Tensors and Relativity: Assignment 6
Problem
Next: About this document
Peter Dunsby Room 310a Applied Mathematics
- (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. - (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.
- (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.] - (5) Calculate the Riemann curvature tensor of the surface of a cylinder. You should find that it is flat.
- (6a) Show that covariant differentiation obeys the usual product rule,
(6b) Prove that
where
. - (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.
If you have any problems please come and see me or contact me by email.
Next: About this document