Fermi–Walker transport

Remarks

The tangent field is always Fermi–Walker transported: since $g(u, u) = -1$ and $g(a, u) = 0$, one gets $\nabla_u u + g(u, u) a - g(a, u) u = a - a - 0 = 0$. For a given vector at $\gamma(\tau_0)$, the linear equation $D_F X/d\tau = 0$ has a unique solution along $\gamma$, and Fermi–Walker transport preserves the inner products $g(X, Y)$. So an orthonormal frame with $e_0 = u$ at one point stays orthonormal with $e_0 = u$ along the curve. If $\gamma$ is a geodesic, then $a = 0$ and Fermi–Walker transport is parallel transport. Non-example: on an accelerated curve, $u$ itself is not parallel transported, since $\nabla_u u = a \ne 0$. In Minkowski space, for a particle in uniform circular motion, a Fermi–Walker transported spatial vector rotates relative to the axes of the inertial frame, which is Thomas precession.