Chapter 3
Mathematical Framework
Abstract This chapter contains a detailed presentation of all the geometrical structures, in the tetrad formalism, that are involved in the computations of this book.
We discuss the fields, equations and symmetries of three distinct manifolds: the line
manifold on which particles’ paths are defined, the space-time manifold on which
the usual space-time fields are defined and the one-particle phase space manifold on
which Boltzmann distributions are defined.
3.1 General Preliminaries
The mathematical framework we are going to discuss in this section mainly involves
three distinct manifolds:
• M: the 4-dimensional space-time manifold on which space-time fields are based,
• L: the 1-dimensional line manifold on which the world-line fields are based,
• T
∗
M: the 8-dimensional cotangent bundle manifold on which the Boltzmann
distributions are based.
These are respectively presented in each one of the following subsections, along
with their associated symmetries and the equations of motion for the fields they host.
In particular, in each case, we provide the description in the tetrad formalism.
Before we jump into the details, let us discuss an important point that has to
do with the mathematical description of the involved physical degrees of freedom
and leads in particular to two complementary perspectives to electromagnetism. In
the cosmological setting, the space-time fields (defined on M) correspond to the
coherent “long” wave-length fluctuations of the underlying quantum fields, which
are therefore well described classically.
1 Here we will consider the gravitational and
electromagnetic contributions, but one could also include extra degrees of freedom
that appear for instance in theories of inflation or dark energy. For coherent long
wave-length and low frequency excitations to arise the corresponding field must be
1 We adopt the effective field theory approach to gravity, so that it makes sense to talk about a
corresponding quantum field, even if it is not fundamental.
© 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_3
21
Mathematical Framework
Abstract This chapter contains a detailed presentation of all the geometrical structures, in the tetrad formalism, that are involved in the computations of this book.
We discuss the fields, equations and symmetries of three distinct manifolds: the line
manifold on which particles’ paths are defined, the space-time manifold on which
the usual space-time fields are defined and the one-particle phase space manifold on
which Boltzmann distributions are defined.
3.1 General Preliminaries
The mathematical framework we are going to discuss in this section mainly involves
three distinct manifolds:
• M: the 4-dimensional space-time manifold on which space-time fields are based,
• L: the 1-dimensional line manifold on which the world-line fields are based,
• T
∗
M: the 8-dimensional cotangent bundle manifold on which the Boltzmann
distributions are based.
These are respectively presented in each one of the following subsections, along
with their associated symmetries and the equations of motion for the fields they host.
In particular, in each case, we provide the description in the tetrad formalism.
Before we jump into the details, let us discuss an important point that has to
do with the mathematical description of the involved physical degrees of freedom
and leads in particular to two complementary perspectives to electromagnetism. In
the cosmological setting, the space-time fields (defined on M) correspond to the
coherent “long” wave-length fluctuations of the underlying quantum fields, which
are therefore well described classically.
1 Here we will consider the gravitational and
electromagnetic contributions, but one could also include extra degrees of freedom
that appear for instance in theories of inflation or dark energy. For coherent long
wave-length and low frequency excitations to arise the corresponding field must be
1 We adopt the effective field theory approach to gravity, so that it makes sense to talk about a
corresponding quantum field, even if it is not fundamental.
© 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_3
21