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 $\Gamma _{bc}^{a}=0$, 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 $p$.

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

See also

• Proper reference frame (flat spacetime)#Proper coordinates or Fermi coordinates
• Geodesic normal coordinates
• Fermi–Walker transport
• Christoffel symbols
• Isothermal coordinates