4.8 Observer Space-Time
77
4.8 Observer Space-Time
The observer space C covers the full light-cone of the observer at the point ˆ
P. We
can, however, consider also the possibility of performing observations at a later time
in order to measure the “drift” of observables [30, 31], and therefore from some
other point ˆ
P
in the future light-cone of ˆ
P. This means that, instead of a single
observer point ˆ
P, we must select a specific observer world-line ˆ
P(τ ), where τ ∈ R
can be conveniently chosen to be the proper time of the observer. To each point ˆ
P(τ )
of this world-line we can then associate a corresponding observer space C(τ ) and
its spectral extension C spec (τ ). This amounts to considering a continuous family of
γ
μ
(ζ, ϑ) maps parametrized by τ , i.e. γ
μ
(τ , ζ, ϑ), so that all C-fields acquire a τ
dependence, just as we did for the ϑ
ˆ
A parametrization in Sect. 4.4.1. The observer
world-line is then given by
ˆ
γ
μ
(τ ) := γ
μ
(τ , 0, ϑ) ,
(4.8.1)
since there is no ϑ
ˆ
A dependence at ζ = 0. In the single observer point case we
considered ˆ
e 0 as the observer 4-velocity, so in the observer world-line case the path
ˆ
γ
μ
(τ ) must be an integral line of the 4-velocity field e
μ
0 (x). If τ is to denote the proper
time of the observer, then the relation is simply
∂ τ ˆ
γ
μ
(τ ) = e
μ
0 ( ˆ
γ(τ )) .
(4.8.2)
Given some reference point ˆ
P, this equation completely determines the world-line
and therefore the corresponding function γ
μ
(τ , ζ, ϑ). Therefore, in general, if ζ = 0
∂ τ γ
μ
(τ , ζ, ϑ) = e
μ
0 (γ(τ , ζ, ϑ)) .
(4.8.3)
Now the space on which γ
μ
(τ , ζ, ϑ) is defined is
O := R × C
(4.8.4)
and we will refer to it as the “observer space-time”. Again, if some fields on that
space also depend on the observed frequency ˆ
ω, then they are actually defined on the
“spectral observer space-time”
O spec := O × R + .
(4.8.5)
We thus have that γ
μ
(τ , ζ, ϑ) probes all the points of space-time that are connected
to the observer world-line by some light-like geodesic, i.e. the image γ(O) is the
observable universe of that observer, by definition. The space O is parametrized by
the proper time τ , log-redshift ζ and angles ϑ
ˆ
A at which the corresponding signal
was observed, i.e. exactly the parameters the observer has access to in practice. It
is important, however, to notice that {τ , ζ, ϑ} is not a coordinate system on M in
77
4.8 Observer Space-Time
The observer space C covers the full light-cone of the observer at the point ˆ
P. We
can, however, consider also the possibility of performing observations at a later time
in order to measure the “drift” of observables [30, 31], and therefore from some
other point ˆ
P
in the future light-cone of ˆ
P. This means that, instead of a single
observer point ˆ
P, we must select a specific observer world-line ˆ
P(τ ), where τ ∈ R
can be conveniently chosen to be the proper time of the observer. To each point ˆ
P(τ )
of this world-line we can then associate a corresponding observer space C(τ ) and
its spectral extension C spec (τ ). This amounts to considering a continuous family of
γ
μ
(ζ, ϑ) maps parametrized by τ , i.e. γ
μ
(τ , ζ, ϑ), so that all C-fields acquire a τ
dependence, just as we did for the ϑ
ˆ
A parametrization in Sect. 4.4.1. The observer
world-line is then given by
ˆ
γ
μ
(τ ) := γ
μ
(τ , 0, ϑ) ,
(4.8.1)
since there is no ϑ
ˆ
A dependence at ζ = 0. In the single observer point case we
considered ˆ
e 0 as the observer 4-velocity, so in the observer world-line case the path
ˆ
γ
μ
(τ ) must be an integral line of the 4-velocity field e
μ
0 (x). If τ is to denote the proper
time of the observer, then the relation is simply
∂ τ ˆ
γ
μ
(τ ) = e
μ
0 ( ˆ
γ(τ )) .
(4.8.2)
Given some reference point ˆ
P, this equation completely determines the world-line
and therefore the corresponding function γ
μ
(τ , ζ, ϑ). Therefore, in general, if ζ = 0
∂ τ γ
μ
(τ , ζ, ϑ) = e
μ
0 (γ(τ , ζ, ϑ)) .
(4.8.3)
Now the space on which γ
μ
(τ , ζ, ϑ) is defined is
O := R × C
(4.8.4)
and we will refer to it as the “observer space-time”. Again, if some fields on that
space also depend on the observed frequency ˆ
ω, then they are actually defined on the
“spectral observer space-time”
O spec := O × R + .
(4.8.5)
We thus have that γ
μ
(τ , ζ, ϑ) probes all the points of space-time that are connected
to the observer world-line by some light-like geodesic, i.e. the image γ(O) is the
observable universe of that observer, by definition. The space O is parametrized by
the proper time τ , log-redshift ζ and angles ϑ
ˆ
A at which the corresponding signal
was observed, i.e. exactly the parameters the observer has access to in practice. It
is important, however, to notice that {τ , ζ, ϑ} is not a coordinate system on M in