- #1

George Keeling

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- TL;DR Summary
- Mysterious alternative 4-velocity conservation equation "from geodesic equation". Normal equation being ##U_\nu U^\nu=-1##

I'm still on section 5.4 of Carroll's book on Schwarzschild geodesics

Carroll says "In addition, we always have another constant of the motion for geodesics: the geodesic equation (together with metric compatibility) implies that the quantity $$

\epsilon=-g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}

$$is constant along the path."

I don't see how that comes from the geodesic equation. But it is very similar to ##U_\nu U^\nu=-1## which comes from the metric equation:$$

-d\tau^2=g_{\mu\nu}dx^\mu dx^\nu\Rightarrow-1=g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=g_{\mu\nu}U^\mu U^\nu=U_\nu U^\nu

$$So ##\epsilon## is just a constant of proportionality between the affine parameter ##\lambda## and the proper time ##\tau##.

What have I missed?

Carroll says "In addition, we always have another constant of the motion for geodesics: the geodesic equation (together with metric compatibility) implies that the quantity $$

\epsilon=-g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}

$$is constant along the path."

I don't see how that comes from the geodesic equation. But it is very similar to ##U_\nu U^\nu=-1## which comes from the metric equation:$$

-d\tau^2=g_{\mu\nu}dx^\mu dx^\nu\Rightarrow-1=g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=g_{\mu\nu}U^\mu U^\nu=U_\nu U^\nu

$$So ##\epsilon## is just a constant of proportionality between the affine parameter ##\lambda## and the proper time ##\tau##.

What have I missed?