Fermi coordinates

Short description: Local coordinates that are adapted to a geodesic

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 [math]\displaystyle{ \gamma=\gamma(\tau) }[/math], [math]\displaystyle{ \tau }[/math] being the proper time along [math]\displaystyle{ \gamma }[/math] in the spacetime [math]\displaystyle{ M }[/math]. Assume that [math]\displaystyle{ p=\gamma(0) }[/math] is the initial point of [math]\displaystyle{ \gamma }[/math].

Fermi coordinates adapted to [math]\displaystyle{ \gamma }[/math] are constructed this way.

Consider an orthonormal basis of [math]\displaystyle{ TM }[/math] with [math]\displaystyle{ e_0 }[/math] parallel to [math]\displaystyle{ \dot\gamma }[/math].

Transport the basis [math]\displaystyle{ \{e_a\}_{a=0,1,2,3} }[/math]along [math]\displaystyle{ \gamma(\tau) }[/math] making use of Fermi-Walker's transport. The basis [math]\displaystyle{ \{e_a(\tau)\}_{a=0,1,2,3} }[/math] at each point [math]\displaystyle{ \gamma(\tau) }[/math] is still orthonormal with [math]\displaystyle{ e_0(\tau) }[/math] parallel to [math]\displaystyle{ \dot\gamma }[/math] 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 [math]\displaystyle{ T }[/math], a neighbourhood of [math]\displaystyle{ \gamma }[/math], emitting all spacelike geodesics through [math]\displaystyle{ \gamma(\tau) }[/math] with initial tangent vector [math]\displaystyle{ \sum_{i=1}^3 v^i e_i(\tau) }[/math], for every [math]\displaystyle{ \tau }[/math].

A point [math]\displaystyle{ q\in T }[/math] has coordinates [math]\displaystyle{ \tau(q),v^1(q),v^2(q),v^3(q) }[/math] where [math]\displaystyle{ \sum_{i=1}^3 v^i e_i(\tau(q)) }[/math] is the only vector whose associated geodesic reaches [math]\displaystyle{ q }[/math] for the value of its parameter [math]\displaystyle{ s=1 }[/math] and [math]\displaystyle{ \tau(q) }[/math] is the only time along [math]\displaystyle{ \gamma }[/math] for that this geodesic reaching [math]\displaystyle{ q }[/math] exists.

If [math]\displaystyle{ \gamma }[/math] itself is a geodesic, then Fermi-Walker's transport becomes the standard parallel transport and Fermi's coordinates become standard Riemannian coordinates adapted to [math]\displaystyle{ \gamma }[/math]. In this case, using these coordinates in a neighbourhood [math]\displaystyle{ T }[/math] of [math]\displaystyle{ \gamma }[/math], we have [math]\displaystyle{ \Gamma^a_{bc}=0 }[/math], all Christoffel symbols vanish exactly on [math]\displaystyle{ \gamma }[/math]. This property is not valid for Fermi's coordinates however when [math]\displaystyle{ \gamma }[/math] 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 [math]\displaystyle{ p }[/math], then the manifold is flat near [math]\displaystyle{ p }[/math].

See also

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