66
4 Observer Space-Time Formalism
This shows again that, even if we set n A = 0 at some ζ value, this quantity will
be turned on by parallel transport. Moreover, note that both the LLT and evolution
equation of n
i
A are independent of n A , which means that we do not need to keep track
of the latter, as only the n
i
A information will turn out to be relevant.
Finally, we must select a definite basis k
a
A (ζ, ϑ) by providing boundary conditions
for it at the observer sky ζ = 0, and in doing so fixing the shift (4.5.3) and rotation
(4.5.4) ambiguities. To that end, we note that we have already privileged a basis in
the tangent space of S, the one given by the dyad (4.3.17), which is nothing but the
unit-normalization of the angular directions ∂ ˆ
A on the sky. For the spatial part n
i
A we
thus pick
ˆ
n
i
A (ϑ) := n
i
A (0, ϑ) = S
ˆ
A
A (ϑ) ∂ ˆ
A ˆ
n
i
(ϑ) ≡ ∂ A ˆ
n
i
(ϑ) ,
(4.5.17)
so that, using (4.3.17) and (4.3.2)
ˆ
n 1 (ϑ) = (cos ϑ cos ϕ, cos ϑ sin ϕ, − sin ϑ) ,
ˆ
n 2 (ϑ) = (− sin ϕ, cos ϕ, 0) .
(4.5.18)
These are the unit-normed longitudinal and latitudinal vectors fields on S, just like
the dyad, but when S is seen as a subset of R
3 . We first note that Eq. (4.5.17) is
consistent because it has the required properties
ˆ
n
i
ˆ
n
i
A ≡ 0 ,
ˆ
n
i
A ˆ
n
i
B ≡ δ AB ,
δ
AB
ˆ
n
i
A ˆ
n
j
B ≡ ˆ
n
i j
ε
AB
ˆ
n
i
A ˆ
n
j
B ≡ ε
i jk
ˆ
n
k
.
(4.5.19)
We must now also make the equality (4.5.17) invariant under LLTs. Put differently,
as in the case of ˆ
n
i
(ϑ), S ˆ
A ˆ
B (ϑ) and S
A
ˆ
A
(ϑ), the ˆ
n
i
A (ϑ) given by Eq. (4.5.18) must now
be the same functions of ϑ
ˆ
A for all observers
˜ ˆ
n
i
A ( ˜
ϑ) = ˆ
n
i
A ( ˜
ϑ) .
(4.5.20)
To achieve this, we first compute the variation of the right-hand side using Eqs.
(4.3.4), (4.3.18) and (4.4.16)
˜
∂ A ˜ ˆ
n
i ( ˜
ϑ) = ˆ
(ϑ) R
B
A (ϑ) ∂ B
ˆ
−1
(ϑ)
ˆ
i
j ˆ
n
j (ϑ) − ˆ
i
0
= R
B
A (ϑ)
( ˆ
⊥ )
i
j (ϑ) ∂ B ˆ
n
j (ϑ)
,
(4.5.21)
and see that it reproduces the transformation (4.5.12) of n
i
A , but along with a Sachs
rotation (4.5.4) given by the LLT-compensating rotation R
A
B (ϑ) of the dyad basis
˜ ˆ
n
i
A ( ˜
ϑ) = R
B
A (ϑ) ( ˆ
⊥ )
i
j (ϑ) ˆ
n
j
B (ϑ) .
(4.5.22)
Thus, with this choice of boundary conditions (4.5.17) we identify the local rotation
symmetry on S (see Eq. (4.3.16)) with the global Sachs rotations along each geodesic
ϑ
ˆ
A (see Eq. (4.5.4)), so that the A indices of n
i
A and of S
ˆ
A
A are now indeed the same.
