Christoffel symbol
/ KRIS-toffel /
In curved or curvilinear space the basis vectors themselves change as you move from point to point — in polar coordinates the radial and angular directions literally point different ways at different places. So when you take the derivative of a vector, part of the change is the vector genuinely changing and part is merely the basis swinging underneath it. The Christoffel symbols are the bookkeeping that separates these two effects: they record exactly how the basis vectors turn as you step in each direction.
Written Gamma^k_{ij} (with three indices), a Christoffel symbol answers: when you move in the direction of coordinate j, how much of the change in the i-th basis vector points along the k-th basis direction? They are computed entirely from the metric and its first derivatives by the formula Gamma^k_{ij} = (1/2) g^{kl} (partial_i g_{jl} + partial_j g_{il} - partial_l g_{ij}), with l summed. They are symmetric in the lower pair (Gamma^k_{ij} = Gamma^k_{ji}) for the standard metric connection. Crucially, despite carrying indices, the Christoffel symbols are NOT a tensor: under a change of coordinates they pick up an extra inhomogeneous term (a second-derivative piece) that no tensor would. This is not a defect — it is exactly the correction needed so that the covariant derivative they build is a tensor.
Christoffel symbols are the workhorses of computational differential geometry and general relativity. They appear in the covariant derivative (the derivative that respects the curved geometry), in the geodesic equation (the straightest-path equation is d^2 x^k/ds^2 + Gamma^k_{ij} (dx^i/ds)(dx^j/ds) = 0, where the Gammas supply the 'fictitious' centrifugal and Coriolis-like terms), and as the building blocks of the Riemann curvature tensor. An honest and important subtlety: because they are not tensors, the Christoffel symbols can be nonzero in flat space (they are nonzero in plain polar coordinates) and can all be made to vanish at any single chosen point by a clever choice of coordinates. So nonzero Gammas do NOT mean the space is curved — only their pattern of derivatives, assembled into the Riemann tensor, detects true curvature.
In flat 2-D polar coordinates (r, theta) the only nonzero Christoffel symbols are Gamma^r_{theta theta} = -r and Gamma^theta_{r theta} = Gamma^theta_{theta r} = 1/r. These nonzero symbols in perfectly flat space are exactly the terms that turn up as the centrifugal acceleration -r (dtheta/dt)^2 and the Coriolis-like (2/r)(dr/dt)(dtheta/dt) when you write Newton's law in polar coordinates.
Even in flat space, polar Christoffel symbols are nonzero — they encode centrifugal and Coriolis terms, not curvature.
Christoffel symbols are not tensors and being nonzero does not prove curvature: in polar coordinates flat space has nonzero Gammas, and at any single point you can choose coordinates making them all vanish. Genuine curvature lives in the Riemann tensor built from their derivatives, which cannot be transformed away.