(a) In the Newtonian weak-field limit, we can write the spacetime metric in the form

$$d s^{2}=-(1+2 \Phi) d t^{2}+(1-2 \Phi) \delta_{i j} d x^{i} d x^{j},$$

where $\delta_{i j} d x^{i} d x^{j}=d x^{2}+d y^{2}+d z^{2}$ and the potential $\Phi(t, x, y, z)$, as well as the velocity $v$ of particles moving in the gravitational field are assumed to be small, i.e.,

$$\Phi, \partial_{t} \Phi, \partial_{x^{i}} \Phi, v^{2} \ll 1$$

Use the geodesic equation for this metric to derive the equation of motion for a massive point particle in the Newtonian limit.

(b) The far-field limit of the Schwarzschild metric is a special case of (*) given, in spherical coordinates, by

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

where now $M / r \ll 1$. For the following questions, state your results to first order in $M / r$, i.e. neglecting terms of $\mathcal{O}\left((M / r)^{2}\right)$.

(i) Let $r_{1}, r_{2} \gg M$. Calculate the proper length $S$ along the radial curve from $r_{1}$ to $r_{2}$ at fixed $t, \theta, \varphi$.

(ii) Consider a massless particle moving radially from $r=r_{1}$ to $r=r_{2}$. According to an observer at rest at $r_{2}$, what time $T$ elapses during this motion?

(iii) The effective velocity of the particle as seen by the observer at $r_{2}$ is defined as $v_{\text {eff }}:=S / T$. Evaluate $v_{\text {eff }}$ and then take the limit of this result as $r_{1} \rightarrow r_{2}$. Briefly discuss the value of $v_{\text {eff }}$ in this limit.