During my special relativity course we defined the covariant derivative of a vector field $ \ \xi^\mu (\tau)$ along a curve $x^\mu(\tau)$ as:
$$ \frac{\mathrm{D}\xi ^\mu}{\mathrm{D} \tau} = \frac{\mathrm{d} \xi ^\mu}{\mathrm{d} \tau} + \Gamma^{\mu}_{\alpha \beta} \, \xi^\alpha \frac{\mathrm{d} x^\beta}{\mathrm{d} \tau} $$
The justification given for this definition was straightforward: the equation of motion for a free particle in general coordinates is
$$\frac{\mathrm{D}u ^\mu}{\mathrm{D} \tau}=0,$$
where $u^\mu$ is the four-velocity of the particle. So, we didn't have to learn a lot about differential geometry for that course. Now, what if I want to compute the covariant derivative of a covector, like $u_\mu$? I've been searching around here and most of what I find doesn't refer to covariant derivatives along a curve. And what does deal with that, is written in ways I honestly don't understand (I don't have a lot of background in differential geometry). From what I've learned from searching about this topic, the following would be incorrect:
$$ \frac{\mathrm{D}\xi _\mu}{\mathrm{D} \tau} = \frac{\mathrm{d} \xi _\mu}{\mathrm{d} \tau} + \Gamma_{\mu \alpha \beta} \, \xi_\alpha \frac{\mathrm{d} x^\beta}{\mathrm{d} \tau} $$
And, even though I haven't seen it written exactly like this, I would guess the correct expression is:
$$ \frac{\mathrm{D}\xi _\mu}{\mathrm{D} \tau} = \frac{\mathrm{d} \xi _\mu}{\mathrm{d} \tau} – \Gamma^\beta _{ \alpha \mu} \, \xi_\beta \frac{\mathrm{d} x^\alpha}{\mathrm{d} \tau} $$
I would like to know if this is correct, and most importantly, why. Hopefully it doesn't require too much background?
Best Answer
Recall that the covariant derivative of a scalar reduces to the ordinary derivative: $\nabla_Xf=X(f).$ If $X$ is the coordinate basis, then $\nabla_\mu f=\partial_\mu f.$ Calculate the covariant derivative of the following scalar: $\langle Y,\omega\rangle=Y^i\omega_i$, which is the contraction of a vector and a covector. You will need to use the product rule. Lastly, let $X$ be a vector field defined along the curve of interest.