Special Relativity

Seen as a particular case of general relativity

In general relativity, we have a relativistic spacetime M, and in a neighborhood of a point we take coordinates xμ. This coordinates give rise to the coordinate frame xμ, which is not necessarily a vielbein. We can choose a reference frame eμ which can even be inertial. If we take a general coordinate transformation y=F(x), in this new coordinates yμ we have a different coordinate frame yμ which is obtained from xμ by the linear transformation (in every point) dF. But the vielbein eμ is the same (what changes is its expression with respect to the corresponding coordinate frame).
Given two different vielbeins, for every point in spacetime we have a Lorentz transformation relating them

In special relativity, the spacetime M is the Minkowski space, given with a preferred choice of coordinates xμ, in such a way that the metric is ημν, and therefore the coordinate frame xμ is a vielbein. Moreover, it is an inertial frame. Now, if we consider, instead of a general coordinate transformation, a transformation y=F(x) belonging to the Poincare group, the new coordinates yμ satisfy the same: the coordinate frame yμ is an inertial frame. Both coordinate frames are related by an element of the Lorentz group. We can call to the coordinates obtained from this way inertial coordinates.