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. [

** (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

** (6) ** Establish the theorem that any

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.