2.1 Measurement and Observer Frame
13
Now that we have explained these geometrical aspects of observation, let us see
what happens if one does not introduce the tetrad vectors ˆ
e a at ˆ
P. In that case, the
only available basis of T ˆ
P M is ∂ μ , i.e. the one induced by some coordinate system
x
μ around ˆ
P. In general, this basis ∂ μ is not orthonormal at ˆ
P, since, by definition
g
∂ μ , ∂ ν
≡ g μν .
(2.1.4)
Thus, if one wishes to interpret ∂ μ | ˆ
P as the frame of some observer at ˆ
P, then the
coordinate system must satisfy
ˆ
g μν = η μν .
(2.1.5)
Put differently, (2.1.5) is required for the coordinate-induced components ˆ
T
ν 1 ...ν n
μ 1 ...μ m
of
some tensor T at ˆ
P to be the ones measured by some observer.
2 The condition (2.1.5)
seems pretty mild a priori, as it only constrains the coordinate system in the infinitesimal vicinity ˆ
I of ˆ
P. In fact, it is not even as constraining as a system of normal
coordinates around ˆ
P, since the latter further requires ∂ μ g νρ | ˆ
P = 0. Nevertheless,
as innocent as this condition may seem, it is not satisfied in the coordinate systems
that are usually employed in cosmology, and the associated perturbation theory, such
as the longitudinal and synchronous gauges. In particular, the practical gauges are
usually defined through global conditions, in contrast to the condition (2.1.5) which
is local.
We conclude that, in the typical coordinate systems employed in cosmology, the
tensor components cannot be interpreted as observables. An important exception
are time-like components, for which one usually explicitly invokes the observer 4velocity ˆ
u, i.e. the analogue of ˆ
e 0 , which therefore allows one to construct observables
such as ˆ
ω ≡ − ˆ
u μ ˆ
k
μ . In the normal subspace to ˆ
u, however, there are no reference
vectors to project onto, so we cannot extract the spatial components of tensors that
are actually being observed. In particular, this means that we do not have access to the
observed angles associated with incoming photons. The only available angles are the
ones extracted from the coordinate-induced components (1.0.2). Since ˆ
g μν = η μν ,
the basis {∂ x , ∂ y , ∂ z }| ˆ
P is neither orthonormal, nor normal to ˆ
u, so (θ, φ) are not the
angles an observer actually uses to parametrize the sky.
In the standard approaches to cosmological observables this problem is not
resolved, but rather hidden under the carpet of cosmological perturbation theory.
One starts by considering the homogeneous and isotropic space-time that appears as
the zeroth-order approximation of g. In the Friedmann-Lemaître-Robertson-Walker
(FLRW) coordinates of that “background” universe, the spatial metric at the observation point is flat, so the tensor components are indeed the ones measured by some
observer, the one at rest in these coordinates. One then parametrizes the observed sky
with the angles used by this observer ¯
θ, ¯
φ. However, when fluctuations are introduced,
their effect is taken into account through deviation angles δθ( ¯
θ, ¯
φ) and δφ( ¯
θ, ¯
φ) relat2 Note that one can reach the present approach by starting with a general ˆ
e a and then choosing the
coordinate system such that ˆ
e
μ
a → δ
μ
a , thus effectively identifying the a and μ indices.
13
Now that we have explained these geometrical aspects of observation, let us see
what happens if one does not introduce the tetrad vectors ˆ
e a at ˆ
P. In that case, the
only available basis of T ˆ
P M is ∂ μ , i.e. the one induced by some coordinate system
x
μ around ˆ
P. In general, this basis ∂ μ is not orthonormal at ˆ
P, since, by definition
g
∂ μ , ∂ ν
≡ g μν .
(2.1.4)
Thus, if one wishes to interpret ∂ μ | ˆ
P as the frame of some observer at ˆ
P, then the
coordinate system must satisfy
ˆ
g μν = η μν .
(2.1.5)
Put differently, (2.1.5) is required for the coordinate-induced components ˆ
T
ν 1 ...ν n
μ 1 ...μ m
of
some tensor T at ˆ
P to be the ones measured by some observer.
2 The condition (2.1.5)
seems pretty mild a priori, as it only constrains the coordinate system in the infinitesimal vicinity ˆ
I of ˆ
P. In fact, it is not even as constraining as a system of normal
coordinates around ˆ
P, since the latter further requires ∂ μ g νρ | ˆ
P = 0. Nevertheless,
as innocent as this condition may seem, it is not satisfied in the coordinate systems
that are usually employed in cosmology, and the associated perturbation theory, such
as the longitudinal and synchronous gauges. In particular, the practical gauges are
usually defined through global conditions, in contrast to the condition (2.1.5) which
is local.
We conclude that, in the typical coordinate systems employed in cosmology, the
tensor components cannot be interpreted as observables. An important exception
are time-like components, for which one usually explicitly invokes the observer 4velocity ˆ
u, i.e. the analogue of ˆ
e 0 , which therefore allows one to construct observables
such as ˆ
ω ≡ − ˆ
u μ ˆ
k
μ . In the normal subspace to ˆ
u, however, there are no reference
vectors to project onto, so we cannot extract the spatial components of tensors that
are actually being observed. In particular, this means that we do not have access to the
observed angles associated with incoming photons. The only available angles are the
ones extracted from the coordinate-induced components (1.0.2). Since ˆ
g μν = η μν ,
the basis {∂ x , ∂ y , ∂ z }| ˆ
P is neither orthonormal, nor normal to ˆ
u, so (θ, φ) are not the
angles an observer actually uses to parametrize the sky.
In the standard approaches to cosmological observables this problem is not
resolved, but rather hidden under the carpet of cosmological perturbation theory.
One starts by considering the homogeneous and isotropic space-time that appears as
the zeroth-order approximation of g. In the Friedmann-Lemaître-Robertson-Walker
(FLRW) coordinates of that “background” universe, the spatial metric at the observation point is flat, so the tensor components are indeed the ones measured by some
observer, the one at rest in these coordinates. One then parametrizes the observed sky
with the angles used by this observer ¯
θ, ¯
φ. However, when fluctuations are introduced,
their effect is taken into account through deviation angles δθ( ¯
θ, ¯
φ) and δφ( ¯
θ, ¯
φ) relat2 Note that one can reach the present approach by starting with a general ˆ
e a and then choosing the
coordinate system such that ˆ
e
μ
a → δ
μ
a , thus effectively identifying the a and μ indices.