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.
holds for any parameter along the curve such that
,
being the proper time
measured along the curve.
[ Note that in two dimensions the Riemann tensor has only one
independent component, so calculate and obtain all other components in terms of it.]
(6b) Prove that
where .
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Peter Dunsby