64
4 Observer Space-Time Formalism
4.5 Full-Sky Sachs Basis
We now wish to complement the n
i
(ζ, ϑ) direction with a 2-dimensional basis of its
normal subspace that is parallel-transported along each geodesic. By decomposing
vectors that are normal to n
i in that basis we will therefore “factor out” the trivial
part of their dynamics which is due to parallel transport. However, parallel transport
is only defined in terms of full Lorentz vectors, whereas the discussed basis will
be purely spatial. We can therefore start by considering two dimensionless Lorentz
vectors k
a
A (ζ, ϑ) on C, with A ∈ {1, 2}, i.e. transforming as
˜
k
a
A ( ˜
ζ, ˜
ϑ) =
a
b (ζ, ϑ) k
b
A (ζ, ϑ) ,
(4.5.1)
under LLTs and invariant under MDs, and also satisfying
∇ ζ k
a
A = 0 ,
k a k
a
A = 0 ,
k Aa k
a
B = δ AB ,
(4.5.2)
where the last two equations are consistently preserved under parallel transport. We
will refer to k
a
A as a “Sachs basis” and to the corresponding indices A, B, C, . . . as
“Sachs indices”. The choice of the latter is not a coincidence, as we will see below.
Observe that Eq. (4.5.2) determines that basis only up to a shift of the form
˜
k
a
A (ζ, ϑ) = k
a
A (ζ, ϑ) + α A (ϑ) k
a
(ζ, ϑ) ,
(4.5.3)
and a rotation of the A indices
˜
k
a
A (ζ, ϑ) = R
B
A (ϑ) k
a
B (ζ, ϑ) ,
R
AB (ϑ) = exp
−α ε
AB (ϑ)
= δ
AB cos α(ϑ) − ε
AB sin α(ϑ) ,
(4.5.4)
where α A and α are independent of ζ in order to maintain parallel transport, so
these are fields on S. We will refer to these transformations as “Sachs shifts” and
“Sachs rotations”, respectively. From the geometrical viewpoint, the shift freedom
corresponds to the 2-parameter family of possible of 2-dimensional space-like normal
subspaces to k
a , while the rotational freedom corresponds to a choice of orthonormal
basis within that subspace.
Now note that
n A := −k
0
A ,
(4.5.5)
is generically non-zero, which means that k
μ
A := e
μ
a k
a
A is not normal to the 4-velocity
of the observer family
n A ≡ e
μ
0 k Aμ = 0 ,
(4.5.6)
because e
μ
0 is not parallel-transported along γ in general. The Sachs shifts allow us
to set n A (ζ, ϑ) = 0 for a given value of ζ, i.e. as a boundary condition, but then the
parallel transport of k
a
A will generically induce n A = 0 at other values. Solving the
algebraic constraints (4.5.2) on k
a
A , we readily derive that it takes the following form
4 Observer Space-Time Formalism
4.5 Full-Sky Sachs Basis
We now wish to complement the n
i
(ζ, ϑ) direction with a 2-dimensional basis of its
normal subspace that is parallel-transported along each geodesic. By decomposing
vectors that are normal to n
i in that basis we will therefore “factor out” the trivial
part of their dynamics which is due to parallel transport. However, parallel transport
is only defined in terms of full Lorentz vectors, whereas the discussed basis will
be purely spatial. We can therefore start by considering two dimensionless Lorentz
vectors k
a
A (ζ, ϑ) on C, with A ∈ {1, 2}, i.e. transforming as
˜
k
a
A ( ˜
ζ, ˜
ϑ) =
a
b (ζ, ϑ) k
b
A (ζ, ϑ) ,
(4.5.1)
under LLTs and invariant under MDs, and also satisfying
∇ ζ k
a
A = 0 ,
k a k
a
A = 0 ,
k Aa k
a
B = δ AB ,
(4.5.2)
where the last two equations are consistently preserved under parallel transport. We
will refer to k
a
A as a “Sachs basis” and to the corresponding indices A, B, C, . . . as
“Sachs indices”. The choice of the latter is not a coincidence, as we will see below.
Observe that Eq. (4.5.2) determines that basis only up to a shift of the form
˜
k
a
A (ζ, ϑ) = k
a
A (ζ, ϑ) + α A (ϑ) k
a
(ζ, ϑ) ,
(4.5.3)
and a rotation of the A indices
˜
k
a
A (ζ, ϑ) = R
B
A (ϑ) k
a
B (ζ, ϑ) ,
R
AB (ϑ) = exp
−α ε
AB (ϑ)
= δ
AB cos α(ϑ) − ε
AB sin α(ϑ) ,
(4.5.4)
where α A and α are independent of ζ in order to maintain parallel transport, so
these are fields on S. We will refer to these transformations as “Sachs shifts” and
“Sachs rotations”, respectively. From the geometrical viewpoint, the shift freedom
corresponds to the 2-parameter family of possible of 2-dimensional space-like normal
subspaces to k
a , while the rotational freedom corresponds to a choice of orthonormal
basis within that subspace.
Now note that
n A := −k
0
A ,
(4.5.5)
is generically non-zero, which means that k
μ
A := e
μ
a k
a
A is not normal to the 4-velocity
of the observer family
n A ≡ e
μ
0 k Aμ = 0 ,
(4.5.6)
because e
μ
0 is not parallel-transported along γ in general. The Sachs shifts allow us
to set n A (ζ, ϑ) = 0 for a given value of ζ, i.e. as a boundary condition, but then the
parallel transport of k
a
A will generically induce n A = 0 at other values. Solving the
algebraic constraints (4.5.2) on k
a
A , we readily derive that it takes the following form