Fermi–Walker transport

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 ${\displaystyle \gamma (s)}$:

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

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

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

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:

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

where ${\displaystyle a^{\tau }}$ and ${\displaystyle \mu }$ are polarization four-vector and magnetic moment, ${\displaystyle u^{\tau }}$ is four-velocity of electron, ${\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}$, ${\displaystyle u^{\tau }a_{\tau }=0}$, and ${\displaystyle F^{\tau \sigma }}$ is the electromagnetic field strength tensor. The right side describes Larmor precession.

A coordinate system co-moving with a particle can be defined. If we take the unit vector ${\displaystyle v^{\mu }}$ 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]

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

${\displaystyle {\frac {{\mathcal {D}}X}{ds}}={\frac {DX}{ds}}+\left(X,{\frac {DV}{ds}}\right){\frac {V}{(V,V)}}-{\frac {(X,V)}{(V,V)}}{\frac {DV}{ds}}-\left(V,{\frac {DV}{ds}}\right){\frac {(X,V)}{(V,V)^{2}}}V,}$[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 ${\displaystyle {\frac {{\mathcal {D}}X}{ds}}={\frac {D_{F}X}{ds}}.}$

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

• 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. Vol. 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.