 Inertial coordinates inside a slowly rotating spherical shell

At linear order in $ϵ=GM/R$ 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 $d^2\mathbf{x}/d{t}^2=2\mathbf{\omega }\times d\mathbf{x}/dt$ 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 $\mathbf{\omega }$ in the original coordinates.