1.Define covariant differentiation as we did in class, namely, first go to normal coordinates, differentiate there and then return to your original coordinates by demanding that DiVj be a tensor. Show that the connection that one derives in this way, ?k
ij , is symmetric in its lower indices.
2.Compute the ? symbols in polar coordinates. Consider the equation for geodesics in these coordinates. Show that you get some curves in polar coordinates which in Cartesian coordinates correspond to straight lines. Be careful to use the affine parameter on the geodesic correctly.
3.(For advanced students.) An important theorem we did not prove in class is that if the Riemann tensor vanishes then the metric can be brought to the form of a flat metric in some open patch. Look for the proof in the books/literature and try to understand it. Sketch the main ideas of the proof.
4.Consider a relativistic particle moving in the Schwarzschild metric. Write the action. Expand in the action in the non-relativistic approximation of weak fields and slow velocities. Show that it coincides with the action you know from classical Newtonian dynamics.
5.An observer at some fixed r ? 2GM/c2 in the Schwarzschild metric is held by a rope from infinity. What is the tension of the rope? What happens to the tension as we bring the observer to the horizon?
6.What happens if we try to continue releasing the rope so that the particle would enter the horizon? Can we at all do that? Note that inside the horizon the fixed r trajectories are space like. So our particle that is hanging on the rope cannot move on such trajectories.