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 
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
With the tetrad thus defined, the contravariant metric of the spacetime is given bymetric coefficients , , , and , the metric can be written as:
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  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: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
Sphere derivatives of spin-weighted functions are given by the action of the operators and its conjugate operator , defined by 
most often taken as the unit metric sphere by
Living Rev. Relativity 15, (2012), 1
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