Normal coordinates

At any spacetime point, normal coordinates make the metric equal to the Minkowski metric and all Christoffel symbols vanish at that point.

Fermi coordinates transport a local frame along a timelike worldline. Along a timelike geodesic, a parallel-transported orthonormal frame defines Fermi normal coordinates in which the metric tensor is Minkowskian and the Christoffel symbols vanish on the worldline. Curvature first appears in terms quadratic in the spatial distance from it, so finite laboratories retain tidal forces.

A Lorentz frame at a spacetime event is an orthonormal tangent basis with metric $\mathrm{diag}(-1,1,1,1)$. Future-oriented frames are related by proper orthochronous Lorentz transformations. Any such basis can be realized at the event by normal coordinates.