defines a null tetrad. It is important to note that there is some ambiguity implicit in the
above assignment, e.g. we can swap the roles of z' and x" (or y") in the above definitions
without changing the character (real or complex) of the null vectors or modifying their
inner products. We will return to this issue later in this section.
For simplicity, we introduce the following notation for our tetrad (borrowed from
C'I 1.in b I-iekhar [29]):
e*, = (1", na, m", ma),
where the tetrad index (i) = {1, 2, 3, 4} = {1, n, m, m}. In a further attempt to avoid
confusion we'll take spacetime indices from the beginning of the alphabet (a, b, c...) and
tetrad indices from later in the alphabet (i, j, k...). Just as the vector index can be raised
or lowered with the spacetime metric
e" gab = 6i~b and e(ij,gab =6t),
we may introduce a similar object for raising and lowering tetrad indices
For a properly normalized (Equation 2-1) null tetrad
0 1 0 0
0 0 0 -1
0 0 -1 0.
It then follows that we can express our spacetime metric as
where 1(,nb) a ~l~ b + bna).