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 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.

With a (-+++) sign convention, this is defined for a vector field X along a curve \gamma(s):

\frac{D_F X}{d s}=\frac{DX}{d s} - (X,\frac{DV}{d s}) V + (X,V)\frac{DV}{d s},

where V is four-velocity, D is the covariant derivative in the Riemannian space, and (,) is scalar product. If

\frac{D_F X}{d s}=0,

the vector field X is Fermi–Walker transported along the curve (see Hawking and Ellis, p. 80). 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[1] for spin precession of electron in an external electromagnetic field can be written as follows:

\frac{D_Fa^{\tau}}{ds} = 2\mu (F^{\tau \lambda} - u^{\tau} u_{\sigma} F^{\sigma \lambda})a_{\lambda},

where a^{\tau} and \mu are polarization four-vector and magnetic moment, u^{\tau} is four-velocity of electron, a^{\tau}a_{\tau} = -u^{\tau}u_{\tau} = -1, u^{\tau} a_{\tau}=0, and F^{\tau \sigma} 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 v^{\mu} 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.[2]

See also

References

  1. ↑ 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).
  2. ↑ Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. p. 170. ISBN 0-7167-0344-0.

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