and
cacceba (T. (5-48)
Depending upon the coordinate labels a and b of h i.e. upon whether h1'
is a scalar, a vector or a tensor, we have corresponding expansion bases, which are
scalar, vector or tensor spherical harmonics. Then, taking the sums over m first in
Eqs. (5-38) and (5-39) using these bases, we simplify the two-index mode-sums to
the one-index mode-sums over only,
h j y [ hact h] (5-49)
K ?h S ry hRt -Z la4I r0h] (5-50)
Ochoab OVch'ab clab c ab
5.4.3 Regularization Parameters
The f-mode singular field ha and its derivative aOchs in Eqs. (5-49) and
(5-50) can be determined fully analytically, and are described by ,.,/,.n:.,/r.:,,n
parameters. The methods to determine the regularization parameters are similar to
that for the scalar field problem. The difference is that hs (or ach b) is treated as
a scalar, a vector or a tensor, depending on the coordinate labels, to comply with
the analysis by R.B--i and Wheeler [33] above, and for each different type we need
develop a different strategy for calculation.
The regularization parameters for hSb can be calculated, for example in the
Lorenz gauge as in Eq. (5-27), in a similar manner to that in Section 4.4. First,
we need to find the functional expressions of hb in the background coordinates
xa = (t, r, 0, 4) using Eqs. (5-27) and Eqs. (4-129)-(4-141) via the transformation
h1 (t, r,0 h (7, XY Z) (% Xa)( X (5-51)
(8A ) ) (5-51)
where A labels THZ coordinates XA = (T, X, Y, Z). Then we categorize the
functions hb (t, r, 0, 0) into the three groups: