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 '19 at 0:32

Riemann normal coordinates (or just normal coordinates for short) are the "granddaddy" of the different normal coordinate constructions: The Riemann normal cordinate system is a system of local coordinates that is errected around a single point of the manifold (in General Relativity, with its space-time manifold, a point is usually called an event).

As coordinate lines, one takes all the geodesics (time-like, null, and space-like alike ;-) ) that intersect this point. The coordinate transformation from the Riemann coordinates to the (arbitrary) background coordinates then follows from a Taylor-series solution ansatz of the corresponding geodesic equation. Its Taylor coefficients can be written in terms of partial derivatives of the Christoffel symbols. One then demands that -- as seen from the Riemann normal coordinate system -- theese geodesics all look like straight lines, i.e. that they be (albeit unphysical) local inertial coordinates. If one now take the above coordinate transformation in Riemann coordinates, one finds that for this, all quadratic and higher Taylor coefficients must vanish in Riemann coordinates. The 1st-order Taylor coefficient is given by the Christoffel symbols, the 2nd-order one by its 1st partial derivative, and so on. This vanishing of the Taylor coefficients then yields the coordinate conditions of the Riemann normal coord. system: The Christoffel symbols and all their higher (symmetrised) derivatives vanish.

Fermi normal coordinates are a slight extension of the construction of Riemann normal coordinates in order to have a local inertial coord. system that is able to describe also physical observers, who always follow time-like world-lines (i.e., they will always move through space-time along their local time-like directions even when they are stationary).

With Fermi coordinates, one expands not only about a single point, but along a whole time-like curve, which is then the world-line of some observer. This is the crucial difference between Riemann and Fermi normal coordinates. In Fermi coordinates, one thus performs a spacial-only (i.e. 3-dimensional) Riemann coordinate expansion, outwards from and orthogonal to the central world-line, i.e. only along the space-like geodesics that intersect the world-line orthogonally (they are orthogonal to the four-velocity, which is the tangent of the central world-line).

So, since Fermi coordinates are based on (a continuous family of spacial-only) Riemann coordinates, they both "have" vansihing Christoffel symbols. More precisely: in Riemann coordinates, the Christoffel symbols vanish, along with all their first and higher (symmetrised) partial derivatives.

In Fermi coordinates, since they involve spacial-only Riemann expansions, only the spacial (lower indices) Christoffel symbols vanish. Correspondingly, at the second and higher orders of the expansion, it is also their totally symmetric first and higher spacial partial derivatives, that are zero.

Now, wether the non-spacial part of the Christoffel symbols also vanishes, depends on wether the central world-line of the construction is also a geodesic, or not.

If it is a geodesic, then also the non-spacial Christoffels vanish (in this case its tangent, i.e. the four-velocity, is parallel transported and the observer following that geodesic world-line is inertial, or freely falling). Conversely, if it is not a geodesic (which means that the observer following the world-line is non-inertial, i.e. he is accelerated and/or his spacial basis vectors are rotating), then the non-spacial Christoffels will not vanish, since they are related to his four-acceleration and rotation.