While the metric is invariant under a Lorentz transformation, the tetrad vectors are
not. In the null tetrad formalism, a Lorentz transformation, which in general is described
by six parameters, is broken up into three classes of tetrad rotations. We will consider
only a tetrad rotation of Type III herel In the language of our Minkowski space example,
this amounts to a boost in the z t plane and a rotation in the x y7 plane. Under such a
transformation
za via
(1 v2 1/2 '
to Vza
(1 v2 4
x" = cos Oxa sin Of ,
y"a = sin Oxa + cos Of ~,
which translates to
(2-16)
~ a iB a
where r = J(1 v)/(1 ) The two transformations can be combined into one using
(2 = reie. Then Equation 2-16 may be summarized by
(2-17)
A quantity, X, is then said to be of type {p, q} if, under Equation 2-17, X ("(97.
Alternatively [9], we may ;?i that X possesses spin weight s = (p q)/2 and boost
weight b = (p + q)/2. The p and q values for the tetrad vectors can be read off from
Equation 2-17. They allow one to determine the spin and boost weights of the spin
1 Descriptions of the other types of tetrad rotation can be found in [30] or [29].