2.6.1 Einstein's New Clothes
The main idea behind our framework is to reorganize the tensors of interest into their
tetrad components. The metric perturbation, for example, has the decomposition
hub = ~'.. "' A b un a~~b + 2hlul(anb) + 2hmmm(,m b)
21hmnja~b) 2htlm76iamb) 2humll~,mb) 2hu-mljamlb) (2-6(3)
+ mmmemb mmmammb,
so that, for example, hit = hub a b. In order for this to be valid within the GHP
formalism, each component of Equation 2-63 must have a well-defined spin- and
boost- weight. Because the background metric (Equation 2-3) is invariant under
a spin-boost (Equation 2-17) it has type {0, 0}, which must also be the type of the
metric perturbation, hab. Therefore the type of the individual components of the metric
perturbation are determined by their tetrad indices:
hit : {2, 2} a {2 2
him : {2, 0} hm : {-2, 0}
him : {0, 2} hm : {0, -2} (2-64)
kmm : {2, -2} kmm : {-2, 2}
hi, : {0,0 kmm,,: {0, 0}.
All of the vectors and tensors we will concern ourselves with can be treated in this way.
It is worthwhile to stop here and take a look at what Equation 2-63 really means.
Comparing with our treatment of Schwarzschild (Equation 1-4), we note that the scalar
parts of the metric are "mixed up" in hu, hin and h,,, all of which have spin weight zero
but differ in boost weight. Similarly, the vector parts are given by him, hm and their
complex conjugates and likewise the tensor pieces are given here by hmm, hmm and hmm.
However, these identifications are completely independent of the background spacetime.
Thus, in a certain sense, Equation 2-63 provides a generalization of the RW mode