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 due to arbitrary spin or rotation of the frame.
It was discovered by Fermi in 1921 and rediscovered by Walker in 1932.
[1] In the theory of Lorentzian manifolds, Fermi–Walker differentiation is a generalization of covariant differentiation.
In general relativity, Fermi–Walker derivatives of the spacelike 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 and non-rotating 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.
sign convention, this is defined for a vector field X along a curve
: where V is four-velocity, D is the covariant derivative, and
is the scalar product.
If then the vector field X is Fermi–Walker transported along the curve.
[2] 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[3] for spin precession of electron in an external electromagnetic field can be written as follows: where
are polarization four-vector and magnetic moment,
is four-velocity of electron,
is the electromagnetic field strength tensor.
The right side describes Larmor precession.
A coordinate system co-moving with a particle can be defined.
If we take the unit vector
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.
(that is, not a light-like vector).
This is defined for a vector field
: Except for the last term, which is new, and basically caused by the possibility that
is not constant, it can be derived by taking the previous equation, and dividing each