Fermi coordinates

{{Short description|Local coordinates that are adapted to a geodesic}} In the mathematical theory of Riemannian geometry, there are two uses of the term '''Fermi coordinates'''. In one use they are local coordinates that are adapted to a geodesic.<ref>{{Cite journal|doi = 10.1063/1.1724316|title = Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry|year = 1963|last1 = Manasse|first1 = F. K.|last2 = Misner|first2 = C. W.|journal = Journal of Mathematical Physics|volume = 4|issue = 6|pages = 735–745|bibcode = 1963JMP.....4..735M}}</ref> In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical.<ref name="z929">{{cite book | last=Lee | first=John M. | title=Introduction to Riemannian Manifolds | publisher=Springer | publication-place=Cham, Switzerland | date=2019-01-02 | isbn=978-3-319-91755-9 | page=136}}</ref><ref>{{Cite journal|arxiv = gr-qc/9402010|doi = 10.1007/BF02108003|title = The physical meaning of Fermi coordinates|year = 1994|last1 = Marzlin|first1 = Karl-Peter|journal = General Relativity and Gravitation|volume = 26|issue = 6|pages = 619–636|bibcode = 1994GReGr..26..619M|s2cid = 17918026}}</ref>

Take a future-directed timelike curve <math>\gamma=\gamma(\tau)</math>, <math>\tau</math> being the proper time along <math>\gamma</math> in the spacetime <math>M</math>. Assume that <math>p=\gamma(0)</math> is the initial point of <math>\gamma</math>. Fermi coordinates adapted to <math>\gamma</math> are constructed this way. Consider an orthonormal basis of <math>TM</math> with <math>e_0</math> parallel to <math>\dot\gamma</math>. Transport the basis <math>\{e_a\}_{a=0,1,2,3}</math>along <math>\gamma(\tau)</math> making use of Fermi–Walker's transport. The basis <math>\{e_a(\tau)\}_{a=0,1,2,3}</math> at each point <math>\gamma(\tau)</math> is still orthonormal with <math>e_0(\tau)</math> parallel to <math>\dot\gamma</math> and is non-rotated (in a precise sense related to the decomposition of Lorentz transformations into pure transformations and rotations) with respect to the initial basis, this is the physical meaning of Fermi–Walker's transport.

Finally construct a coordinate system in an open tube <math>T</math>, a neighbourhood of <math>\gamma</math>, emitting all spacelike geodesics through <math>\gamma(\tau)</math> with initial tangent vector <math>\sum_{i=1}^3 v^i e_i(\tau)</math>, for every <math>\tau</math>. A point <math> q\in T</math> has coordinates <math> \tau(q),v^1(q),v^2(q),v^3(q)</math> where <math>\sum_{i=1}^3 v^i e_i(\tau(q))</math> is the only vector whose associated geodesic reaches <math>q</math> for the value of its parameter <math>s=1</math> and <math>\tau(q)</math> is the only time along <math>\gamma</math> for that this geodesic reaching <math>q</math> exists.

If <math>\gamma</math> itself is a geodesic, then Fermi–Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to <math>\gamma</math>. In this case, using these coordinates in a neighbourhood <math>T</math> of <math>\gamma</math>, we have <math>\Gamma^a_{bc}=0</math>, all Christoffel symbols vanish exactly on <math>\gamma</math>. This property is not valid for Fermi's coordinates however when <math>\gamma</math> is not a geodesic. Such coordinates are called '''Fermi coordinates''' and are named after the Italian physicist Enrico Fermi. The above properties are only valid on the geodesic. The Fermi-coordinates adapted to a null geodesic is provided by Mattias Blau, Denis Frank, and Sebastian Weiss.<ref>{{Cite journal |last1=Blau |first1=Matthias |last2=Frank |first2=Denis |last3=Weiss |first3=Sebastian |date=2006 |title=Fermi coordinates and Penrose limits | journal= Class. Quantum Grav. |arxiv = hep-th/0603109 |doi= 10.1088/0264-9381/23/11/020 |volume=23 |issue=11 |pages=3993–4010 |bibcode=2006CQGra..23.3993B |s2cid=3109453 }}</ref> Notice that, if all Christoffel symbols vanish near <math>p</math>, then the manifold is flat near <math>p</math>.

In the Riemannian case at least, Fermi coordinates can be generalized to an arbitrary submanifold.<ref name="z929"/>

==References== {{Reflist}}

{{DEFAULTSORT:Fermi Coordinates}} Category:Riemannian geometry Category:Coordinate systems in differential geometry