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 a 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.[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:
where and are polarization four-vector and magnetic moment, is four-velocity of electron, , , and is the electromagnetic field strength tensor. The right side describes Larmor precession.
Co-moving coordinate systems
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.[3]
Generalised Fermi–Walker differentiation
Fermi–Walker differentiation can be extended for any , this is defined for a vector field along a curve :
where .
If , then
and
See also
Notes
- Hawking & Ellis 1973, p. 80
- Bargmann, Michel & Telegdi 1959
- Misner, Thorne & Wheeler 1973, p. 170
- Kocharyan (2004). "Geometry of Dynamical Systems". arXiv:astro-ph/0411595.
References
- Bargmann, 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.CS1 maint: ref=harv (link).
- Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics. 2 (4th ed.). Butterworth–Heinemann. ISBN 0-7506-2768-9.CS1 maint: ref=harv (link)
- Misner, Charles W.; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0
- Hawking, Stephen W.; Ellis, George F.R. (1973), The Large Scale Structure of Space-time, Cambridge University Press, ISBN 0-521-09906-4
- Kocharyan A.A. (2004). Geometry of Dynamical Systems. arXiv:astro-ph/0411595.