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 $$\gamma=\gamma(\tau)$$, $$\tau$$ 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. Notice that, if all Christoffel symbols vanish near $$p$$, then the manifold is flat near $$p$$.