 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2018/1/b/solution

In general coordinates the basis changes along the world line. Differentiating only the components of a vector omits that basis change, and does not transform as a vector under a position-dependent coordinate transformation. The appropriate rate is the covariant derivative along a curve:

\frac{D V ^{α}}{d λ} = \frac{d V ^{α}}{d λ} + Γ^{α}_{β γ} \frac{d x ^{β}}{d λ} V^{γ} .

Γ^{α}_{β γ} = \frac{1}{2} g^{α δ} (\partial_{β} g_{δ γ} + \partial_{γ} g_{δ β} - \partial_{δ} g_{β γ}) .

a^{α} = \frac{D u ^{α}}{d τ} = \frac{d u ^{α}}{d τ} + Γ^{α}_{β γ} u^{β} u^{γ} .

In Cartesian coordinates of an inertial frame in flat spacetime, the connection coefficients vanish and the ordinary component derivative happens to suffice. In curved spacetime it suffices at a chosen event in a local inertial frame, but not generally along an extended trajectory.

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