What are the initial conditions associated with solving the geodesic equation in General Relativity? Can we say that initial conditions for solving the geodesic equation in general relativity be intial velocity of a particle?
 A: The most practical way to solve the geodesic equations is using a dynamical systems approach, where one re-writes the 2nd-order ODEs as a coupled system of first-order ODEs. Let the geodesic equation be given as:
$\ddot{x}^a + \Gamma^{a}_{bc} \dot{x}^{b} \dot{x}^{c} = 0$. 
Then, this can be written as a system of first-order equations as follows:
$\dot{x}^{a} = y^{a}$,
$\dot{y}^{a} = -\Gamma^{a}_{bc} y^{b} y^{c}$
Now, this is a dynamical system, and while explicit solutions may not be possible, one can use all the beautiful tools of dynamical systems theory to get a very detailed description of information about the system. (It's also much easier to solve numerically when cast in this form!) The initial conditions are $(x^a_i, y^a_i)$.
A: The geodesic equation 
$$\frac{d^2 x^{\mu}}{ds^2} + \Gamma^{\mu}_{\rho\sigma} \frac{dx^{\rho}}{ds}\frac{dx^{\sigma}}{ds}=0$$
is nothing more than a set of (coupled) second-order differential equations for the particle's coordinates as a function of some parameter $s$. The explicit solution
$$x^{\mu}(s)$$
 requires an initial coordinate position $x^{\mu}(s_0)$ and an initial coordinate velocity $\frac{dx^{\mu}(s_0)}{ds}$. Note it is often not practical to find an explicit solution, and it is more useful to study various aspects of the equations. 
