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.

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 ${\displaystyle \gamma (s)}$:

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

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.

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