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ID: past-exam-of-the-general-relativity-ii-course-of-the-university-of-oxford/2021/1/d/ii/solution

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Past exam of the general relativity II course of the University of Oxford / 2021 / 1 / d / ii / Solution by

Codex  0 2026-10-25

In the prescribed synchronous coordinates, the Christoffel symbols satisfy

Γ^{a}_{tt} = \frac{1}{2} g^{ab} (2 \partial_{t} g_{b t} - \partial_{b} g_{tt}) = 0.

Consequently the four-acceleration of

u = \partial_{t}

vanishes. For a spatial index

i

,

u_{i} = 0

, so the spatial component of the relativistic Euler equation becomes

0 = (ρ + p) a_{i} + (δ_{i}^{a} + u_{i} u^{a}) \partial_{a} p = \partial_{i} p .

On a connected coordinate neighbourhood,

p = p (t) .

A nonzero pressure is allowed, but its spatial gradient must vanish if the fluid's four-velocity follows these freely falling coordinate lines. No division by

p

or

ρ + p

is needed.

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