COMENIUS UNIVERSITY IN BRATISLAVA
FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS

DUALITY OPERATOR IN TELEPARALLEL GRAVITY
MASTER’S THESIS

2024

Bc. Marek Horňák

COMENIUS UNIVERSITY IN BRATISLAVA
FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS

DUALITY OPERATOR IN TELEPARALLEL GRAVITY
MASTER’S THESIS

Study program: | Theoretical physics
---|---
Field: | Physics
Department: | Department of Theoretical Physics
Supervisor: | Mgr. Martin Krššák, Dr.rer.nat.

Bratislava 2024

Bc. Marek Horňák

### Acknowledgements

I would like to express my thanks to my supervisor, Mgr. Martin Krššák,
Dr.rer.nat., for all the helpful discussions, insights, advice and all his
patience throughout the journey. I am also grateful to my friends and family
for their unwavering support and encouragement.

### Declaration

I declare that I have written this thesis on my own under the guidance of my
supervisor and using the literature listed in the bibliography.

Bratislava 2024

Bc. Marek Horňák

### Abstract

Author: | Bc. Marek Horňák
---|---
Title: | Duality Operator in Teleparallel Gravity
University: | Comenius University in Bratislava
Faculty: | Faculty of Mathematics, Physics and Informatics
Department: | Department of Theoretical Physics
Supervisor: | Mgr. Martin Krššák, Dr.rer.nat.
City: | Bratislava
Date: | 28.8.2024
Number of pages: | LABEL:LastPage
Type of thesis: | Master’s thesis

Teleparallel gravity is a theory of gravity which replaces the Levi-Civita
connection by a teleparallel connection – a metric-compatible connection with
vanishing curvature. Teleparallel equivalent of general relativity (TEGR) is a
special case from this class of theories, and its dynamics is governed by
equations which are equivalent to Einstein field equations of general
relativity. In 2009 Lucas and Pereira presented the idea of a new operator
that would allow us to express the action of TEGR in a form reminiscent of the
Yang-Mills action for a gauge theory, for which torsion is the field strength.
The authors constructed the operator as a duality operator acting on torsion
2-forms and curvature 2-forms, taking into account all the contractions of
these forms with the volume form of the spacetime manifold. The definition of
this operator for other forms, however, remained unclear. In this thesis, we
seek to give a mathematically consistent definition of the operator, and we
follow the results published by Lucas and Pereira. We show that the approach
taken by Lucas and Pereira does not lead to a consistent definition of this
operator, because it results in the operator with the desired properties
either being non-unique or non-existent.

Keywords:

duality operator, teleparallel gravity, tetrad formalism, torsion

###### Contents

  1. 1 Introduction
  2. 2 Metric-affine geometries
    1. 2.1 Linear connection
    2. 2.2 Special cases of metric-affine geometries
    3. 2.3 Connection decomposition
  3. 3 Tetrads
    1. 3.1 Holonomic and anholonomic frames
    2. 3.2 Orthonormal frames
    3. 3.3 Connection, torsion and curvature forms
    4. 3.4 General teleparallel connection
  4. 4 Teleparallel gravity
    1. 4.1 Action of teleparallel gravity
    2. 4.2 Equations of motion
    3. 4.3 Alternative form of action
  5. 5 New duality operator
    1. 5.1 Duality
    2. 5.2 Dual torsion
      1. 5.2.1 Rules for possible contractions
      2. 5.2.2 Duality condition
      3. 5.2.3 Additional rule
    3. 5.3 Dual connection
      1. 5.3.1 Possible contractions and parametrizations
      2. 5.3.2 Duality condition
      3. 5.3.3 Using the additional rule
      4. 5.3.4 Transformation of dual connection
    4. 5.4 Dual curvature
      1. 5.4.1 Possible contractions and parametrization
      2. 5.4.2 Duality condition
      3. 5.4.3 Applying the additional rule
      4. 5.4.4 Dual RLC curvature
  6. 6 Applications
    1. 6.1 Teleparallel action as Yang-Mills action for gauge theory
      1. 6.1.1 Teleparallel gravity as gauge theory – motivation
      2. 6.1.2 Gauge theory of translation group
    2. 6.2 Teleparallel gravity as analogue of electrodynamics
  7. 7 Conclusions
  8. A Calculation of dual torsion
  9. B Dual connection

##  Chapter 1 Introduction

General relativity is an exceptionally successful theory of gravity. To this
day, it stood several observational tests on almost all scales of physics and
remains together with quantum field theory one of the pillars of modern
physics.

Albert Einstein formulated the field equations of general relativity in 1915.
The fundamental idea of the theory is that gravity is a manifestation of the
curvature of spacetime. It entirely disposed of the notion of gravitational
force and described the motion of small bodies in the gravitational field as
the natural motion of the bodies in the curved spacetime. The source of the
curvature of spacetime is energy and momentum, as expressed by the Einstein
field equations.

In 1920s, Einstein – who, of course, could not have been at that time aware of
the need to modify his newly-established theory of gravity – was attempting to
develop a unified theory of gravity and electromagnetism based on the use of
tetrads (orthonormal frames) and distant parallelism (teleparallelism). After
publishing eight papers on the topic in the Sitzungsberichte of the Prussian
Academy, he abandoned his idea because it did not lead to his ultimate goal of
unification of electromagnetism and gravity [1]. Nevertheless, Einstein’s work
involved several important novelties, namely using tetrads instead of the
metric tensor and considering other than Riemannian geometries to describe
gravity. In 1960s and 1970s it became clear that Einstein teleparallelism can
be a successful theory of gravity only, i.e. without attempts to incorporate
electromagnetism [2].

The development of quantum mechanics in 1920s and then quantum field theory in
the following years started to expose some of the limitations of general
relativity. Another challenge for general relativity came with the
observations of the accelerating expansion of the universe [3]. One possible
approach to explain these phenomena is making modifications in the matter
sector of the theory, which requires modifications of the Standard Model of
particle physics. However, it may also be the case that it is not necessary to
significantly restructure the Standard Model (the matter section), but that
the gravitational section requires further revisiting. Thus, alternative
theories of gravity are of considerable interest, because, instead of changing
the matter content of the theory, an alternative theory of gravity may
potentially provide an explanation for phenomena which within the current
cosmological paradigm based on general relativity are ascribed to the presence
of dark energy and dark matter [4, 5].

Our current understanding of elementary particles, which make up matter, and
three of the four fundamental interactions of nature – electromagnetic, strong
and weak – is captured by the Standard Model of particle physics. It is a
quantum field theory in which all the interactions are mediated by so-called
gauge bosons111The gauge bosons of Standard Model are photons (which mediate
the electromagnetic interaction), eight gluons (for the strong interaction)
and $W^{+},W^{-}$ and $Z^{0}$ bosons (they mediate the weak interaction)..
Standard Model is an example of a gauge theory.

Gauge theories have a special place among the field theories. One of the
reasons is that they can be quantized. Gauge bosons are particles which
correspond to gauge fields, which are fields that have to be introduced into a
theory to make a global symmetry of this theory to be local. In other words,
gauge theories have some unphysical degrees of freedom. The same physical
fields correspond to several different configurations of the gauge fields.
This is a kind of redundancy in the theoretical description of reality.

Examples of gauge theories include the Standard Model and electrodynamics
(classical, as well as quantum), which is a $U(1)$-gauge theory – which means
that it is invariant under $U(1)$ local transformations – and the gauge bosons
of this theory are photons. Of great importance for particle physics are Yang-
Mills theories, which are non-abelian gauge theories. An example of a Yang-
Mills theory is quantum chromodynamics (QCD), which is a $SU(3)$-gauge theory.

General relativity is coordinate-independent, which means that the laws are
invariant under changes of coordinates. In other words, the equations take the
same form in any coordinates. This, however, does not make it a gauge theory
from the mathematical point of view, because conceptually this is a different
kind of redundancy in the physical model. On the other hand, all the other
fundamental forces of physics are described by gauge theories, so it is
natural to expect that gravity too can be described in the framework of a
gauge theory.

There exist several modified theories of gravity, and also attempts to
conceptualize general relativity as a gauge theory. An interesting possibility
that has been gaining momentum in the recent decades is teleparallel gravity.
The interest in teleparallelism was revived in 1960s and 1970s by the works of
Møller concerning the definition of the energy-momentum for gravity [6, 7, 8],
and then by Pellegrini and Plebański [9]. Another impulse for revisiting
Einstein’s teleparalellism came from the work of Hayashi and Nakano on gauge
aspects of gravity [10].

While in general relativity the gravitational interaction is realized via the
curvature of the Levi-Civita connection, which is metric-compatible and
torsion-free, in teleparallel gravity the interaction is an effect of torsion
of a teleparallel connection, that is, a metric-compatible connection with
zero curvature. Torsion in this case acts as a force, which similarly to the
Lorentz force equation of electromagnetism appears as an effective force term
on the right-hand side of the equation of motion of a freefalling particle
[11].

One of the many theories that this torsion-based approach has produced is the
teleparallel equivalent of general relativity (TEGR), which is dynamically
equivalent to general relativity, meaning that they cannot be distinguished
through classical experiments. However, they are conceptually quite different
and in scenarios such as quantum gravity the theories may exhibit different
behaviors.

In 1967 Hayashi and Nakano published a paper [10] which discussed teleparallel
gravity as gauge theory of translations only, instead of gauging the Lorentz
or Poincaré group. The Lagrangian density of the theory is quadratic in
torsion, which is identified as the field strength of the translational group.
Later, Cho has shown [12] that requiring covariance of the field equations
restricts the form of the translational Lagrangian to the Lagrangian of TEGR,
which Einstein considered already in 1929 [13]. TEGR may thus be regarded as a
gauge theory of translations.

The action of TEGR may be rewritten in a form revealing a striking similarity
with the Yang-Mills action for a gauge theory with field strength
$\mathcal{F}$:

$\mathcal{S}_{YM}=-\int_{\mathcal{M}}{\operatorname{Tr}\left(\mathcal{F}\wedge\star\mathcal{F}\right)}.$
(1.1)

The Hodge $\star$ operator, however, has to be replaced by a different
operator, which we shall denote
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$.
It turns out that if
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
is to be an operator, then
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}T=-T$,
where $T$ is the torsion. This is true for the Hodge $\star$ operator too:
$\star\star T=-T$.

In the paper titled “Hodge Dual for Soldered Bundles” [14], Lucas and Pereira
suggest that this
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
actually is an operator222Lucas and Pereira denote it simply by $\star$, but
we would like to make the distinction between this newly-proposed operator and
the Hodge operator clear., and that its existence is a consequence of the
presence of the solder form on the frame bundle. Their primary condition is
that the operator be a duality operator, so that
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}=\pm
1$. However, the details of the definition of this operator are unclear.

The aim of this thesis is to present the fundamental features of teleparallel
gravity. Later, we explore the possibility of a proper and mathematically
consistent definition of the
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
duality operator on soldered bundles presented in [14, 15, 16].

In chapter 2, we start in a more general setting of metric-affine geometries.
We derive the formula for decomposition of the coefficients of a general
affine connection on a Riemannian manifold into the sum of three terms.

Chapter 3 contains the description of the tetrad formalism and the definition
of connection 1-forms, torsion 2-forms and curvature 2-forms. We also show
that the general teleparallel spin connection may be expressed as
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}=\left(\Lambda^{-1}\right)^{a}_{\phantom{a}c}\partial_{\mu}\Lambda^{c}_{\phantom{c}b}$
for some $\Lambda\in SO(1,3)$.

In chapter 4 we finally turn our attention to teleparallel gravity. We derive
the action of TEGR, which leads to equations of motion. Then we rewrite the
action in a form resembling the Yang-Mills action of a gauge theory.

Chapter 5 explores the possibility of a proper definition of the new duality
operator on soldered bundles introduced by Lucas and Pereira [14]. Their
construction of this operator is not entirely clear, so in this chapter we
state the rules for its construction in an attempt to be consistent. These
rules are motivated by expecting this new operator to be similar to the Hodge
star operator and also by observing the results of Lucas and Pereira. We
proceed to determine how the operator behaves when applied on torsion 2-forms,
connection 1-forms and curvature 2-forms.

The final chapter, chapter 6, is dedicated to potential applications of this
new duality operator for soldered bundles. We also give a general idea behind
the gauge structure of TEGR.

#### Notation

The signature of a Lorentzian metric is $(-,+,+,+)$ and the Minkowski metric
is $\eta=\operatorname{diag}\left(-1,+1,+1,+1\right)$. Objects related to the
Levi-Civita connection will by denoted by a circle over the symbol, e.g.
$\accentset{\circ}{\nabla},\accentset{\circ}{\Gamma}^{\alpha}_{\mu\nu},\accentset{\circ}{R}$,
while objects related to the metric teleparallel connection will be denoted by
a full circle over the symbol, e.g.
$\accentset{\bullet}{\nabla},\accentset{\bullet}{\Gamma}^{\alpha}_{\mu\nu},\accentset{\bullet}{R}$.
Throughout the entire thesis, we use units in which the speed of light is
$c=1$.

##  Chapter 2 Metric-affine geometries

The mathematical formalism of general relativity appears to only require a
differentiable spacetime manifold with with one geometric object defined on it
– the metric tensor $g$. A differentiable manifold endowed with a metric
tensor is called Riemannian manifold. However, a closer investigation reveals
that there is another geometric structure needed in order to define covariant
derivative, which is necessary to define the curvature tensor. This structure
is called affine connection. In general relativity, the choice of a connection
is special in the sense that it is completely dependent on the metric, and
nothing else. In a more general geometric context, however, a connection on
the manifold is independent of the metric. Even non-Riemannian manifolds may
have a connection defined on them.

###  2.1 Linear connection

For the basics of differential geometry we follow here the exposition and
notation from the textbook by Marián Fecko [17].

