Fermi coordinates

Coordinates in which the components of the metric tensor of a Riemannian space, computed along the points of some curve, are the same as the components of the metric tensor of a Euclidean space in Cartesian coordinates. The concept of Fermi coordinates can be generalized pseudo-Riemannian spaces, for which they were also first introduced by E. Fermi (1922) [1]. He showed that in a neighbourhood of a sufficiently small segment of a sufficiently regular time-like curve on a sufficiently regular pseudo-Riemannian manifold of Lorentzian signature there actually are Fermi coordinates.

References

[1]	E. Fermi, "Sopra i fenomeni che avvengono in vinicinanza di una linea ovaria" Atti R. Accad. Lincei Rend. Cl. Sci. Fis. Mat. Nat. , 31 (1922) pp. 21–51
[2]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

Comments

Using properties of the exponential mapping restricted to the normal bundle of a curve of fixed causal character and without self-intersections, the existence of Fermi coordinates along any compact piece of the curve can be shown in an arbitrary pseudo-Riemannian space.