Fermi coordinates
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. In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical. Take a future-directed timelike curve formula_1, formula_2 being the proper time along formula_3 in the spacetime formula_4. Assume that formula_5 is the initial point of formula_3. Fermi coordinates adapted to formula_3 are constructed this way. Consider an orthonormal basis of formula_8 with formula_9 parallel to formula_10. Transport the  basis formula_11along formula_12 making use of Fermi-Walker's transport. The basis formula_13 at each point formula_12 is still orthonormal with formula_15 parallel to formula_10 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 formula_17, a neighbourhood of formula_3, emitting all spacelike geodesics through formula_12 with initial tangent vector formula_20, for every formula_2. A point formula_22 has coordinates formula_23 where formula_24 is the only  vector whose associated geodesic reaches formula_25 for the value of its parameter formula_26 and formula_27 is the only time along formula_3 for that this geodesic reaching formula_25 exists. If formula_3 itself is a geodesic, then Fermi-Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to formula_3. In this case, using these coordinates in a neighbourhood formula_17 of formula_3, we have formula_34, all Christoffel symbols vanish exactly on formula_3. This property is not valid for Fermi's coordinates however when formula_3 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. Notice that, if all Christoffel symbols vanish near formula_37, then the manifold is flat near formula_37.