126
5 General-Relativistic Matrix Kinetic Theory
while the dimensionful ones have an extra ˆ
3
(ϑ) factor. One can also express ˆ
P AB
in terms of the Stokes parameters
ˆ
P
AB
≡ ˆ
Qσ
AB
+ + ˆ
U σ
AB
× ,
ˆ
Q :=
1
2
σ
AB
+
ˆ
P AB ,
ˆ
U :=
1
2
σ
AB
×
ˆ
P AB ,
(5.11.12)
where the σ +,× matrices have been defined in Eq. (4.7.27). In terms of the complex
combination
ˆ
P := ˆ
Q + i ˆ
U ,
(5.11.13)
the transformation (5.11.11) reads
˜ ˆ
P( ˜
ϑ, ˜ ˆ
ω) = e
2iα(ϑ) ˆ
P(ϑ, ˆ
ω) ,
(5.11.14)
where α is the angle in LLT-compensating local rotation R
A
B (ϑ) (see Eq. (4.3.16)).
Finally, it is also conventional to express ˆ
I in terms of some effective “temperature”
distribution ˆ
T through the photon Bose-Einstein distribution
ˆ
I (ϑ, ˆ
ω) ≡
2
exp
ˆ
ω
ˆ
T (ϑ, ˆ
ω)
− 1
,
(5.11.15)
even though temperature is a macroscopic variable which therefore cannot depend
on the microscopic momenta
k.
We next decompose these fields in the basis of (spin-weighted) spherical harmonics (see for instance [22] for a description). In the presence of tensors, such as
ˆ
P
AB , this construction involves the covariant derivative on S with respect to the local
rotations appearing in (5.11.11), so we invoke the spin connection w
AB
ˆ
A
(ϑ) on that
space, i.e. the one for which the dyad S
A
ˆ
A
(ϑ) is torsion-free. Given the dimensionality
of S, we can write
w
AB
ˆ
A
≡ ε
AB
w ˆ
A ,
(5.11.16)
so the covariant derivative reads
∇ ˆ
A
ˆ
X
A
:= ∂ ˆ
A
ˆ
X
A
+ w ˆ
A ε
A
C
ˆ
X
C
,
(5.11.17)
and we will actually use the Sachs-indexed one ∇ A := S
ˆ
A
A ∇ ˆ
A . With Eq. (4.3.17) and
the 2-dimensional analogue of Eq. (3.2.32), we then find
w ˆ
A = −δ
ϕ
ˆ
A
cos ϑ ,
w A := S
ˆ
A
A w ˆ
A = −δ
2
A cot ϑ .
(5.11.18)
Pay attention to the fact that, under a generic LLT at the observer, the frequency and
angles mix, leading in particular to the rule (4.3.7) and thus
5 General-Relativistic Matrix Kinetic Theory
while the dimensionful ones have an extra ˆ
3
(ϑ) factor. One can also express ˆ
P AB
in terms of the Stokes parameters
ˆ
P
AB
≡ ˆ
Qσ
AB
+ + ˆ
U σ
AB
× ,
ˆ
Q :=
1
2
σ
AB
+
ˆ
P AB ,
ˆ
U :=
1
2
σ
AB
×
ˆ
P AB ,
(5.11.12)
where the σ +,× matrices have been defined in Eq. (4.7.27). In terms of the complex
combination
ˆ
P := ˆ
Q + i ˆ
U ,
(5.11.13)
the transformation (5.11.11) reads
˜ ˆ
P( ˜
ϑ, ˜ ˆ
ω) = e
2iα(ϑ) ˆ
P(ϑ, ˆ
ω) ,
(5.11.14)
where α is the angle in LLT-compensating local rotation R
A
B (ϑ) (see Eq. (4.3.16)).
Finally, it is also conventional to express ˆ
I in terms of some effective “temperature”
distribution ˆ
T through the photon Bose-Einstein distribution
ˆ
I (ϑ, ˆ
ω) ≡
2
exp
ˆ
ω
ˆ
T (ϑ, ˆ
ω)
− 1
,
(5.11.15)
even though temperature is a macroscopic variable which therefore cannot depend
on the microscopic momenta
k.
We next decompose these fields in the basis of (spin-weighted) spherical harmonics (see for instance [22] for a description). In the presence of tensors, such as
ˆ
P
AB , this construction involves the covariant derivative on S with respect to the local
rotations appearing in (5.11.11), so we invoke the spin connection w
AB
ˆ
A
(ϑ) on that
space, i.e. the one for which the dyad S
A
ˆ
A
(ϑ) is torsion-free. Given the dimensionality
of S, we can write
w
AB
ˆ
A
≡ ε
AB
w ˆ
A ,
(5.11.16)
so the covariant derivative reads
∇ ˆ
A
ˆ
X
A
:= ∂ ˆ
A
ˆ
X
A
+ w ˆ
A ε
A
C
ˆ
X
C
,
(5.11.17)
and we will actually use the Sachs-indexed one ∇ A := S
ˆ
A
A ∇ ˆ
A . With Eq. (4.3.17) and
the 2-dimensional analogue of Eq. (3.2.32), we then find
w ˆ
A = −δ
ϕ
ˆ
A
cos ϑ ,
w A := S
ˆ
A
A w ˆ
A = −δ
2
A cot ϑ .
(5.11.18)
Pay attention to the fact that, under a generic LLT at the observer, the frequency and
angles mix, leading in particular to the rule (4.3.7) and thus