Re-parametrization of Geodesics: Can You Confirm?

[bloby]

Hello.

If I find a solution of the geodesic equation and I change the parametrization, the new function does not
satisfy this equation for a general re-parametrization. But the world line is the same.

Can you confirm it: does it come from the fact that we usually choose $\nabla_VV=0$ instead of
$\nabla_VV=fV$for the geodesic equation?

 

[PAllen]

Hello.

If I find a solution of the geodesic equation and I change the parametrization, the new function does not
satisfy this equation for a general re-parametrization. But the world line is the same.

Can you confirm it: does it come from the fact that we usually choose $\nabla_VV=0$ instead of
$\nabla_VV=fV$for the geodesic equation?

Yes. Only parameters related by a linear function will satisfy the simple form of geodesic equation. Further, invariant interval (proper time or distance) is one of the possible solutions, so only e.g. aτ+b will work as another parametrization and still satisfy this equation.

 

[bloby]

Ok, thank you.