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 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 by
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:
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, Sphere derivatives of spin-weighted functions are given by the action of the operators
and its conjugate operator
, defined by [28
]
![]() |
most often taken as the unit metric sphere by
![]() |
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
![]() This work is licensed under a Creative Commons License. E-mail us: |