|c denotes partial differentiation wrt coordinate xc and ||c denotes covariant differentiation wrt coordinate xc .
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.
2. Consider two observers, Jane and Susan, both in the vacuum region surrounding a 15 compact spherical object of geometrized mass m and radius less than 4m, but greater than 2m.
Jane is in a circular orbit of radius r = 4m. Susan is fired radially from the surface of the compact object with a velocity that is less than the escape velocity. Through careful calculation, the timing of the launch of Susan is such that when she arrives at r = 4m, Jane is just arriving there as well in her circular orbit. At this spacetime event common to both Jane and Susan they synchronize their respective clocks. Susan continues on her journey radially away from the system, and eventually begins to fall back towards the system. Suppose that by the appropriate choice of the initial speed of Susan, when she falls back to r = 4m, Jane happens to be at the same location in her orbit, and Susan and Jane are again at the same spacetime event, and Jane has completed N orbits in the interim. At this second common point in their two world lines, they compare their clocks and determine the difference in their two clocks' times.
Calculate elapsed time on the clocks of each of Jane and Susan, the difference in the times on the two clocks, and the maximum value of Jane's distance from the centre of the spherical object.
Now consider a third observer, Lucy, equipped with an appropriate rocket so that she hovers at the position in Jane's orbit throughout the flight of Susan away from Jane until they meet again as described above. Calculate the elapsed time on Lucy's clock through this period of Susan's flight and compare this to those of Susan and Jane. Provide an explanation of the various elapsed times, and why they differ, if they do in fact differ.
3. Suppose that ξ is a Killing vector, so that it satisfies the equation:
ξa||b + ξb||a = 0.
You have previously shown that ξ satisfies the equation:
ξa b c = Rd ξd .
Derive the maximum number of independent Killing vector fields that can exist in a spacetime of dimension N . Compare the number of Killing vectors that exist in some of the metrics we have considered to the maximum and try to break the isometry group into smaller parts for these spacetimes. Also, for each spacetime you consider, analyze the isometry group for significant subspaces, and comment briefly on how these might have helped in arriving at the particular spacetime.