General Relativity: An Introduction for Physicists · Problem 3.1

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · basis-vectors-coordinate-transformation

Problem

Show that, in general,

e_a = g_{ab}e^b

and

e^a = g^{ab}e_b

. Show also that, under a coordinate transformation,

e'_a = \frac{\partial x^b}{\partial x'^a} e_b \qquad \text{and} \qquad e'^a = \frac{\partial x'^a}{\partial x^b} e^b.