General Relativity Assignment #2

Deadline Date To Be Announced In Class

Part One

(1)        Show that covariant differentiation commutes with contraction by checking that    (2)        The line elements of 3-D manifold  in Cartesian, cylindrical polar, and spherical polar coordinates are given respectively by  (a)  ,  (b)  ,  (c)    Find  in each case.  (3)        Find the geodesic equation for  in cylindrical polars. [Hint: Use the results of the previous question to compute the metric connection components and then substitute these into the geodesic equation of motion.]  (4)        Show that the Einstein tensor satisfies  . [Note: Prove it both ways]  Part Two

(5)        Show that the Weyl Tensor always satisfies  . Show that this works for all pairs of indices.  (6)        Establish the theorem that any 2-dimensional Riemannian manifold is conformally flat in the case of a metric of signature 0 (i.e. at any point the metric can be reduced to the diagonal form  ).  Hint: Use null curves as coordinate curves. This means that you should change to new coordinates as follows. Let  . Constrain these coordinates to satisfy the relations  . Then show that the line element reduces to  and, finally, to show conformal flatness, introduce new coordinates  and  .          Part Three

(7)        Consider the 4-D spherically symmetric line element given by    where  and  are arbitrary functions t and r.    (i) Find  .  (ii) Use the expressions in (i) to calculate  . Remember that it is symmetric.  (iii) Calculate the Riemann curvature tensor  . You can use the symmetry relation  to simplify the calculation.    (iv) Calculate  .  (v) Calculate  .  NOTE: ALL logical steps should be justified to get full marks.