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- TL;DR Summary
- Trying to understand a trick Carroll uses to derive the geodesic equation by extremizing proper time

On pages 106-107 of Spacetime & Geometry, Carroll derives the geodesic equation by extremizing the proper time functional. He writes:

What I am unclear on is the step in 3.47. I understand that the four velocity is normalized to -1 for timelike paths, but if the value of f is fixed, how can we vary it at all?

What I am unclear on is the step in 3.47. I understand that the four velocity is normalized to -1 for timelike paths, but if the value of f is fixed, how can we vary it at all?