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 wordline, even not geodesical. [2]

Take[3] 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. For example, if all Christoffel symbols vanish near ${\displaystyle p}$, then the manifold is flat near ${\displaystyle p}$.

See also

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

References

1. Manasse and Misner , Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry. Journal of Mathematical Physics 4:6, 1963.
2. K.-P. Marzlin, "On the physical meaning of Fermi coordinates",.
3. V.Moretti, discussion on