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'''Fermi–Walker transport''' is a process in [[general relativity]] used to define a [[coordinate system]] or [[Frame of reference|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 <math>\gamma(s)</math>:

:<math>\frac{D_F X}{d s}=\frac{DX}{d s} - (X,\frac{DV}{d s}) V + (X,V)\frac{DV}{d s},</math>

where ''V'' is four-velocity, ''D'' is the covariant derivative in the [[Riemannian manifold|Riemannian space]], and ''(,)'' is scalar product. If
:<math>\frac{D_F X}{d s}=0,</math>
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]]<ref>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).</ref> for spin precession of electron in an external electromagnetic field can be written as follows:
:<math>\frac{D_Fa^{\tau}}{ds} = 2\mu (F^{\tau \lambda} - u^{\tau} u_{\sigma} F^{\sigma \lambda})a_{\lambda},</math>
where <math>a^{\tau}</math> and <math>\mu</math> are polarization four-vector and magnetic moment, <math>u^{\tau}</math> is four-velocity of electron, <math>a^{\tau}a_{\tau} = -u^{\tau}u_{\tau} = -1</math>, <math>u^{\tau} a_{\tau}=0</math>, and <math>F^{\tau \sigma}</math> 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 <math> v^{\mu} </math> 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.<ref>{{cite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | page=170 | isbn=0-7167-0344-0}}</ref>

==See also==
*[[Basic introduction to the mathematics of curved spacetime]]
*[[Enrico Fermi]]
*[[Transition from Newtonian mechanics to General relativity]]

==References==
{{Reflist}}

==Textbooks==
*{{cite book | author=Landau, L. D. and Lifshitz, E. M.| title=Classical Theory of Fields (Fourth Revised English Edition) | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}

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[[Category:Mathematical methods in general relativity]]

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