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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 {\displaystyle \gamma =\gamma } , τ {\displaystyle \tau } being the proper time along γ {\displaystyle \gamma } in the spacetime M {\displaystyle M}. Assume that p = γ {\displaystyle p=\gamma } 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 }}}.