I'm working in spherical coordinates and I want to transport a vector for a radial velocity over an interval with $dr\ne 0$ but all other increments zero. The formula I have found for parallel transport is $$v_{r + dr}^\mu \approx v_r^\mu - \Gamma _{\nu \alpha }^\mu v_r^\nu d{x^\alpha }$$ I understand the four vector has elements $\gamma(c, u, 0, 0)$ where $u$ is my radial velocity. Looking at the equation for $v_{r+dr}^1$ I get
$$v_{r + dr}^1 \approx v_r^1 - \Gamma _{01}^1v_r^0d{x^1} - \Gamma _{11}^1v_r^1d{x^1}$$ where, clearly, $dx^1=dr$ (and no other Christoffel symbols apply because other $v^\nu$ are zero). Here's my problem. Terms $v_r^0dx^1$ and $v_r^1dx^1$ have the same dimensionality but the two Christoffel symbols do not. What have I misunderstood?
(Dimensionality of $\Gamma _{01}^1$ is $T^{-1}$ but of $ \Gamma _{11}^1$ is $L^{-1}$)
