Re-parametrization of Geodesics: Can You Confirm?

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SUMMARY

The discussion confirms that when re-parametrizing solutions of the geodesic equation, only linear functions maintain the validity of the equation. Specifically, the choice of \nabla_VV=0 is crucial, as it ensures that the geodesic equation holds true under linear transformations. Non-linear re-parametrizations, such as \nabla_VV=fV, do not satisfy the geodesic equation, although they represent the same world line. The invariant interval, such as proper time or distance, is a valid solution for re-parametrization.

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bloby
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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=fVfor the geodesic equation?
 
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bloby said:
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=fVfor 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.
 
Ok, thank you.
 

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