4.7 Observables from Localized Sources
69
Just as ∂ ζ γ
μ , this is a vector under MDs, i.e. in the passive and active versions we
have
∂ ˆ
A ˜
γ
μ
=
∂ ˜
x
μ
∂x ν (γ) ∂ ˆ
A γ
ν
,
δ ξ ∂ ˆ
A γ
μ
= [∂ ν ξ
μ
](γ) ∂ ˆ
A γ
ν
+ O(ξ
2
) ,
(4.7.2)
respectively. Therefore, reminding that ∂ A := S
ˆ
A
A ∂ ˆ
A , the following quantity
J
A
B := k
A
a e
a
μ ∂ B γ
μ
,
(4.7.3)
is invariant under MDs. It is also invariant under Sachs shifts, thanks to Eq. (4.4.12),
which here translates into k μ ∂ ˆ
A γ
μ
= 0, so it is consistently independent of n A
J
A
B ≡ n
A
i e
i
μ ∂ B γ
μ
.
(4.7.4)
Finally, it transforms linearly under LLTs, which is non-trivial because the angular
derivative ∂ ˆ
A mixes with ∂ ζ and ∂ ˆ
ω according to Eq. (4.4.16). In this case, however,
the non-angular derivatives cancel out because k
A
a e
a
μ ∂ ζ γ
μ
∼ k
A
a k
a
≡ 0 and ∂ ˆ
ω γ
μ
≡ 0.
Thus, in the PLD-compensated case we have
˜
J AB ( ˜
ζ, ˜
ϑ) = R
C
A (ϑ) R
D
B (ϑ) ˆ
(ϑ) J C D (ζ, ϑ) ,
(4.7.5)
while in the ALD-compensated case we have
δ θ J AB = −κ ∂ ζ J AB + ˆ
n
i ˆ
θ
0i J AB − α [ε AC J C B + ε BC J AC ] + O(θ
2
) .
(4.7.6)
In particular, note that J AB is only sensitive to LLTs at ˆ
P. The boundary condition
(4.4.5) implies
ˆ
J AB (ϑ) := J AB (0, ϑ) ≡ 0 ,
(4.7.7)
It will be useful to use a matrix notation J for J AB in what follows. This 2 × 2 matrix
is the “Jacobi map” [8, 17–27] which relates the physical observed angular deviation
on the sky S
A
ˆ
A
dϑ
ˆ
A to the corresponding physical vector normal to n
i in the source’s
rest-frame
n
A
i e
i
μ d ϑ γ
μ
≡ J
A
B
S
B
ˆ
A
dϑ
ˆ
A
.
(4.7.8)
For instance, the physical area at γ(ζ, ϑ) normal to n
i
(ζ, ϑ) corresponding to the
observed solid angle element
d :=
1
2
ε AB
S
A
ˆ
A
dϑ
ˆ
A
∧
S
B
ˆ
B
dϑ
ˆ
B
≡ sin ϑ dϑ ∧ dϕ ,
(4.7.9)
is given by