Fermi Walker vs. Fermi transport

Question

A vector field $f^\mu$ is said to be Fermi-Walker transported along a curve $\gamma$ parametrized with $\tau$ if the following holds $$\frac{\mathrm{D}}{\mathrm{d}\tau}f^\mu = -(a^\mu v^\nu - a^\nu v^\mu) f_\nu,$$ where $v^\mu$ is the tangent vector $\gamma$ and $a^\mu$ is its derivative. The uppercase D denotes the covariant derivative. This is the usual transport law for nonrotating tetrads. However, in "Introduction to General Relativity" Lewis Ryder mentions the Fermi transport for which $$\frac{\mathrm{D}}{\mathrm{d}\tau}f^\mu = v^\mu a^\nu f_\nu.$$ What is the significance of this transport? Are there any applications of it?

Answer 1

In his original work, Fermi considered only vectors $f^{\mu}$ which are orthogonal to the curve $f^{\mu} v_{\mu} = 0$. His analysis is relevant to the spin or photon polarization vectors which are orthogonal to the four-velocity by definition.

Walker generalized Fermi's work to vectors which are not necessarily orthogonal to the velocity. (Thus the second term in the Fermi-Walker transport is not identically $0$).

This fact and more historical remarks on Fermi's work on accelerated bodies in general relativity, the Fermi-Walker transport and their connections to modern works on spinning bodies in general relativity as well as references to their original work can be found in the following article by: Bini and Jantzen.