Solving Geodesic Eq.: Mysterious Conservation Eq. (Sec. 5.4 Carroll)

[George Keeling]

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?

 

[Orodruin]

He is treating both null geodesics and timelike geodesics. For timelike geodesics you can take $\lambda = \tau$ and get $\epsilon = 1$, but not for null geodesics. For null geodesics, $\epsilon = 0$.

 

[vanhees71]

It's only valid for affine parameters $\lambda$, but you can show that you always can parametrize geodesics with affine parameters.

 

[George Keeling]

@vanhees71 and @Orodruin are right and I forgot to explicitly say that for null paths $d\tau=0$. So both variants of the equation are correct with $\epsilon = 1, \epsilon = 0$ for timelike and null paths and they still follow from the metric equation. In full:$$
-d\tau^2=g_{\mu\nu}dx^\mu dx^\nu\Rightarrow-\frac{d\tau^2}{d\lambda^2}=g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}
$$Timelike: $\lambda=\tau$ $$
-1=g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}
$$Null: $d\tau=0$$$
0=g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}
$$
I still don't need the geodesic equation to get to these!

 

[vanhees71]

This also follows from the fact that one possible Lagrangian (afaik the most convenient one) for the geodesic equation is
$$L=\frac{1}{2} g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}.$$
Since this is not explicitly dependent on the world-line parameter $\lambda$ (derivatives wrt. $\lambda$ are denoted with a dot), the corresponding "Hamliltonian" is again $L$ itself, i.e., $L=\text{const}$ in this form of the action principle for geodesics you get automatically a parametrization with an affine parameter. For timelike (spacelike) geodesics you can simply set $L=\pm 1$ and for lightlike ones $L=0$.

The Euler-Lagrange equations are the usual geodesic equations for an affine parametrization,
$$\mathrm{D}_{\lambda} x^{\mu}=\ddot{x}^{\mu} + {\Gamma^{\mu}}_{\nu \rho} \dot{x}^{\nu} \dot{x}^{\rho}$$
with
$$\Gamma_{\mu \nu \rho}=\frac{1}{2} (\partial_{\nu} g_{\mu \rho} + \partial_{\rho} g_{\mu \nu} -\partial_{\mu} g_{\nu \rho}), \quad {\Gamma^{\sigma}}_{\nu \rho} =g^{\mu \sigma} \Gamma_{\mu \nu \rho}.$$