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.