# Confusion in Gravitation Foundation and Frontiers?

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

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 $$$$\tag{1} [u,\xi] = 0 \quad\text{or}\quad \mathcal{L}_{u} \xi=0$$$$ or explicitly, $$\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$$, we have $$$$\tag{2} \xi = \frac{\partial x^i}{\partial v} \frac{\partial}{\partial x^i} \equiv \xi^i \partial_i, \quad\quad u = \frac{\partial x^i}{\partial\tau} \frac{\partial}{\partial x^i} \equiv u^i \partial_i.$$$$ Substituting (2) into (1), we obtain $$$$\tag{3} u^i \partial_i \xi^j - \xi^i \partial_i u^j = 0.$$$$ From the definition of covariant derivative, we have $$$$u^i \nabla_i \xi^j = u^i \partial_i \xi^j + \Gamma^j_{ik} u^i \xi^k \quad\text{and}\quad \xi^i \nabla_i u^j = \xi^i \partial_i u^j + \Gamma^j_{ik} \xi^i u^k$$$$ For Riemann connection, we have $$\Gamma^j_{ik} = \Gamma^j_{ki}$$, then $$\Gamma^j_{ik} u^i \xi^k = \Gamma^j_{ik} \xi^i u^k$$. We finally obtain $$$$u^i \nabla_i \xi^j - \xi^i \nabla_i u^j = u^i \partial_i \xi^j - \xi^i \partial_i u^j = 0.$$$$