# Difference between Fermi and Riemann normal coordinates

1. What is the difference between Fermi normal coordinates and Riemann normal coordinates?

2. Which one of them is related to the vanishing of the Christoffel symbols?

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 $T_pM$ with $e_0$ parallel to $\dot{\gamma}$.

Transport the basis $\{e_a\}_{a=0,1,2,3}$ along $\gamma= \gamma(\tau)$ 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 $\dot{\gamma}(\tau)$ 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 a 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 $\sum_{i=1}^3 v^i(q) e_i(\tau(q))$ 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 $\Gamma_{ab}^c =0$ exactly on $\gamma$. This property is not valid for Fermi's coordinates however when $\gamma$ is not a geodesic.

A modern reference on the subject is http://arxiv.org/abs/gr-qc/9402010 where also rotating coordinates are discussed.

• I am confused by answer. Because wiki said that Fermi coordinate is adapted to timelike geodesics $\gamma$ and you said that when $\gamma$ is a geodesic then it becomes the Riemann coordinate. Then they are the same thing by your answer and wikipedia's definition. I think Fermi coordinate is a generalization of Riemann coordinate such that the Christoffel symbol vanishes not only at a point but along the whole geodesic. en.wikipedia.org/wiki/Fermi_coordinates – mastrok Apr 12 '18 at 15:56
• In my answer I refer to Riemannian coordinates adapted to $\gamma$ which means that the Christoffel symbols vanishes along the curve (it must be a geodesics). This type of coordinates is known since long time before relativity and Fermi's work! Fermi coordinates instead are strictly related to relativity and concern the transport of a non-rotating frame along a timelike curve, that is another idea with some overlap with that of Riemannian coordinates. My impression on the wikipedia page is that it is written into a bit sloppy way. – Valter Moretti Apr 12 '18 at 17:59
• I also think that there are different viewpoints on the terminology in the literature. – Valter Moretti Apr 12 '18 at 18:02
• @ValterMoretti nice reference! Did you notice that the author keeps calling Enrico Fermi as M. Fermi? :-) – magma Jun 28 at 0:32