Derive the equations of motion of the observer

Question

|c denotes partial differentiation wrt coordinate x^c and ||c denotes covariant differentiation wrt coordinate x^c .

1. Consider an observer falling freely towards the centre of a Schwarzschild vacuum 15 region of spacetime, that is, falling radially inwards. Using the usual choice for the Schwarzschild coordinates, (t, r), derive the equations of motion of this observer, and in particular derive equations that give the proper time, τ , of the observer and the radial coordinate of the observe, r. Consider the three cases:

(a) The observer falls from from rest at a radial position r = r0.
(b) The observer falls from rest at r = ∞.
(c) The observer falls from infinity, where the radial velocity was v∞ relative to a locally static observer at r = ∞.
In some cases, your solution can be given in parametric form.