Paper 1, Section II, 37E
Consider the de Sitter metric
where is a constant.
(a) Write down the Lagrangian governing the geodesics of this metric. Use the Euler-Lagrange equations to determine all non-vanishing Christoffel symbols.
(b) Let be a timelike geodesic parametrized by proper time with initial conditions at ,
where the dot denotes differentiation with respect to and is a constant. Assuming both and to be future oriented, show that at ,
(c) Find a relation between and along the geodesic of part (b) and show that for a finite value of . [You may use without proof that
(d) Briefly interpret this result.
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