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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?

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The second one is the same just for vectors orthogonal to the curve. My guess is that Ryder is considering the world line of an observer and transporting spacial vectors along it hence the first term is zero. –  MBN Dec 3 '13 at 10:57
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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.

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