The Weyl tensor $C_{\alpha \beta \gamma \delta}$ may be defined (in $n=4$ spacetime dimensions) as

$C_{\alpha \beta \gamma \delta}=R_{\alpha \beta \gamma \delta}-\frac{1}{2}\left(g_{\alpha \gamma} R_{\beta \delta}+g_{\beta \delta} R_{\alpha \gamma}-g_{\alpha \delta} R_{\beta \gamma}-g_{\beta \gamma} R_{\alpha \delta}\right)+\frac{1}{6}\left(g_{\alpha \gamma} g_{\beta \delta}-g_{\alpha \delta} g_{\beta \gamma}\right) R$

where $R_{\alpha \beta \gamma \delta}$ is the Riemann tensor, $R_{\alpha \beta}$ is the Ricci tensor and $R$ is the Ricci scalar.

(a) Show that $C_{\beta \alpha \delta}^{\alpha}=0$ and deduce that all other contractions vanish.

(b) A conformally flat metric takes the form

$$g_{\alpha \beta}=e^{2 \omega} \eta_{\alpha \beta},$$

where $\eta_{\alpha \beta}$ is the Minkowski metric and $\omega$ is a scalar function. Calculate the Weyl tensor at a given point $p$. [You may assume that $\partial_{\alpha} \omega=0$ at $p$.]

(c) The Schwarzschild metric outside a spherically symmetric mass (such as the Sun, Earth or Moon) is

$$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\left(1-\frac{2 M}{r}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$$

(i) Calculate the leading-order contribution to the Weyl component $C_{t r t r}$ valid at large distances, $r \gg 2 M$, beyond the central spherical mass.

(ii) What physical phenomenon, known from ancient times, can be attributed to this component of the Weyl tensor at the location of the Earth? [This is after subtracting off the Earth's own gravitational field, and neglecting the Earth's motion within the solar system.] Briefly explain why your answer is consistent with the Einstein equivalence principle.