######  Definition 2.1.1.

Affine connection $\nabla$ on a differentiable manifold $\mathcal{M}$ is a
mathematical structure such that to any vector field $W$ on $\mathcal{M}$ it
assigns an operator $\nabla_{W}$, the covariant derivative in the direction of
$W$, with the following properties:

  1. 1.

it is a linear operator on the tensor algebra, and it preserves the type of
the tensor

$\displaystyle\nabla_{W}:\mathcal{T}^{p}_{q}\left(\mathcal{M}\right)\longrightarrow\mathcal{T}^{p}_{q}\left(\mathcal{M}\right),$
$\displaystyle\nabla_{W}\left(A+\lambda
B\right)=\nabla_{W}A+\lambda\nabla_{W}B,$ $\displaystyle
A,B\in\mathcal{T}^{p}_{q}\left(\mathcal{M}\right),\lambda\in\mathbb{R}.$

  2. 2.

its behavior on the tensor product obeys Leibniz rule

$\displaystyle\nabla_{W}\left(A\otimes
B\right)=\left(\nabla_{W}A\right)\otimes B+A\otimes\left(\nabla_{W}B\right),$
$\displaystyle
A\in\mathcal{T}^{p}_{q}\left(\mathcal{M}\right),B\in\mathcal{T}^{p^{\prime}}_{q^{\prime}}\left(\mathcal{M}\right).$

  3. 3.

on type $\binom{0}{0}$ tensors (functions) it gives

$\displaystyle\nabla_{W}\psi=W\psi,$
$\displaystyle\psi\in\mathcal{F}\left(\mathcal{M}\right)\equiv\mathcal{T}^{0}_{0}\left(\mathcal{M}\right).$

  4. 4.

it commutes with contractions

$\displaystyle\nabla_{W}\circ C=C\circ\nabla_{W},$

where $C$ is any contraction.

  5. 5.

it is $\mathcal{F}$-linear in $W$, i.e.

$\displaystyle\nabla_{V+fW}=\nabla_{V}+f\nabla_{W},$ $\displaystyle
V,W\in\mathcal{T}^{1}_{0}\left(\mathcal{M}\right),f\in\mathcal{F}\left(\mathcal{M}\right).$

Let $\mathcal{M}$ be a smooth manifold with a metric tensor $g$ and an affine
connection $\nabla$ and let $\partial_{\mu}$ be a coordinate basis on
$\mathcal{M}$. We denote $\Gamma^{\rho}_{\nu\mu}$ the Christoffel symbols (of
the second kind) for the connection:
$\nabla_{\partial_{\mu}}\partial_{\nu}=\Gamma^{\rho}_{\nu\mu}\partial_{\rho}$.
The Christoffel symbols of the first kind are then defined by
$\Gamma_{\rho\nu\mu}\coloneqq g_{\rho\lambda}\Gamma^{\lambda}_{\nu\mu}$.

######  Definition 2.1.2.

Curvature $R$ is a tensor of type $\binom{1}{3}$, i.e.
$R\in\mathcal{T}^{1}_{3}\left(\mathcal{M}\right)$, defined by

$R\left(\alpha,U,V,W\right)=\langle\alpha,\left(\left[\nabla_{U},\nabla_{V}\right]-\nabla_{\left[U,V\right]}\right)W\rangle.$
(2.1)

The components of curvature with respect to the basis $\partial_{\mu}$ are

$R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}=\partial_{\mu}\Gamma^{\alpha}_{\beta\nu}-\partial_{\nu}\Gamma^{\alpha}_{\beta\mu}+\Gamma^{\alpha}_{\lambda\mu}\Gamma^{\lambda}_{\beta\nu}-\Gamma^{\alpha}_{\lambda\nu}\Gamma^{\lambda}_{\beta\mu}.$
(2.2)

######  Definition 2.1.3.

Torsion $T$ is a tensor of type $\binom{1}{2}$, i.e.
$T\in\mathcal{T}^{1}_{2}\left(\mathcal{M}\right)$, defined by

$T\left(\alpha,U,V\right)=\langle\alpha,\nabla_{U}V-\nabla_{V}U-\left[U,V\right]\rangle.$
(2.3)

The components of torsion with respect to $\partial_{\mu}$ are

$T^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\nu\mu}-\Gamma^{\rho}_{\mu\nu}.$ (2.4)

######  Definition 2.1.4.

Non-metricity $Q$ is a tensor of type $\binom{0}{3}$, i.e.
$Q\in\mathcal{T}^{0}_{3}\left(\mathcal{M}\right)$, defined by

$Q\left(U,V,W\right)=\left(\nabla_{U}g\right)\left(V,W\right).$ (2.5)

The components of non-metricity with respect to $\partial_{\mu}$ are

$Q_{\mu\nu\rho}=\nabla_{\mu}g_{\nu\rho}=\partial_{\mu}g_{\nu\rho}-\Gamma^{\lambda}_{\nu\mu}g_{\lambda\rho}-\Gamma^{\lambda}_{\rho\mu}g_{\nu\lambda}.$
(2.6)

###  2.2 Special cases of metric-affine geometries

Metric-affine geometries are concerned with the study of geometry on smooth
manifolds equipped with a metric tensor and an affine connection. For a
general metric-affine setting, any of the tensorial quantities defined in the
previous section may be nonzero. It may also happen that all of them are
nonvanishing. However, several special cases of metric-affine geometries are
obtained by requiring that some of these tensors be zero [18, 19].

  * •

Flat connection: $R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\equiv 0$
If both torsion and non-metricity are present, this case is known as (general)
teleparallel geometry. [20]

  * •

Symmetric connection: $T^{\rho}_{\phantom{\rho}\mu\nu}\equiv 0$
In this case, $\Gamma^{\rho}_{\nu\mu}=\Gamma^{\rho}_{\mu\nu}$. This case is
considered, e.g., in Weyl geometry used in Weyl gravity.

  * •

Metric-compatible connection: $Q_{\mu\nu\rho}\equiv 0$
Geometry with metric-compatible connection is known as Riemann-Cartan
geometry. Such connections are used in Poincaré gauge theory. [18]

  * •

Metric teleparallel geometry: $R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\equiv
0$, $Q_{\mu\nu\rho}\equiv 0$
We will be primarily concerned with this case. From now, we will refer to this
case as teleparallel geometry. [16]

  * •

Symmetric teleparallelism: $R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\equiv 0$,
$T^{\rho}_{\phantom{\rho}\mu\nu}\equiv 0$ [21]

  * •

RLC connection 111“RLC” in the name of the connection stands for “Riemann-
Levi-Civita”, but is also a pun used by Marián Fecko in his book [17].:
$T^{\rho}_{\phantom{\rho}\mu\nu}\equiv 0$, $Q_{\mu\nu\rho}\equiv 0$
The most well-known class of geometries, which is used in the description of
gravity in general relativity. We will denote the Christoffel symbols of the
RLC connection by a circle: $\accentset{\circ}{\Gamma}^{\rho}_{\mu\nu}$.

  * •

(Locally) Minkowski space: $R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\equiv 0$
, $T^{\rho}_{\phantom{\rho}\mu\nu}\equiv 0$, $Q_{\mu\nu\rho}\equiv 0$
Note that here (as well as in the previous cases) only local geometry of the
space is being discussed. Its global topology may be different from that of
Minkowski space. Consider, for example [17], the metric induced from embedding
of torus $T^{4}$ in $\mathbb{R}^{8}$ with metric tensor

$g=-dx^{1}\otimes dx^{1}-dx^{2}\otimes dx^{2}+dx^{3}\otimes
dx^{3}+\dots+dx^{8}\otimes dx^{8}.$ (2.7)

On $T^{4}$ we use coordinates $\tau,\alpha,\beta,\gamma$ so that

$\displaystyle x^{1}=\cos\tau,$ $\displaystyle x^{2}=\sin\tau,$ $\displaystyle
x^{3}=\cos\alpha,$ $\displaystyle x^{4}=\sin\alpha,$ $\displaystyle
x^{5}=\cos\beta,$ $\displaystyle x^{6}=\sin\beta,$ $\displaystyle
x^{7}=\cos\gamma,$ $\displaystyle x^{8}=\sin\gamma.$

The metric induced on $T^{4}$ is then

$g=-d\tau\otimes d\tau+d\alpha\otimes d\alpha+d\beta\otimes
d\beta+d\gamma\otimes d\gamma.$ (2.8)

Obviously, the RLC connection of $T^{4}$ with this metric has zero curvature,
while the global topology of $T^{4}$ is different from Minkowski space
$\mathbb{R}^{4}$.

Throughout the text, we will denote the quantities related to the (metric)
teleparallel connection by a full circle over the symbol, e.g. the
teleparallel connection will be $\accentset{\bullet}{\nabla}$.

###  2.3 Connection decomposition

The Christoffel symbols of a RLC connection on $\mathcal{M}$ may be expressed
in terms of the metric tensor $g_{\mu\nu}$ and its derivatives as

$\accentset{\circ}{\Gamma}^{\rho}_{\mu\nu}=\dfrac{1}{2}g^{\rho\lambda}\left(g_{\lambda\mu,\nu}+g_{\lambda\nu,\mu}-g_{\mu\nu,\lambda}\right).$
(2.9)

We may expect that for a general connection we will obtain a similar result,
but now containing terms that depend on torsion and non-metricity. We start
with the three equations

$\displaystyle Q_{\mu\nu\rho}$
$\displaystyle=g_{\rho\nu,\mu}-\Gamma^{\lambda}_{\nu\mu}g_{\lambda\rho}-\Gamma^{\lambda}_{\rho\mu}g_{\nu\lambda},$
$\displaystyle Q_{\nu\mu\rho}$
$\displaystyle=g_{\rho\mu,\nu}-\Gamma^{\lambda}_{\mu\nu}g_{\lambda\rho}-\Gamma^{\lambda}_{\rho\nu}g_{\mu\lambda},$
$\displaystyle-Q_{\rho\nu\mu}$
$\displaystyle=-g_{\mu\nu,\rho}+\Gamma^{\lambda}_{\nu\rho}g_{\lambda\mu}+\Gamma^{\lambda}_{\mu\rho}g_{\nu\lambda}.$

Taking the sum and using (2.9), we get

$\displaystyle Q_{\mu\nu\rho}+Q_{\nu\mu\rho}-Q_{\rho\nu\mu}$
$\displaystyle=2g_{\rho\lambda}\accentset{\circ}{\Gamma}^{\lambda}_{\mu\nu}-g_{\lambda\rho}\left(\Gamma^{\lambda}_{\mu\nu}+\Gamma^{\lambda}_{\nu\mu}\right)$
(2.10)
$\displaystyle+g_{\lambda\mu}\left(\Gamma^{\lambda}_{\nu\rho}-\Gamma^{\lambda}_{\rho\nu}\right)+g_{\lambda\nu}\left(\Gamma^{\lambda}_{\mu\rho}-\Gamma^{\lambda}_{\rho\mu}\right),$
(2.11)

and finally, using (2.4)

$\displaystyle Q_{\mu\nu\rho}+Q_{\nu\mu\rho}-Q_{\rho\nu\mu}$
$\displaystyle=2g_{\rho\lambda}\accentset{\circ}{\Gamma}^{\lambda}_{\mu\nu}-2g_{\lambda\rho}\Gamma^{\lambda}_{\mu\nu}-g_{\lambda\rho}T^{\lambda}_{\mu\nu}+g_{\lambda\mu}T^{\lambda}_{\rho\nu}+g_{\lambda\nu}T^{\lambda}_{\rho\mu}.$
(2.12)

Rearranging and raising the index $\rho$

$\displaystyle\Gamma^{\rho}_{\mu\nu}$
$\displaystyle=\accentset{\circ}{\Gamma}^{\rho}_{\mu\nu}+\dfrac{1}{2}\left(-T^{\rho}_{\phantom{\rho}\mu\nu}+T^{\phantom{\mu}\rho}_{\mu\phantom{\rho}\nu}+T^{\phantom{\nu}\rho}_{\nu\phantom{\rho}\mu}\right)-\dfrac{1}{2}\left(Q_{\mu\nu}^{\phantom{\mu\nu}\rho}+Q_{\nu\mu}^{\phantom{\nu\mu}\rho}-Q^{\rho}_{\phantom{\rho}\nu\mu}\right).$
(2.13)

We see that the connection (or more precisely, the Christoffel symbols of the
connection) may be decomposed and expressed as [19]

$\Gamma^{\rho}_{\mu\nu}=\accentset{\circ}{\Gamma}^{\rho}_{\mu\nu}+K^{\rho}_{\phantom{\rho}\mu\nu}+L^{\rho}_{\phantom{\rho}\mu\nu},$
(2.14)

where we have introduced the contortion

$K^{\rho}_{\phantom{\rho}\mu\nu}\coloneqq\dfrac{1}{2}\left(T^{\phantom{\mu}\rho}_{\mu\phantom{\rho}\nu}+T^{\phantom{\nu}\rho}_{\nu\phantom{\rho}\mu}-T^{\rho}_{\phantom{\rho}\mu\nu}\right)$
(2.15)

and the disformation

$L^{\rho}_{\phantom{\rho}\mu\nu}\coloneqq\dfrac{1}{2}\left(Q^{\rho}_{\phantom{\rho}\mu\nu}-Q^{\phantom{\mu}\rho}_{\mu\phantom{\rho}\nu}-Q^{\phantom{\nu}\rho}_{\nu\phantom{\rho}\mu}\right).$
(2.16)

Both are tensors of type $\binom{1}{2}$. Note that while disformation is
symmetric, contortion is neither symmetric, nor antisymmetric in the lower
indices.

##  Chapter 3 Tetrads

General relativity is essentialy just geometry on a Riemannian manifold. In
order to do any calculations on a manifold we need to use coordinates,
although the results of such calculations remain independent of the choice of
coordinates. In the standard formulation of general relativity, all tensors
are expressed with respect to coordinate bases and Einstein field equations
are also expressed using a coordinate basis. However, it is possible, and
sometimes even convenient, to express them with respect to a non-coordinate
basis.

