# Tangent space symmetries in general relativity and teleparallelism

Tom Lawrence<EMAIL_ADDRESS>Ronin Institute for Independent
Scholarship, 127 Haddon Place
Montclair, New Jersey 07043-2314, United States of America

((Day Month Year); (Day Month Year))

###### Abstract

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold
induce homogeneous linear transformations on its tangent spaces. We see that a
pseudo-orthonormal frame in a given tangent space is the basis for a set of
Riemann normal coordinates. A Lorentz subgroup of the general linear
transformations preserves this pseudo-orthonormality. We borrow techniques
from the methodology of non-linear realizations to analyze this group-subgroup
structure.

‘Parallel maps’ are used to relate tangent space at different points.
‘Parallelisms’ across a finite region of the manifold may be built up from
them. These are used to define Weitzenböck connections and Levi-Civita
connections.

This provides a new formulation of teleparallel gravity, in which the tetrad
field is viewed as a field-valued group element relating the coordinate basis
to the frame basis used in defining a parallelism. This formulation separates
the metric degrees of freedom from those associated with the choice of
parallelism. The group element can be combined by matrix multiplication with
Lorentz transformations of frame or with other Jacobian matrices. We show how
this facilitates a new understanding of inertial forces and local Lorentz
transformations.

The analysis is also applied to translations of the coordinates. If they are
constant across spacetime, this has no effect on the tangent space bases. If
the translation parameters become fields, they induce general linear
transformations of the coordinate basis; however, the tetrad components can
only be expressed in terms of translations on a flat spacetime.

###### keywords:

Teleparallelism; non-linear realization; tangent space.

##  1 Introduction

We start this paper with some historical background on the main concepts,
focusing on how they were first used in, or emerged from, physics research.
General relativity (GR) is covered very briefly, as it so well known. It is
assumed that readers will have at least some basic familiarity with
teleparallelism. This subject is covered to the extent needed to set out the
main debates and concepts addressed in this paper. Non-linear realizations are
covered in more detail, because few researchers in GR or teleparallel gravity
are likely to have a solid understanding of both this subject. (These topics
are covered roughly in chronological order, splitting the narrative on
teleparallelism either side of that on non-linear realizations.)

This introductory section then concludes by summarizing the main concepts in
this paper, the notation used and the structure of the paper.

###  1.1 General relativity

Einstein spent many years developing GR, completing it with Hilbert in 1915.
It takes spacetime to be a pseudo-Riemannian manifold. This means that tangent
spaces admit a ten-component metric. These ten components are the basic
variables in the gravitational sector of the theory. More generally, in an
$N$-dimensional spacetime, there would be $\frac{1}{2}N(N+1)$ of these.
Neighbouring tangent spaces are compared using the Levi-Civita connection,
allowing covariant derivatives to be defined. This connection is uniquely
defined on a given spacetime for a given coordinate system and is constructed
from the metric - or more precisely, its derivatives and its inverse. Its
field strength is the Riemann curvature of the spacetime, which represents the
gravitational field. A contraction of this tensor, the Ricci tensor, is non-
zero wherever matter is present, and its relation to another tensor describing
the matter distribution provides the field equation of the theory.

###  1.2 Teleparallelism - the early years

When GR was developed, the only other fundamental interaction that was
recognized by physics was electromagnetism. Einstein was keen to extend the
theory to incorporate electromagnetism in a geometric way. He tried a number
of different approaches to this[1].

One approach was ‘Fernparallelismus’, translated as ‘distant parallelism’ and
now known as ‘teleparallelism’[2]. He noted that an $N$-bein field has $N^{2}$
independent components. The components of the metric can be written as
functions of these, but then there are $\frac{1}{2}N(N-1)$ degrees of freedom
contained in the $N$-bein field which describe invariances of the metric[3].
His idea was that these additional degrees of freedom could be used in
describing electromagnetism. In defining an $N$-bein field across the
spacetime manifold, he needed to use a new type of connection, which he
discovered had already been investigated by Weitzenböck, Cartan and others[2,
4].

This approach was unsuccessful in its aim and it appears that papers on the
subject were only published sporadically in the decades following the Second
World War. While not strictly a paper on teleparallelism, the work of
Utiyama[5] is worth noting in the development of the theory of the ‘tetrad’,
as the four-dimensional $N$-bein field has come to be known111In the usual
formulation of teleparallel gravity, there is a pseudo-orthonormal frame
associated with each point in spacetime. Each ‘leg’ of this vielbein or tetrad
can be decomposed in the coordinate basis, giving rise to a 16-component
matrix, which carries the field degrees of freedom. The frame basis and the
16-component matrix are both sometimes referred to as the ‘tetrad’, as the 16
fields of the latter are seen as components of the former. In this paper, they
are viewed as fundamentally separate objects: the matrix is an element of the
general linear group, while the frame basis is an element of its carrier
space. We only use the word ‘tetrad’ to make contact with the usual
formulation, and when we use it, we mean the field-valued matrix.. His aim was
to provide a common gauge theory description of gravity, electromagnetism and
the theory of Yang and Mills[6] which had been published the previous year.
Utiyama considered gravity as a gauge theory of the Lorentz group, although he
saw general relativity as also having a separate symmetry under coordinate
transformations.

