Tensors and Relativity: Assignment 6

Problem

(1) Consider the integral

evaluated along some path connecting fixed points A and B in spacetime. Here f is regarded as a function of 8 independent variables and . By varying the path , show that A is extremized for

where . These are the Euler- Lagrange equations.
(2) Prove that the geodesic equation

holds for any parameter along the curve such that , being the proper time measured along the curve.

Hint!
(3) Using the usual coordinate transformations from Cartesian to spherical polars, calculate the metric on the surface of a sphere of unit radius. Find the inverse metric.
Hint!
(4) Calculate the Riemann curvature tensor of the surface of a sphere of unit radius using the result of the previous problem.
[ Note that in two dimensions the Riemann tensor has only one independent component, so calculate and obtain all other components in terms of it.]

Hint!
(5) Calculate the Riemann curvature tensor of the surface of a cylinder. You should find that it is flat.
Hint!
(6a) Show that covariant differentiation obeys the usual product rule,

(6b) Prove that

where .

Hint!
(7) Show that if and are parallel- transported along a curve, then is constant along the curve. Deduce that a geodesic that is spacelike/timelike/null somewhere, remains so everywhere.
Hint!