###  3.1 Holonomic and anholonomic frames

Let $e_{a}$ be a frame defined in some region $\mathcal{O}\subset\mathcal{M}$
(i.e. a basis of $T_{p}\mathcal{M}$ in each $p\in\mathcal{O}$) and let
$x^{\alpha}$ be coordinates in some region
$\tilde{\mathcal{O}}\subset\mathcal{M}$ such that
$\mathcal{O}\cap\tilde{\mathcal{O}}\neq\emptyset$. As $e_{a}$ and
$\partial_{\alpha}$ are both bases, there exists a (non-singular) matrix
$e_{a}^{\phantom{a}\mu}(x)$ such that

$e_{a}=e_{a}^{\phantom{a}\mu}(x)\partial_{\mu}.$ (3.1)

Similarly, we define the matrix $e^{a}_{\phantom{a}\mu}(x)$, which relates the
corresponding dual bases:

$e^{a}=e^{a}_{\phantom{a}\mu}(x)dx^{\mu}.$ (3.2)

The condition of duality gives us

$\displaystyle e^{a}_{\phantom{b}\mu}e_{b}^{\phantom{b}\mu}=\delta^{a}_{b},$
$\displaystyle
e_{a}^{\phantom{a}\mu}e^{a}_{\phantom{a}\nu}=\delta^{\mu}_{\nu}.$ (3.3)

Now we have two frames in a certain region: the coordinate frame
$\partial_{\alpha}$ and the frame $e_{a}$. The components of a tensor $t$ with
respect to the coordinate frame $\partial_{\alpha}$ are

$t^{\alpha\dots\beta}_{\phantom{\alpha\dots\beta}\gamma\dots\delta}=t(dx^{\alpha},\dots,dx^{\beta};\partial_{\gamma},\dots,\partial_{\delta}),$
(3.4)

while its components with respect to the frame $e_{a}$ are

$t^{a\dots b}_{\phantom{a\dots b}c\dots
d}=t(e^{a},\dots,e^{b};e_{c},\dots,e_{d}).$ (3.5)

We may wish to express the components with mixed indices, e.g.

$t^{\alpha\dots\beta a\dots b}_{\phantom{\alpha\dots\beta a\dots
b}\gamma\dots\delta c\dots
d}=t(dx^{\alpha},\dots,dx^{\beta},e^{a},\dots,e^{b};\partial_{\gamma},\dots,\partial_{\delta},e_{c},\dots,e_{d}).$
(3.6)

Strictly speaking, these are not the components of the tensor (3.4). The
components of a tensor are defined by the tensor evaluated on vectors of one
basis and covectors of the corresponding dual basis – like in (3.4) – but here
we are using two bases $\partial_{\alpha}$ and $e_{a}$.

It is clear that an upper Greek index may be changed to a Latin index by
contracting with $e^{a}_{\phantom{a}\mu}$:

$t^{\dots a\dots}_{\phantom{\dots
a\dots}\dots\dots}=e^{a}_{\phantom{a}\mu}t^{\dots\mu\dots}_{\phantom{\dots\mu\dots}\dots\dots},$
(3.7)

an upper Latin index may be changed to a Greek index by contracting with
$e_{a}^{\phantom{a}\mu}$:

$t^{\dots\mu\dots}_{\phantom{\dots\mu\dots}\dots\dots}=e_{a}^{\phantom{a}\mu}\,t^{\dots
a\dots}_{\phantom{\dots a\dots}\dots\dots},$ (3.8)

etc.

We define the coefficients of anholonomy $c^{a}_{bc}(x)$ for a given frame
$e_{a}$ by the relation

$\left[e_{b},e_{c}\right]=c^{a}_{bc}(x)e_{a}.$ (3.9)

For any coordinate frame,

$\left[\partial_{\alpha},\partial_{\beta}\right]=0.$ (3.10)

That means that the coefficients of anholonomy for a coordinate frame vanish.
Therefore, these frames are holonomic. The reverse statement is also true: if,
for a given frame $e_{a}$, it holds that $\left[e_{a},e_{b}\right]=0$, then
$e_{a}$ is a coordinate frame, that is, in some neighbourhood of any point
there exist local coordinates $x^{a}$ such that $e_{a}=\partial_{a}$. Frames
for which $c^{a}_{bc}(x)\neq 0$ are called anholonomic [17].

Any two bases $E_{a},\tilde{E}_{b}$ of the same 4-dimensional real vector
space $L$ are related via some $A\in GL\left(4,\mathbb{R}\right)$:

$E_{a}=A^{b}_{\phantom{b}a}\tilde{E}_{b}.$ (3.11)

As any two frames $e_{a},\tilde{e}_{a}$ on a 4-dimensional manifold
$\mathcal{M}$ are at any given point $x\in\mathcal{M}$ just two bases of the
tangent space $T_{x}\mathcal{M}$, which is a real vector space, the frames are
related by local $GL\left(4,\mathbb{R}\right)$ transformation:

$e_{a}(x)=A^{b}_{\phantom{b}a}(x)\tilde{e}_{b}(x).$ (3.12)

###  3.2 Orthonormal frames

Suppose that $\mathcal{M}$ is a Riemannian manifold with metric $g$. A special
class of frames on $\mathcal{M}$ are the orthonormal frames. These frames are
called vielbeins, and in the case $\dim\mathcal{M}=4$ they are called
vierbeins or tetrads [22]. We will adopt the following convention: we will
denote general frames by $e_{a}$, while tetrads by $h_{a}$. Authors commonly
use the term tetrad to refer to any of
$h_{a},h^{a},h^{a}_{\phantom{a}\mu},h_{a}^{\phantom{a}\mu}$.

By definition,

$g(h_{a},h_{b})=\eta_{ab},$ (3.13)

which means that

$\displaystyle g=\eta_{ab}h^{a}\otimes h^{b},$ $\displaystyle
g_{\mu\nu}=\eta_{ab}h^{a}_{\phantom{a}\mu}h^{b}_{\phantom{b}\nu}.$ (3.14)

Of course, we can lower the Greek indices by contraction with $g_{\mu\nu}$ and
raise them by contracting with $g^{\mu\nu}$. However, Latin indices are
lowered by $\eta_{ab}$ and raised by $\eta^{ab}$.

An important corollary of (3.14) is that we can change our viewpoint. We
started this section by assuming that a metric $g$ is given on $\mathcal{M}$.
Then we defined tetrads as orthonormal frames. However, we may also start from
a tetrad $h_{a}$ on $\mathcal{M}$ and then use (3.14) to construct the metric
$g$ for which $h_{a}$ is a tetrad.

Let $h_{a},\tilde{h}_{a}$ be two tetrads on $\mathcal{M}$. The tetrads are
frames, so they are related by a local $GL\left(4,\mathbb{R}\right)$
transformation

$h_{a}(x)=A^{b}_{\phantom{b}a}(x)\tilde{h}_{b}(x).$ (3.15)

However, $h_{a}$ and $\tilde{h}_{b}$ are tetrads, therefore

$\eta_{ab}=g(h_{a},h_{b})=A^{c}_{\phantom{c}a}A^{d}_{\phantom{d}b}g(\tilde{h}_{c},\tilde{h}_{d})=\left(A^{T}\eta
A\right)_{ab}.$ (3.16)

Thus, assuming that the signature of $g$ is $(1,3)$, two tetrads
$h_{a},\tilde{h}_{a}$ with the same orientation111From now on we will always
assume that our tetrads are all right-handed. are always related by a matrix
$\Lambda(x)\in SO(1,3)$ [17], i.e. by a local Lorentz transformation [22].

###  3.3 Connection, torsion and curvature forms

Let $\mathcal{M}$ be a manifold, $\nabla$ a connection on $\mathcal{M}$ and
let $e_{a}$ be a frame defined in some region $\mathcal{O}\subset\mathcal{M}$.
The following definitions adhere to the exposition and notation from [17].

######  Definition 3.3.1.

The connection 1-forms
$\omega^{a}_{\phantom{a}b}\in\Omega^{1}\left(\mathcal{M}\right)$ with respect
to the frame $e_{a}$ are defined by

$\omega^{a}_{\phantom{a}b}(V)e_{a}\coloneqq\nabla_{V}e_{b}.$ (3.17)

######  Definition 3.3.2.

The curvature 2-forms
$\Omega^{a}_{\phantom{a}b}\in\Omega^{2}\left(\mathcal{M}\right)$ with respect
to the frame $e_{a}$ are defined by

$\Omega^{a}_{\phantom{a}b}\coloneqq\dfrac{1}{2}R^{a}_{\phantom{a}b\mu\nu}dx^{\mu}\wedge
dx^{\nu}.$ (3.18)

We also define

$\displaystyle\omega_{ab}\coloneqq g_{ac}\omega^{c}_{\phantom{c}b}$
$\displaystyle\Omega_{ab}\coloneqq g_{ac}\Omega^{c}_{\phantom{a}b}.$ (3.19)

######  Definition 3.3.3.

The torsion 2-forms $T^{a}\in\Omega^{2}\left(\mathcal{M}\right)$ with respect
to the frame $e_{a}$ are defined by

$T^{a}\coloneqq\dfrac{1}{2}T^{a}_{\phantom{a}\mu\nu}dx^{\mu}\wedge dx^{\nu}.$
(3.20)

Further in this thesis, we will consider a connection which is compatible with
a metric tensor of signature $(1,3)$ on a 4-dimensional manifold. In that
case, if $\omega^{a}_{\phantom{a}b}$ are the 1- forms of this connection with
respect to a tetrad (i.e. orthonormal frame) $h_{a}$, the fields
$\omega^{a}_{\phantom{a}b\mu}(x)$ are commonly called spin connection [17].
The connection 1-forms may be viewed as a single 1-form $\omega$ with values
in the Lie algebra $\mathfrak{so}(1,3)$ of the Lorentz group.

######  Theorem 3.3.1.

Let $\nabla$ be a metric-compatible connection on $\mathcal{M}$,
$\dim\mathcal{M}=4$, for metric of signature $\left(1,3\right)$. The 1-forms
of this connection $\omega^{a}_{\phantom{a}b}$ with respect to tetrad $h_{a}$
constitute a matrix from $\mathfrak{so}(1,3)$ algebra.

###### Proof.

Metric compatibility of a connection means that parallel transport preserves
lengths and angles between vectors. Let us consider a vector field $V$, choose
a point $p$ and consider the integral curve $\gamma(t)$ passing through it so
that $\gamma(0)=p$, and let us parallel-transport vector $h_{a}$ of the tetrad
from point $\gamma(\varepsilon)$ to point $p$ along $\gamma$. We denote the
result of this parallel transport $h_{a}^{\parallel}(p)$. The covariant
derivative is

$\left.\nabla_{V}h_{a}\right|_{p}=\lim_{\varepsilon\rightarrow
0}\dfrac{h_{a}^{\parallel}(p)-h_{a}(p)}{\varepsilon}.$ (3.21)

Because of metricity of the connection,
$h_{a}^{\parallel}(p)=\Lambda^{b}_{\phantom{b}a}(\varepsilon)h_{b}(p)$, where
$\Lambda\in SO(1,3)$ and $\Lambda(0)=\mathbb{I}$. Therefore by Definition
3.3.1

$\left.\omega^{b}_{\phantom{b}a}(V)h_{b}\right|_{p}=\lim_{\varepsilon\rightarrow
0}\dfrac{\Lambda^{b}_{\phantom{b}a}(\varepsilon)-\delta^{b}_{a}}{\varepsilon}h_{b}(p),$
(3.22)

and so $\omega^{b}_{\phantom{b}a}(V)$ must be components of a matrix from the
Lie algebra $\mathfrak{so}(1,3)$ of the Lorentz group [17]. ∎

Cartan structure equations allow us to express the torsion 2-forms and
curvature 2-forms with respect to frame (tetrad) $h_{a}$ using the coframe
(cotetrad) $h^{a}$ and the connection 1-forms $\omega^{a}_{\phantom{a}b}$:

$\displaystyle T^{a}$ $\displaystyle=dh^{a}+\omega^{a}_{\phantom{a}b}\wedge
h^{b},$ (3.23) $\displaystyle\Omega^{a}_{\phantom{a}b}$
$\displaystyle=d\omega^{a}_{\phantom{a}b}+\omega^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}.$
(3.24)

###  3.4 General teleparallel connection

######  Theorem 3.4.1.

Let $\accentset{\bullet}{\nabla}$ be a metric teleparallel connection on
$\mathcal{M}$, $\dim\mathcal{M}=4$, for metric of signature
$\left(1,3\right)$. There exists some (local) orthonormal frame (tetrad)
$h_{a}$ on $\mathcal{M}$ such that the forms
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}$ of this connection with
respect to $h_{a}$ vanish, that is

$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}=0.$ (3.25)

###### Proof.

First, we prove an auxiliary statement [17]:

Let $\omega$ be a connection 1-form on $O\mathcal{M}$ (the bundle of all
orthonormal right-handed frames on $\mathcal{M}$) and $\Omega=D\omega=0$ its
curvature. Then there exists a (local) section
$\sigma:\mathcal{M}\supset\mathcal{O}\longrightarrow O\mathcal{M}$ such that
$\sigma^{*}\omega=0$. [17]

To prove this auxiliary statement, recall that $\omega$ is a 1-form whose
component forms $\omega^{i}$ determine the horizontal distribution
$\mathcal{D}$ on $O\mathcal{M}$, that is, a vector $U\in\mathcal{D}$ if and
only if $\omega^{i}(U)=0$. Further, by Frobenius theorem $\mathcal{D}$ is
integrable if and only if $d\omega^{i}(U,V)=0$ for every $U,V\in\mathcal{D}$.
Equivalently, $\mathcal{D}$ is integrable if and only if
$D\omega=\operatorname{hor}d\omega=0$. That is because
$\operatorname{hor}d\omega(U,V)=d\omega(\operatorname{hor}U,\operatorname{hor}V)$,
and $\operatorname{hor}U$ and $\operatorname{hor}V$ are (by definition)
horizontal vectors for any vectors $U,V$ on $O\mathcal{M}$. Now, because
$\Omega\equiv D\omega=0$, the horizontal distribution determined by $\omega$
is integrable. Consider an integral submanifold $\mathcal{S}\subset
O\mathcal{M}$ given by distribution $\mathcal{D}$, and a section
$\sigma:\mathcal{M}\supset\mathcal{O}\longrightarrow\mathcal{S}$. Then
$\sigma$ is a horizontal section, which means that for any vector $W$ tangent
to $\mathcal{M}$ we have $\sigma^{*}\omega(W)=\omega(\sigma_{*}W)=0$ because
$\sigma_{*}W$ is tangent to $\mathcal{S}$, hence horizontal. Thus
$\sigma^{*}\omega=0$.