This view of these symmetries as separate has persisted until today, but we
show in this paper that this is not actually the case. Lorentz transformations
form a subgroup of the general linear transformations induced by changes of
coordinates. Teleparallel papers of the 1960s and 1970s seem to regard them as
separate transformations for the purpose of calculations (see [7, 8] and
references therein), although Pellegrini and Plebanski[7] talk about “the
group of coordinate transformations” and its “subgroup of constant Lorentz
rotations of the tetrad”.

###  1.3 Coset space decompositions and non-linear realizations

Non-linear realizations became an active area of research in the 1960s. This
started with the non-linear sigma model, introduced by Gell-Mann and Lévy in
1960[9]. In this model, a four-component multiplet of Lorentz scalar fields
was constrained by fixing the norm of the multiplet. This allowed them to
eliminate one component from the Lagrangian. This elimination left the
Lagrangian depending non-linearly on the remaining components. (The resulting
field space was diffeomorphic to $S^{3}$, parametrized by the three remaining
field components, although this was not stated in the paper.)

This was followed by many papers in which the Lagrangian was invariant under
an internal symmetry group - typically a chiral group - but was constructed
non-linearly from the fields on which the transformations act. In 1968,
Coleman _et al_[10] identified a common feature of this research as crucial to
further analysis: that the fields transformed linearly only under a subgroup
of these symmetries and transformed in a non-linear way under the remainder of
the group. (See references within [10] for the earlier papers.)

This paper showed how it was possible to classify all such non-linear
realizations, presenting a comprehensive analysis based on group theory. A
companion paper provided the general method for constructing Lagrangians with
the required symmetries, based on covariant derivatives[11]. The following
year, a paper by Salam and Strathdee[12] showed how this theory was applicable
whenever a Lie group symmetry was spontaneously broken to a continuous
subgroup222This was also shown in the specific case of chiral SU(3) by
Honerkamp[13].. (Such spontaneous symmetry breaking in field theory was first
proposed by Goldstone[14] in 1960 - the same year as the Gell-Mann and Lévy
paper, and indeed, in the same journal.)

This body of theory has been developed further since 1969, notably in
Meetz[15], Isham[16, 17], Balachandran _et al_[18] and Boulware and Brown[19].
However, the necessary background for the analysis in the remainder of this
paper can all be found in Coleman _et al_[10] and Salam and Strathdee[12].

The starting point of Coleman _et al_ was to note that a subgroup $H$ of a
group $G$ could be used to partition $G$ into cosets of the form $gH$, where
$g\in G$. Now if $G$ and $H$ are both linear Lie groups, we may write any
element of $G$ in the form of an exponential of a matrix in its Lie algebra,
and similarly for any element of $H$. However, if $h\in H$, then the cosets
$gH$ and $ghH$ are identical. It is then easy to see that we can uniquely
write $gH$ in the form

$gH=\mathrm{e}^{\mathrm{i}\theta^{a}M_{a}}\left\\{\mathrm{e}^{\mathrm{i}\theta^{A}M_{A}}\;\forall\,\theta^{A}\right\\}$
(1)

where $M_{A}$ are the generators of $H$, $M_{a}$ are the remaining generators
of $G$ (now called the ‘broken generators’), the $\theta$ are group parameters
associated with these various generators and repeated indices are summed over.
The first factor here is a representative of the coset, denoted $L$:

$L=\mathrm{e}^{\mathrm{i}\theta^{a}M_{a}}\,.$ (2)

Then, because each element of $G$ is in only one coset, it can be uniquely
decomposed in the form

$g=Lh\,.$ (3)

A Lorentz scalar multiplet of $G$ can be constrained either ‘by hand’ (as in
the non-linear sigma model) or through a potential with degenerate minima
constructed from the multiplet (as in the Goldstone mechanism). The vacuum
manifold/constrained field space is then diffeomorphic to $G/H$, where $H$ is
the subgroup under which a point in this field space is invariant. This field
space is then parametrized by the coset space parameters $\theta^{a}$ \- that
is, there is one ‘Goldstone boson’ for each of these parameters.

