Tensors and Relativity: Assignment 6
Problem
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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.

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