Chapter 2
Motivation
Abstract In this chapter we motivate the use of the tetrad formulation of differential
geometry for computations in cosmology. We show that the introduction of a tetrad
is necessary, at least at the observation point, for defining cosmological observables
in an exact and coordinate-independent way, unless one works with unconventional
coordinate systems, that are also inconvenient for practical purposes. We then discuss
several conceptual and computational advantages of working with a tetrad field in
general.
2.1 Measurement and Observer Frame
In cosmological observations we measure the light of remote sources, both localized and diffuse. Denoting by M the space-time manifold, the measurement event
takes place in a neighborhood ˆ
I ⊂ M of the observer’s position ˆ
P ∈ M. We will
generically use a hat to denote evaluation at that point, or to denote quantities that
are only defined there.
1 For all practical purposes in cosmology, ˆ
I can be considered
to be of infinitesimal extent, i.e. just enough to give us access to the tangent space
T ˆ
P M. The information of cosmological observables is ultimately contained in the
momentum and polarization of the incoming photons, so what we measure are the
components of some tensors at ˆ
P, a statement which only makes sense with respect
to some basis of the tangent space T ˆ
P M. As shown in Appendix 6.1, for a basis
of T ˆ
P M to correspond to the frame with respect to which an observer is making
measurements at ˆ
P, it must be orthonormal with respect to the metric tensor g at ˆ
P.
This basis is then referred to as the “observer frame”.
So let us consider a set of four vectors ˆ
e a ∈ T ˆ
P M labeled by a ∈ {0, 1, 2, 3} and
satisfying the orthonormality condition
ˆ
g
ˆ
e a , ˆ
e b
≡ η ab ,
(2.1.1)
1 This notation is chosen such that it does not clog too much the equations and is inspired by the
fact that ˆ
P is the tip of the observer’s light-cone.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_2
11
Motivation
Abstract In this chapter we motivate the use of the tetrad formulation of differential
geometry for computations in cosmology. We show that the introduction of a tetrad
is necessary, at least at the observation point, for defining cosmological observables
in an exact and coordinate-independent way, unless one works with unconventional
coordinate systems, that are also inconvenient for practical purposes. We then discuss
several conceptual and computational advantages of working with a tetrad field in
general.
2.1 Measurement and Observer Frame
In cosmological observations we measure the light of remote sources, both localized and diffuse. Denoting by M the space-time manifold, the measurement event
takes place in a neighborhood ˆ
I ⊂ M of the observer’s position ˆ
P ∈ M. We will
generically use a hat to denote evaluation at that point, or to denote quantities that
are only defined there.
1 For all practical purposes in cosmology, ˆ
I can be considered
to be of infinitesimal extent, i.e. just enough to give us access to the tangent space
T ˆ
P M. The information of cosmological observables is ultimately contained in the
momentum and polarization of the incoming photons, so what we measure are the
components of some tensors at ˆ
P, a statement which only makes sense with respect
to some basis of the tangent space T ˆ
P M. As shown in Appendix 6.1, for a basis
of T ˆ
P M to correspond to the frame with respect to which an observer is making
measurements at ˆ
P, it must be orthonormal with respect to the metric tensor g at ˆ
P.
This basis is then referred to as the “observer frame”.
So let us consider a set of four vectors ˆ
e a ∈ T ˆ
P M labeled by a ∈ {0, 1, 2, 3} and
satisfying the orthonormality condition
ˆ
g
ˆ
e a , ˆ
e b
≡ η ab ,
(2.1.1)
1 This notation is chosen such that it does not clog too much the equations and is inspired by the
fact that ˆ
P is the tip of the observer’s light-cone.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_2
11