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.{{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}} In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical.{{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}}{{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}}

Take a future-directed timelike curve $\gamma=\gamma(\tau)$ ,

$\tau$ being the proper time along $\gamma$ in the spacetime $M$ .

Assume that $p=\gamma(0)$ is the initial point of $\gamma$ . Fermi coordinates adapted to $\gamma$ are constructed this way. Consider an orthonormal basis of $TM$ with $e_0$ parallel to $\dot\gamma$ . Transport the basis $\{e_a\}_{a=0,1,2,3}$ along $\gamma(\tau)$ making use of Fermi–Walker's transport. The basis $\{e_a(\tau)\}_{a=0,1,2,3}$ at each point $\gamma(\tau)$ is still orthonormal with $e_0(\tau)$

parallel to $\dot\gamma$ 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 $T$ , a neighbourhood of $\gamma$ , emitting all spacelike geodesics through $\gamma(\tau)$ with initial tangent vector $\sum_{i=1}^3 v^i e_i(\tau)$ , for every $\tau$ . A point $q\in T$ has coordinates $\tau(q),v^1(q),v^2(q),v^3(q)$ where $\sum_{i=1}^3 v^i e_i(\tau(q))$ is the only vector whose associated geodesic reaches $q$ for the value of its parameter $s=1$ and $\tau(q)$ is the only time along $\gamma$ for that this geodesic reaching $q$ exists.

If $\gamma$ itself is a geodesic, then Fermi–Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to $\gamma$ . In this case, using these coordinates in a neighbourhood $T$ of $\gamma$ , we have $\Gamma^a_{bc}=0$ , all Christoffel symbols vanish exactly on $\gamma$ . This property is not valid for Fermi's coordinates however when $\gamma$ 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.{{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

}} Notice that, if all Christoffel symbols vanish near $p$ , then the manifold is flat near $p$ .

In the Riemannian case at least, Fermi coordinates can be generalized to an arbitrary submanifold.

References

Category:Riemannian geometry

Category:Coordinate systems in differential geometry

Fermi coordinates

See also

References