Using the parametrization of the coset space (1), Coleman _et al_ analyzed the
transformation of these fields. If we start with a coset $L_{0}H$, the action
of another element $g$ on it is as follows:

$g:L_{0}H=\mathrm{e}^{\mathrm{i}\theta^{a}M_{a}}\left\\{\mathrm{e}^{\mathrm{i}\theta^{A}M_{A}}\;\forall\,\theta^{A}\right\\}\mapsto
L^{\prime}H=\mathrm{e}^{\mathrm{i}\theta^{\prime
a}M_{a}}\left\\{\mathrm{e}^{\mathrm{i}\theta^{A}M_{A}}\;\forall\,\theta^{A}\right\\}\,.$
(4)

Thus $L_{0}$ is mapped to a new coset space representative

$L^{\prime}=\mathrm{e}^{\mathrm{i}\theta^{\prime a}(\theta^{a},g)M_{a}}\,.$
(5)

In particular, under the action of an element of $H$, they showed that

$h:L_{0}\mapsto L^{\prime}=hL_{0}h^{-1}\,.$ (6)

They also showed that $L^{-1}$ can be used to reduce any other multiplet of
$G$ in the system to a multiplet of $H$ only, but we do not reproduce this
analysis here because it will not be needed in this paper.

The coset space representative $L$ also turned out to be crucial to
constructing covariant derivatives. From it, one can construct a derivative
based on the Maurer-Cartan form, $L^{-1}\partial_{\mu}L$. This takes values in
the algebra of $G$. It has a Cartan decomposition into the subgroup and coset
space parts of the Lie algebra:

$L^{-1}\partial_{\mu}L=2\mathrm{i}(v^{A}_{\mu}M_{A}+a^{a}_{\mu}M_{a})\,.$ (7)

It turns out that the $a^{a}_{\mu}$ are the covariant derivatives for the
Goldstone bosons:

$D_{\mu}M^{a}=a^{a}_{\mu}\,,$ (8)

while the $v^{A}_{\mu}$ are the parameters of the connections involved in the
covariant derivatives of any other fields in the system:

$D_{\mu}\psi=\partial_{\mu}\psi+\mathrm{i}v^{A}_{\mu}S(M_{A})\psi\,,$ (9)

where $S(M_{A})$ is the representation of the subgroup generators appropriate
to the multiplet $\psi$.

In later sections of this paper, we apply some of the above theory to tangent
space symmetries. It should be noted that this is far from the first time that
this theory has been applied to spacetime symmetries. Indeed, Salam and
Strathdee provided a companion paper to that mentioned above, in which $G$ was
the conformal group and $H$ was its Lorentz subgroup[20]. Since then, there
have been many others (see, for example, [21, 22, 23, 24]). Indeed, in recent
decades, there have even been papers on non-linear realizations in
teleparallel gravity[25, 26, 27] and symmetric teleparallel gravity[28]. Many
of these papers have had an explicit aim of establishing a non-linearly
realized gauge symmetry, often as a route to quantization. The analysis in the
current paper is quite different. We merely observe the presence of a group-
subgroup structure, in which the metric degrees of freedom are associated with
the coset space parameters, and borrow techniques from the theory of non-
linear realizations for analysing it. There is actually a non-linear
realization hidden within this structure, but it is not utilized in the
analysis in this paper.

###  1.4 Teleparallelism - recent decades

Since the late 1970s, teleparallelism has undergone a revolution. This has
resulted in the Teleparallel Equivalent of General Relativity (TEGR) and
several families of ‘modified’ theories. Details of their genesis can be found
in a 2013 review by Maluf[29]. All of them use the Weitzenböck connection,
which has zero field strength but does have a torsion.

It is a scalar constructed from this torsion which is used in the action of
TEGR, in place of the Einstein-Hilbert action. The action is varied with
respect to the tetrad and results in the same field equations as GR.

In the modified theories, the Lagrangian contains torsion scalars which are
more general than that used in TEGR. It is now known the universe is expanding
and that the rate of this expansion is accelerating. Cosmological observations
are becoming increasingly precise, revealing growing detail about this
expansion and about the early epochs of the universe. These do not fit easily
within GR as it stands and researchers are looking for consistent teleparallel
theories which fit the observations.

Teleparallel theories of gravity are also seen as having other advantages over
GR.

Firstly, they engage directly with spacetime symmetries, as is explored in
this paper. The connections contained in covariant derivatives therefore
appear to be gauge fields for these symmetries. The other fundamental
interactions are also described in terms of gauged symmetries, and quantized
on that basis. These similarities have provided a basis for working towards a
quantum field theory for gravity.

There is some debate, however, as to which symmetries are gauged in
teleparallel theory. Hayashi and Nakano[30], for example, took a different
view from Utiyama (see above), publishing a paper in 1978 which proposed that
gravity can be expressed as a gauge theory of translations. This was based on
the construction of a derivative operator which is covariant under local
translations. From this point onwards, it has often been stated that
teleparallel gravity constitutes a gauge theory of translations. Recent
publications to this effect by Aldrovandi and Pereira[31] and Pereira and
Obukhov [32] have been disputed by Fontanini _et al_[33, 34]. They point to
the role of the general linear group as the structure group of the tangent
bundle. Hohmann _et al_[35], meanwhile, do not view general linear
transformations and translations as separate, stating that “A displacement
transformation can be seen, from the passive perspective, as a general
coordinate transformation. Obviously, such a transformation is included in the
GL (General Linear) transformations. In a GL gauge connection, the
displacement component is precisely that which has neither curvature nor
torsion”.

