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Tensors and Relativity: Chapter 6

# The variational method for geodesics

We now apply the variational technique to compute the geodesics  for a given metric.

For a curved spacetime, the proper time is defined to be Remember in flat spacetime  it was just Therefore the proper time between two points A and B along an arbitrary timelike curve is so we can write the Lagrangian  as and the action becomes varying the action  we get the Euler- Lagrange equations : Now and Since we get and so the Euler- Lagrange equations become: Multiplying by we obtain Using we get Multiplying by gives Now Using the above result gives us so we get the geodesic equation again This is the equation of motion for a particle moving on a timelike geodesic in curved spacetime. Note that in a local inertial frame i.e. where , the equation reduces to which is the equation of motion for a free particle .

The geodesic equation preserves its form if we parameterize the curve by any other parameter such that for constants a and b. A parameter which satisfies this condition is said to be affine.