Confusion in Gravitation Foundation and Frontiers?

Question

Taken from Gravitation: Foundations and Frontiers by T. Padmanabhan Pg $197$

Consider a bunch of particles moving along geodesics in a spacetime. Let $x_i = x_i (\tau,v)$ denote the family of geodesics, where $\tau$ represents an affine parameter along the geodesics and $v$ labels a particular geodesic. The vector $$n_i = (\partial x_i / \partial v) \delta v \equiv v_i$$ $\delta v$ denotes the deviation between two neighboring geodesics parametrized by the values $v$ and $v +dv$. From the definition of the covariant derivative and the relation $(\partial u_i / \partial v) = \partial v_i / \partial τ$ (where $u_i$ is the tangent vector to the geodesic), it follows that:

$$ v^k \nabla_k u^i = u^k \nabla_k v^i$$

I'm still confused about this passage. Can someone provide an explicit example of family of curves dependent of $v$? And secondly, how does one go from $$(\partial u_i / \partial v) = \partial v_i / \partial τ$$ to $$ v^k \nabla_k u^i = u^k \nabla_k v^i$$?

Answer 1

Consider the surface of the Earth. It is a 2-sphere, i.e. a 2-dimensional manifold, named $M$. The collection of all longitude lines is a bunch of geodesics, each line is labeled by its longitude ($v$). Each point on a specific line is labeled by its latitude ($\tau$). So every point on the surface of the Earth is labeled by a pair of number $(v, \tau)$, which serves as a good coordinate system of the surface. The coordinate basis of the tangent bundle $TM$ is then $\{\xi \equiv\frac{\partial}{\partial v}, u \equiv \frac{\partial}{\partial\tau}\}$, and we have for a coordinate basis \begin{equation}\tag{1} [u,\xi] = 0 \quad\text{or}\quad \mathcal{L}_{u} \xi=0 \end{equation} or explicitly, $\frac{\partial}{\partial v} \frac{\partial}{\partial \tau} = \frac{\partial}{\partial \tau}\frac{\partial}{\partial v}$.

In an arbitrary coordinate system $(x^{i})$ of $M$, we have \begin{equation}\tag{2} \xi = \frac{\partial x^i}{\partial v} \frac{\partial}{\partial x^i} \equiv \xi^i \partial_i, \quad\quad u = \frac{\partial x^i}{\partial\tau} \frac{\partial}{\partial x^i} \equiv u^i \partial_i. \end{equation} Substituting (2) into (1), we obtain \begin{equation}\tag{3} u^i \partial_i \xi^j - \xi^i \partial_i u^j = 0. \end{equation} From the definition of covariant derivative, we have \begin{equation} u^i \nabla_i \xi^j = u^i \partial_i \xi^j + \Gamma^j_{ik} u^i \xi^k \quad\text{and}\quad \xi^i \nabla_i u^j = \xi^i \partial_i u^j + \Gamma^j_{ik} \xi^i u^k \end{equation} For Riemann connection, we have $\Gamma^j_{ik} = \Gamma^j_{ki}$, then $\Gamma^j_{ik} u^i \xi^k = \Gamma^j_{ik} \xi^i u^k$. We finally obtain \begin{equation} u^i \nabla_i \xi^j - \xi^i \nabla_i u^j = u^i \partial_i \xi^j - \xi^i \partial_i u^j = 0. \end{equation}