In this paper, we look at translations mainly as changes of coordinates. It is
shown that such passive translations do not act directly on the tangent
bundle. Global translations do not act on the tangent spaces at all. The
action on the tangent space induced by local translations is simply that of
the general linear group, albeit represented as a displacement of the basis
rather than a contraction with the basis. This is a subtle advance on the
statement made by Hohmann _et al_ \- we find that _any_ general linear
transformation on a specific tangent space can be expressed in terms of
translations. However, expressing a tetrad _field_ in terms of a local
translation does constrain its derivatives. Consequently, the Weitzenböck
connection can only be constructed from translation parameters in a flat
region of spacetime. Nonetheless, the relationship between local translations
and the general linear group allows one to write, for example, the connection
along a geodesic in terms of the derivatives of the translation parameters.

The displacement of the basis provides a minimal coupling to the derivatives
of the translation parameters, in line with the covariant derivative operator
of Hayashi and Nakano[30]. However, an inhomogeneous displacement of the
vector components cannot be induced by a coordinate transformation. Allowing
for such a transformation would constitute extending general relativity.

This would all seem to support many of the arguments advanced by Fontanini _et
al_. However, it may not preclude teleparallel gravity being a gauge theory of
translations expressed as _point transformations_. We touch on this only
briefly at the end of this paper and note that it is an area worthy of further
study.

Secondly, these theories are particularly useful for calculations concerning
conservation of energy density, due to the way in which the energy-momentum
density of the gravitation field may be separated out from inertial effects.
Deep and informative analysis on this issue was carried out by Aldrovandi _et
al_[36].

While most of their analysis seems valid, it suggests that when the
teleparallel spin connection has a non-zero value, this is associated purely
with inertial effects. This is a claim often found in the literature. For
example, Krs̆s̆ak and Saridakis[37] state that “various spin connections
represent different inertial effects for the observer”, while Golovnev _et
al_[38] state that “the spin connection is then associated solely with
inertia”. We show in Sections 4 and 5 that this is not in fact the case. The
spin connection depends on the choice of frame _used to define the
parallelism_. Inertial effects, by contrast, are due to the choice of local
inertial frame for the observer, which is entirely determined by the
coordinate system.

###  1.5 Overview and structure of this paper

This paper looks at the way in which changes of coordinates on a pseudo-
Riemannian manifold form an infinite dimensional group, and induce homogeneous
linear transformations of the coordinate basis on each tangent space. This can
be seen as a homomorphism from the covariance group to the structure group of
the tangent bundle, a general linear group. Pseudo-orthonomality is preserved
under a Lorentz subgroup of this general linear group.

Recently, there has been considerable debate in the teleparallel gravity
research community about this Lorentz subgroup of symmetries - its physical
meaning and whether the degrees of freedom in the Lorentz connection represent
physical fields[37, 38, 39, 40]. A summary of recent papers and the current
state of understanding can be found in a paper by Bejarano _et al_[40]).

The approach of this paper is to note that the $N^{2}$ degrees of freedom
identified by Einstein - in our case 16 - are those of the general linear
group. We then separate them into those which are contained in the metric and
those which are associated with the Lorentz subgroup using a decomposition
analogous to (3). In outline, we will do this as follows.

We denote the general linear group $J$ and its Lorentz subgroup $I$. An
arbitrary element $j$ can then be decomposed as

$j=li\,,$ (10)

where $i\in I$ and $l$ is a representative of the coset space $J/I$. Thus if
we write the generators of $I$ as $M_{\mu\nu}$, the remaining generators of
$J$ as $m_{\mu\nu}$, and their corresponding parameters $\Omega^{\mu\nu}$ and
$\omega^{\mu\nu}$ respectively, we have

$l=\mathrm{e}^{\mathrm{i}\omega^{\mu\nu}m_{\mu\nu}}$ (11)

and

$i=\mathrm{e}^{\mathrm{i}\Omega^{\mu\nu}M_{\mu\nu}}\,.$ (12)

The group $J$ is used to carry out changes of basis on the tangent space. Its
elements are matrices which act on a basis by contraction to map it to another
basis. This includes (pseudo-orthonormal) frame bases (which, as we will see,
can be considered locally as coordinate bases for a set of Riemann normal
coordinates). Thus in a specific tangent space, the transformation $j_{0}$
between a given orthonormal frame and a given coordinate basis can be
decomposed as

$j_{0}=l_{0}i_{0}\,,$ (13)

with $l_{0}$ and $i_{0}$ taking the forms (11) and (12) above.

To construct covariant derivatives, we need a coordinate-independent way of
comparing vectors in different tangent spaces. We do this by choosing a
‘parallel map’ from one tangent space into a second, which is linear over the
space and preserves pseudo-orthonormality. This defines a frame in the second
space as the image of the frame in the first. By constructing such a map to
every tangent space in a coordinate neighbourhood, we build up a
‘parallelism’, with the frame bases forming a frame field. If the frame field
is continuous over the neighbourhood, the connection associated with this
parallelism is a Weitzenböck connection.

