# Fermi–Walker transport

# 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 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.

`With asign convention, this is defined for a vector field`

*X*along a curve:`where`

*V*is four-velocity,*D*is the covariant derivative, andis the scalar product. Ifthen the vector field *X* is Fermi–Walker transported along the curve^{[1]}. 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^{[2]} for spin precession of electron in an external electromagnetic field can be written as follows:

`whereandare polarization four-vector andmagnetic moment,is four-velocity of electron,,, andis theelectromagnetic field strength tensor. The right side describesLarmor precession.`

Co-moving coordinate systems

`A coordinate system co-moving with a particle can be defined. If we take the unit vectoras defining an axis in the co-moving coordinate system, then any system transforming with proper time is said to be undergoing Fermi Walker transport.`

^{[3]}Generalised Fermi–Walker differentiation

`Fermi–Walker differentiation can be extended for any, this is defined for a vector fieldalong a curve:`

`where.`

`If, then`

^{[4]}

See also

Basic introduction to the mathematics of curved spacetime

Enrico Fermi

Transition from Newtonian mechanics to general relativity

## References

*The Large Scale Structure of Space-time*, Cambridge University Press, ISBN 0-521-09906-4, p. 80

*Phys. Rev. Lett*. APS.

**2**(10): 435. Bibcode:1959PhRvL...2..435B. doi:10.1103/PhysRevLett.2.435.

*Gravitation*, W. H. Freeman, ISBN 0-7167-0344-0, p. 170