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
γ = γ ( τ )
being the proper time along
is the initial point of
Fermi coordinates adapted to
Consider an orthonormal basis of
Transport the basis
γ ( τ )
making use of Fermi–Walker's transport.
γ ( τ )
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
, emitting all spacelike geodesics through
γ ( τ )
with initial tangent vector
is the only vector whose associated geodesic reaches
for that this geodesic reaching
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
, all Christoffel symbols vanish exactly on
This property is not valid for Fermi's coordinates however when
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.