54
4 Observer Space-Time Formalism
=
1
ω 0
,
(4.2.5)
in which case the log-redshift equation becomes trivial
∂ λ ζ = 1 ,
(4.2.6)
so that
λ = ˆ
λ + ζ .
(4.2.7)
Setting for definiteness ˆ
λ ≡ 0, we get that in this gauge the log-redshift ζ is the
parametrization of the geodesic. This is very convenient since this is the parametrization the actual observer uses in practice. More precisely, the observer uses z ≡ e
ζ
− 1,
but any differential equation given in terms of ζ can be written in terms of z straightforwardly. This gauge is of course not defined for space-times or regions where 0
can go through zero, so from now on we are constraining our field of applications to
cosmology with mild inhomogeneity and anisotropy.
1
The evolution Eqs. (4.2.1) and (4.2.3) now become
∂ ζ γ
μ
=
e
μ
0 − n
i e
μ
i
0
,
(4.2.8)
∂ ζ n
i
= n
i j j
0
,
(4.2.9)
or, in terms of k
a
∂ ζ γ
μ
=
e
μ
a k
a
ω 0
,
∂ ζ k
a
= −
ω
a
0
.
(4.2.10)
Note that Eq. (4.2.5) and the resulting (4.2.7) are invariant under MDs and local
rotations, but not under local boosts, because the right-hand sides transform nontrivially. Thus, every local boost must now be compensated by some LD in order to
preserve the gauge condition (4.2.5), i.e. the latter actually breaks both symmetries
down to a combination of the two. Another way to see this is that, now that we use
an observer-dependent parameter ζ, L gets reparametrized under local boosts.
For the passive case, the compensating PLD is simply the one maintaining the
relation in Eq. (4.2.7), i.e. λ must transform as in Eq. (4.1.7). To check this at the
level of the gauge condition (4.2.5), we apply both a PLD and a LLT on the inverse
quantities for convenience
∂ ˜
λ
∂λ
−1
=
1
ω
a
0
abc +
d
b ∂ c da
k
b k
c
.
(4.2.11)
1 For generic space-times, a computationally convenient parametrization would rather be = ω −1 ,
because then Eqs. (4.2.1), (4.2.2) and (4.2.3) become independent of ω.
4 Observer Space-Time Formalism
=
1
ω 0
,
(4.2.5)
in which case the log-redshift equation becomes trivial
∂ λ ζ = 1 ,
(4.2.6)
so that
λ = ˆ
λ + ζ .
(4.2.7)
Setting for definiteness ˆ
λ ≡ 0, we get that in this gauge the log-redshift ζ is the
parametrization of the geodesic. This is very convenient since this is the parametrization the actual observer uses in practice. More precisely, the observer uses z ≡ e
ζ
− 1,
but any differential equation given in terms of ζ can be written in terms of z straightforwardly. This gauge is of course not defined for space-times or regions where 0
can go through zero, so from now on we are constraining our field of applications to
cosmology with mild inhomogeneity and anisotropy.
1
The evolution Eqs. (4.2.1) and (4.2.3) now become
∂ ζ γ
μ
=
e
μ
0 − n
i e
μ
i
0
,
(4.2.8)
∂ ζ n
i
= n
i j j
0
,
(4.2.9)
or, in terms of k
a
∂ ζ γ
μ
=
e
μ
a k
a
ω 0
,
∂ ζ k
a
= −
ω
a
0
.
(4.2.10)
Note that Eq. (4.2.5) and the resulting (4.2.7) are invariant under MDs and local
rotations, but not under local boosts, because the right-hand sides transform nontrivially. Thus, every local boost must now be compensated by some LD in order to
preserve the gauge condition (4.2.5), i.e. the latter actually breaks both symmetries
down to a combination of the two. Another way to see this is that, now that we use
an observer-dependent parameter ζ, L gets reparametrized under local boosts.
For the passive case, the compensating PLD is simply the one maintaining the
relation in Eq. (4.2.7), i.e. λ must transform as in Eq. (4.1.7). To check this at the
level of the gauge condition (4.2.5), we apply both a PLD and a LLT on the inverse
quantities for convenience
∂ ˜
λ
∂λ
−1
=
1
ω
a
0
abc +
d
b ∂ c da
k
b k
c
.
(4.2.11)
1 For generic space-times, a computationally convenient parametrization would rather be = ω −1 ,
because then Eqs. (4.2.1), (4.2.2) and (4.2.3) become independent of ω.