Lorentz force in curved spacetime

I am trying to derive the equation for Lorentz force mentioned in the following Wikipedia article -

https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime

viz., $$\frac{d p_{\alpha}}{d t} \, = \, \Gamma^{\beta}_{\alpha \gamma} \, p_{\beta} \, \frac{d x^{\gamma}}{d t} \, + \, q \, F_{\alpha \gamma} \, \frac{d x^{\gamma}}{d t}.$$

Starting from the modified geodesic equation, (based on Wald's Eq. 4.3.2, https://en.wikipedia.org/wiki/Geodesics_in_general_relativity) and using $p^\alpha = m \frac{dx^\alpha}{dt}$, we arrive at,

$$\frac{d p^\mu}{dt} = -\Gamma^{\mu}_{\alpha \beta} ~p^\alpha\frac{d x^\beta}{dt} + q ~F^{\mu}_{\alpha}\frac{d x^\alpha}{dt}.$$ Lower $p^\mu$ to $p_\alpha$ by multiplying both sides with $g_{\gamma \mu}$ to get - $$\frac{d p_\gamma}{dt} = -g_{\gamma \mu}\Gamma^{\mu}_{\alpha \beta} ~p^\alpha \frac{d x^\beta}{dt} + q~F_{\gamma \alpha}\frac{d x^\alpha}{dt},\\ \mathrm{or , \quad}\frac{d p_\gamma}{dt} = -\Gamma_{\gamma \alpha \beta} ~p^\alpha \frac{d x^\beta}{dt} + q~F_{\gamma \alpha}\frac{d x^\alpha}{dt}.$$ Relabeling $\gamma$ as $\alpha$ and $\alpha$ as $\gamma$, $$\frac{d p_\alpha}{dt} = -\Gamma_{\alpha \gamma \beta} ~p^\gamma \frac{d x^\beta}{dt} + q~F_{\alpha \gamma}\frac{d x^\gamma}{dt},$$ which is not the same as the first equation even if we lower $p^\gamma$. What am I missing?

1) The path parameter should be stated $\tau$ (proper time).
3) The metric g to raise/lower indices should be applied to a tensor, but the connection $\Gamma$ is not a tensor.
• Thanks for your answer. I agree that $t$ should be replaced by $\tau$ but what I am not able to follow from your reply is how should one arrive at the first equation from the second (if at all it's possible)? Is there any reference for the first equation like there is for the second (Wald's book)? Jan 1, 2018 at 19:03
You are missing the fact that you cannot lower those indices separately since single components of that equation are NOT tensors. Indeed the correct expression is obtained by the formula (with the Levi-Civita connection): $$(\nabla_{u}u)_{\mu}=u^{\sigma}\partial_{\sigma}u_{\mu}-\Gamma^{\beta}_{\mu\sigma}u_{\beta}u^{\sigma}=0$$ but in that case you are treating that as acting over the, in your case, covariant vector $u$. (You can do that since you are working in a metric manifold and the metric $g$ allows you to "connect/indentify" vectors and covectors). Be careful that the $u$ in the covariant derivative is a vector and all is applied to a covector $u$. Since $$u^{\sigma}\partial_{\sigma}u_{\mu}=\ddot{x}_{\mu}$$ and the Lorentz contribute goes on the right hand side of the geo eq. then you get the same first equation.