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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?

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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.

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    $\begingroup$ Thanks for the quick response! Can you explain how you got from the first step to the second step (mainly where the two came from)? Also, what was your intuition to attacking the problem in this way? $\endgroup$
    – John Smith
    Feb 20, 2022 at 8:45
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    $\begingroup$ $\nabla$ 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. $\endgroup$
    – Prahar
    Feb 20, 2022 at 8:54
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    $\begingroup$ Ah, I see now that you raised and lowered an index in the first term. As someone just starting out with this stuff, how would you recommend learning it? I’ve just been trying my best to go slow reading the book and then attempting the problems but there aren’t any solutions, so I don’t know if I’m doing it right. Is there a better way? And I really appreciate the help! $\endgroup$
    – John Smith
    Feb 20, 2022 at 9:04
  • $\begingroup$ @JohnSmith - That is by far the best way. GR requires a LOT of long and tedious calculations during the learning process. Don't shy away from long (like 4-5 pages of algebra per calculation) calculations! This is a required step for you to gain intuition. Do as many problems as possible! Your mind will twist and turn as you learn, but this is also required. $\endgroup$
    – Prahar
    Feb 20, 2022 at 12:33
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    $\begingroup$ @JohnSmith - Carroll is a good book to start learning. Wald has some more advanced topics so I suggest reading that as a second pass. Poisson's book teaches you how to do calculations in GR. $\endgroup$
    – Prahar
    Feb 20, 2022 at 12:35

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