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 $\_$along $\gamma(\tau)$ making use of Fermi-Walker's transport.
The basis $\_$ 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 $\sum_^3 v^i e_i(\tau)$, for every $\tau$.

A point $q\in T$ has coordinates $\tau(q),v^1(q),v^2(q),v^3(q)$ where $\sum_^3 v^i e_i(\tau(q))$ 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 $\Gamma^a_=0$, 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 $p$.

See also

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

References

{{DEFAULTSORT:Fermi Coordinates
Riemannian geometry
Coordinate systems in differential geometry