I recently studied these things in Lee's "Riemannian manifolds: an introduction to curvature", so it might help me as well as you if I write down a summary of it here. (I'm writing indices as i,j,k rather than with greek letters because i,j,k is easier to type).
We need to distinguish between "a vector field" and "a vector field along a curve". Let \gamma:[a,b]\rightarrow M be a smooth curve in a manifold M. A vector field on an open subset U of M takes each p\in U to a tangent vector at p. A vector field along \gamma is a function that takes each t\in[a,b] to a tangent vector at \gamma(t). The most obvious example of vector field along \gamma is its velocity, defined by
\dot\gamma(t)=\gamma_*D_t
where \gamma_* is the pushforward of \gamma and D_t is the operator that takes f:[a,b]\rightarrow R to f'(t). Note that this is just the definition of the tangent vector of \gamma at the point \gamma(t).
\dot\gamma(t)f=D_t(f\circ\gamma)=(f\circ\gamma)'(t)
If the manifold is equipped with a connection, we can use it to define a covariant derivative of vector fields along \gamma.
The book explains it well.
\gamma is said to be a geodesic if the covariant derivative along \gamma of \dot\gamma is 0, i.e. if
0=D\dot\gamma(t)=(\nabla_V V)_{\gamma(t)}
for all t, where D is the covariant dervative operator associated with \gamma, V is any vector field along \gamma such that V_{\gamma(t)}=\dot\gamma(t) for all t. Such a V is said to be an extension of \dot\gamma. The second equality is explained in the book as well.
The definition of a connection tells us that
\nabla_XY=X^i\nabla_{\partial_i}(Y^j\partial_j)=X^i(\partial_i Y^j\partial_j+Y^j\Gamma^k_{ij}\partial_k)=(XY^k+\Gamma^k_{ij}X^iY^j)\partial_k
so
0=(\nabla_V V)_{\gamma(t)}^k=V_{\gamma(t)}V^k+\Gamma^k_{ij}(\gamma(t))V^i(\gamma(t))V^j(\gamma(t))
But we have
V_{\gamma(t)}V^k=\dot\gamma(t)V^k=(V^k\circ\gamma)'(t)=\frac{d}{dt}V^k(\gamma(t))
This is what needs to be zero for the components of the tangent vector to be constant along gamma. So the definition of a geodesic is telling us that this happens if and only if the Christoffel symbols \Gamma^k_{ij}(\gamma(t))[/tex] all vanish in the coordinate system we're using.<br />
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If we restrict ourselves to metric compatible symmetric connections (that includes any metric of GR, and of course the Euclidean metric as well), the Christoffel symbols can be expressed as<br />
<br />
\Gamma^k_{ij}=\frac 1 2 g^{kl}(\partial_i g_{jl}+\partial_j g_{il} -\partial_l g_{ij})<br />
<br />
So they all vanish if the components of the metric are constant in the coordinate system we're using. The components of the Minkowski metric in any global inertial coordinate system are of course just g_{ij}(p)=\eta_{ij} for all p, so the Christoffel symbols vanish, and the components of the tangent vector of a geodesic are constant along the geodesic.