Any continuous frame field can be used to construct a Weitzenböck connection.
They are all related by local Lorentz transformations. This is a crucial
difference between Levi-Civita and Weitzenböck connections. To calculate the
functional form of a Levi-Civita connection on a given spacetime, one needs
only to choose a coordinate system. For a Weitzenböck connection, by contrast,
one must also choose a parallelism.

Once a coordinate system and a parallelism have been chosen, this specifies
the coordinate basis and frame basis at every point. (13) can then be used to
decompose the transformation between them, with $j_{0},l_{0},i_{0}$ promoted
to fields.

We will denote the coordinate basis $\mathbf{e}_{\mu}$ and the frame basis
associated with the parallelism $\hat{\mathbf{n}}_{\mu}$, with the hat
denoting orthonormality, so that

$\mathbf{e}_{\mu}=(j_{0})_{\mu}{}^{\xi}\hat{\mathbf{n}}_{\xi}\,.$ (14)

While it is possible to identify these bases with differential operators, as
is done in the formalism of differential forms, this is not necessary for the
analysis in this paper. We stick to the more abstract form above, as this
better represents the bases as belonging to the carrier space of $J$.
Crucially, $\mathbf{e}_{\mu}$ and $\hat{\mathbf{n}}_{\mu}$ belong to the
_same_ carrier space - they are bases on the same tangent space, and for this
reason they carry the same index. This is a major difference from the usual
tetrad formulation of teleparallel gravity, in which the frame basis usually
carries a Latin index and the coordinate basis a Greek one. This means that
$j_{0},l_{0},i_{0}$ have two indices of the same type, as appropriate to group
elements, and they can be combined with other group elements by matrix
multiplication. This facilitates changing coordinates or changing paralellism
and is what gives this formulation its power. We will explain the relation
between this formulation and the usual tetrad one shortly in this subsection.

As we will see, the decomposition (13) allows us to define an ‘intermediate’
basis, $\hat{\mathbf{k}}_{\xi}$. The resulting situation can then be
schematically represented as in Fig. 1.

Figure 1: Action of $j_{0}$

A change of coordinates induces an action $j(u)$ on $\mathbf{e}_{\mu}$ and
therefore acts on $j_{0}$ from the left. This will change both the $l_{0}$ and
$i_{0}$ factors. A change of parallelism is achieved by applying $i(u)$ to
$\hat{\mathbf{n}}_{\mu}$ and therefore acts on $j_{0}$ from the right. This
changes just the $i_{0}$ factor.

This decomposition is helpful when working with the Weitzenböck connection,
because, as we shall see, its associated Lorentz connection depends only upon
$i_{0}$ and its derivatives and hence upon the $\Omega^{\mu\nu}$ fields.
Similarly, the metric is a function of $l_{0}$ and hence the $\omega^{\mu\nu}$
fields only. These will transform as a non-linear realization of $J$. However,
in practice, it is easier to work with $i_{0}$ and $l_{0}$ than
$\Omega^{\mu\nu}$ and $\omega^{\mu\nu}$ and these group parameters are not
needed at all in the analysis in this paper. The expressions (11) and (12) are
implicitly assumed throughout this paper, but these forms are never explicitly
used. (They could be important, though, when trying to identify explicit forms
for the ‘tetrad fields’ in particular situations, as in [41].)

This deliberate separation of these degrees of freedom at the outset
distinguishes this paper from the existing literature on teleparallelism. As
noted by Golovnev _et al_ , until recently, most research has been carried out
in the ‘Weitzenböck gauge’, in which the Weitzenböck spin connection vanishes.
This happens when the frame field used to define the Weitzenböck
connection333The Weitzenböck connection is described in qualitative terms
above and defined more precisely in Section 4. Some authors, such as Bejarano
_et al_[40], use the phrase ‘Weitzenböck connection’ to mean this connection
specifically in the Weitzenböck gauge. This has probably arisen from the
practice of working in this gauge by default. By contrast, we use it in the
wider sense given by Pereira[42]. is such that $i_{0}$ is constant (a global
Lorentz transformation). These are referred to as ‘proper frames’ by Krs̆s̆ak
and Saridakis, while Golovnev _et al_ call this a ‘pure-tetrad approach’.
(‘Proper frames’ therefore differ from $\hat{\mathbf{k}}_{\xi}$ by at most a
global Lorentz transformation.) Golovnev _et al_ comment that “Often the pure-
tetrad approach is assumed explicitly, often it is not clear what is assumed’.

