Christoffel Symbols in Flat Space-Time Does it make sense to talk about Christoffel symbols in flat space time? Do they have non-zero values? I understand that the Christoffel symbols appear as an indication of curvature in space. So, are they non-existent in flat space-time?
 A: Yes, it makes sense to talk about Christoffel symbols in flat spacetime. Every coordinate system has associated Christoffel symbols. On Minkowski spacetime in the standard coordinates, the Christoffel symbols are all zero. But in different coordinates (e.g., spherical coordinates), they will not be zero. The Christoffel symbols contain information about the intrinsic curvature of the spacetime and about the "curvature of the coordinates".
A: As frakbak explained, one has a notion of Christoffel symbols in flat spacetime, as they basically record information about derivatives of the metric tensor with respect to different indices, each. It makes kind of sense, since changes of the metric tensor describe local changes of a scalar product which encodes information about changes of projections to a new coordinate basis, featureing a new tangent plane. While such changes of the projection to a new coordinate basis are inevitable in an arbritrarily curved spacetime, you can just deliberately provide the covariant derivative with such unnatural changes in flat spacetime by changing to an "inappropriate", curved coordinate system.  
