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. 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 γ = γ ( τ ) {\displaystyle \gamma =\gamma (\tau )} , τ {\displaystyle \tau } being the proper time along γ {\displaystyle \gamma } in the spacetime M {\displaystyle M} . Assume that p = γ ( 0 ) {\displaystyle p=\gamma (0)} is the initial point of γ {\displaystyle \gamma } . Fermi coordinates adapted to γ {\displaystyle \gamma } are constructed this way. Consider an orthonormal basis of T M {\displaystyle TM} with e 0 {\displaystyle e_{0}} parallel to γ ˙ {\displaystyle {\dot {\gamma }}} . Transport the basis { e a } a = 0 , 1 , 2 , 3 {\displaystyle \{e_{a}\}_{a=0,1,2,3}} along γ ( τ ) {\displaystyle \gamma (\tau )} making use of Fermi–Walker's transport. The basis { e a ( τ ) } a = 0 , 1 , 2 , 3 {\displaystyle \{e_{a}(\tau )\}_{a=0,1,2,3}} at each point γ ( τ ) {\displaystyle \gamma (\tau )} is still orthonormal with e 0 ( τ ) {\displaystyle e_{0}(\tau )} parallel to γ ˙ {\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 T {\displaystyle T} , a neighbourhood of γ {\displaystyle \gamma } , emitting all spacelike geodesics through γ ( τ ) {\displaystyle \gamma (\tau )} with initial tangent vector ∑ i = 1 3 v i e i ( τ ) {\displaystyle \sum _{i=1}^{3}v^{i}e_{i}(\tau )} , for every τ {\displaystyle \tau } . A point q ∈ T {\displaystyle q\in T} has coordinates τ ( q ) , v 1 ( q ) , v 2 ( q ) , v 3 ( q ) {\displaystyle \tau (q),v^{1}(q),v^{2}(q),v^{3}(q)} where ∑ i = 1 3 v i e i ( τ ( q ) ) {\displaystyle \sum _{i=1}^{3}v^{i}e_{i}(\tau (q))} is the only vector whose associated geodesic reaches q {\displaystyle q} for the value of its parameter s = 1 {\displaystyle s=1} and τ ( q ) {\displaystyle \tau (q)} is the only time along γ {\displaystyle \gamma } for that this geodesic reaching q {\displaystyle q} exists. If γ {\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 γ {\displaystyle \gamma } . In this case, using these coordinates in a neighbourhood T {\displaystyle T} of γ {\displaystyle \gamma } , we have Γ b c a = 0 {\displaystyle \Gamma _{bc}^{a}=0} , all Christoffel symbols vanish exactly on γ {\displaystyle \gamma } . This property is not valid for Fermi's coordinates however when γ {\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. 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 p {\displaystyle p} , then the manifold is flat near p {\displaystyle p} . In the Riemannian case at least, Fermi coordinates can be generalized to an arbitrary submanifold.