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.

  • $\begingroup$ 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 $\endgroup$ – mastrok Apr 12 '18 at 15:56
  • $\begingroup$ 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. $\endgroup$ – Valter Moretti Apr 12 '18 at 17:59
  • $\begingroup$ I also think that there are different viewpoints on the terminology in the literature. $\endgroup$ – Valter Moretti Apr 12 '18 at 18:02
  • $\begingroup$ @ValterMoretti nice reference! Did you notice that the author keeps calling Enrico Fermi as M. Fermi? :-) $\endgroup$ – magma Jun 28 '19 at 0:32

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