The variational method for geodesics
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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 :
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.