Now we use this auxiliary statement to prove our proposition. First, the
metricity of $\accentset{\bullet}{\nabla}$ is necessary to ensure that the
connection can be constructed as a connection on $O\mathcal{M}$. Second, we
know that the curvature forms
$\accentset{\bullet}{\Omega}^{a}_{\phantom{a}b}=0$. However,
$\accentset{\bullet}{\Omega}^{a}_{\phantom{a}b}\in\Omega^{2}\left(\mathcal{M}\right)$,
while in the auxiliary statement,
$\Omega\in\Omega^{2}(O\mathcal{M},\operatorname{Ad})$. We have to show that
the curvature $\Omega\in\Omega^{2}(O\mathcal{M},\operatorname{Ad})$
corresponding to
$\accentset{\bullet}{\Omega}^{a}_{\phantom{a}b}\in\Omega^{2}\left(\mathcal{M}\right)$
is also zero. We will use equation

$\Omega=d\omega+\dfrac{1}{2}\left[\omega\wedge\omega\right],$ (3.26)

where $\omega$ and $\Omega$ are matrix forms, i.e. $\omega^{a}_{\phantom{a}b}$
and $\Omega^{a}_{\phantom{a}b}$ are the components of $4\times 4$ matrices,
and we will treat them as such. Let us remark that
$\left[\omega\wedge\omega\right]=\omega^{a}\wedge\omega^{b}\left[E_{a},E_{b}\right]$,
that is, the product of the component forms $\omega^{a}$ is the exterior
product, and the basis vectors $E_{a}$ of the Lie algebra are treated as
elements of algebra. Hence222$E^{a}_{b}$ is a matrix with components
$\left(E^{a}_{b}\right)^{c}_{d}=\delta^{a}_{d}\delta^{c}_{b}$.

$\displaystyle\Omega^{a}_{\phantom{a}b}E^{b}_{a}$
$\displaystyle=d\omega^{a}_{\phantom{a}b}E^{b}_{a}+\dfrac{1}{2}\omega^{a}_{\phantom{a}c}\wedge\omega^{d}_{\phantom{d}b}\left[E^{c}_{a},E^{b}_{d}\right]$
$\displaystyle=d\omega^{a}_{\phantom{a}b}E^{b}_{a}+\dfrac{1}{2}\omega^{a}_{\phantom{a}c}\wedge\omega^{d}_{\phantom{d}b}\left(E^{c}_{a}E^{b}_{d}-E^{b}_{d}E^{c}_{a}\right)$
$\displaystyle=d\omega^{a}_{\phantom{a}b}E^{b}_{a}+\dfrac{1}{2}\omega^{a}_{\phantom{a}c}\wedge\omega^{d}_{\phantom{d}b}\left(\delta^{c}_{d}E^{b}_{a}-\delta^{b}_{a}E^{c}_{d}\right)$
$\displaystyle=d\omega^{a}_{\phantom{a}b}E^{b}_{a}+\dfrac{1}{2}\left(\omega^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}E^{b}_{a}-\omega^{a}_{\phantom{a}c}\wedge\omega^{d}_{\phantom{d}a}E^{c}_{d}\right)$
$\displaystyle=d\omega^{a}_{\phantom{a}b}E^{b}_{a}+\dfrac{1}{2}\left(\omega^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}-\omega^{c}_{\phantom{c}b}\wedge\omega^{a}_{\phantom{a}c}\right)E^{b}_{a}$
$\displaystyle=\left(d\omega^{a}_{\phantom{a}b}+\omega^{a}_{\phantom{a}c}\wedge\omega^{c}_{\phantom{c}b}\right)E^{b}_{a}.$

Applying pull-back $\sigma^{*}$ gives

$\sigma^{*}\Omega^{a}_{\phantom{a}b}=d\left(\sigma^{*}\omega^{a}_{\phantom{a}b}\right)+\left(\sigma^{*}\omega^{a}_{\phantom{a}c}\right)\wedge\left(\sigma^{*}\omega^{c}_{\phantom{c}b}\right)=d\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}+\accentset{\bullet}{\omega}^{a}_{\phantom{a}c}\wedge\accentset{\bullet}{\omega}^{c}_{\phantom{c}b}=\accentset{\bullet}{\Omega}^{a}_{\phantom{a}b}.$
(3.27)

Here $\accentset{\bullet}{\Omega}^{a}_{\phantom{a}b}$ are curvature 2-forms
with respect to the tetrad $h_{a}$ that corresponds to the section $\sigma$.
However, curvature is a tensor, so
$\accentset{\bullet}{\Omega}^{a}_{\phantom{a}b}=0$ with respect to any tetrad.
Therefore $\sigma^{*}\Omega^{a}_{\phantom{a}b}=0$ for any $\sigma$, which
means $\Omega^{a}_{\phantom{a}b}=0$. By the auxiliary statement, there exists
a local section $\tilde{\sigma}$ such that
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}\equiv\tilde{\sigma}^{*}\omega^{a}_{\phantom{a}b}=0$333This
gauge is known as the Weitzenböck gauge [19].. The section $\tilde{\sigma}$
corresponds to some tetrad $\tilde{h}_{a}$ on $\mathcal{M}$. ∎

######  Corollary 3.4.2.

Let $\accentset{\bullet}{\nabla}$ be a metric teleparallel connection on
$\mathcal{M}$, $\dim\mathcal{M}=4$, for metric of signature
$\left(1,3\right)$, and let $\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}$
be the 1-forms of this connection with respect to tetrad $h_{a}$. There exists
a matrix 0-form $\Lambda\in\Omega^{0}\left(\mathcal{M},SO(1,3)\right)$ such
that [19]

$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}=\left(\Lambda^{-1}\right)^{a}_{\phantom{a}c}d\Lambda^{c}_{\phantom{c}b}.$
(3.28)

###### Proof.

Let $\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}$ be the connection 1-forms
with respect to tetrad $h_{a}$ and
$\tilde{\accentset{\bullet}{\omega}}^{a}_{\phantom{a}b}$ the connection
1-forms with respect to tetrad $\tilde{h}_{a}$. The tetrads are related by a
local Lorentz transformation

$h_{a}(x)=\Lambda^{b}_{\phantom{b}a}(x)\tilde{h}_{b},$ (3.29)

where $\Lambda(x)\in SO(1,3)$, and the connection 1-forms are transformed as
[17]

$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}=\left(\Lambda^{-1}\right)^{a}_{\phantom{a}c}\tilde{\accentset{\bullet}{\omega}}^{c}_{\phantom{c}d}\Lambda^{d}_{\phantom{d}b}+\left(\Lambda^{-1}\right)^{a}_{\phantom{a}c}d\Lambda^{c}_{\phantom{c}b}.$
(3.30)

Now suppose that $\tilde{h}_{a}$ is a tetrad with respect to which
$\tilde{\accentset{\bullet}{\omega}}^{a}_{\phantom{a}b}=0$. Then

$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}=\left(\Lambda^{-1}\right)^{a}_{\phantom{a}c}d\Lambda^{c}_{\phantom{c}b},$
(3.31)

and we can view $\Lambda(x)$ as a matrix 0-form. ∎

We have thus found the connection 1-forms of a general metric teleparallel
connection. Note that the the theorem and its corollary above may be restated
for a manifold $\mathcal{M}$ of any dimension and a metric of any signature.

##  Chapter 4 Teleparallel gravity

###  4.1 Action of teleparallel gravity

The dynamics of general relativity is governed by the Einstein field
equations, which can be obtained by varying action
$\mathcal{S}=\accentset{\circ}{\mathcal{S}}+\mathcal{S}_{m}$, where
$\mathcal{S}_{m}$ is the matter action and $\accentset{\circ}{\mathcal{S}}$
the Einstein-Hilbert action

$\accentset{\circ}{\mathcal{S}}=\frac{1}{2\kappa}\int_{\mathcal{M}}{\accentset{\circ}{R}\sqrt{-g}\,d^{4}x}.$
(4.1)

We used the standard notation $g=\det(g_{\mu\nu})$, $\kappa=8\pi G$, where $G$
is the gravitational constant, and $\accentset{\circ}{R}$ is the Ricci scalar,
also called scalar curvature

$\accentset{\circ}{R}\left(g_{\mu\nu}\right)\equiv\accentset{\circ}{R}^{\alpha\beta}_{\phantom{\alpha\beta}\alpha\beta}$
(4.2)

Variation of $\mathcal{S}$ leads to Einstein field equations (without the
cosmological constant)

$\accentset{\circ}{G}_{\mu\nu}=\kappa\Theta_{\mu\nu}.$ (4.3)

where
$\accentset{\circ}{G}_{\mu\nu}=\accentset{\circ}{R}_{\mu\nu}-\frac{1}{2}\accentset{\circ}{R}g_{\mu\nu}$
is the Einstein tensor,
$\accentset{\circ}{R}_{\mu\nu}=\accentset{\circ}{R}^{\lambda}_{\phantom{\lambda}\mu\lambda\nu}$,
$g_{\mu\nu}$ is the metric and $\Theta_{\mu\nu}$ is the energy-momentum tensor
[22].

To obtain an alternative mathematical description of gravity which is based on
teleparallelism, we have to find the action for this theory [16]. The question
is: What is our starting point? We know that general relativity is in good
agreement with observations and experimental evidence. Therefore, our
criterion for the construction of the Lagrangian of teleparallel gravity will
be that the theory must agree with the results of general relativity. That
means that the dynamical content of the equations of motion derived from the
Lagrangian of teleparallel gravity must be the same as that of equations
derived from the Einstein-Hilbert action, i.e. the Einstein field equations.

We start [16, 19] by writing the the curvature tensor for a teleparallel
connection111The result of this calculation is independent of the choice of
frame, therefore we can do it in a holonomic frame $\partial_{\alpha}$.:

$\accentset{\bullet}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}=\partial_{\mu}\accentset{\bullet}{\Gamma}^{\alpha}_{\beta\nu}-\partial_{\nu}\accentset{\bullet}{\Gamma}^{\alpha}_{\beta\mu}+\accentset{\bullet}{\Gamma}^{\alpha}_{\lambda\mu}\accentset{\bullet}{\Gamma}^{\lambda}_{\beta\nu}-\accentset{\bullet}{\Gamma}^{\alpha}_{\lambda\nu}\accentset{\bullet}{\Gamma}^{\lambda}_{\beta\mu}.$
(4.4)

By (2.14), the (metric) teleparallel connection decomposes as

$\accentset{\bullet}{\Gamma}^{\rho}_{\mu\nu}=\accentset{\circ}{\Gamma}^{\rho}_{\mu\nu}+\accentset{\bullet}{K}^{\rho}_{\phantom{\rho}\mu\nu},$
(4.5)

leading to

$\displaystyle\accentset{\bullet}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}$
$\displaystyle=\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}+\partial_{\mu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\nu}-\partial_{\nu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\mu}+\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}+$
(4.6)
$\displaystyle+\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\circ}{\Gamma}^{\lambda}_{\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\circ}{\Gamma}^{\lambda}_{\beta\mu}+\accentset{\circ}{\Gamma}^{\alpha}_{\lambda\mu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}-\accentset{\circ}{\Gamma}^{\alpha}_{\lambda\nu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}$
$\displaystyle=\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}+\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}+$
$\displaystyle+\left(\partial_{\mu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\circ}{\Gamma}^{\lambda}_{\beta\mu}+\accentset{\circ}{\Gamma}^{\alpha}_{\lambda\mu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}\right)-\left(\partial_{\nu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\mu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\circ}{\Gamma}^{\lambda}_{\beta\nu}+\accentset{\circ}{\Gamma}^{\alpha}_{\lambda\nu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}\right).$

Adding a zero term in the form
$\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\lambda}\accentset{\circ}{\Gamma}^{\lambda}_{\mu\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\lambda}\accentset{\circ}{\Gamma}^{\lambda}_{\nu\mu}$
allows us to simplify:

$\displaystyle\accentset{\bullet}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}$
$\displaystyle=\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}+\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}+$
$\displaystyle+\left(\partial_{\mu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\circ}{\Gamma}^{\lambda}_{\beta\mu}+\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}\accentset{\circ}{\Gamma}^{\alpha}_{\lambda\mu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\lambda}\accentset{\circ}{\Gamma}^{\lambda}_{\nu\mu}\right)$
$\displaystyle-\left(\partial_{\nu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\mu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\circ}{\Gamma}^{\lambda}_{\beta\nu}+\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}\accentset{\circ}{\Gamma}^{\alpha}_{\lambda\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\lambda}\accentset{\circ}{\Gamma}^{\lambda}_{\mu\nu}\right)$
$\displaystyle=\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}+\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\mu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\nu}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\nu}\accentset{\bullet}{K}^{\lambda}_{\phantom{\lambda}\beta\mu}+\accentset{\circ}{\nabla}_{\mu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\nu}-\accentset{\circ}{\nabla}_{\nu}\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\beta\mu}.$

The curvature of teleparallel connection vanishes, which implies that the
scalar curvature

$0=\accentset{\circ}{R}+\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\alpha}\accentset{\bullet}{K}^{\lambda\beta}_{\phantom{\lambda\beta}\beta}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\beta}\accentset{\bullet}{K}^{\lambda\beta}_{\phantom{\lambda\beta}\alpha}+\accentset{\circ}{\nabla}_{\alpha}\accentset{\bullet}{K}^{\alpha\beta}_{\phantom{\alpha\beta}\beta}-\accentset{\circ}{\nabla}_{\beta}\accentset{\bullet}{K}^{\alpha\beta}_{\phantom{\alpha\beta}\alpha}.$
(4.7)

