Frame fields in general relativity

Remarks

The dual coframe is the quadruple of one-forms $\theta^0, \ldots, \theta^3$ with $\theta^a(e_b) = \delta^a_b$, and then $g = \eta_{ab} \, \theta^a \otimes \theta^b$, summed over $a, b$. Example: for the Schwarzschild metric $g = -f \, dt^2 + f^{-1} dr^2 + r^2(d\theta^2 + \sin^2\theta \, d\phi^2)$ with $f = 1 - 2m/r$, on the region $r > 2m$, $0 < \theta < \pi$, the static frame is $e_0 = f^{-1/2} \partial_t$, $e_1 = f^{1/2} \partial_r$, $e_2 = r^{-1} \partial_\theta$, $e_3 = (r \sin\theta)^{-1} \partial_\phi$; for example $g(e_0, e_0) = f^{-1} \cdot (-f) = -1$. Non-example: the coordinate fields $\partial_t, \partial_r, \partial_\theta, \partial_\phi$ there are not a frame field, since $g(\partial_\theta, \partial_\theta) = r^2 \ne 1$ in general. Two frame fields on $U$ differ at each point by a Lorentz transformation, $e'_b = \Lambda^a{}_b \, e_a$. The integral curves of $e_0$ are the world lines of the observers that the frame field describes.