I am currently trying to compute the Riemann tensor components for the Alcubierre metric, and already, on the computation of the first component, I'm running into some issues.
The trouble component in question is $R^x_{\;txt}$ , so I've started with the formula: $$ R^x_{\;txt} = \partial_x\Gamma^x_{\;tt} -\partial_t\Gamma^x_{\;xt}+\Gamma^x_{\;x\mu}\Gamma^\mu_{\;tt}-\Gamma^x_{\;t\mu}\Gamma^\mu_{\;xt} $$ Using the Christoffel symbols provided by Mueller and Grave's Catalogue of Spacetimes, I started computing the first term, $\partial_x\Gamma^x_{\;tt}$: $$ \begin{align} \partial_x\Gamma^x_{\;tt} &= \partial_x\frac{f^3f_xv_s^4-c^2ff_xv_s^2-c^2f_tv_s}{c^2} \\ &=\frac1{c^2}\Big(\partial_xf^3f_xv_s^4-\partial_xc^2ff_xv_s^2-\partial_xc^2f_tv_s\Big) \end{align} $$ And, again isolating the first term and computing further: $$ \partial_xf^3f_xv_s^4 = f_xv_s^4\partial_xf^3 + f^3v_s^4\partial_xf_x + f^3f_x\partial_xv_s^4 $$ If my math is correct.
The area I'm having trouble is in the partial derivative $\partial_xv_s$. I can't quite understand how to compute it. For reference, $v_s$ is defined by Alcubierre as a function of time: $$ v_s(t) = \frac{dx_s(t)}{dt} $$ and $x_s(t)$ is simply described as an "arbitrary function of time", describing the trajectory of the hypothetical spacecraft in the scenario of the metric.
I don't quite see how the actual $x$ axis relates to this function, and so I'm at a loss as to how I should treat a derivative of the function with respect to $x$. What am I missing?