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. It was discovered by Fermi in 1921 and rediscovered by Walker in 1932.^[1]

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

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

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.^[2] 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^[3] 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.

Co-moving coordinate systems

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.^[4]

Generalised Fermi–Walker differentiation

Fermi–Walker differentiation can be extended for any ${\displaystyle V}$ where ${\displaystyle (V,V)\neq 0}$ (that is, not a light-like vector). This is defined for a vector field ${\displaystyle X}$ along a curve ${\displaystyle \gamma (s)}$:

${\displaystyle {\frac {{\mathcal {D}}X}{ds}}={\frac {DX}{ds}}+\left(X,{\frac {DV}{ds}}\right){\frac {V}{(V,V)}}-{\frac {(X,V)}{(V,V)}}{\frac {DV}{ds}}-\left(V,{\frac {DV}{ds}}\right){\frac {(X,V)}{(V,V)^{2}}}V,}$^[5]

Except for the last term, which is new, and basically caused by the possibility that ${\displaystyle (V,V)}$ is not constant, it can be derived by taking the previous equation, and dividing each ${\displaystyle V^{2}}$ by ${\displaystyle (V,V)}$.

If ${\displaystyle (V,V)=-1}$, then we recover the Fermi–Walker differentiation:

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

See also

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

Notes

References

• Bargmann, V.; Michel, L.; Telegdi, V. L. (1959). "Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field". Physical Review Letters. 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.