 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2017/1/c/solution

T^{α β} = ρ u^{α} u^{β}

, with

u^{α} u_{α} = - c^{2}

0 = \nabla_{α} (ρ u^{α} u^{β}) = u^{β} \nabla_{α} (ρ u^{α}) + ρ u^{α} \nabla_{α} u^{β} .

Contract with

u_{β}

u_{β} u^{α} \nabla_{α} u^{β} = \frac{1}{2} u^{α} \nabla_{α} (u_{β} u^{β}) = 0.

Consequently

- c^{2} \nabla_{α} (ρ u^{α}) = 0.

This is the covariant mass conservation equation. Substituting it back, wherever

ρ > 0

, gives

u^{α} \nabla_{α} u^{β} = 0.

u^{α} = d x^{α} / d τ

τ

is an affine parameter, and explicitly

\frac{d ^{2} x ^{β}}{d τ ^{2}} + Γ^{β}_{α γ} \frac{d x ^{α}}{d τ} \frac{d x ^{γ}}{d τ} = 0.

The absence of pressure removes the force produced by pressure gradients; gravity is encoded by the Levi-Civita connection. At a point with

ρ = 0

this calculation makes no claim about a fluid velocity.

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