en.wikipedia.org

Fermi–Walker transport - Wikipedia

️Thu Dec 14 2023

From Wikipedia, the free encyclopedia

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 due 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

[edit]

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

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

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

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

then the vector field X is Fermi–Walker transported along the curve.^[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:

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

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

[edit]

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.^[4]

Generalised Fermi–Walker differentiation

[edit]

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

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

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

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

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

^ Bini, Donato; Jantzen, Robert T. (2002). "Circular Holonomy, Clock Effects and Gravitoelectromagnetism: Still Going Around in Circles After All These Years". Nuovo Cimento B. 117 (9–11): 983–1008. arXiv:gr-qc/0202085. Archived from the original on 2023-12-14. Retrieved 2023-12-14.
^ Hawking & Ellis 1973, p. 80
^ Bargmann, Michel & Telegdi 1959
^ Misner, Thorne & Wheeler 1973, p. 170
^ Kocharyan, A. A. (2004). "Geometry of Dynamical Systems". arXiv:astro-ph/0411595.

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.