a) geodesic paths are force free paths so are inertial frames where SR works. [1 mark]

Any 2 from the list: Radial free fall, circular orbits, elliptical precessing orbits, hyperbolic paths. [1 mark]

b) covariant derivative of the metric tensor

∂g_{ab}/∂x^{c} = \frac{∂}{∂x^{c}} (e⃗_{a}.e⃗_{b})

= \frac{∂e⃗_{a}}{∂x^{c}} .e⃗_{b} + e⃗_{a}.\frac{∂e⃗_{b}}{∂x^{c}} = Γ_{ac}^{d}e⃗_{d}.e⃗_{b} + Γ_{bc}^{d}e⃗_{a}.e⃗_{d} = Γ_{ac}^{d}g_{db} + Γ_{bc}^{d}g_{ad}

\frac{∂g_{ab}}{∂x^{c}} − Γ_{ac}^{d}g_{db} − Γ_{bc}^{d}g_{ad} = 0

[1 mark]

The LHS has the same form as the second order covariant derivative so g_{ab;c} = 0. [1 mark]

c) proper distance is measured by measuring the ends at the same time so

ds^{2} = R^{2}(t)[\frac{dr^{2}}{(1 − kr^{2})}]

ds = R(t)\frac{dr}{(1 − kr^{2})^{1/2}}

put one at zero, the other at r_{e} and at time t = 0 (close by)

d_{prop} = R(0) ∫_{0}^{r_{e}} \frac{dr}{(1 − kr^{2})^{1/2}} ≈ R(0)r_{e}

[1 mark]

v_{prop} = \frac{d}{dt}d_{prop} ≈ Ṙ(0)r_{e} = \frac{Ṙ(0)}{R(0)}r_{e}R(0) = H(0)d_{prop}

[1 mark]

d) Parallel transport of a vector - change neither the magnitude or direction ie slide the vector from one point to another. In curved space the vector can swing around, following the curvature of the space [1 mark]

to do differentiation of vectors/tensors we need to compare the components at one point with them at a separate point at small distance away - so need to parallel transport the vector to compare it.[1 mark]

e) curvature=gravitation. describe the source of gravity by the stress energy tensor. This is second order with covariant derivative of zero. [1 mark] Describe curvature by the reimman curvature tensor - 4th order! contract to second order ricci tensor - but this has non-zero divergance. So use ricci tensor and curvature scalar normalised metric to get a valid tensor equations. [1 mark]