Defining torsion scalar [19]

$\accentset{\bullet}{T}=\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\alpha}\accentset{\bullet}{K}^{\lambda\beta}_{\phantom{\lambda\beta}\beta}-\accentset{\bullet}{K}^{\alpha}_{\phantom{\alpha}\lambda\beta}\accentset{\bullet}{K}^{\lambda\beta}_{\phantom{\lambda\beta}\alpha}$
(4.8)

and noticing

$\displaystyle\accentset{\bullet}{K}^{\alpha\beta}_{\phantom{\alpha\beta}\beta}-\accentset{\bullet}{K}^{\beta\alpha}_{\phantom{\alpha\beta}\beta}$
$\displaystyle=\dfrac{1}{2}\left(\accentset{\bullet}{T}^{\beta\alpha}_{\phantom{\beta\alpha}\beta}+\accentset{\bullet}{T}^{\phantom{\beta}\alpha\beta}_{\beta\phantom{\alpha\beta}}-\accentset{\bullet}{T}^{\alpha\beta}_{\phantom{\alpha\beta}\beta}-\accentset{\bullet}{T}^{\alpha\beta}_{\phantom{\beta\alpha}\beta}-\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta\phantom{\alpha\beta}}+\accentset{\bullet}{T}^{\beta\alpha}_{\phantom{\alpha\beta}\beta}\right)$
(4.9)
$\displaystyle=-2\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta\phantom{\alpha\beta}},$

we can express the scalar curvature of the RLC connection as

$\accentset{\circ}{R}=-\accentset{\bullet}{T}-\accentset{\circ}{\nabla}_{\alpha}\left(\accentset{\bullet}{K}^{\alpha\beta}_{\phantom{\alpha\beta}\beta}-\accentset{\bullet}{K}^{\beta\alpha}_{\phantom{\beta\alpha}\beta}\right)=-\accentset{\bullet}{T}+2\accentset{\circ}{\nabla}_{\alpha}\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta}.$
(4.10)

We will adopt the handy notation $h=\lvert\det(h^{a}_{\phantom{a}\mu})\rvert$.
Looking at (3.14) we see that

$\sqrt{-g}=\sqrt{-\det(g_{\mu\nu})}=\sqrt{-\det(\eta_{ab})\det(h^{a}_{\phantom{a}\mu})\det(h^{b}_{\phantom{a}\nu})}=\lvert\det(h^{a}_{\phantom{a}\mu})\rvert=h.$

All this should serve as a good motivation for postulating the following:

######  Definition 4.1.1.

The action of teleparallel gravity is

$\accentset{\bullet}{\mathcal{S}}=-\dfrac{1}{2\kappa}\int_{\mathcal{M}}{\accentset{\bullet}{T}h\,d^{4}x}.$
(4.11)

To be more precise, we should call this the action of teleparallel equivalent
of general relativity (TEGR) [19], but some authors (e.g. Aldrovandi and
Pereira [16]) actually mean TEGR when they speak of teleparallel gravity.

The equations of motion may be obtained through the variation of this action.
It is vitally important to realize that the dynamics of the theory with this
action is by construction equivalent to the dynamics of general relativity, in
which the dynamics is determined by the Einstein-Hilbert action. The reason is
that
$\left(2\kappa\right)^{-1}\int_{\mathcal{M}}{2\accentset{\circ}{\nabla}_{\alpha}\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta}\sqrt{-g}\,d^{4}x}$
constitutes only a surface term, which is irrelevant when varying the action.
To show this, we use the formula [22]

$\accentset{\circ}{\nabla}_{\mu}V^{\mu}=\dfrac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}V^{\mu}\right),$
(4.12)

which allows us to write

$\dfrac{1}{2\kappa}\int_{\mathcal{M}}{2\accentset{\circ}{\nabla}_{\alpha}\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta\phantom{\alpha\beta}}\sqrt{-g}\,d^{4}x}=\dfrac{1}{2\kappa}\int_{\mathcal{M}}{2\partial_{\alpha}\left(\sqrt{-g}\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta\phantom{\alpha\beta}}\right)d^{4}x}.$
(4.13)

Now let us rewrite the torsion scalar in terms of the torsion
$\accentset{\bullet}{T}^{\lambda}_{\alpha\beta}$. Using (4.8) and (2.15) we
have

$\displaystyle\accentset{\bullet}{T}$
$\displaystyle=\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\alpha\lambda}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}-\dfrac{1}{4}\left(\accentset{\bullet}{T}^{\phantom{\lambda}\alpha}_{\lambda\phantom{\alpha}\beta}+\accentset{\bullet}{T}^{\phantom{\beta}\alpha}_{\beta\phantom{\alpha}\lambda}-\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\lambda\beta}\right)\left(\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\beta\lambda}\alpha}+\accentset{\bullet}{T}^{\phantom{\alpha}\lambda\beta}_{\alpha\phantom{\lambda\beta}}-\accentset{\bullet}{T}^{\lambda\beta}_{\phantom{\lambda\beta}\alpha}\right)$
$\displaystyle=\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\alpha\lambda}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}-\dfrac{1}{4}\left(\accentset{\bullet}{T}_{\lambda\alpha\beta}+\accentset{\bullet}{T}_{\beta\alpha\lambda}-\accentset{\bullet}{T}_{\alpha\lambda\beta}\right)\left(\accentset{\bullet}{T}^{\beta\lambda\alpha}+\accentset{\bullet}{T}^{\alpha\lambda\beta}-\accentset{\bullet}{T}^{\lambda\beta\alpha}\right)$
$\displaystyle=\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\alpha\lambda}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}-\dfrac{1}{4}\accentset{\bullet}{T}_{\lambda\alpha\beta}\left(\accentset{\bullet}{T}^{\beta\lambda\alpha}+\accentset{\bullet}{T}^{\alpha\lambda\beta}-\accentset{\bullet}{T}^{\lambda\beta\alpha}\right)-$
$\displaystyle-\dfrac{1}{4}\accentset{\bullet}{T}_{\beta\alpha\lambda}\left(\accentset{\bullet}{T}^{\beta\lambda\alpha}+\accentset{\bullet}{T}^{\alpha\lambda\beta}-\accentset{\bullet}{T}^{\lambda\beta\alpha}\right)+\dfrac{1}{4}\accentset{\bullet}{T}_{\alpha\lambda\beta}\left(\accentset{\bullet}{T}^{\beta\lambda\alpha}+\accentset{\bullet}{T}^{\alpha\lambda\beta}-\accentset{\bullet}{T}^{\lambda\beta\alpha}\right).$

Notice that

$\accentset{\bullet}{T}_{\beta\alpha\lambda}\left(\accentset{\bullet}{T}^{\beta\lambda\alpha}+\accentset{\bullet}{T}^{\alpha\lambda\beta}-\accentset{\bullet}{T}^{\lambda\beta\alpha}\right)=\accentset{\bullet}{T}_{\lambda\alpha\beta}\left(\accentset{\bullet}{T}^{\lambda\beta\alpha}+\accentset{\bullet}{T}^{\alpha\beta\lambda}-\accentset{\bullet}{T}^{\beta\lambda\alpha}\right),$

so the first two terms with the parantheses cancel. What remains is

$\displaystyle\accentset{\bullet}{T}$
$\displaystyle=\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\alpha\lambda}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}+\dfrac{1}{4}\accentset{\bullet}{T}_{\alpha\lambda\beta}\left(\accentset{\bullet}{T}^{\beta\lambda\alpha}+\accentset{\bullet}{T}^{\alpha\lambda\beta}-\accentset{\bullet}{T}^{\lambda\beta\alpha}\right)$
(4.14)
$\displaystyle=\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\alpha\lambda}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\beta\lambda}\beta}+\dfrac{1}{4}\accentset{\bullet}{T}_{\lambda\alpha\beta}\left(\accentset{\bullet}{T}^{\beta\alpha\lambda}+\accentset{\bullet}{T}^{\lambda\alpha\beta}-\accentset{\bullet}{T}^{\alpha\beta\lambda}\right)$
(4.15)
$\displaystyle=\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\alpha\lambda}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\beta\lambda}\beta}+\dfrac{1}{4}\accentset{\bullet}{T}_{\lambda\alpha\beta}\left(\accentset{\bullet}{T}^{\beta\alpha\lambda}+\accentset{\bullet}{T}^{\lambda\alpha\beta}+\accentset{\bullet}{T}^{\beta\alpha\lambda}\right)$
(4.16)
$\displaystyle=\dfrac{1}{4}\accentset{\bullet}{T}_{\lambda\alpha\beta}\accentset{\bullet}{T}^{\lambda\alpha\beta}+\dfrac{1}{2}\accentset{\bullet}{T}_{\lambda\alpha\beta}\accentset{\bullet}{T}^{\beta\alpha\lambda}-\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\lambda\alpha}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}$
(4.17)
$\displaystyle=\dfrac{1}{4}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\alpha\beta}\accentset{\bullet}{T}_{\lambda}^{\phantom{\lambda}\alpha\beta}+\dfrac{1}{2}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\alpha\beta}\accentset{\bullet}{T}_{\phantom{\beta\alpha}\lambda}^{\beta\alpha}-\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\lambda\alpha}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}.$
(4.18)

The Lagrangian of teleparallel gravity is therefore

$\accentset{\bullet}{\mathcal{L}}=-\dfrac{h}{2\kappa}\left(\dfrac{1}{4}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\alpha\beta}\accentset{\bullet}{T}_{\lambda}^{\phantom{\lambda}\alpha\beta}+\dfrac{1}{2}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\alpha\beta}\accentset{\bullet}{T}_{\phantom{\beta\alpha}\lambda}^{\beta\alpha}-\accentset{\bullet}{T}^{\alpha}_{\phantom{\alpha}\lambda\alpha}\accentset{\bullet}{T}^{\beta\lambda}_{\phantom{\mu\lambda}\beta}\right).$
(4.19)

###  4.2 Equations of motion

We are now ready to derive the equations determining the dynamics in
teleparallel gravity. Before we can even approach the variation of the action,
we need to clarify what the variables of the Lagrangian are. As we have
mentioned in the first chapter, the fundamental variables on manifold
$\mathcal{M}$ are metric $g$ and connection $\accentset{\bullet}{\nabla}$,
therefore the Lagrangian naturally depends on the tetrad
$h^{a}_{\phantom{a}\mu}$ (from which we construct the metric by (3.14)) and
the spin connection $\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$.
Writing the Euler-Lagrange equations

$\partial_{\sigma}\dfrac{\partial\accentset{\bullet}{\mathcal{L}}}{\partial\left(\partial_{\sigma}h^{a}_{\phantom{a}\rho}\right)}-\dfrac{\partial\accentset{\bullet}{\mathcal{L}}}{\partial
h^{a}_{\phantom{a}\rho}}=0$ (4.20)

for the variation of the Lagrangian given in the section above with respect to
the tetrad, we arrive at the equations

$\partial_{\sigma}\left(h\accentset{\bullet}{S}_{a}^{\phantom{a}\rho\sigma}\right)-\kappa
h\accentset{\bullet}{\mathcal{J}}_{a}^{\phantom{a}\rho}=\kappa
h\Theta_{a}^{\phantom{a}\rho},$ (4.21)

where
$\Theta_{a}^{\phantom{a}\rho}=-\dfrac{1}{h}\dfrac{\delta\mathcal{L}_{s}}{\delta
h^{a}_{\phantom{a}\rho}}$ is the matter energy-momentum tensor and
$\mathcal{L}_{s}$ is the Lagrangian of a general source field [11]. The other
two objects appearing in eq. (4.21) are the so-called superpotential

$\accentset{\bullet}{S}_{a}^{\phantom{a}\rho\sigma}=\dfrac{1}{2}\left(\accentset{\bullet}{T}_{\phantom{\sigma\rho}a}^{\sigma\rho}+\accentset{\bullet}{T}_{a}^{\phantom{a}\rho\sigma}-\accentset{\bullet}{T}_{\phantom{\rho\sigma}a}^{\rho\sigma}\right)-h_{a}^{\phantom{a}\sigma}\accentset{\bullet}{T}_{\phantom{\theta\rho}\theta}^{\theta\rho}+h_{a}^{\phantom{a}\rho}\accentset{\bullet}{T}_{\phantom{\theta\sigma}\theta}^{\theta\sigma}$
(4.22)

and the so-called gauge current

$\accentset{\bullet}{\mathcal{J}}_{a}^{\phantom{a}\rho}=\dfrac{1}{\kappa}h_{a}^{\phantom{a}\lambda}\accentset{\bullet}{S}_{c}^{\phantom{c}\nu\rho}\accentset{\bullet}{T}^{c}_{\phantom{c}\nu\lambda}+\dfrac{h_{a}^{\phantom{a}\rho}}{h}\accentset{\bullet}{\mathcal{L}}+\dfrac{1}{\kappa}\accentset{\bullet}{\omega}^{c}_{\phantom{c}a\sigma}\accentset{\bullet}{S}_{c}^{\phantom{c}\rho\sigma}.$
(4.23)

Now half the work is done, but we have yet to execute the variation with
respect to the other variable of the Lagrangian, the spin connection
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$. For this, we will resort
to a dirty trick [23, 24]. In the previous section, we have shown that

$\accentset{\bullet}{\mathcal{L}}\left(h^{a}_{\phantom{a}\rho},\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}\right)=\accentset{\circ}{\mathcal{L}}\left(h^{a}_{\phantom{a}\rho}\right)-\dfrac{1}{\kappa}\partial_{\alpha}\left(h\accentset{\bullet}{T}^{\phantom{\beta}\beta\alpha}_{\beta\phantom{\alpha\beta}}\right),$
(4.24)

where

$\accentset{\circ}{\mathcal{L}}\left(h^{a}_{\phantom{a}\rho}\right)=\frac{1}{2\kappa}h\accentset{\circ}{R}\left(h^{a}_{\phantom{a}\rho}\right)\equiv\frac{1}{2\kappa}\sqrt{-g}\accentset{\circ}{R}\left(g_{\mu\nu}\right)$
(4.25)

is the Einstein-Hilbert Lagrangian, which, as we know, does not depend on spin
connection $\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$, but only on
the metric, and therefore (by (3.14)) on the tetrad $h^{a}_{\phantom{a}\rho}$.
Due to the fact that the second term on the right is just a 4-divergence, it
will only contribute a surface term in the action. However, surface terms are
irrelevant when it comes to variation of the action, so variations of
$\accentset{\bullet}{\mathcal{S}}$ with respect to
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$ are identically vanishing,
hence they give no equation [11].

