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 $\gamma(s)$:

$\frac{D_F X}{d s}=\frac{DX}{d s} - (X,\frac{DV}{d s}) V + (X,V)\frac{DV}{d s},$

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

$\frac{D_F X}{d s}=0,$

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:

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

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 is said to be undergoing Fermi Walker transport.

See also

• 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.