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 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 $\gamma (s)$ :

{\frac {D_{F}X}{ds}}={\frac {DX}{ds}}-(X,{\frac {DV}{ds}})V+(X,V){\frac {DV}{ds}},

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

{\frac {D_{F}X}{ds}}=0,

then 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_{F}a^{{\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 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 $v^{{\mu }}$ 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]

Notes

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: 435. doi:10.1103/PhysRevLett.2.435. (Subscription required (help)). .

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.

Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0

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Fermi–Walker transport

Fermi–Walker differentiation

Co-moving coordinate systems

See also

Notes

References