However, in recent years, it has been realized that working in the Weitzenböck
gauge violates Lorentz symmetry, which is seen as undesirable. Some authors
have tried to remedy this (for example [37, 38, 39]), as described by Bejarano
_et al_[40], by taking the theories in the Weitzenböck gauge and supplementing
them with a non-zero Lorentz connection, to ‘covariantize’ them. These have
started to reveal some of the structure described in this paper. However, much
of this research has been hampered by two things. The first is not
understanding the Lorentz group as a subgroup of the structure group on the
tangent bundle. The second is confusion about its relation to inertial effects
- which can only be understood when the group theory aspects are clear.
Consequently, these papers have adopted a wide variety of concepts,
terminology and symbols, which are inconsistent across papers and sometimes
within papers.

For example, Bejarano _et al_ use $\mathbf{e}_{a}$ for a generic frame basis,
with components $e_{a}^{\mu}$ (when the coordinate basis is $\partial_{\mu}$).
Golovnev _et al_ use $e_{a}^{\mu}$ for a ‘proper tetrad’. Böhmer _et al_[39],
on the other hand, reserve $e_{a}$ for a frame basis on Minkowski (flat)
spacetime, specializing further to $e^{\prime}_{a}$ for a holonomic frame on
this spacetime, and use $h^{a}{}_{\mu}$ for a ‘non-trivial tetrad field’.
Krs̆s̆ak and Saridakis[37] use $h^{a}{}_{\mu}$ for a generic tetrad, but also
introduce a ‘reference tetrad’ $h^{a}_{(r)}{}_{\mu}$, based on the limit
$G\rightarrow 0$. Most authors simply reverse the positions of indices to
denote the inverse tetrad, but some use a different symbol - for example
Hohmann _et al_[41] use $\theta^{a}{}_{\mu}$ for the tetrad and
$e_{a}{}^{\mu}$ for the inverse tetrad.

Given the inconsistency of notation and its unsuitability for the formulation
of teleparallel gravity contained in this paper, I have decided to stick to
the notation I have used for many years prior to learning of these debates.
The notation adopted in this paper emphasizes, and is appropriate to, a group
theory approach. Inverse group elements, for example, are denoted with a
suffix of $-1$ rather than just changing the position of the indicies.
(Another example of this difference is that in this paper, a Lorentz factor
$i_{0}$ is included in the Weitzenböck spin connection from the start; any
further Lorentz transformation of the frame is then denoted $i$. In the
existing literature, by contrast, $\Lambda^{a}{}_{b}$ is often used for any
Lorentz transformation; if it is used in the Weitzenböck spin connection it
may differ from the $i_{0}$ here by a global factor.)

However, this paper does follow the notation of Pereira[42] in using a dot
above a connection or covariant derivative to specify that it is a Weitzenböck
connection or covariant derivative. Similarly, circles above connections or
covariant derivatives denote that they are Levi-Civita ones.

Indices are usually used explicitly, except where their suppression makes a
point of group theory clearer. Greek letters are used for indices, but we
avoid using the letter $\tau$ for this purpose - this is exclusively used to
denote proper time (and on occasion is used as a subscript in this context).

Where it is necessary to specify which coordinate system a set of tensor
components relates to, this will be done by putting it in brackets in a
superscript or subscript. For example, the components of a vector $\mathbf{V}$
in a coordinate system $u^{\prime\mu}$ will be written
$V^{\mu}_{(u^{\prime})}$.

In this analysis, it is very easy to confuse the value of a function at a
point with its functional form. Consequently, we will often state explicitly
which point a function is evaluated at, if this is what we are doing.

The language of fibre bundles is largely avoided, as the approach is intended
to emphasize firstly how the group transformations are induced by changes of
coordinate, and secondly the physical interpretation of these transformations.

The theory in this paper assumes a pseudo-Riemannian spacetime, in which
observations are made by a classical (point-like) observer. Measurements of
field configurations taking values in a Hilbert space are explicitly excluded
from the analysis. It proceeds from this point with geometric arguments, which
are independent of the action being used. While it is phrased in terms of a
four-dimensional spacetime, the theory is also valid for a higher-dimensional
spacetime.

While the theory presented above summarizes much of the content of Sections 2
to 4, there are many subtleties and traps for the unwary in this analysis. We
therefore take the arguments step-by-step from first principles. It may seem
unnecessarily slow and basic to some readers, but this is what has led to the
insights in this paper into the current debates, and it seems prudent to err
on the side of caution to avoid further misunderstandings.

Section 2 looks at the covariance group of coordinate transformations. It
finds the induced action of this group on the coordinate bases and vector
components in a tangent space _at a single point_. The Jacobian matrix is seen
to be an element of a general linear group. (This analysis will be crucial
when looking at translations in Section 6.) The metric is then defined. The
pseudo-orthogonal invariance group of the Minkowski metric is used to
partition the general linear group into cosets and this leads to a natural
decomposition of the change of frame.

This analysis is then extended to a curve in Section 3, allowing one to define
connections and covariant derivatives. The transformation properties of these
are identified. It is then extended further to a four-dimensional chart on the
spacetime in Section 4. Choices of connection and their associated covariant
derivative are then available and we look at the features of these (in
particular, the Weitzenböck and Levi-Civita connections) and the relationships
between them. In particular, we look at what happens on a geodesic and how
Riemann normal coordinates may be applied, as well as field strengths of the
connections and their associated Lorentz connections. We consider two
different decompositions of the Weitzenböck connection: that using the coset
space decomposition and that using its contorsion.

