Christoffel symbols track how coordinates twist across a curved space
On a curved surface, the directions you call north and east quietly rotate as you move. Christoffel symbols are the bookkeeping for that drift, a three-dimensional grid of numbers at every point. In Einstein's general relativity they take on a startling job: they behave like the gravitational force field itself.
Elwin Bruno Christoffel, who lived from 1829 to 1900, gave his name to these quantities. They describe a metric connection, the rule that links neighbouring points of a surface or higher-dimensional manifold on which distances can be measured. Ideas such as parallel transport, covariant derivatives and geodesics can be defined without any metric, but once distances exist, those ideas become tied to the actual shape of the space, and Christoffel symbols express that link in concrete coordinates.
For a given metric there are infinitely many possible connections, yet exactly one is free of torsion: the Levi-Civita connection. Physicists, especially those working in general relativity, almost always choose it, using so-called holonomic coordinates in which torsion disappears. Even in ordinary flat Euclidean space the symbols have meaning, recording how local coordinate directions shift from place to place, as they do when polar coordinates are used on a flat plane.
In a space of n dimensions, the symbols form an n by n by n array of real numbers at each point, labelled with three indices. They are not quite a tensor. Under linear changes of coordinates they transform the way tensor components do, but under general smooth changes of coordinates they do not, which is why they can vanish in one coordinate system and not in another.
Their practical value lies in calculation. The Riemann curvature tensor, which captures how curved a space truly is, can be built from nothing more than these symbols plus their first derivatives. Within Einstein's theory the metric tensor plays the part of a gravitational potential, while the connection acts as the force field. When the coordinates share a symmetry with the metric, many of the entries drop to zero, which greatly simplifies the arithmetic.
Source: Christoffel symbols
