Spin four-vector ($s^{\mu }$)

The spin of a small torque-free test gyroscope is represented by a spacelike vector orthogonal to its four-velocity, $s^{\mu }{u}_{\mu }=0$. Its temporal component vanishes in its own rest frame. Along a geodesic, $D{s}^{\mu }/d\tau =0$: parallel transport preserves the orthogonality and the invariant spin magnitude. For an accelerated observer the corresponding nonrotating transport is Fermi-Walker transport.

For the orthogonal parametrization $s^{\mu }=\gamma (\mathbf{s}\cdot \mathbf{v}/c,\mathbf{s})$, torque-free acceleration along a fixed direction keeps the longitudinal laboratory component constant and the transverse component proportional to $1/\gamma$. The invariant proper-spin length remains constant; these changing three-components are not a torque-induced loss of spin.