Christoffel symbol (${\Gamma }_{ij}^k$)

In a coordinate frame, the Levi-Civita connection is determined byThe coefficients ${\Gamma }_{ij}^k$ are the Christoffel symbols. They are not the components of a tensor because a coordinate change contributes second derivatives.

For a nondegenerate metric tensor and its Levi-Civita connection, contracting the Christoffel symbol formula gives ${\Gamma }^{\rho }{}_{\rho \mu }=\frac{1}{2}{g}^{ab}{\partial }_{\mu }{g}_{ab}$. The Jacobi determinant derivative formula identifies this with ${\partial }_{\mu }\log \sqrt{|\det g|}$. The volume density, rather than the determinant without an absolute value, works in every fixed metric signature.

Forwrite dots and primes for $t$- and $r$-derivatives. Up to symmetry in the lower indices, the nonzero connection coefficients are