60
4 Observer Space-Time Formalism
through ˆ
P.
2 We can therefore label these geodesics by ϑ
ˆ
A , leading to a bundle of
paths γ(ζ, ϑ) generating the light-cone of ˆ
P. Simply put, we are parametrizing the
geodesics by their boundary data ˆ
n
i in the ˆ
e a frame, which are directly the observed
angles. Along with the redshift parameter z (or ζ), these are the fundamental observables the actual observer uses to parametrize events on the light-cone in practice. As
a consequence, any L-field X (ζ) acquires a ϑ
ˆ
A -dependence X (ζ, ϑ) which, just as
γ
μ
(ζ, ϑ), effectively makes it a field on
C := L × S .
(4.4.1)
We will refer to this space as the “observer space”. If we are also interested in spectral
distributions, then these are fields of the form X (ζ, ϑ, ˆ
ω), thus living on the “spectral
observer space”
C spec := L × S spec ≡ C × R + .
(4.4.2)
Let us now pay closer attention to the boundary conditions at the observer position
ˆ
P. By construction, the image of C under the γ map is the observer light-cone
γ(C) ⊂ M, i.e. the subspace of M spanned by all the light-like geodesics attached
to ˆ
P, so in particular
lim
ζ→0
γ(ζ, ϑ) ≡ ˆ
P ,
∀ ϑ
ˆ
A
.
(4.4.3)
Consequently, the space-time fields evaluated on C obey
lim
ζ→0
X (γ(ζ, ϑ)) ≡ X ( ˆ
P) ,
∀ ϑ
ˆ
A
.
(4.4.4)
Since the image γ(C) has conical topology, the γ map is continuous at ζ = 0, but not
differentiable there. More precisely, its angular derivative is well-defined
lim
ζ→0
∂ ˆ
A γ
μ
(ζ, ϑ) ≡ 0 ,
∀ ϑ
ˆ
A
,
(4.4.5)
because of (4.4.3) and the fact that the cone is smooth in these directions. However,
∂ ζ γ
μ is multivalued at ζ = 0
lim
ζ→0
∂ ζ γ
μ
(ζ, ϑ) = ∂ ζ γ
μ
(0, ϑ) ,
(4.4.6)
since ∂ ζ γ
μ
(0, ϑ) is precisely the boundary information ˆ
n
i
(ϑ) ∼ ˆ
e
i
μ ∂ ζ γ
μ
(0, ϑ) by
construction, and also
lim
ζ→0
n
i
(ζ, ϑ) = ˆ
n
i
(ϑ) .
(4.4.7)
2 This is not the case for time-like geodesics, where one needs all of the three ˆ
k i numbers to
distinguish among all possible geodesics at ˆ
P. Indeed, in the light-like case the geodesics are
constrained to lie on the light-cone, whereas in the time-like case they probe its interior, which has
one more dimension.
4 Observer Space-Time Formalism
through ˆ
P.
2 We can therefore label these geodesics by ϑ
ˆ
A , leading to a bundle of
paths γ(ζ, ϑ) generating the light-cone of ˆ
P. Simply put, we are parametrizing the
geodesics by their boundary data ˆ
n
i in the ˆ
e a frame, which are directly the observed
angles. Along with the redshift parameter z (or ζ), these are the fundamental observables the actual observer uses to parametrize events on the light-cone in practice. As
a consequence, any L-field X (ζ) acquires a ϑ
ˆ
A -dependence X (ζ, ϑ) which, just as
γ
μ
(ζ, ϑ), effectively makes it a field on
C := L × S .
(4.4.1)
We will refer to this space as the “observer space”. If we are also interested in spectral
distributions, then these are fields of the form X (ζ, ϑ, ˆ
ω), thus living on the “spectral
observer space”
C spec := L × S spec ≡ C × R + .
(4.4.2)
Let us now pay closer attention to the boundary conditions at the observer position
ˆ
P. By construction, the image of C under the γ map is the observer light-cone
γ(C) ⊂ M, i.e. the subspace of M spanned by all the light-like geodesics attached
to ˆ
P, so in particular
lim
ζ→0
γ(ζ, ϑ) ≡ ˆ
P ,
∀ ϑ
ˆ
A
.
(4.4.3)
Consequently, the space-time fields evaluated on C obey
lim
ζ→0
X (γ(ζ, ϑ)) ≡ X ( ˆ
P) ,
∀ ϑ
ˆ
A
.
(4.4.4)
Since the image γ(C) has conical topology, the γ map is continuous at ζ = 0, but not
differentiable there. More precisely, its angular derivative is well-defined
lim
ζ→0
∂ ˆ
A γ
μ
(ζ, ϑ) ≡ 0 ,
∀ ϑ
ˆ
A
,
(4.4.5)
because of (4.4.3) and the fact that the cone is smooth in these directions. However,
∂ ζ γ
μ is multivalued at ζ = 0
lim
ζ→0
∂ ζ γ
μ
(ζ, ϑ) = ∂ ζ γ
μ
(0, ϑ) ,
(4.4.6)
since ∂ ζ γ
μ
(0, ϑ) is precisely the boundary information ˆ
n
i
(ϑ) ∼ ˆ
e
i
μ ∂ ζ γ
μ
(0, ϑ) by
construction, and also
lim
ζ→0
n
i
(ζ, ϑ) = ˆ
n
i
(ϑ) .
(4.4.7)
2 This is not the case for time-like geodesics, where one needs all of the three ˆ
k i numbers to
distinguish among all possible geodesics at ˆ
P. Indeed, in the light-like case the geodesics are
constrained to lie on the light-cone, whereas in the time-like case they probe its interior, which has
one more dimension.