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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:
whereVis four-velocity,Dis the covariant derivative, andis the scalar product. If

then 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

See also

  • Basic introduction to the mathematics of curved spacetime

  • Enrico Fermi

  • Transition from Newtonian mechanics to general relativity

References

[1]
Citation Linkopenlibrary.orgHawking, Stephen W.; Ellis, George F.R. (1973), The Large Scale Structure of Space-time, Cambridge University Press, ISBN 0-521-09906-4, p. 80
Oct 1, 2019, 4:07 AM
[2]
Citation Linkopenlibrary.orgBargmann, V.; Michel, L.; Telegdi, V. L. (1959). "Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field". Phys. Rev. Lett. APS. 2 (10): 435. Bibcode:1959PhRvL...2..435B. doi:10.1103/PhysRevLett.2.435.
Oct 1, 2019, 4:07 AM
[3]
Citation Linkopenlibrary.orgMisner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0, p. 170
Oct 1, 2019, 4:07 AM
[4]
Citation Link//arxiv.org/abs/astro-ph/0411595Kocharyan (2004). "Geometry of Dynamical Systems". arXiv:astro-ph/0411595.
Oct 1, 2019, 4:07 AM
[5]
Citation Linkui.adsabs.harvard.edu1959PhRvL...2..435B
Oct 1, 2019, 4:07 AM
[6]
Citation Linkdoi.org10.1103/PhysRevLett.2.435
Oct 1, 2019, 4:07 AM
[7]
Citation Linkarxiv.orgastro-ph/0411595
Oct 1, 2019, 4:07 AM
[8]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Oct 1, 2019, 4:07 AM