Fermi normal coordinates

Summary

Fermi normal coordinates are local coordinates around a whole timelike curve in spacetime, adapted to an observer moving along it. The time coordinate is the proper time of the observer. The space coordinates are measured along geodesics orthogonal to the curve, in the directions of a non-rotating frame. Along a geodesic they make the metric equal to the Minkowski metric, with vanishing first derivatives. So they realise the equivalence principle along a whole world line, not only at one event. The corrections to the flat metric at second order are given by the curvature, which describes tidal forces.