General Relativity: An Introduction for Physicists · Problem 3.5

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · affine-connection-coordinate-transformation

Problem

Show that, under a coordinate transformation, the affine connection transforms as

\Gamma'^a{}_{bc} = \frac{\partial x'^a}{\partial x^d} \frac{\partial x^f}{\partial x'^b} \frac{\partial x^g}{\partial x'^c} \Gamma^d{}_{fg} - \frac{\partial x^d}{\partial x'^b} \frac{\partial x^f}{\partial x'^c} \frac{\partial^2 x'^a}{\partial x^d \partial x^f}.