= Inertial coordinates inside a slowly rotating spherical shell
{title2=$T=(1-\epsilon)t,\quad\mathbf X=(1+\epsilon)\mathcal R_z(-\omega t)\mathbf x$}
At linear order in $\epsilon=GM/R$ and the leading rotation term, this coordinate change takes the interior shell metric to <Minkowski spacetime>. Consequently the retained interior <Riemann curvature tensor> vanishes although inertial axes rotate relative to distant reference axes. A freely moving particle has $\mathbf X=\mathbf X_0+\mathbf V T$, while its original coordinates obey $d^2\mathbf x/dt^2=2\boldsymbol\omega\times d\mathbf x/dt$ to first rotation order. Terms of order $\omega^2$ would include centrifugal acceleration and are outside this approximation. A <gyroscope> at rest to the retained order has its parallel-transported spin precessing at $\boldsymbol\omega$ in the original coordinates.
Back to article page
