### 2.2 Bondi coordinates and null tetrad

Proceeding with our examination of the properties of , we introduce, in the neighborhood of , what is known as a Bondi coordinate system: . In this system, , the Bondi time, labels the null surfaces, is the affine parameter along the null geodesics of the constant surfaces and , the complex stereographic coordinate labeling the null geodesics of . To reach , we simply let , so that has coordinates . The time coordinate , the topologically portion of , labels ‘cuts’ of . The stereographic coordinate accounts for the topological generators of the portion of , i.e., the null generators of . The choice of a Bondi coordinate system is not unique, there being a variety of Bondi coordinate systems to choose from. The coordinate transformations between any two, known as Bondi–Metzner–Sachs (BMS) transformations or as the BMS group, are discussed later in this section.

Associated with the Bondi coordinates is a (Bondi) null tetrad system, (). The first tetrad vector is the tangent to the geodesics of the constant null surfaces given by [60]

The second null vector is normalized so that:
In Bondi coordinates, we have [60]
for functions and to be determined, and . At , is tangent to the null generators of .

The tetrad is completed with the choice of a complex null vector , which is itself orthogonal to both and , initially tangent to the constant cuts at and parallel propagated inward on the null geodesics. It is normalized by

Once more, in coordinates, we have [60]
for some and to be determined. All other scalar products in the tetrad are to vanish.

With the tetrad thus defined, the contravariant metric of the spacetime is given by

In terms of the metric coefficients , , , and , the metric can be written as:
We thus have the spacetime metric in terms of the metric coefficients.

There remains the issue of both coordinate and tetrad freedom, i.e., local Lorentz transformations. Most of the time we work in one arbitrary but fixed Bondi coordinate system, though for special situations more general coordinate systems are used. The more general transformations are given, essentially, by choosing an arbitrary slicing of , written as with labeling the slices. To keep conventional coordinate conditions unchanged requires a rescaling of . It is also useful to be able to shift the origin of by with arbitrary .

The tetrad freedom of null rotations around , performed in the neighborhood of , will later play a major role. For an arbitrary function on , the null rotation about the vector  [60] is given by

Eventually, by the appropriate choice of the function , the new null vector, , can be made into the tangent vector of an asymptotically shear-free NGC.

A second type of tetrad transformation is the rotation in the tangent plane, which keeps and fixed:

This latter transformation provides motivation for the concept of spin weight. A quantity is said to have spin-weight if, under the transformation, Eq. (2.13), it transforms as

An example would be to take a vector on , say , and form the spin-weight-one quantity, .

Comment: For later use we note that has spin weight, .

For each , spin- functions can be expanded in a complete basis set, the spin- harmonics, or spin- tensor harmonics, (cf. Appendix C).

A third tetrad transformation, the boosts, are given by

These transformations induce the idea of conformal weight, an idea similar to spin weight. Under a boost transformation, a quantity, , will have conformal weight if

Sphere derivatives of spin-weighted functions are given by the action of the operators and its conjugate operator , defined by [28]

where the function is the conformal factor defining the conformal sphere metric,

most often taken as the unit metric sphere by