Fermi–Walker transport

Mathematical technique in general relativity

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 γ ( s ) {\displaystyle \gamma (s)} :

D F X d s = D X d s − ( X , D V d s ) V + ( X , V ) D V d 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

D F X d s = 0 , {\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:

D F a τ d s = 2 μ ( F τ λ − u τ u σ F σ λ ) a λ , {\displaystyle {\frac {D_{F}a^{\tau }}{ds}}=2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },}

where a τ {\displaystyle a^{\tau }} and μ {\displaystyle \mu } are polarization four-vector and magnetic moment, u τ {\displaystyle u^{\tau }} is four-velocity of electron, a τ a τ = − u τ u τ = − 1 {\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1} , u τ a τ = 0 {\displaystyle u^{\tau }a_{\tau }=0} , and F τ σ {\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 v μ {\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 V {\displaystyle V} where ( V , V ) ≠ 0 {\displaystyle (V,V)\neq 0} (that is, not a light-like vector). This is defined for a vector field X {\displaystyle X} along a curve γ ( s ) {\displaystyle \gamma (s)} :

D X d s = D X d s + ( X , D V d s ) V ( V , V ) − ( X , V ) ( V , V ) D V d s − ( V , D V d s ) ( X , V ) ( V , V ) 2 V , {\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 ( V , V ) {\displaystyle (V,V)} is not constant, it can be derived by taking the previous equation, and dividing each V 2 {\displaystyle V^{2}} by ( V , V ) {\displaystyle (V,V)} .

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

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

See also

Notes

  1. ^ 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.
  2. ^ Hawking & Ellis 1973, p. 80
  3. ^ Bargmann, Michel & Telegdi 1959
  4. ^ Misner, Thorne & Wheeler 1973, p. 170
  5. ^ Kocharyan, A. A. (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". 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.