Sections 2 to 4 therefore work carefully through the geometric and group
theoretic conceptual structure of the analysis. In Section 5, we turn to how
this can be used to advance our understanding of physics, focusing on the
experience of a classical point-like observer. We start by correcting a
misunderstanding that has, quite reasonably, appeared in the literature. We
show that, perhaps surprisingly, inertial forces such as centrifugal forces
cannot be represented as local Lorentz transformations. (This does not require
any of the mathematical framework of the preceding sections - the
demonstration is a simple calculation on a flat spacetime, which is then
explained in the context of the geodesic equation of GR.) We then proceed to
the main topic of this section, the physical interpretation of the Lorentz
gauge freedom in teleparallel theories. We show that the choice of parallelism
has no effect on the experience of an observer, but this choice is affected by
changes of coordinate, which do also affect the experience of the observer.
The Lorentz gauge freedom is just a mathematical representation of the freedom
to choose the parallelism used to define the Weitzenböck connection.

In Section 6 we go back to the covariance group and focus on coordinate
transformations in its Poincaré subgroup - in particular, translations of the
coordinates. We look at the map induced on the bases by translations, firstly
by ones with constant parameters and then by ones with spacetime-dependent
parameters. We look at how these relate to the Jacobian matrices. We invert
this to find the transformations of the vector components. We see that
inhomogeneous transformations of the vector field components cannot result
from coordinate transformations. However, they can result from translations as
point transformations, as we show in an appendix to the paper.

Finally, the paper is summarized in Section 7.

##  2 The tangent space at a point

Following GR, we assume spacetime to be a pseudo-Riemannian manifold. We take
Nakahara’s definition[43], which can be summarized as follows. A manifold is a
topological space which can be covered by open sets known as ‘coordinate
neighbourhoods’. Each coordinate neighbourhood has a mapping onto
$\mathbb{R}^{N}$ that provides coordinates for the neighbourhood, and where
they overlap, the relations between the two sets of coordinates are infinitely
differentiable - we shall go slightly further and assume they are analytic. A
pseudo-Riemannian manifold is simply one which admits a pseudo-Riemannian
metric.

If $u^{\xi}$ and $u^{\prime\xi}$ are two sets of curvilinear coordinates on a
coordinate neighbourhood $\Omega$ of a manifold $\mathcal{M}$, the analyticity
assumption tells us that we may write

$u^{\prime\xi}=f^{\xi}+f^{\xi}{}_{\rho}u^{\rho}+f^{\xi}{}_{\rho\kappa}u^{\rho}u^{\kappa}+\ldots$
(15)

where the coefficients $f^{\xi}{}_{\rho\kappa\ldots}$ are real and symmetric
on their lower indices and independent of the coordinates.

This can be viewed as a map $a$:

$a:u^{\xi}\mapsto u^{\prime\xi}\,.$ (16)

This can be combined with a map $b$ to a third coordinate system:

$b:u^{\prime\xi}\mapsto u^{\prime\prime\xi}\,,$ (17)

so that

$ba:u^{\xi}\mapsto u^{\prime\prime\xi}\,.$ (18)

Each such map is assumed to be invertible. It is possible to combine three
such maps and we easily see that

$c(ba)=(cb)a\,.$ (19)

With the rule for combining these maps, the set of all such maps forms a
group, sometimes called the covariance group, which is known to be isomorphic
to the diffeomorphism group $\mathrm{Diff}(\mathcal{M})$. From (15), it is
clear this is an infinite-dimensional group, with the parameters
$f^{\xi},f^{\xi}{}_{\rho},f^{\xi}{}_{\rho\kappa},\ldots$.

General relativity and teleparallel theories of gravity are constructed to be
generally covariant, meaning that equations can be expressed in forms
independent of the precise coordinate system being used. This is achievable
despite the complexity of the relationship between $u^{\xi}$ and
$u^{\prime\xi}$, because general relativity is expressed in terms of tangent
vectors, tensors and connections. The fact that the spacetime is a manifold
admitting a metric means that it approximates to flat spacetime at each point.
This allows one to define a tangent space at each point, the elements of which
are vectors. By taking outer products of the tangent spaces and their duals,
one can define tensors of higher rank. These have much simpler transformation
laws than the underlying coordinates do.

