# Fermi–Walker transport

(Redirected from Fermi–Walker differentiation)

Fermi–Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not to arbitrary spin or rotation of the frame.

## Fermi–Walker differentiation

In the theory of Lorentzian manifolds, Fermi–Walker differentiation is a generalization of covariant differentiation. In general relativity, Fermi–Walker derivatives of the spacelike unit vector fields in a frame field, taken with respect to the timelike unit vector field in the frame field, are used to define non-inertial but nonspinning frames, by stipulating that the Fermi–Walker derivatives should vanish. In the special case of inertial frames, the Fermi–Walker derivatives reduce to covariant derivatives.

With a ${\displaystyle (-+++)}$ sign convention, this is defined for a vector field X along a curve ${\displaystyle \gamma (s)}$:

${\displaystyle {\frac {D_{F}X}{ds}}={\frac {DX}{ds}}-(X,{\frac {DV}{ds}})V+(X,V){\frac {DV}{ds}},}$

where V is four-velocity, D is the covariant derivative in the Riemannian space, and (,) is scalar product. If

${\displaystyle {\frac {D_{F}X}{ds}}=0,}$

the vector field X is Fermi–Walker transported along the curve (see Hawking and Ellis, p. 80). Vectors perpendicular to the space of four-velocities in Minkowski spacetime, e.g., polarization vectors, under Fermi–Walker transport experience Thomas precession.

Using the Fermi derivative, the Bargmann–Michel–Telegdi equation[1] for spin precession of electron in an external electromagnetic field can be written as follows:

${\displaystyle {\frac {D_{F}a^{\tau }}{ds}}=2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },}$

where ${\displaystyle a^{\tau }}$ and ${\displaystyle \mu }$ are polarization four-vector and magnetic moment, ${\displaystyle u^{\tau }}$ is four-velocity of electron, ${\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}$, ${\displaystyle u^{\tau }a_{\tau }=0}$, and ${\displaystyle F^{\tau \sigma }}$ is electromagnetic field-strength tensor. The right side describes Larmor precession.

## Co-moving coordinate systems

A coordinate system co-moving with a particle can be defined. If we take the unit vector ${\displaystyle v^{\mu }}$ as defining an axis in the co-moving coordinate system, then any system transforming with proper time is said to be undergoing Fermi Walker transport.[2]

## References

1. ^ V. Bargmann, L. Michel, and V. L. Telegdi, Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field, Phys. Rev. Lett. 2, 435 (1959).
2. ^ Misner, Charles; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. p. 170. ISBN 0-7167-0344-0.

## Textbooks

• Landau, L. D.; Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.