Tensors and Relativity: Assignment 7

Peter Dunsby Room 310a Applied Mathematics

• Show that for the Schwarzschild metric the non- vanishing Christoffel symbols [ with G=c=1 ] are:

  

  Hint!

• Deduce that the non- vanishing components of the Riemann tensor are:

  

  

  plus those obtained by symmetries of the Riemann Tensor. Confirm that [ as required ].

  Hint!

• Show that the energy equation for orbits in a Schwarzschild spacetime is

  

  and deduce that the orbital equation is

  

  where .

  [ Notation as in the lecture notes; this exercise fills in the algebra.]

  Derive the orbital equation in Newtonian theory and show that it has the solution

  

  Use the arguement in the lecture notes to show that the GR solution is

  

  Show that the solution for a Photon orbit () is

  

  where b is the impact parameter.

  Hint!

If you have any problems please come and see me or contact me by email.