###  4.3 Alternative form of action

Notice that the Lagrangian of teleparallel gravity may be rewritten in a
different form [16]. First, we factor out
$\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\mu\nu}$

$\displaystyle\accentset{\bullet}{\mathcal{L}}$
$\displaystyle=-\dfrac{h}{2\kappa}\left(\dfrac{1}{4}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\mu\nu}\accentset{\bullet}{T}_{\lambda}^{\phantom{\lambda}\mu\nu}+\dfrac{1}{2}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\mu\nu}\accentset{\bullet}{T}_{\phantom{\nu\mu}\lambda}^{\nu\mu}-\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\mu\lambda}\accentset{\bullet}{T}_{\phantom{\nu\mu}\nu}^{\nu\mu}\right)$
(4.26)
$\displaystyle=-\dfrac{h}{2\kappa}\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\mu\nu}\left(\dfrac{1}{4}\accentset{\bullet}{T}_{\lambda}^{\phantom{\lambda}\mu\nu}+\dfrac{1}{2}\accentset{\bullet}{T}_{\phantom{\nu\mu}\lambda}^{\nu\mu}-\delta^{\nu}_{\lambda}\accentset{\bullet}{T}_{\phantom{\rho\mu}\rho}^{\rho\mu}\right)$
$\displaystyle=-\dfrac{h}{2\kappa}\eta_{ab}h^{b}_{\phantom{b}\lambda}\accentset{\bullet}{T}^{a}_{\phantom{a}\mu\nu}\left(\dfrac{1}{4}\accentset{\bullet}{T}^{\lambda\mu\nu}+\dfrac{1}{2}\accentset{\bullet}{T}^{\nu\mu\lambda}-g^{\nu\lambda}\accentset{\bullet}{T}_{\phantom{\rho\mu}\rho}^{\rho\mu}\right)$
$\displaystyle\equiv-\dfrac{h}{2\kappa}\eta_{ab}h^{b}_{\phantom{b}\lambda}\accentset{\bullet}{T}^{a}_{\phantom{a}\mu\nu}\accentset{\bullet}{H}^{\mu\nu\lambda}.$

Next, we use Kronecker deltas to change the indices on
$\accentset{\bullet}{H}$ and use the fact that
$\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\mu\nu}=\accentset{\bullet}{T}^{\lambda}_{\phantom{\lambda}\left[\mu\nu\right]}$

$\displaystyle\accentset{\bullet}{\mathcal{L}}$
$\displaystyle=-\dfrac{h}{2\kappa}\eta_{ab}h^{b}_{\phantom{b}\lambda}\accentset{\bullet}{T}^{a}_{\phantom{a}\mu\nu}\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}\accentset{\bullet}{H}^{\alpha\beta\lambda}$
(4.27)
$\displaystyle=-\dfrac{h}{2\kappa}\eta_{ab}h^{b}_{\phantom{b}\lambda}\accentset{\bullet}{T}^{a}_{\phantom{a}\mu\nu}\delta^{\left[\mu\right.}_{\alpha}\delta^{\left.\nu\right]}_{\beta}\accentset{\bullet}{H}^{\alpha\beta\lambda}.$

Using the identity
$\varepsilon^{\mu\nu\gamma\sigma}\varepsilon_{\alpha\beta\gamma\sigma}=4\delta^{\left[\mu\right.}_{\alpha}\delta^{\left.\nu\right]}_{\beta}$
[17] this further leads to

$\displaystyle\accentset{\bullet}{\mathcal{L}}$
$\displaystyle=-\dfrac{h}{2\kappa}\eta_{ab}h^{b}_{\phantom{b}\lambda}\accentset{\bullet}{T}^{a}_{\phantom{a}\mu\nu}\dfrac{1}{4}\varepsilon^{\mu\nu\gamma\sigma}\varepsilon_{\alpha\beta\gamma\sigma}\accentset{\bullet}{H}^{\alpha\beta\lambda}$
(4.28)
$\displaystyle=-\dfrac{h}{2\kappa}6\eta_{ab}h^{b}_{\phantom{b}\lambda}\accentset{\bullet}{T}^{a}_{\phantom{a}\mu\nu}\varepsilon_{\gamma\sigma\alpha\beta}\accentset{\bullet}{H}^{\alpha\beta\lambda}\dfrac{1}{4!}\varepsilon^{\mu\nu\gamma\sigma}$
$\displaystyle=-\dfrac{h}{2\kappa}\eta_{ab}h^{b}_{\phantom{b}\lambda}6\accentset{\bullet}{T}^{a}_{\phantom{a}\left[\mu\nu\right.}\varepsilon_{\left.\gamma\sigma\right]\alpha\beta}\accentset{\bullet}{H}^{\alpha\beta\lambda}\dfrac{1}{4!}\varepsilon^{\mu\nu\gamma\sigma}.$

Interestingly,
$\accentset{\bullet}{T}^{a}_{\phantom{a}\left[\mu\nu\right.}\varepsilon_{\left.\gamma\sigma\right]\alpha\beta}\accentset{\bullet}{H}^{\alpha\beta\lambda}$
would almost look like the exterior product of the torsion forms
$\accentset{\bullet}{T}^{a}$ with some other form, were it not for the fact
that at this point we do not know if
$\varepsilon_{\gamma\sigma\alpha\beta}\accentset{\bullet}{H}^{\alpha\beta\lambda}$
is a form. In order to be more explicit, let us rewrite
$6=\dfrac{(2+2)!}{2!2!}$ and define

$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)^{b}_{\gamma\sigma}=h^{b}_{\phantom{b}\lambda}h\varepsilon_{\gamma\sigma\alpha\beta}\left(\dfrac{1}{4}\accentset{\bullet}{T}^{\lambda\alpha\beta}+\dfrac{1}{2}\accentset{\bullet}{T}^{\beta\alpha\lambda}-g^{\beta\lambda}\accentset{\bullet}{T}_{\phantom{\rho\alpha}\rho}^{\rho\alpha}\right).$
(4.29)

Then

$\displaystyle\accentset{\bullet}{\mathcal{L}}$
$\displaystyle=-\dfrac{1}{2\kappa}\eta_{ab}\dfrac{(2+2)!}{2!2!}\accentset{\bullet}{T}^{a}_{\phantom{a}\left[\mu\nu\right.}\varepsilon_{\left.\gamma\sigma\right]\alpha\beta}h^{b}_{\phantom{b}\lambda}h\accentset{\bullet}{H}^{\alpha\beta\lambda}\dfrac{1}{4!}\varepsilon^{\mu\nu\gamma\sigma}$
(4.30)
$\displaystyle\equiv-\frac{1}{2\kappa}\eta_{ab}\left(\accentset{\bullet}{T}^{a}\wedge\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}^{b}\right)_{\mu\nu\gamma\sigma}\dfrac{1}{4!}\varepsilon^{\mu\nu\gamma\sigma}.$

We see that $\accentset{\bullet}{H}$ is a tensor, while
$h\varepsilon_{\gamma\sigma\alpha\beta}=\omega_{\gamma\sigma\alpha\beta}$ is
the (metric-compatible) volume form (see Definition 5.1.3), and therefore
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}^{b}$
is a contraction of two tensors, thus a tensor. Moreover, this tensor is
antisymmetric (in the lower indices), so it is also a 2-form. Hence the use of
the exterior product $\wedge$ is justified.

As
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}^{b}$
are 2-forms and the underlying manifold $\mathcal{M}$ is 4-dimensional, it
must be that for any $a,b$

$\displaystyle\left(\accentset{\bullet}{T}^{a}\wedge\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}^{b}\right)$
$\displaystyle\propto\omega,$ (4.39)
$\displaystyle\left(\accentset{\bullet}{T}^{a}\wedge\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}^{b}\right)_{\mu\nu\gamma\sigma}$
$\displaystyle\equiv\left(\accentset{\bullet}{T}^{a},\accentset{\bullet}{T}^{b}\right)_{g}\omega_{\mu\nu\gamma\sigma}.$
(4.48)

Recalling that222$\operatorname{sgn}g$ is the sign of
$\det\left(g_{\mu\nu}\right)$.
$\omega^{\mu\nu\gamma\sigma}=\dfrac{\operatorname{sgn}g}{h}\varepsilon^{\mu\nu\gamma\sigma}=-\dfrac{1}{h}\varepsilon^{\mu\nu\gamma\sigma}$,
we can now rewrite the action

$\displaystyle\accentset{\bullet}{\mathcal{S}}$
$\displaystyle=\int_{\mathcal{M}}{\accentset{\bullet}{\mathcal{L}}\,d^{4}x}$
(4.49)
$\displaystyle=\int_{\mathcal{M}}{\frac{1}{2\kappa}\eta_{ab}\left(\accentset{\bullet}{T}^{a},\accentset{\bullet}{T}^{b}\right)_{g}\omega_{\mu\nu\gamma\sigma}\dfrac{1}{4!}\omega^{\mu\nu\gamma\sigma}h\,d^{4}x}$
$\displaystyle=\int_{\mathcal{M}}{\frac{1}{2\kappa}\eta_{ab}\left(\accentset{\bullet}{T}^{a},\accentset{\bullet}{T}^{b}\right)_{g}\omega}$
$\displaystyle=\int_{\mathcal{M}}{\frac{1}{2\kappa}\eta_{ab}\left(\accentset{\bullet}{T}^{a}\wedge\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}^{b}\right)}.$

The action of teleparallel gravity expressed in this form resembles the Yang-
Mills action for a gauge theory for which torsion $\accentset{\bullet}{T}^{a}$
is the field strength [16].

##  Chapter 5 New duality operator

###  5.1 Duality

In the last section, we have introduced a new symbol
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$,
but it remains unclear what this symbol represents, and whether it represents
a well-defined operator. The similarity of (4.49) with the Yang-Mills action
for a gauge theory, in which
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
is replaced by $\star$, leads us to turn our attention to the Hodge operator.
Here we give a brief review of the Hodge star operator and its most important
properties [17].

######  Definition 5.1.1.

A volume form $\omega$ on a $n$-dimensional vector space $L$ is any nonzero
$n$-form in $L$.
Let $L$ be, in addition, an oriented vector space with a metric tensor $g$,
and let $h_{a}$ be its right-handed orthonormal basis. Then the metric-
compatible volume form is

$\omega=h^{1}\wedge\dots\wedge h^{n}.$ (5.1)

######  Definition 5.1.2.

Let $L$ be a $n$-dimensional vector space with metric $g$ and orientation $o$
and let $\omega$ be the metric-compatible volume form. Hodge star operator is
a map $\star\colon\Lambda^{p}L^{*}\longrightarrow\Lambda^{n-p}L^{*}$,
$\alpha\mapsto\star\alpha$ defined by

$(\star\alpha)_{a\dots b}\coloneqq\dfrac{1}{p!}\alpha^{c\dots d}\omega_{a\dots
bc\dots d},$ (5.2)

where $\alpha^{c\dots d}\equiv g^{cr}\dots g^{ds}\alpha_{r\dots s}$.

We see immediately that $\star$ is a linear operator on $\Lambda^{p}L^{*}$.
Perhaps the most important property of the Hodge operator is

$\star\star=\operatorname{sgn}g(-1)^{p(n+1)},$ (5.3)

i.e. it is a duality operator.

######  Definition 5.1.3.

A volume form $\omega$ on an $n$-dimensional manifold $\mathcal{M}$ is any
$n$-form which is nonzero everywhere on $\mathcal{M}$.
Let $\mathcal{M}$ be, in addition, an orientable111From now on, we will always
assume this is the case. manifold with a metric tensor $g$, and let $h_{a}$ be
a right-handed tetrad. Then the metric-compatible volume form is

$\omega=h^{1}\wedge\dots\wedge h^{n}.$ (5.4)

Interestingly, it turns out that
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)=-\accentset{\bullet}{T}$.
To show this, we first have to specify what we mean by applying
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
on
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}$,
because so far we have only defined the action of
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
on $\accentset{\bullet}{T}$. This step, however, is rather trivial. Looking
more closely at the definition of
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}$
in (5.5) it is obvious that, algebraically, there is no difference between
$\accentset{\bullet}{T}$ and
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}$
– both are antisymmetric in the lower indices, and $\accentset{\bullet}{T}$ is
not special in any way. Therefore, for the second application of
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
we use the same formula:

$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)^{b}_{\gamma\sigma}=h^{b}_{\phantom{b}\lambda}h\varepsilon_{\gamma\sigma\alpha\beta}\left[\dfrac{1}{4}\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)^{\lambda\alpha\beta}+\dfrac{1}{2}\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)^{\beta\alpha\lambda}-g^{\beta\lambda}\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)_{\phantom{\rho\alpha}\rho}^{\rho\alpha}\right].$
(5.5)

We will skip the tedious calculation, but we will approach the problem from
the opposite side, as this will enable us to discover something more
interesting along the way and the result will make it obvious that the
statement above is indeed true. We attempt to construct a duality operator
using all possible contractions of $\accentset{\bullet}{T}$ with the volume
form $\omega$, following the suggestion of Lucas and Pereira [14].

