Let us begin by considering a small perturbation, hab, of the Schwarzschild geometry.
Thus our spacetime metric is
9ab = gb a Lb,
(11)
where
gfbd ads b I _1)t- 2Ii _'d ~ 28 in2 tii' 2 _2)
is the Schwarzschild metric in Schwarzschild coordinates. Putting Equation 1-1 into the
Einstein equations and keeping only terms linear in hub leads us to the perturbed Einstein
equations :
1 1 1
~ab ~c c b ~a b cc c ~(a b)c + ab ~c c dd ~c d cd) = 0,
2 2 2
(1-3)
where V, is the derivative operator compatible with the background geometry 1-2 and
the indices are raised and lowered with the background metric. Henceforth we will refer to
Sab aS the Einstein tensor, and the expression to the right of it as the Einstein equations
(dropping the qualifier "perturbed" for brevity).
Essentially every perturbative analysis of the Schwarzschild spacetime makes
extensive use of its spherical symmetry. The first step in this direction is to decompose
the components of the metric perturbation into scalar, vector and tensor harmonics.
Heuristically, we write
'U2
respectively and the subscripts
81 S2 'Ul
hab 2 3 'U2
vl v2 t+4
vl v2 t
where s, v and t stand for scalar, vector and tensor,
distinguish between the various scalars and vectors.
Consider the metric of the two-sphere:
_15)
yABd A XB d2 Sin2 d2