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

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

where V is four-velocity, D is the covariant derivative, and \( {\displaystyle (\cdot ,\cdot )} \) is the scalar product. If

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

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:

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

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
Main article: Proper reference frame (flat spacetime) § Comoving tetrads

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.[3]

Generalised Fermi–Walker differentiation

Fermi–Walker differentiation can be extended for any V, this is defined for a vector field X along a curve \( \gamma (s) \):

\( {\displaystyle {\frac {{\mathcal {D}}X}{ds}}={\frac {DX}{ds}}+\left(X,{\frac {DV}{ds}}\right){\frac {V}{(V,V)}}-{\frac {(X,V)}{(V,V)}}{\frac {DV}{ds}}-\left(V,{\frac {DV}{ds}}\right){\frac {(X,V)}{(V,V)^{2}}}V,}[4]

where \( {\displaystyle (V,V)\neq 0} \).

If \( {\displaystyle (V,V)=-1}, , \) then

\( {\displaystyle \left(V,{\frac {DV}{ds}}\right)={\frac {1}{2}}{\frac {d}{ds}}(V,V)=0\ ,} \) and \( {\displaystyle {\frac {{\mathcal {D}}X}{ds}}={\frac {D_{F}X}{ds}}.} \)
See also

Basic introduction to the mathematics of curved spacetime
Enrico Fermi
Transition from Newtonian mechanics to general relativity

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..

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

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.

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