###  5.2 Dual torsion
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}$

Now, let us be more specific. The $\star$-operator in Definition 5.1.2
includes all possible contractions of $\alpha_{c\dots d}$ with $\omega_{a\dots
bc\dots d}$. However, torsion is a tensor with one upper index and two lower
indices in which it is antisymmetric. Therefore, there exist possible
contractions which would be omitted if we used Definition 5.1.2:

$\left(\star\accentset{\bullet}{T}\right)^{a}_{\alpha\beta}=\dfrac{1}{2}\omega_{\alpha\beta\mu\nu}\accentset{\bullet}{T}^{a\mu\nu}.$
(5.6)

If we choose to view torsion as a 2-form, we have to keep in mind that it is a
vector-valued form and $\accentset{\bullet}{T}^{a}$ are the component 2-forms.
Thus, the index $a$ is not a spacetime index, so the Hodge operator “ignores
it”.

Before trying to find all the possible contractions, let us first formulate
some rules, which will be also useful later.

####  5.2.1 Rules for possible contractions

We have already mentioned that the connection 1-form
$\accentset{\bullet}{\omega}$, as well as the curvature 2-form $\Omega$, is a
Lie algebra-valued form. They are matrix forms, the matrices being $4\times
4$, and the component forms are
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}$ and
$\Omega^{a}_{\phantom{a}b}$. Torsion may be viewed as a 2-form
$\accentset{\bullet}{T}$ with values in the Lie algebra of the group
$\mathbb{T}_{4}$ of translations in 4 dimensions – the component
forms222Although we will only calculate
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}T$
and
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}{\omega}$
for the case of teleparallel connection, from the treatment it will be easily
seen that the calculation is valid for a general metric-compatible connection.
are $\accentset{\bullet}{T}^{a}$. We will call indices $a,b$ the algebraic
indices of these forms, as they pertain to the algebra in which the forms take
values. $\mathbb{R}$-valued forms are tensors, so the forms
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b}$, $\Omega^{a}_{\phantom{a}b}$
and $\accentset{\bullet}{T}^{a}$ will have additional indices, which we will
call tensorial indices, e.g. in $\Omega^{a}_{\phantom{a}b\mu\nu}$ the
algebraic indices are $a,b$, and the tensorial indices $\mu,\nu$ can be
changed to $c,d$ using tetrad $h_{c}^{\phantom{c}\mu}$.

We will work in a tetrad basis, and all the components of forms will be
expressed with respect to the same tetrad $h_{a}$. Suppose $\alpha$ is a
$p$-form with $q$ algebraic indices. When determining all possible
contractions of $\alpha$ with the volume form $\varepsilon_{abcd}$, we will
abide by the following rules:

  1. 1.

$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\alpha\right)$
has $q$ algebraic indices and $(n-p)$ tensorial indices.
This requirement ensures that
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
is similar to $\star$, which behaves this way when applied on $p$-forms with
$q$ algebraic indices (these algebraic indices have nothing to do with the
manifold or coordinates).

  2. 2.

The tensorial indices of
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\alpha\right)$
must come from the volume form $\varepsilon_{abcd}$.
The motivation behind this rule also lies in our desire for
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
to be similar to $\star$. This rule ensures that
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\alpha$
will be a $(n-p)$-form.

  3. 3.

The behavior of
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\alpha\right)$
in the $q$ algebraic indices is the same as the
behavior of $\alpha$ in its $q$ algebraic indices.
For example, for the connection
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}bc}$ it holds that
$\accentset{\bullet}{\omega}_{abc}=-\accentset{\bullet}{\omega}_{bac}$ (it is
pseudo-antisymmetric in $a,b$), so
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)^{a}_{\phantom{a}bcde}$
must also be pseudoantisymmetric in $a,b$.

####  5.2.2 Duality condition

All the possible contractions of torsion with the volume form according to the
rules stated above are333The result of this calculation is independent of the
choice of frame, therefore we can work in a holonomic frame
$\partial_{\alpha}$. :
$\omega_{\alpha\beta\mu\nu}\accentset{\bullet}{T}^{\gamma\mu\nu},\omega_{\alpha\beta\mu\nu}\accentset{\bullet}{T}^{\mu\gamma\nu}$
and
$\omega_{\alpha\beta\mu\nu}\accentset{\bullet}{T}^{\lambda\mu}_{\phantom{\lambda\mu}\lambda}g^{\nu\gamma}$.

We try to construct an operator
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
using these contractions:

$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)^{\gamma}_{\phantom{\gamma}\alpha\beta}=\omega_{\alpha\beta\mu\nu}\left(a\accentset{\bullet}{T}^{\gamma\mu\nu}+b\accentset{\bullet}{T}^{\mu\gamma\nu}+c\accentset{\bullet}{T}^{\lambda\mu}_{\phantom{\lambda\mu}\lambda}g^{\nu\gamma}\right),$
(5.7)

and we will require that
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}\right)=-\accentset{\bullet}{T}$.
In Appendix A we show that this condition leads us to two equations

$\displaystyle 4a^{2}+2ab+2ac-2bc$ $\displaystyle=+1,$ (5.8) $\displaystyle
b^{2}+2ab-2ac+2bc$ $\displaystyle=0.$ (5.9)

Taking the sum of these equations, we obtain the following:

$+1=4a^{2}+4ab+b^{2}=(2a+b)^{2},$ (5.10)

hence

$b=-2a\pm 1.$ (5.11)

Combining the last equation with eq. (5.9) yields

$c=\pm\dfrac{b}{2\left(a-b\right)}=\dfrac{\mp 2a+1}{6a\mp 2}.$ (5.12)

We see that there are infinitely many possible constructions of
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$-operator.
Choosing $a=\frac{1}{2}$ and using the upper sign in (5.11) leads to $b=c=0$,
so that
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}=\star$
acts exactly the same way as the Hodge operator. Choosing $a=\frac{1}{4}$ and
again using the upper sign in (5.11) leads to $b=\frac{1}{2},c=-1$, which is
exactly444It is easy to see, that
$\varepsilon_{\gamma\sigma\alpha\beta}\accentset{\bullet}{T}^{\beta\alpha\lambda}=-\varepsilon_{\gamma\sigma\beta\alpha}\accentset{\bullet}{T}^{\beta\alpha\lambda}=-\varepsilon_{\gamma\sigma\alpha\beta}\accentset{\bullet}{T}^{\alpha\beta\lambda}=\varepsilon_{\gamma\sigma\alpha\beta}\accentset{\bullet}{T}^{\alpha\lambda\beta}$.
the operator constructed in a different way in the previous section!

####  5.2.3 Additional rule

To make the construction of
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}$
unique, we might apply an additional assumption. Looking at Definition 5.1.2,
we can see that the summation implied by the repeated indices in
$(\star\alpha)_{a\dots b}=\frac{1}{p!}\alpha^{c\dots d}\omega_{a\dots bc\dots
d}$ contains only one independent term, because in each term in the sum the
indices $c,...,d$ may be rearranged in ascending order. For example, for a
3-form in 5D,

$\displaystyle(\star\alpha)_{13}$
$\displaystyle=\dfrac{1}{3!}\alpha^{cde}\omega_{13cde}$
$\displaystyle=\dfrac{1}{3!}\left(\alpha^{245}\omega_{13245}\right.$
$\displaystyle+\left.\alpha^{425}\omega_{13425}+\alpha^{254}\omega_{13254}\right.$
$\displaystyle\left.+\alpha^{452}\omega_{13452}+\alpha^{524}\omega_{13524}+\alpha^{542}\omega_{13542}\right)$
$\displaystyle=\alpha^{245}\omega_{13245}.$

Therefore, the factor $\frac{1}{p!}$ prevents double-counting.

If we apply a similar rule for
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$,
that is, we include each independent term the same number of times, we have to
require $b=2a$, because $\accentset{\bullet}{T}^{\gamma\mu\nu}$ is
antisymmetric in $\mu,\nu$, while in general there is no relationship between
$\accentset{\bullet}{T}^{\mu\gamma\nu}$ and
$\accentset{\bullet}{T}^{\nu\gamma\mu}$. This rule does not tell us anything
about $c$ and its relationship with $a$, because
$\accentset{\bullet}{T}^{\lambda\mu}_{\phantom{\lambda\mu}\lambda}g^{\nu\gamma}$
is a different tensor from $\accentset{\bullet}{T}^{\gamma\mu\nu}$. Using this
additional rule, we obtain

$\displaystyle a=\pm\frac{1}{4},$ $\displaystyle b=\pm\frac{1}{2},$
$\displaystyle c=\mp 1,$ (5.13)

hence
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{T}$
is now given uniquely, up to a sign. Notice that the solution with the upper
sign corresponds to (5.5), which is what we were after.

###  5.3 Dual connection
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}$

Now we would like to see whether there are any other objects on which we can
apply our would-be operator
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$.
Until now, we have only found one such object, the torsion (or any 2-form with
1 algebraic index, i.e. vector-valued 2-form), which means that
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
does not yet deserve the noble title “operator”. We can start by looking at
the case of connection $\accentset{\bullet}{\omega}$.

####  5.3.1 Possible contractions and parametrizations

The whole procedure will be very similar to our treatment of the case of
torsion from the previous section. Connection
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$ has 3 indices, and
$\accentset{\bullet}{\omega}_{ab\mu}=-\accentset{\bullet}{\omega}_{ba\mu}$.
The result should be a 3-form, and the three indices must be inherited from
the volume form, which has therefore only 1 index left for contractions.
Hence, the only possible terms are
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}bc}\varepsilon^{b}_{\phantom{b}def}$,
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}bc}\varepsilon^{c}_{\phantom{c}def}$
and $\accentset{\bullet}{\omega}^{af}_{\phantom{af}f}\varepsilon_{bcde}$:

$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)^{a}_{\phantom{a}bcde}\equiv
a\left(\accentset{\bullet}{\omega}^{a}_{\phantom{a}fb}\varepsilon^{f}_{\phantom{f}cde}-\accentset{\bullet}{\omega}^{\phantom{bf}a}_{bf}\varepsilon^{f}_{\phantom{f}cde}\right)+b\accentset{\bullet}{\omega}^{a}_{\phantom{a}bf}\varepsilon^{f}_{\phantom{f}cde}+c\left(\accentset{\bullet}{\omega}^{af}_{\phantom{af}f}\varepsilon_{bcde}-\accentset{\bullet}{\omega}^{\phantom{b}f}_{b\phantom{f}f}\varepsilon^{a}_{\phantom{a}cde}\right).$
(5.14)

The terms in brackets are antisymmetrised so that pseudo-antisymmetry of
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$ is passed on to
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)^{a}_{\phantom{a}b\mu\nu\lambda}$.

Now, as we wish to apply
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$
on
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}$,
and
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\equiv\accentset{\bullet}{\Xi}$
is a 3-form, we have to repeat the same exercise we did for
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}b\mu}$, but now for
$\accentset{\bullet}{\Xi}^{a}_{\phantom{a}b\mu\nu\lambda}$. Three indices of
the volume form may be contracted. The only possible terms are
$\accentset{\bullet}{\Xi}^{a}_{\phantom{a}bcde}\varepsilon^{cde}_{\phantom{cde}f}$,
$\accentset{\bullet}{\Xi}^{a}_{\phantom{a}bcde}\varepsilon^{bcd}_{\phantom{bcd}f}$,
$\accentset{\bullet}{\Xi}^{a}_{\phantom{a}bcde}\varepsilon^{\phantom{a}bc}_{a\phantom{bc}f}$,
then
$\accentset{\bullet}{\Xi}^{ae}_{\phantom{ae}cde}\varepsilon^{cd}_{\phantom{cd}fg}$,
$\accentset{\bullet}{\Xi}^{a}_{\phantom{a}bcae}\varepsilon^{bc}_{\phantom{bc}fg}$,
$\accentset{\bullet}{\Xi}^{a\phantom{b}c}_{\phantom{a}b\phantom{c}ce}\varepsilon^{\phantom{a}b}_{a\phantom{b}fg}$
(however, the last of these is zero), and finally
$\accentset{\bullet}{\Xi}^{de}_{\phantom{de}cde}\varepsilon^{c}_{\phantom{c}fgh}$,
$\accentset{\bullet}{\Xi}^{c\phantom{bc}d}_{\phantom{c}bc\phantom{d}d}\varepsilon^{b}_{\phantom{b}fgh}$
(but the last one is zero):

$\displaystyle\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\Xi}\right)^{a}_{\phantom{a}bf}$
$\displaystyle\equiv\alpha\accentset{\bullet}{\Xi}^{a}_{\phantom{a}bcde}\varepsilon^{cde}_{\phantom{cde}f}+\beta\left(\accentset{\bullet}{\Xi}^{a}_{\phantom{a}cdeb}-\accentset{\bullet}{\Xi}^{\phantom{bcde}a}_{bcde}\right)\varepsilon^{cde}_{\phantom{cde}f}+\gamma\accentset{\bullet}{\Xi}^{c\phantom{deb}a}_{\phantom{c}deb}\varepsilon^{\phantom{c}de}_{c\phantom{de}f}+$
(5.15)
$\displaystyle+\mu\left(\accentset{\bullet}{\Xi}^{ae}_{\phantom{ae}cde}\varepsilon^{cd}_{\phantom{cd}bf}-\accentset{\bullet}{\Xi}^{\phantom{b}e}_{b\phantom{e}cde}\varepsilon^{cda}_{\phantom{cda}f}\right)+\nu\left(\accentset{\bullet}{\Xi}^{e\phantom{cde}a}_{\phantom{e}cde}\varepsilon^{cd}_{\phantom{cd}bf}-\accentset{\bullet}{\Xi}^{e}_{\phantom{e}cdeb}\varepsilon^{cda}_{\phantom{cda}f}\right)+$
$\displaystyle+\lambda\accentset{\bullet}{\Xi}^{de}_{\phantom{de}cde}\varepsilon^{ca}_{\phantom{ca}bf}.$

####  5.3.2 Duality condition

Now, we have too many constants – nine, in fact – and only two conditions:

$\displaystyle\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)$
$\displaystyle=\accentset{\bullet}{\omega},$ (5.32)
$\displaystyle\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\Xi}\right)$
$\displaystyle=\accentset{\bullet}{\Xi}.$ (5.49)

We cannot expect that these conditions determine the constants uniquely. As
the calculations would be tedious and prone to error if done by hand, we
resort to using xTensor, a Mathematica package from a suite of packages called
xAct.

