Fermi–Walker transport

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.

This is defined for a vector field X along a curve $\gamma (s)$:

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

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

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

the vector field X is Fermi-Walker transported along the curve (see Hawking and Ellis, p. 80). Vectors tangent 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 for spin precession of electron in an external electromagnetic field can be written as follows:

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

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

Co-moving coordinate systems

A coordinate system co-moving with the particle can be defined. If we take the unit vector $v^{\mu }$ as defining an axis in the co-moving coordinate system, then any system transforming with proper time as Equation 1. is said to be undergoing Fermi Walker transport.

See also

• Albert Einstein
• Basic introduction to the mathematics of curved spacetime
• Enrico Fermi
• Transition from Newtonian mechanics to General relativity

Textbooks

• Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.{{cite book}}: CS1 maint: multiple names: authors list (link)