a} U^{b}

removes every term proportional to

g_{ab}

. Since

G_{ab} = T_{ab}

and

U^{a} U^{b} \nabla_{a} (Φ \nabla_{b} Φ) = \frac{d}{d λ} (Φ Φ^{'}) = (Φ^{'})^{2} + Φ Φ^{''},

one obtains

(1 - ξ Φ^{2}) T_{ab} U^{a} U^{b} = (1 - 2 ξ) (Φ^{'})^{2} - 2 ξ Φ Φ^{''} .

Put

D = 1 - ξ Φ^{2}

\int_{- \infty}^{\infty} T_{ab} U^{a} U^{b} d λ = [- \frac{2 ξ Φ Φ ^{'}}{D}]_{- \infty}^{\infty} + \int_{- \infty}^{\infty} \frac{1 + ξ ( 4 ξ - 1 ) Φ ^{2}}{D ^{2}} (Φ^{'})^{2} d λ .

The stated endpoint condition kills the boundary term. If

ξ < 0

, then

D > 0

and

ξ (4 ξ - 1) > 0

, so the remaining integrand is nonnegative. Consequently this nonminimally coupled scalar satisfies the averaged null energy condition:

\int_{- \infty}^{\infty} T_{ab} U^{a} U^{b} d λ \geq 0 .

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