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.¹ In a second, more general one, they are local coordinates that are adapted to any world line, even not geodesical.²³

Take a future-directed timelike curve ${\displaystyle \gamma =\gamma (\tau )}$, ${\displaystyle \tau }$ being the proper time along ${\displaystyle \gamma }$ in the spacetime ${\displaystyle M}$. Assume that ${\displaystyle p=\gamma (0)}$ is the initial point of ${\displaystyle \gamma }$. Fermi coordinates adapted to ${\displaystyle \gamma }$ are constructed this way. Consider an orthonormal basis of ${\displaystyle TM}$ with ${\displaystyle e_{0}}$ parallel to ${\displaystyle {\dot {\gamma }}}$. Transport the basis ${\displaystyle \{e_{a}\}_{a=0,1,2,3}}$along ${\displaystyle \gamma (\tau )}$ making use of Fermi–Walker's transport. The basis ${\displaystyle \{e_{a}(\tau )\}_{a=0,1,2,3}}$ at each point ${\displaystyle \gamma (\tau )}$ is still orthonormal with ${\displaystyle e_{0}(\tau )}$ parallel to ${\displaystyle {\dot {\gamma }}}$ 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 ${\displaystyle T}$, a neighbourhood of ${\displaystyle \gamma }$, emitting all spacelike geodesics through ${\displaystyle \gamma (\tau )}$ with initial tangent vector ${\displaystyle \sum _{i=1}^{3}v^{i}e_{i}(\tau )}$, for every ${\displaystyle \tau }$. A point ${\displaystyle q\in T}$ has coordinates ${\displaystyle \tau (q),v^{1}(q),v^{2}(q),v^{3}(q)}$ where ${\displaystyle \sum _{i=1}^{3}v^{i}e_{i}(\tau (q))}$ is the only vector whose associated geodesic reaches ${\displaystyle q}$ for the value of its parameter ${\displaystyle s=1}$ and ${\displaystyle \tau (q)}$ is the only time along ${\displaystyle \gamma }$ for that this geodesic reaching ${\displaystyle q}$ exists.

If ${\displaystyle \gamma }$ itself is a geodesic, then Fermi–Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to ${\displaystyle \gamma }$. In this case, using these coordinates in a neighbourhood ${\displaystyle T}$ of ${\displaystyle \gamma }$, we have ${\displaystyle \Gamma _{bc}^{a}=0}$, all Christoffel symbols vanish exactly on ${\displaystyle \gamma }$. This property is not valid for Fermi's coordinates however when ${\displaystyle \gamma }$ 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.⁴ Notice that, if all Christoffel symbols vanish near ${\displaystyle p}$, then the manifold is flat near ${\displaystyle p}$.

In the Riemannian case at least, Fermi coordinates can be generalized to an arbitrary submanifold.²