4.4 Observer Space
61
Consequently, for space-time fields evaluated on the geodesic
lim
ζ→0
∂ ˆ
A X (γ(ζ, ϑ)) = lim
ζ→0
∂ ˆ
A γ
μ
[∂ μ X ](γ(ζ, ϑ)) ≡ 0 ,
∀ ϑ
ˆ
A
,
(4.4.8)
but, given (4.2.10),
lim
ζ→0
∂ ζ X (γ(ζ, ϑ)) = lim
ζ→0
∂ ζ γ
μ
[∂ μ X ](γ(ζ, ϑ)) =
−1
0 [∂ X ](γ(ζ, ϑ)) = 0 .
(4.4.9)
Note that these conditions are not inconsistent, because the γ
μ
(ζ, ϑ) are fields on C,
which is a 3-cylinder (4.4.1), not a 3-cone, so they can have an angular dependence
for all ζ, i.e. including at the ζ = 0 value. It is only the image γ(C) that has a conical
topology in M because of the boundary condition in Eq. (4.4.3). The latter then
only affects the way in which geodesics are glued together, i.e. their ϑ
ˆ
A -dependence
given in Eq. (4.4.5). A crucial requirement for this construction is that the angular
parameters ϑ
ˆ
A are not part of some coordinate system x
μ on M, but a parametrization of ∂ ζ γ
μ data in T ˆ
P M. Indeed, if the ϑ
ˆ
A were angular coordinates on M, i.e.
parametrizing the cone γ(C) instead of the cylinder C, then they would necessarily
be ill-defined at ˆ
P.
Finally, note the following important property, which actually holds for all λparametrizations of the light-cone (λ, ϑ
ˆ
A
), i.e. not only the log-redshift one λ = ζ
corresponding to Eq. (4.2.5). Using the geodesic light-like deviation equations
∂ γ
μ
= e
μ
a k
a
,
∇ k
a
≡ ∂ k
a
+
a
bμ ∂ γ
μ k
b
= 0 ,
(4.4.10)
one obtains straightforwardly
∂
k a e
a
μ ∂ ˆ
A γ
a
= 0 .
(4.4.11)
The boundary condition in Eq. (4.4.5) then implies that this quantity is zero everywhere, and therefore that ∂ ζ γ
μ and ∂ ˆ
A γ
μ are orthonormal
g μν ∂ ζ γ
μ
∂ ˆ
A γ
ν
= 0 .
(4.4.12)
4.4.2 Caustic Resolution
From the previous paragraph we understand that the 3-cylinder construction is clearly
distinct from the observational coordinate [1, 2] and geodesic light-cone coordinate
[2–16] formalisms, where the angles associated with incoming light-rays are part
of a specific coordinate system on M. Although these angles are not defined at
the observer, they are an unambiguous parametrization of the incoming light-like
geodesics because they are constant along these paths, by construction. There is,
61
Consequently, for space-time fields evaluated on the geodesic
lim
ζ→0
∂ ˆ
A X (γ(ζ, ϑ)) = lim
ζ→0
∂ ˆ
A γ
μ
[∂ μ X ](γ(ζ, ϑ)) ≡ 0 ,
∀ ϑ
ˆ
A
,
(4.4.8)
but, given (4.2.10),
lim
ζ→0
∂ ζ X (γ(ζ, ϑ)) = lim
ζ→0
∂ ζ γ
μ
[∂ μ X ](γ(ζ, ϑ)) =
−1
0 [∂ X ](γ(ζ, ϑ)) = 0 .
(4.4.9)
Note that these conditions are not inconsistent, because the γ
μ
(ζ, ϑ) are fields on C,
which is a 3-cylinder (4.4.1), not a 3-cone, so they can have an angular dependence
for all ζ, i.e. including at the ζ = 0 value. It is only the image γ(C) that has a conical
topology in M because of the boundary condition in Eq. (4.4.3). The latter then
only affects the way in which geodesics are glued together, i.e. their ϑ
ˆ
A -dependence
given in Eq. (4.4.5). A crucial requirement for this construction is that the angular
parameters ϑ
ˆ
A are not part of some coordinate system x
μ on M, but a parametrization of ∂ ζ γ
μ data in T ˆ
P M. Indeed, if the ϑ
ˆ
A were angular coordinates on M, i.e.
parametrizing the cone γ(C) instead of the cylinder C, then they would necessarily
be ill-defined at ˆ
P.
Finally, note the following important property, which actually holds for all λparametrizations of the light-cone (λ, ϑ
ˆ
A
), i.e. not only the log-redshift one λ = ζ
corresponding to Eq. (4.2.5). Using the geodesic light-like deviation equations
∂ γ
μ
= e
μ
a k
a
,
∇ k
a
≡ ∂ k
a
+
a
bμ ∂ γ
μ k
b
= 0 ,
(4.4.10)
one obtains straightforwardly
∂
k a e
a
μ ∂ ˆ
A γ
a
= 0 .
(4.4.11)
The boundary condition in Eq. (4.4.5) then implies that this quantity is zero everywhere, and therefore that ∂ ζ γ
μ and ∂ ˆ
A γ
μ are orthonormal
g μν ∂ ζ γ
μ
∂ ˆ
A γ
ν
= 0 .
(4.4.12)
4.4.2 Caustic Resolution
From the previous paragraph we understand that the 3-cylinder construction is clearly
distinct from the observational coordinate [1, 2] and geodesic light-cone coordinate
[2–16] formalisms, where the angles associated with incoming light-rays are part
of a specific coordinate system on M. Although these angles are not defined at
the observer, they are an unambiguous parametrization of the incoming light-like
geodesics because they are constant along these paths, by construction. There is,