PeterDonis said:
I'm not sure exactly what conditions are preserved by Fermi-Walker transport as opposed to parallel transport. In some special cases, at least, F-W transport preserves orthogonality of particular pairs of vectors; for example, it preserves the orthogonality of the basis vectors ##\partial_t## and ##\partial_x## in Rindler coordinates along the worldline of an object with constant proper acceleration.
In special relativity the Fermi-Walker transport is most easily defined as a description of the transport of arbitrary vectors along a time like curve such that the tetrad transported by an observer on this curve (it's an arbitrarily accelerated observer) such that his spatial basis vectors are not rotating.
This description in words is not very precise, but it must be made clear by the mathematical formulation. Without loss of generality we'll parametrize the world line by it's proper time, ##\tau##, such that the four-velocity of the observer is normalized
$$u^{\mu} = \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=\mathrm{d}_{\tau} x^{\mu}, \quad u_{\mu} u^{\mu}=1.$$
Then the momentary proper acceleration of the observer
$$a^{\mu}=\mathrm{d}_{\tau} u^{\mu}$$
is perpendicular to ##u^{\mu}## since
$$u_{\mu} u^{\mu}=1 \; \Rightarrow \; u_{\mu} \mathrm{d}_{\tau} u^{\mu}=u_{\mu} a^{\mu}=0.$$
So it defines a space-like direction.
Now you can define instantaneous inertial restframes of the observer by introducing a special tetrad, i.e., four orthonormal basis vectors as a function of ##\tau##, where always ##\boldsymbol{e}_0=\boldsymbol{u}##. Now there are a lot of possibilities to choose at any proper time three spacelike Minkowski-orthonormal unit vectors orthnormal to ##\boldsymbol{u}##. What we, however, usually want is a tetrad where the observer has his spatial basis vectors not rotated against each other. To achieve this we take an arbitrary Minkowski-orthonormal frame and think the corresponding tetrad to be transported along the worldline such that
$$\boldsymbol{e}_{\mu} \cdot \boldsymbol{e}_{\nu}=\eta_{\mu \nu}, \quad \boldsymbol{e}_0=\boldsymbol{u}.$$
Now we have to establish the "non-rotating property" of the spatial tetrad vectors. To do this we define the Fermi-Walker transport. It's heuristically easily found in the following way. Take an arbitrary vector ##\boldsymbol{V}## along the trajectory and define ##\boldsymbol{V}(\tau)## as the transport of ##\boldsymbol{V}(\tau=0)## along the curve such that when going from ##\tau## an infinitesimal step further to ##\tau+\mathrm{d} \tau## such that it undergoes only an infinitesimal
rotation free Lorentz boost in the ##\boldsymbol{u}##-##\boldsymbol{a}## plane. This means that
$$\mathrm{d} V^{\mu}=\mathrm{d} \tau (a^{\mu} u^{\nu}-u^{\mu} a^{\nu}) V_{\nu}$$
or the Fermi-Walker transport is defined by the differential equation
$$\mathrm{d}_{\tau} V^{\mu}=(a^{\mu} u^{\nu}-u^{\mu} a^{\nu}) V_{\nu}.$$
Since the infinitesimal change corresponds just to a Lorentz trafo in the ##\boldsymbol{u}##-##\boldsymbol{a}## plane, which is a plane spanned by a time-like and a space-like vector, and thus the Lorentz trafo is a rotation-free boost. It also has the other properties we like. Indeed, the Fermi-Walker transport of ##\boldsymbol{u}## gives
$$\mathrm{d}_{\tau} u^{\mu} = a^{\mu},$$
because ##u_{\mu} u^{\mu}=1## and ##u_{\mu} a^{\mu}=0##.
Further if ##V## and ##W## are Fermi-Walker transported we have
$$\mathrm{d}_{\tau} (\boldsymbol{V} \cdot \boldsymbol{W})=(\mathrm{d}_{\tau} \boldsymbol{V}) \cdot \boldsymbol{W} + \boldsymbol{V} \cdot \mathrm{d}_{\tau} \boldsymbol{W}\\=\boldsymbol{a} \cdot \boldsymbol{W} \boldsymbol{u} \cdot \boldsymbol{V} - \boldsymbol{u} \cdot \boldsymbol{W} \boldsymbol{a} \cdot \boldsymbol{V} + \boldsymbol{a} \cdot \boldsymbol{V} \boldsymbol{u} \cdot \boldsymbol{W} - \boldsymbol{u} \cdot \boldsymbol{V} \boldsymbol{a} \cdot \boldsymbol{W}=0,$$
i.e., the Fermi-Walker transport also leaves the Minkowski product between arbitrary vectors unchanged.
So if you start with an arbitrary tetrad at ##\tau=0## and Fermi-Walker transport it along the worldline it stays a tetrad and from one infinitesimal time step to another you have only a rotation free Lorentz boost.
An important physical application is the Thomas precession. Take a particle with spin (historically it was about an electron moving around an atomic nucleus) which is moving along its world line accelerated by an arbitrary force which doesn't apply a torque to the spin. The corresponding equation of motion for the spin is just that of the Fermi-Walker transport of this spin. It turns out that the spin nevertheless precesses, i.e., it rotates, and that's due to the fact that the composition of two Lorentz boosts (if not in the same spatial direction) leads to a Lorentz trafo that's not rotation free, i.e., it consists of a Lorentz boost followed by a rotation. For an electron moving with constant angular velocity along a circle the Thomas-precession frequency is ##\omega_{\text{Thomas}}=(\gamma-1)\omega##.
It is also very easy to extend this now to the curved pseudo-Riemannian spacetime of GR. Here, everywhere, where I wrote ##\mathrm{d}_{\tau}## one has to write a covariant derivative ##\mathrm{D}_{\tau}##, defined by
$$\mathrm{D}_{\tau} V^{\mu} = \mathrm{d}_{\tau} V^{\mu} + {\Gamma^{\mu}}_{\nu \rho} V^{\nu} u^{\rho}.$$
Particularly the proper acceleration is defined by
$$a^{\mu} = \mathrm{D}_{\tau} u^{\mu}.$$
With this you have a Fermi-Walker transport in curved space time with basically the same geometrical "local" meaning as in flat Minkowski space. It also is clear that Fermi-Walker transport along a time-like geodesic is identical with the parallel transport along this geodesic, because in this case ##a^{\mu}=0##. The Fermi-Walker transport can be used to derive the geodesic precession of a spinning top in non-rotating spacetimes (like the Schwarzschild metric) as well as the Lense-Thirring effect on a spinning top in rotating spacetimes (like the Kerr metric).