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 :

[4]

where .

If , then

and

See also

Notes

  1. Hawking & Ellis 1973, p. 80
  2. Bargmann, Michel & Telegdi 1959
  3. Misner, Thorne & Wheeler 1973, p. 170
  4. Kocharyan (2004). "Geometry of Dynamical Systems". arXiv:astro-ph/0411595.

References


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