Tensors and Relativity: Chapter 6

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  tex2html_wrap_inline3639 is defined to be

equation3017

Remember in flat spacetime  it was just

equation3019

Therefore the proper time between two points A and B along an arbitrary timelike curve is

eqnarray3021

so we can write the Lagrangian  as

equation3025

and the action becomes

equation3030

varying the action  we get the Euler- Lagrange equations :

equation3033

Now

equation3035

and

equation3037

Since

equation3039

we get

equation3041

and

equation3043

so the Euler- Lagrange equations become:

equation3045

Multiplying by tex2html_wrap_inline3645 we obtain

equation3047

Using

equation3049

we get

equation3051

Multiplying by tex2html_wrap_inline3647 gives

equation3053

Now

eqnarray3055

Using the above result gives us

eqnarray3057

so we get the geodesic equation again 

equation3059

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 tex2html_wrap_inline3649 , the equation reduces to

equation3061

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 tex2html_wrap_inline3612 such that

equation3063

for constants a and b. A parameter which satisfies this condition is said to be affine.