habilitation`a diriger les recherches - LUTH - Observatoire de Paris

386 Kerr solution in Dirac gauge (Vasset et al. 2009)
the h rr spherical component. Let us note that these relations are more tractable when using a scalar
spherical harmonics decomposition (introduced in Sec. 13.4.2) for all fields. Indeed, an angular Laplace
operator acting on a field reduces then to a simple algebraic operation on every spherical harmonics
component. Inverse relations can also be computed to retrieve the classical components of h ij from
spherical harmonics quantities.
We now derive the main variables related to our study: with the divergence-free decomposition
h ij = D i V j + D j V i + h ij
T
, and Dih ij
T = 0, a choice for three quantities defined from hij and verifying:
can be expressed as the following scalar fields (see [340]):
h ij
T
C = ∂(h − hrr )
∂r
= 0 ⇒ A = B = C = 0, (13.47)
B = ∂W
∂r
− 3hrr
r
A = ∂X
∂r
µ
− , (13.48)
r
∆θϕW η h − hrr
− − + , (13.49)
2r
r 4r
h ∂W W
+ − 2∆θϕ + (13.50)
r ∂r r
These quantities can also be decomposed onto a scalar spherical harmonics basis. The equivalence
in (13.47) is achieved up to boundary conditions.
To show how the quantities A, B and C behave with respect to the Laplace operator, we shall
suppose in the following that the tensor h ij is the solution of a Poisson equation of the type ∆h ij = S ij .
We can deduce a scalar elliptic system verified by A, B and C as:
∆A = AS
∆B − C
= BS
(13.52)
2r2 (13.51)
∆C + 2C 8∆θϕB
+
r2 r2 = CS, (13.53)
Where AS, BS and CS are the corresponding quantities associated with the source Sij . A simple
way of decoupling the last two elliptic equations is to define the variables ˜ B =
l,m ˜ BlmYlm and
˜C =
l,m ˜ ClmYlm with:
˜B lm = B lm + Clm
Thus, we can write an equivalent system for (13.51,13.52,13.53) as:
,
2(l + 1)
(13.54)
˜C lm = C lm − 4lB lm . (13.55)
∆A = AS, (13.56)
˜∆ ˜ B = ˜ BS, (13.57)
∆ ∗ ˜ C = ˜ CS, (13.58)
With the following elliptic operators defined for each spherical harmonic index l: