How can I prove that the covariant acceleration is normal to the velocity?

Question

I'm trying to show that the covariant acceleration is normal to the velocity for a massive particle. I think that I understand why this has to happen because I understand the covariant acceleration as the derivative of the "tangent component" of the velocity but I haven't been able to show it mathematically.

The covariant acceleration is defined as:

$$ a^{\mu} = \frac{D u^{\mu}}{D \tau}$$

And I want to prove that:

$$u_{\mu} a^{\mu} = 0$$

What I've done so far is to expand this equation to get something where some terms cancel and where I have any idea of how to proceed. I got:

$$ u_{\mu} a^{\mu} = g_{\mu \nu} \left( u^{\nu} \frac{d u^{\mu}}{d \tau} + \Gamma^{\mu}_{\lambda \rho} u^{\nu} u^{\lambda} u^{\rho} \right)$$

But I don't know how to continue from here. I'm pretty sure that it is a very elementary question but I've been stuck with it for a couple of hours.

Is this the right approach? How can I conclude?

I'm posting this question here because the notation that I'm using arises from a course in General Relativity but I don't know if I should post it in Math SE instead. Thanks.

Answer 1

I will post the answer that I got using Javier's hint just for the sake of completeness.

$$u_{\mu} u^{\mu} = -1$$

Taking derivatives in this equation:

$$\frac{D u_{\mu}}{D \tau} u^\mu + u_\mu \frac{D u^\mu}{D \tau} = 0 \implies 2 u_\mu a^\mu = 0 \implies u_\mu a^\mu = 0$$