4.5 Full-Sky Sachs Basis
67
geodesic reaching the observer. By fixing the boundary condition (4.5.17) we are
fixing the freedom of independent Sachs rotations by matching them to the local
rotations on S, thus making them act solely as compensators of LLTs at the observer
point ˆ
P. For a given observer ˆ
e a , we have thus fully determined an orthonormal basis
{n
i
, n
i
A } for the spatial part of Lorentz vectors on all of C. This basis is adapted to
the factorization of that space L × S in that n
i generates the propagation direction,
while the n
i
A generate the transverse directions.
Let us now consider the boundary conditions for ˆ
n A (ϑ) := n A (0, ϑ). We first
remind that, since n A will not appear in any of the physical observables, its value is
actually irrelevant. Nevertheless, as a matter of aesthetics, one might be interested in
considering the privileged choice ˆ
n A = 0, meaning that ˆ
k
μ
A is normal to the observer
velocity ˆ
e
μ
0 . However, this condition is not preserved under local boosts at ˆ
P, because
the LLT of ˆ
n A is not linear (see Eq. (4.5.12)). With the compensating Sachs rotation
that is now required it reads
˜ ˆ
n A ( ˜
ϑ) = R
B
A (ϑ)
ˆ
(ϑ) ˆ
n B (ϑ) − ˆ
0
i ˆ
n
i
B (ϑ)
.
(4.5.23)
Nevertheless, remember that n A also transforms non-linearly under the Sachs shift
transformations (4.5.8) which, just like the Sachs rotations, can now depend on the
observed angles ϑ
ˆ
A . We can therefore use these shifts to compensate the non-linear
part of (4.5.23) and the required parameter is
α A (ϑ) = − ˆ
ω
−1 ˆ
0
i R
B
A (ϑ) ˆ
n
i
B (ϑ) .
(4.5.24)
With this, the boundary condition that makes the Sachs basis k
a
A purely spatial at the
observer
ˆ
n A (ϑ) = 0 ,
(4.5.25)
is preserved under LLTs. Thus, just as for the Sachs rotations, the Sachs shift freedom
can be completely fixed by acting as a compensator of LLTs in order to preserve
some condition. Nevertheless, when defining observables, it is still useful to check
invariance under these shifts to make sure that they do not depend on n A .
4.6 Transformation Rule Summary
To conclude this subsection, let us summarize the LLT transformations of the fields
on C. We remind that our construction has fixed the LD, Sachs rotation and Sachs
shift symmetries, with these transformations now being induced by LLTs in order
to preserve the conditions (4.2.5), (4.3.17) and (4.5.25), respectively. In the PLDcompensated case we have
67
geodesic reaching the observer. By fixing the boundary condition (4.5.17) we are
fixing the freedom of independent Sachs rotations by matching them to the local
rotations on S, thus making them act solely as compensators of LLTs at the observer
point ˆ
P. For a given observer ˆ
e a , we have thus fully determined an orthonormal basis
{n
i
, n
i
A } for the spatial part of Lorentz vectors on all of C. This basis is adapted to
the factorization of that space L × S in that n
i generates the propagation direction,
while the n
i
A generate the transverse directions.
Let us now consider the boundary conditions for ˆ
n A (ϑ) := n A (0, ϑ). We first
remind that, since n A will not appear in any of the physical observables, its value is
actually irrelevant. Nevertheless, as a matter of aesthetics, one might be interested in
considering the privileged choice ˆ
n A = 0, meaning that ˆ
k
μ
A is normal to the observer
velocity ˆ
e
μ
0 . However, this condition is not preserved under local boosts at ˆ
P, because
the LLT of ˆ
n A is not linear (see Eq. (4.5.12)). With the compensating Sachs rotation
that is now required it reads
˜ ˆ
n A ( ˜
ϑ) = R
B
A (ϑ)
ˆ
(ϑ) ˆ
n B (ϑ) − ˆ
0
i ˆ
n
i
B (ϑ)
.
(4.5.23)
Nevertheless, remember that n A also transforms non-linearly under the Sachs shift
transformations (4.5.8) which, just like the Sachs rotations, can now depend on the
observed angles ϑ
ˆ
A . We can therefore use these shifts to compensate the non-linear
part of (4.5.23) and the required parameter is
α A (ϑ) = − ˆ
ω
−1 ˆ
0
i R
B
A (ϑ) ˆ
n
i
B (ϑ) .
(4.5.24)
With this, the boundary condition that makes the Sachs basis k
a
A purely spatial at the
observer
ˆ
n A (ϑ) = 0 ,
(4.5.25)
is preserved under LLTs. Thus, just as for the Sachs rotations, the Sachs shift freedom
can be completely fixed by acting as a compensator of LLTs in order to preserve
some condition. Nevertheless, when defining observables, it is still useful to check
invariance under these shifts to make sure that they do not depend on n A .
4.6 Transformation Rule Summary
To conclude this subsection, let us summarize the LLT transformations of the fields
on C. We remind that our construction has fixed the LD, Sachs rotation and Sachs
shift symmetries, with these transformations now being induced by LLTs in order
to preserve the conditions (4.2.5), (4.3.17) and (4.5.25), respectively. In the PLDcompensated case we have