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.