possibility that m" and m" are in fact not the null vectors associated with the background
spacetime, but rather just two (complex) null vectors tangent to S. In that case 1" and n"
will then be identified with the two null directions orthogonal to S. We will use the same
notation regardless of whether the tetrad is globally defined or just in some neighborhood
of S and the application at hand will dictate the appropriate interpretation. The metric's
other role as the projector into S can be realized by simply raising one index
,b --mb ~b (2
For example consider some vector, v", defined in the spacetime:
v" = I 2a" + val"a vmm" v~mm"
The restriction of v" to S is simply given by the projection
b,', vm ~b
which generalizes to n-indexed tensors by projecting each index individually. We can then
carry out covariant differentiation within S by simply taking the full covariant derivative
and projecting back into S with aqb. The final object we introduce is the Levi-Civita
symbol on S, which, in tetrad language takes the following form:
Eab Eluab = i@memb memb). (33)
These are all the tools necessary for what follows.
We begin by considering the projection of vectors defined in the spacetime onto S. To
identify the odd and even parity pieces we start by decomposing a general vector on S
~a = Gab~b even Eab~b odd
-ma(8(eve i'Codd a evn od