# Regular vs. General Geodesic Equation

I'm reading Wald, and I've just got up to the Geodesic equation: $$T^a \nabla_a T^b = 0.\tag{1}$$

Right after, Wald says that "one might require only that the tangent vector to the curve point in the same direction as itself when parallel propagated, and not demand that it maintain the same length", which yields: $$T^a \nabla_a T^b = \alpha T^b.\tag{2}$$

How can we start from the second equation, assume that the tangent to the curve $$T^a$$ has constant length, and reach the first equation?

I've tried something like the folllowing:

We have $$T^a \nabla_a T^b = \alpha T^b$$ and know that $$T^a$$ has a constant length, so we can say $$g_{ab}T^aT^b = K$$. In a coordinate system $$\psi$$, we can rewrite the first equation as $$\frac{dT^\mu}{dt} + \Gamma^\mu_{\sigma \nu}T^\sigma T^\nu = \alpha T^\mu$$

$$\Rightarrow \frac{dT^\mu}{dt} + (g^{\sigma \nu}g_{\sigma \nu})\Gamma^\mu_{\sigma \nu}T^\sigma T^\nu = \alpha T^\mu$$

$$\Rightarrow \frac{dT^\mu}{dt} + g^{\sigma \nu}\Gamma^\mu_{\sigma \nu}(g_{\sigma \nu}T^\sigma T^\nu) = \alpha T^\mu$$

$$\Rightarrow \frac{dT^\mu}{dt} + g^{\sigma \nu}\Gamma^\mu_{\sigma \nu}K = \alpha T^\mu$$

I'm stuck at this point. I've tried plugging in the Christoffel symbols in terms of the metric but didn't see any simplification. Any tips on how to proceed?

Length of the tangent vector is given by $$T^a T_a$$. Its change along the geodesic is then $$T^a \nabla_a ( T^b T_b) = 2 T^a \nabla_a T^b T_b = 2 \alpha T^b T_b.$$ Thus, if $$\alpha=0$$, then the length remains unchanged along the geodesic.
• $\nabla$ is a derivative operator so you can use the product rule, $\nabla_a ( T^b T_b ) = \nabla_a T^b T_b + T^b \nabla_a T_b = 2 T^b \nabla_a T_b$. My intuition came largely from the fact that I've been doing this for 10 years now. But if I was attempting to think about this from your perspective, I would say this. The statement says that some quantity is constant along the curve. Changes along the curve are captured by the tangential derivative so I decided to take the tangential derivative of that quantity to see what happens. Feb 20, 2022 at 8:54