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} .

For the metric-compatible torsion-free connection, the Christoffel symbols are determined by the metric tensor and its first derivatives:

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}) .

Thus the ship's four-acceleration is

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.

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version