Remember that the Sachs rotations are independent of ζ, in order to preserve parallel
transport, but they can depend on ϑ
ˆ
A , i.e. they can be different for each light-like
4 Observer Space-Time Formalism
This shows again that, even if we set n A = 0 at some ζ value, this quantity will
be turned on by parallel transport. Moreover, note that both the LLT and evolution
equation of n
i
A are independent of n A , which means that we do not need to keep track
of the latter, as only the n
i
A information will turn out to be relevant.
Finally, we must select a definite basis k
a
A (ζ, ϑ) by providing boundary conditions
for it at the observer sky ζ = 0, and in doing so fixing the shift (4.5.3) and rotation
(4.5.4) ambiguities. To that end, we note that we have already privileged a basis in
the tangent space of S, the one given by the dyad (4.3.17), which is nothing but the
unit-normalization of the angular directions ∂ ˆ
A on the sky. For the spatial part n
i
A we
thus pick
ˆ
n
i
A (ϑ) := n
i
A (0, ϑ) = S
ˆ
A
A (ϑ) ∂ ˆ
A ˆ
n
i
(ϑ) ≡ ∂ A ˆ
n
i
(ϑ) ,
(4.5.17)
so that, using (4.3.17) and (4.3.2)
ˆ
n 1 (ϑ) = (cos ϑ cos ϕ, cos ϑ sin ϕ, − sin ϑ) ,
ˆ
n 2 (ϑ) = (− sin ϕ, cos ϕ, 0) .
(4.5.18)
These are the unit-normed longitudinal and latitudinal vectors fields on S, just like
the dyad, but when S is seen as a subset of R
3 . We first note that Eq. (4.5.17) is
consistent because it has the required properties
ˆ
n
i
ˆ
n
i
A ≡ 0 ,
ˆ
n
i
A ˆ
n
i
B ≡ δ AB ,
δ
AB
ˆ
n
i
A ˆ
n
j
B ≡ ˆ
n
i j
ε
AB
ˆ
n
i
A ˆ
n
j
B ≡ ε
i jk
ˆ
n
k
.
(4.5.19)
We must now also make the equality (4.5.17) invariant under LLTs. Put differently,
as in the case of ˆ
n
i
(ϑ), S ˆ
A ˆ
B (ϑ) and S
A
ˆ
A
(ϑ), the ˆ
n
i
A (ϑ) given by Eq. (4.5.18) must now
be the same functions of ϑ
ˆ
A for all observers
˜ ˆ
n
i
A ( ˜
ϑ) = ˆ
n
i
A ( ˜
ϑ) .
(4.5.20)
To achieve this, we first compute the variation of the right-hand side using Eqs.
(4.3.4), (4.3.18) and (4.4.16)
˜
∂ A ˜ ˆ
n
i ( ˜
ϑ) = ˆ
(ϑ) R
B
A (ϑ) ∂ B
ˆ
−1
(ϑ)
ˆ
i
j ˆ
n
j (ϑ) − ˆ
i
0
= R
B
A (ϑ)
( ˆ
⊥ )
i
j (ϑ) ∂ B ˆ
n
j (ϑ)
,
(4.5.21)
and see that it reproduces the transformation (4.5.12) of n
i
A , but along with a Sachs
rotation (4.5.4) given by the LLT-compensating rotation R
A
B (ϑ) of the dyad basis
˜ ˆ
n
i
A ( ˜
ϑ) = R
B
A (ϑ) ( ˆ
⊥ )
i
j (ϑ) ˆ
n
j
B (ϑ) .
(4.5.22)
Thus, with this choice of boundary conditions (4.5.17) we identify the local rotation
symmetry on S (see Eq. (4.3.16)) with the global Sachs rotations along each geodesic
ϑ
ˆ
A (see Eq. (4.5.4)), so that the A indices of n
i
A and of S
ˆ
A
A are now indeed the same.
Remember that the Sachs rotations are independent of ζ, in order to preserve parallel
transport, but they can depend on ϑ
ˆ
A , i.e. they can be different for each light-like