Fermi coordinates explained
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.[1] In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical.[2] [3]
Take a future-directed timelike curve
,
being the proper time along
in the spacetime
. Assume that
is the initial point of
. Fermi coordinates adapted to
are constructed this way. Consider an orthonormal basis of
with
parallel to
. Transport the basis
along
making use of
Fermi–Walker's transport. The basis
at each point
is still orthonormal with
parallel to
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
, a neighbourhood of
, emitting all spacelike geodesics through
with initial tangent vector
, for every
. A point
has coordinates
\tau(q),v1(q),v2(q),v3(q)
where
is the only vector whose associated geodesic reaches
for the value of its parameter
and
is the only time along
for that this geodesic reaching
exists.
If
itself is a geodesic, then Fermi–Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to
. In this case, using these coordinates in a neighbourhood
of
, we have
, all
Christoffel symbols vanish exactly on
. This property is not valid for Fermi's coordinates however when
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.
[4] Notice that, if all Christoffel symbols vanish near
, then the manifold is flat near
.
In the Riemannian case at least, Fermi coordinates can be generalized to an arbitrary submanifold.[2]
See also
Notes and References
- 10.1063/1.1724316. Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry. 1963. Manasse. F. K.. Misner. C. W.. Journal of Mathematical Physics. 4. 6. 735–745. 1963JMP.....4..735M.
- Book: Lee, John M. . Introduction to Riemannian Manifolds . Springer . Cham, Switzerland . 2019-01-02 . 978-3-319-91755-9 . 136.
- gr-qc/9402010. 10.1007/BF02108003. The physical meaning of Fermi coordinates. 1994. Marzlin. Karl-Peter. General Relativity and Gravitation. 26. 6. 619–636. 1994GReGr..26..619M. 17918026.
- Blau . Matthias . Frank . Denis . Weiss . Sebastian . 2006 . Fermi coordinates and Penrose limits . Class. Quantum Grav. . hep-th/0603109 . 10.1088/0264-9381/23/11/020 . 23 . 11 . 3993–4010. 2006CQGra..23.3993B . 3109453 .
