4.3 Observer Sky
59
This choice is not invariant under neither LLTs nor local rotations, because both lead
to a non-diagonal matrix in general. We have again broken two symmetries down to
a combination of them that preserves the gauge condition. Moreover, here too we
still have the full LLT freedom, as any such transformation can be compensated by
some local rotation. The transformation which preserves (4.3.17) is therefore of the
form
˜
S
ˆ
A
A ( ˜
ϑ) = R
B
A (ϑ) ˆ
(ϑ)
∂ ˜
ϑ
ˆ
A
∂ϑ
ˆ
B
(ϑ) S
ˆ
B
B (ϑ) ,
(4.3.18)
where now the the compensating angle α(ϑ) in R
A
B (ϑ) is entirely determined by the
Lorentz generator ˆ
θ ab , such that
˜
S
A
ˆ
A
( ˜
ϑ) = S
A
ˆ
A
( ˜
ϑ) .
(4.3.19)
To show that such a compensating local rotation exists, we just need to isolating R
B
A
in Eq. (4.3.18)
R
A
B (ϑ) = ˆ
−1
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ)
∂ ˜
ϑ
ˆ
A
∂ϑ
ˆ
B
( ˜
ϑ) S
B
ˆ
B
(ϑ) ,
(4.3.20)
and show that it is a rotation matrix indeed, i.e. R R
T
= id. Using (4.3.11), we get
R
C
A (ϑ) R
C
B (ϑ) = ˆ
−2
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ) ˜
S
ˆ
B
B ( ˜
ϑ)
∂ϑ
ˆ
C
∂ ˜
ϑ
ˆ
A
( ˜
ϑ)
∂ϑ
ˆ
D
∂ ˜
ϑ
ˆ
B
( ˜
ϑ) S
C
ˆ
C
(ϑ) S
C
ˆ
D
(ϑ)
≡ ˆ
−2
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ) ˜
S
ˆ
B
B ( ˜
ϑ)
∂ϑ
ˆ
C
∂ ˜
ϑ
ˆ
A
( ˜
ϑ)
∂ϑ
ˆ
D
∂ ˜
ϑ
ˆ
B
( ˜
ϑ) S ˆ
C ˆ
D (ϑ)
≡ ˆ
−2
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ) ˜
S
ˆ
B
B ( ˜
ϑ) ˜
S ˆ
A ˆ
B ( ˜
ϑ) ≡ δ AB .
(4.3.21)
Thus, we still have the LLT freedom at ˆ
P, we must only be aware that this symmetry
now induces a compensating local rotation on the tangent space of S, just as it induces
a compensating LD on L, and both transformations are controlled by ˆ
a
b and ˆ
n
i
(ϑ).
4.4 Observer Space
4.4.1 Bundle of Geodesics
From (4.2.8) and (4.2.9) we see that γ
μ
(ζ) is uniquely determined by the boundary data ˆ
n
i and ˆ
γ = ˆ
P, i.e. it is independent of ˆ
ω. This means that the points of S
are in a one-to-one correspondence with the light-like geodesic paths γ ⊂ M going
59
This choice is not invariant under neither LLTs nor local rotations, because both lead
to a non-diagonal matrix in general. We have again broken two symmetries down to
a combination of them that preserves the gauge condition. Moreover, here too we
still have the full LLT freedom, as any such transformation can be compensated by
some local rotation. The transformation which preserves (4.3.17) is therefore of the
form
˜
S
ˆ
A
A ( ˜
ϑ) = R
B
A (ϑ) ˆ
(ϑ)
∂ ˜
ϑ
ˆ
A
∂ϑ
ˆ
B
(ϑ) S
ˆ
B
B (ϑ) ,
(4.3.18)
where now the the compensating angle α(ϑ) in R
A
B (ϑ) is entirely determined by the
Lorentz generator ˆ
θ ab , such that
˜
S
A
ˆ
A
( ˜
ϑ) = S
A
ˆ
A
( ˜
ϑ) .
(4.3.19)
To show that such a compensating local rotation exists, we just need to isolating R
B
A
in Eq. (4.3.18)
R
A
B (ϑ) = ˆ
−1
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ)
∂ ˜
ϑ
ˆ
A
∂ϑ
ˆ
B
( ˜
ϑ) S
B
ˆ
B
(ϑ) ,
(4.3.20)
and show that it is a rotation matrix indeed, i.e. R R
T
= id. Using (4.3.11), we get
R
C
A (ϑ) R
C
B (ϑ) = ˆ
−2
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ) ˜
S
ˆ
B
B ( ˜
ϑ)
∂ϑ
ˆ
C
∂ ˜
ϑ
ˆ
A
( ˜
ϑ)
∂ϑ
ˆ
D
∂ ˜
ϑ
ˆ
B
( ˜
ϑ) S
C
ˆ
C
(ϑ) S
C
ˆ
D
(ϑ)
≡ ˆ
−2
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ) ˜
S
ˆ
B
B ( ˜
ϑ)
∂ϑ
ˆ
C
∂ ˜
ϑ
ˆ
A
( ˜
ϑ)
∂ϑ
ˆ
D
∂ ˜
ϑ
ˆ
B
( ˜
ϑ) S ˆ
C ˆ
D (ϑ)
≡ ˆ
−2
(ϑ) ˜
S
ˆ
A
A ( ˜
ϑ) ˜
S
ˆ
B
B ( ˜
ϑ) ˜
S ˆ
A ˆ
B ( ˜
ϑ) ≡ δ AB .
(4.3.21)
Thus, we still have the LLT freedom at ˆ
P, we must only be aware that this symmetry
now induces a compensating local rotation on the tangent space of S, just as it induces
a compensating LD on L, and both transformations are controlled by ˆ
a
b and ˆ
n
i
(ϑ).
4.4 Observer Space
4.4.1 Bundle of Geodesics
From (4.2.8) and (4.2.9) we see that γ
μ
(ζ) is uniquely determined by the boundary data ˆ
n
i and ˆ
γ = ˆ
P, i.e. it is independent of ˆ
ω. This means that the points of S
are in a one-to-one correspondence with the light-like geodesic paths γ ⊂ M going