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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 :
so the Euler- Lagrange equations become:
Multiplying by we obtain
Multiplying by gives
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.