In Appendix B, we obtain 10 equations from these conditions, but then we
discover that only 6 are independent. We divide them into two triples of
equations. The first triple contains no $c$-term

$\displaystyle 1$
$\displaystyle=\left(2a+b\right)\left(6\alpha+4\beta-2\gamma\right),$ (5.50)
$\displaystyle 0$
$\displaystyle=a\left(6\alpha+4\beta-2\gamma-2\mu+\nu\right)+b\left(-2\mu+\nu\right),$
(5.51) $\displaystyle 0$
$\displaystyle=a\left(4\mu-2\nu+2\lambda\right)-b\lambda,$ (5.52)

and the other triple is

$\displaystyle 0$
$\displaystyle=4a\beta+2b\beta+c\left(-6\alpha+10\beta+4\gamma\right),$ (5.53)
$\displaystyle 0$
$\displaystyle=a\left(6\alpha+2\beta-2\mu\right)-2b\mu+c\left(-2\mu-2\nu\right),$
(5.54) $\displaystyle 0$
$\displaystyle=4a\gamma+2b\gamma+c\left(8\beta+8\gamma\right),$ (5.55)

The very first equation tells us that we can not have $a=b=0$ (which means
that the other two equations must be linearly dependent), and that $b\neq-2a$.
For what follows, it is useful to notice that also $b\neq-a$: if $b=-a$ were
true, then eq. (5.51) would reduce to
$0=a\left(6\alpha+4\beta-2\gamma\right)$, which leads to either $a=b=0$, in
contradiction with (5.50), or $6\alpha+4\beta-2\gamma=0$, in contradiction
with (5.50).

Now, let us express everything in terms of $a,b$ and $c$. First, we take the
first three independent equations and simplify them by denoting $A\equiv
6\alpha+4\beta-2\gamma$ and $B\equiv-2\mu+\nu$:

$\displaystyle 1$ $\displaystyle=\left(2a+b\right)A,$ (5.56) $\displaystyle 0$
$\displaystyle=a\left(A+B\right)+bB,$ (5.57) $\displaystyle 0$
$\displaystyle=a\left(-2B+2\lambda\right)-b\lambda.$ (5.58)

Multiplying (5.57) by $\left(2a+b\right)$ leads to

$0=a+B\left(a+b\right)\left(2a+b\right),$ (5.59)

and multiplying (5.58) by $\left(a+b\right)\left(2a+b\right)$ gives

$0=2a^{2}+\left(2a-b\right)\left(a+b\right)\left(2a+b\right)\lambda.$ (5.60)

Having $b=2a$ would lead to $a=0$, but in that case the solution would
actually have to be $a=b=0$, which is impossible. Therefore, we have

$\displaystyle A$ $\displaystyle=\dfrac{1}{2a+b},$ (5.61) $\displaystyle B$
$\displaystyle=-\dfrac{a}{\left(a+b\right)\left(2a+b\right)},$ (5.62)
$\displaystyle\lambda$
$\displaystyle=-\dfrac{2a^{2}}{\left(2a-b\right)\left(a+b\right)\left(2a+b\right)}.$
(5.63)

Now we can attempt to express $\alpha,\beta,\gamma,\mu,\nu$ in terms of
$a,b,c$ using equations

$\displaystyle 0$
$\displaystyle=-6c\alpha+\left(4a+2b+10c\right)\beta+4c\gamma,$ (5.64)
$\displaystyle 0$ $\displaystyle=8c\beta+\left(4a+2b+8c\right)\gamma,$ (5.65)
$\displaystyle A$ $\displaystyle=6\alpha+4\beta-2\gamma,$ (5.66)
$\displaystyle 3a\alpha+a\beta$ $\displaystyle=\left(a+b+c\right)\mu+c\nu,$
(5.67) $\displaystyle B$ $\displaystyle=-2\mu+\nu.$ (5.68)

The first three equations do not contain $\mu,\nu$, so we can perform Gauss
elimination to express $\alpha,\beta$ and $\gamma$, and then express $\mu,\nu$
from the last two equations555If the denominator in the expression for
$\gamma$ is zero, the system has no solution.:

$\displaystyle\alpha$
$\displaystyle=\frac{1}{6\left(2a+b\right)}-\dfrac{2}{3}\beta+\dfrac{1}{3}\gamma,$
(5.69) $\displaystyle\beta$ $\displaystyle=-\dfrac{2a+b+4c}{4c}\gamma,$ (5.70)
$\displaystyle\gamma$
$\displaystyle=-\dfrac{2c}{(2a+b)^{2}+24c^{2}+11c(2a+b)}\cdot\dfrac{c}{2a+b},$
(5.71) $\displaystyle\mu$
$\displaystyle=\dfrac{a\left(3\alpha+\beta\right)-Bc}{a+b+3c},$ (5.72)
$\displaystyle\nu$
$\displaystyle=B+2\dfrac{a\left(3\alpha+\beta\right)-Bc}{a+b+3c}.$ (5.73)

This is a solution only in case $c\neq 0$ and $a+b+3c\neq 0$.

In case $c\neq 0$ and $a+b+3c=0$ we have $\alpha$, $\beta$ and $\gamma$ given
by (5.69), (5.70) and (5.71). In this case, we get a condition

$a\left(\dfrac{1}{2\left(2a+b\right)}-\beta+\gamma\right)=-\dfrac{ac}{\left(a+b\right)\left(2a+b\right)},$
(5.74)

which means that we can not write the solution in terms of $a,b,c$, but we
have to replace one of them by $\mu$ or $\nu$, and then express the other one
from equation (5.68).

In case $c=0$ the solution is

$\displaystyle\alpha$ $\displaystyle=\frac{1}{6\left(2a+b\right)},$ (5.75)
$\displaystyle\beta$ $\displaystyle=0,$ (5.76) $\displaystyle\gamma$
$\displaystyle=0,$ (5.77) $\displaystyle\mu$
$\displaystyle=\dfrac{a}{2\left(a+b\right)\left(2a+b\right)},$ (5.78)
$\displaystyle\nu$ $\displaystyle=0.$ (5.79)

####  5.3.3 Using the additional rule

The additional rule discussed in section 5.2.3 would give us

$\displaystyle\beta$ $\displaystyle=(-1)^{\xi}3\alpha,$ (5.80)
$\displaystyle\gamma$ $\displaystyle=3\alpha,$ (5.81) $\displaystyle\nu$
$\displaystyle=(-1)^{\zeta}2\mu,$ (5.82)

where $\xi,\zeta\in\\{0,1\\}$. However, looking at eq. (5.66) we see that we
can not have any of $\alpha,\beta,\gamma$ equal to $0$, as in that case
$\alpha=\beta=\gamma=0$, while $A\neq 0$, which contradicts (5.66). According
to (5.65), $\beta\neq-\gamma$, otherwise we have

$0=-8c\gamma+\left(4a+2b+8c\right)\gamma=2\left(2a+b\right)\gamma,$ (5.83)

which is impossible. Thus $\beta=\gamma=3\alpha$. Then, subtracting (5.65)
from (5.64) gives

$0=-6c\alpha+2c\beta-4c\beta=-12c\alpha,$ (5.84)

leading to $c=0$. This reduces eq. (5.65) to

$0=2\left(2a+b\right)\gamma,$ (5.85)

again, an impossibility. Therefore, imposing the additional rule leads to a
system of equations which has no solution.

####  5.3.4 Transformation of dual connection

We have to address one more issue: Is this construction of the
$\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}$-operator
as an operator acting on connection forms canonical? That is, we are
interested in its behavior under local Lorentz transformations.

The problem is that not all contractions are canonical tensor operations, e.g
the contraction implied by summation through $b$ in
$\accentset{\bullet}{\omega}^{a}_{\phantom{a}bc}\varepsilon^{b}_{\phantom{b}def}$
is, strictly speaking, not a contraction of two tensors.

Changing the tetrad

$h_{a}\longrightarrow\tilde{h}_{a}=\Lambda^{b}_{\phantom{b}a}(x)h_{b}$ (5.86)

leads to the transformation [17]

$\displaystyle\eta^{ab}$
$\displaystyle\longrightarrow\eta^{cd}\left(\Lambda^{-1}\right)^{a}_{\phantom{a}c}\left(\Lambda^{-1}\right)^{b}_{\phantom{b}d}=\eta^{ab},$
(5.87) $\displaystyle\varepsilon_{abcd}$
$\displaystyle\longrightarrow\varepsilon_{efgh}\Lambda^{e}_{\phantom{e}a}\Lambda^{f}_{\phantom{f}b}\Lambda^{g}_{\phantom{g}c}\Lambda^{h}_{\phantom{h}d}=\left(\det\Lambda\right)\varepsilon_{abcd}=\varepsilon_{abcd},$
(5.88) $\displaystyle\accentset{\bullet}{\omega}^{a}_{\phantom{a}bc}$
$\displaystyle\longrightarrow\tilde{\accentset{\bullet}{\omega}}^{a}_{\phantom{a}bc}=\left(\Lambda^{-1}\right)^{a}_{\phantom{a}d}\accentset{\bullet}{\omega}^{d}_{\phantom{d}ef}\Lambda^{e}_{\phantom{e}b}\Lambda^{f}_{\phantom{f}c}+\tilde{h}_{c}^{\phantom{c}\mu}\left(\Lambda^{-1}\right)^{a}_{\phantom{a}d}\partial_{\mu}\Lambda^{d}_{\phantom{d}b}.$
(5.89)

The first term on the right in (5.89) is exactly what should be there if
$\accentset{\bullet}{\omega}$ were a tensor of type $\binom{1}{2}$. Therefore,
to make the following diagram

${\accentset{\bullet}{\omega}}$${\tilde{\accentset{\bullet}{\omega}}}$${\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}{\accentset{\bullet}{\omega}}}$${\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}{\tilde{\accentset{\bullet}{\omega}}}}$${\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}}$${\Huge{\Lambda}}$${\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}}$${\Lambda}$

commutative, where $\Lambda$ is a local Lorentz transformation, the dual
connection has to respond to the local Lorentz transformation in the following
manner:

$\displaystyle\begin{split}\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)^{a}_{\phantom{a}bcde}\longrightarrow&\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\tilde{\accentset{\bullet}{\omega}}\right)^{a}_{\phantom{a}bcde}=a\left(\tilde{\accentset{\bullet}{\omega}}^{a}_{\phantom{a}fb}-\tilde{\accentset{\bullet}{\omega}}^{p}_{\phantom{p}fq}\eta_{bp}\eta^{aq}\right)\eta^{fg}\varepsilon_{gcde}+\dots\\\
&=\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)^{k}_{\phantom{k}lmnr}\left(\Lambda^{-1}\right)^{a}_{\phantom{a}k}\Lambda^{l}_{\phantom{l}b}\Lambda^{m}_{\phantom{m}c}\Lambda^{n}_{\phantom{n}d}\Lambda^{r}_{\phantom{r}e}+\\\
&+a\left[\tilde{h}_{b}^{\phantom{c}\mu}\left(\Lambda^{-1}\right)^{a}_{\phantom{a}k}\left(\partial_{\mu}\Lambda^{k}_{\phantom{k}f}\right)-\eta_{bp}\eta^{aq}\tilde{h}_{q}^{\phantom{c}\mu}\left(\Lambda^{-1}\right)^{p}_{\phantom{a}k}\left(\partial_{\mu}\Lambda^{k}_{\phantom{k}f}\right)\right]\eta^{fg}\varepsilon_{gcde}+\dots\\\
&=\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)^{k}_{\phantom{k}lmnr}\left(\Lambda^{-1}\right)^{a}_{\phantom{a}k}\Lambda^{l}_{\phantom{l}b}\Lambda^{m}_{\phantom{m}c}\Lambda^{n}_{\phantom{n}d}\Lambda^{r}_{\phantom{r}e}+\\\
&+a\left[\delta^{q}_{b}\delta^{a}_{p}-\eta_{bp}\eta^{aq}\right]\tilde{h}_{q}^{\phantom{c}\mu}\left(\Lambda^{-1}\right)^{p}_{\phantom{p}k}\left(\partial_{\mu}\Lambda^{k}_{\phantom{k}f}\right)\eta^{fg}\varepsilon_{gcde}+\dots\end{split}$

This just means that the dual connection
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)$
has a rather strange trasformation rule. However, there is no problem with
that, because the connection 1-form $\accentset{\bullet}{\omega}$ itself
transforms in an unusual manner. It means that the dual connection
$\left(\mathbin{\mathchoice{\ooalign{$\displaystyle\vbox{\hbox{\scalebox{0.77778}{$\displaystyle\bigcirc$}}}$\cr$\displaystyle\star$\cr}}{\ooalign{$\textstyle\vbox{\hbox{\scalebox{0.77778}{$\textstyle\bigcirc$}}}$\cr$\textstyle\star$\cr}}{\ooalign{$\scriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptstyle\bigcirc$}}}$\cr$\scriptstyle\star$\cr}}{\ooalign{$\scriptscriptstyle\vbox{\hbox{\scalebox{0.77778}{$\scriptscriptstyle\bigcirc$}}}$\cr$\scriptscriptstyle\star$\cr}}}\accentset{\bullet}{\omega}\right)$
is a different 3-form depending on the basis in which we are working.

This peculiarity is not present in the case of torsion
$\accentset{\bullet}{T}$ or curvature $\Omega$, as all of their indices are
essentially “tensorial” in nature – $\Omega^{a}_{\phantom{a}bdc}$ can be
thought of as components of a tensor. However, this problem reappears for
general $\mathfrak{so}(1,3)$-valued 2-forms or translation algebra-valued
2-forms, in which case we do not know how they transform with respect to the
algebraic (“non-tensorial”) indices.

###  5.4 Dual curvature