Null Tetrad

The tetrad formalism is a transformed coordinate approach to general relativity (GR) that replaces the coordinate basis by the local basis for the tangent bundle, which is less restricted. It is constituted by a set of four linearly independent vector fields known as a tetrad or vierbein. Usually, specific tetrad basis, which is a set of four independent vector fields, is used to represent a tensor in tetrad formalism. This basis spans the four-dimensional (4D) vector tangent space at each point in spacetime. Therefore, at any point P on the curve, we can present an orthogonal frame of three unit space-like vectors

which are all orthogonal to vα (which is unit tangent vector), a time-like vector and we define

These four vectors (i = 0, 1, 2, 3) are known to form a frame or tetrad at P.

Treating as a 4×4 matrix at P, we can define its inverse (called the dual basis or dual tetrad), which follows

Actually, are four linearly independent vector fields.

We introduce a new matrix gij defined by

which is known as frame metric.

Here, are linearly independent and the global metric, is nonsingular. As a result the matrix gij is nonsingular and hence invertible, i.e., gij exists, which is contravariant frame metric. Note that

Exactly the same way as metric tensor, we can raise and lower the tensor indices with the frame metric gij.

Eq. (10.2) yields the inverse relation as

Note that tetrad vectors determine the linear differential forms

and from this the metric takes the form

For a simple physical interpretation of the frame, we can think as the four velocity of an observer whose world line is C and three space-like vectors (i = 1, 2, 3) are rectangular coordinate vectors (such as usual cartesian basis) at P. For different tetrad, one gets different frame metric.