Fermi–Walker transport#Fermi.E2.80.93Walker 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 $(-+++)$ sign convention, this is defined for a vector field X along a curve $\backslash gamma(s)$:

:$\backslash frac\{D\_F\; X\}\{d\; s\}=\backslash frac\{DX\}\{d\; s\}\; -\; \backslash left(X,\backslash frac\{DV\}\{d\; s\}\backslash right)\; V\; +\; (X,V)\backslash frac\{DV\}\{d\; s\},$

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

:$\backslash frac\{D\_F\; X\}\{d\; s\}=0,$

then the vector field {{math|X}} is Fermi–Walker transported along the curve.{{harvnb|Hawking|Ellis|1973|p=80}} 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{{harvnb|Bargmann|Michel|Telegdi|1959}} for spin precession of electron in an external electromagnetic field can be written as follows:

:$\backslash frac\{D\_Fa^\{\backslash tau\}\}\{ds\}\; =\; 2\backslash mu\; (F^\{\backslash tau\; \backslash lambda\}\; -\; u^\{\backslash tau\}\; u\_\{\backslash sigma\}\; F^\{\backslash sigma\; \backslash lambda\})a\_\{\backslash lambda\},$

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

{{main|Proper reference frame (flat spacetime)#Comoving tetrads}}

A coordinate system co-moving with a particle can be defined. If we take the unit vector $v^\{\backslash mu\}$ 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.{{harvnb|Misner|Thorne|Wheeler|1973|p=170}}

Fermi–Walker differentiation can be extended for any $V$ where $(V,V)\backslash ne0$ (that is, not a light-like vector). This is defined for a vector field $X$ along a curve $\backslash gamma(s)$:

:$\backslash frac\{\backslash mathcal\; D\; X\}\{d\; s\}=\backslash frac\{D\; X\}\{d\; s\}$

+ \left(X,\frac{DV}{d s}\right)\frac{V}{(V,V)}

- \frac{(X,V)}{(V,V)}\frac{DV}{d s}

- \left(V,\frac{DV}{d s}\right)\frac{(X,V)}{(V,V)^2} V ,{{cite arXiv|eprint=astro-ph/0411595 |title=Geometry of Dynamical Systems|date=2004 |last1=Kocharyan |first1=A. A. }}

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:

$\backslash left(V,\backslash frac\{DV\}\{d\; s\}\backslash right)=\backslash frac\{1\}\{2\}\backslash frac\{d\}\{ds\}(V,V)=0\backslash \; ,$ and $\backslash frac\{\backslash mathcal\{D\}\; X\}\{d\; s\}=\backslash frac\{D\_F\; X\}\{d\; s\}\; .$

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

{{Reflist}}

• {{cite journal|first1=V.|last1=Bargmann|authorlink1=Valentine Bargmann|first2=L.|last2=Michel|first3=V. L.|last3=Telegdi|title=Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field|journal=Physical Review Letters|volume=2|issue=10|page=435|year=1959|doi=10.1103/PhysRevLett.2.435 |bibcode=1959PhRvL...2..435B}}.
• {{cite book|last1=Landau|first1=L.D.|authorlink1=Lev Landau|last2=Lifshitz|first2=E.M.|authorlink2=Evgeny Lifshitz|title=The Classical Theory of Fields|series=Course of Theoretical Physics|volume=2|edition=4th|publisher=Butterworth–Heinemann|isbn=0-7506-2768-9|year=2002|orig-year=1939}}
• {{cite book|first1=Charles W.|last1=Misner|author-link=Charles W. Misner|first2=Kip S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler|author3-link=John A. Wheeler|title=Gravitation|publisher= W. H. Freeman|date=1973|isbn=0-7167-0344-0|title-link=Gravitation (book)}}
• {{cite book |first1=Stephen W.|last1=Hawking|author-link=Stephen Hawking|first2=George F.R.|last2=Ellis|author2-link=George Francis Rayner Ellis|title=The Large Scale Structure of Space-time|publisher= Cambridge University Press|date=1973|isbn=0-521-09906-4}}
• {{cite arXiv |eprint=astro-ph/0411595 |last1=Kocharyan |first1=A. A. |title=Geometry of Dynamical Systems |date=2004 }}

{{Physics-footer}}

{{DEFAULTSORT:Fermi-Walker transport}}

Category:Mathematical methods in general relativity