Chapter 10 - Rotating Black Holes

Summary

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 v^α (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 g_ij defined by

which is known as frame metric.

Here, are linearly independent and the global metric, is nonsingular. As a result the matrix g_ij is nonsingular and hence invertible, i.e., g^ij 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 g_ij.

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.