Vielbein

Definition

Also called vierbein, in the case $n = 4$ , or tetrad or frame field or frame of reference.
Coming from general relativity. It is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on spacetime. In other words, is a section of the orthonormal frame bundle of the spacetime. It is the same as (or, at least, related to) a moving frame. The timelike unit vector field is often denoted by ${\vec{e}}_{0}$ , and the three spacelike unit vector fields by ${\vec{e}}_{1}$ , ${\vec{e}}_{2}$ , and ${\vec{e}}_{3}$ . All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.

Physical interpretation

Frame fields of a Lorentzian manifold always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer. The triad may be thought of as defining the spatial coordinate axes of a local laboratory frame, which is valid very near the observer's worldline.

Inertial frames

Some reference frames or vielbeins are preferable to others. Particularly in vacuum or electrovacuum solutions, the physical experience of inertial observers—those who feel no forces—may be of special interest. The mathematical characterization of an inertial frame is straightforward: the integral curves of the timelike unit vector field must define geodesics. In other words, its acceleration vector must vanish:

\nabla_{{\vec{e}}_{0}} {\vec{e}}_{0} = 0

It is also often desirable to ensure that the spatial triad carried by each observer does not rotate. In this case, the triad can be viewed as being gyrostabilized. The condition for a nonspinning inertial frame is similarly straightforward:

\nabla_{{\vec{e}}_{0}} {\vec{e}}_{j} = 0, j = 1, 2, 3

This condition means that, as we move along the worldline of each observer, their spatial triad is parallel-transported. Nonspinning inertial frames hold a special place in general relativity, as they are as close as possible in a curved Lorentzian manifold to the Lorentz frames used in special relativity (which are special cases of nonspinning inertial frames in the Minkowski vacuum).
In general we can only obtain inertial frames at a point. See Riemann normal coordinates.

More terminology

When using particular coordinates $x^{μ}$ , the frame $e_{μ} = \frac{\partial}{\partial x^{μ}}$ is called the coordinate frame (also known as the coordinate basis or associated basis); any other frame is then called a non-coordinate frame. A holonomic frame is a coordinate frame in some coordinates (though perhaps not the ones being used); this condition is equivalent to requiring that $[e_{μ}, e_{ν}] = 0$ (see canonical form of commuting vector fields). An anholonomic frame is then a frame that cannot be derived from any coordinate chart in its region of definition. Using a non-coordinate frame suited to a specific problem is sometimes called the method of moving frames.

Particular case: special relativity

See special relativity#Seen as a particular case of general relativity theory