The vectors tangent to the curves of increasing $u^{0},u^{1}$, $u^{2},u^{3}$
at a point $A$ form a basis for the tangent space $T_{A}\mathcal{M}$, denoted
$\mathbf{e}_{\mu}|_{A}$ \- the ‘coordinate basis’ for $u^{\mu}$. The _value_
of a vector field at $A$ may then be written as a linear sum of this
coordinate basis. Using $|_{A}$ to denote the value of a quantity at $A$, we
have

$\mathbf{V}|_{A}\in
T_{A}\mathcal{M}=V_{(u)}^{\mu}|_{A}\mathbf{e}_{\mu}|_{A}\,.$ (20)

Indeed, it can be written as a linear sum of any set of four independent
vectors in the space. In particular, it may be written as a linear sum of a
second coordinate basis:

$\mathbf{V}|_{A}=V_{(u)}^{\mu}|_{A}\mathbf{e}_{\mu}|_{A}=V_{(u^{\prime})}^{\nu}|_{A}\mathbf{e^{\prime}}_{\nu}|_{A}\,.$
(21)

We find the relations between the two bases by considering two neighbouring
points, $A$ and $B$. If they are separated by an infinitesimal interval, the
displacement is a vector in $T_{A}\mathcal{M}$. This may be written in the two
coordinate systems as

$du^{\mu}|_{A}\,\mathbf{e}_{\mu}|_{A}=du^{\prime\nu}|_{A}\,\mathbf{e^{\prime}}_{\nu}|_{A}\,.$
(22)

Now $B$ has coordinates $u^{\xi}=du^{\xi}$ and $u^{\prime\xi}+du^{\prime\xi}$.
From (15), these are related by

$u^{\prime\xi}+du^{\prime\xi}=f^{\xi}+f^{\xi}{}_{\rho}(u^{\rho}+du^{\rho})+f^{\xi}{}_{\rho\kappa}(u^{\rho}+du^{\rho})(u^{\kappa}+du^{\kappa})+\ldots$
(23)

so

$du^{\prime\xi}=f^{\xi}{}_{\rho}du^{\rho}+f^{\xi}{}_{\rho\kappa}u^{\rho}du^{\kappa}+f^{\xi}{}_{\rho\kappa}du^{\rho}u^{\kappa}+\ldots\,,$
(24)

while we can find from first principles that

$\left.\frac{\partial u^{\prime\xi}}{\partial
u^{\rho}}\right|_{A}=f^{\xi}{}_{\rho}+f^{\xi}{}_{\rho\kappa}u^{\kappa}+f^{\xi}{}_{\kappa\rho}u^{\kappa}+\ldots\,.$
(25)

Comparing these last two equations, we find, unsurprisingly, that

$du^{\prime\xi}=\left.\frac{\partial u^{\prime\xi}}{\partial
u^{\rho}}\right|_{A}du^{\rho}\,.$ (26)

Substituting this into (22) gives us

$du^{\mu}|_{A}\,\mathbf{e}_{\mu}|_{A}=du^{\mu}|_{A}\left.\frac{\partial
u^{\prime\nu}}{\partial u^{\mu}}\right|_{A}\mathbf{e^{\prime}}_{\nu}|_{A}\,.$
(27)

This same transformation law is valid for any vector:

$V_{(u)}^{\mu}|_{A}\mathbf{e}_{\mu}|_{A}=V_{(u)}^{\mu}|_{A}\left.\frac{\partial
u^{\prime\nu}}{\partial u^{\mu}}\right|_{A}\mathbf{e^{\prime}}_{\nu}|_{A}\,.$
(28)

We can see this as a transformation of either the basis:

$\mathbf{e}_{\mu}|_{A}=\left.\frac{\partial u^{\prime\nu}}{\partial
u^{\mu}}\right|_{A}\mathbf{e^{\prime}}_{\nu}|_{A}$ (29)

or the components:

$V_{(u^{\prime})}^{\mu}|_{A}=V_{(u)}^{\mu}|_{A}\left.\frac{\partial
u^{\prime\nu}}{\partial u^{\mu}}\right|_{A}\,.$ (30)

Thus while the coordinate transformation (15) and the expression for the
Jacobian matrix are (possibly infinite) series of polynomial terms, the actual
rules for transforming bases (29) and vector components (30) are simple
homogeneous linear equations. This simplification is arguably the greatest
advantage of working with Riemannian or pseudo-Riemannian manifolds. It should
be stressed here that the symmetries we are concerned with in this paper are
those under _coordinate transformations_. These relate to different ways of
breaking the _value_ of a vector into components, associated with coordinate
systems. Nonetheless, each such component is itself a field, and these may
satisfy symmetries under _point transformations_ \- we return to this subject
in Section 6.

The Jacobian matrices for transforming between bases _at $A$_ are invertible
$4\times 4$ real matrices. These are matrices of _values_ , not _functions_ \-
the matrix in (29) can be viewed as taking a ‘slice’ of the derivative matrix
at $A$. The set of all such matrices thus forms a group $J_{A}$ which is
isomorphic to $GL(4,\mathbb{R})$.

Two sequential coordinate transformations result in a product of Jacobian
matrices, which is the group rule for the group $J_{A}$:

$\left.\frac{\partial u^{\prime\prime\nu}}{\partial
u^{\prime\mu}}\right|_{A}\left.\frac{\partial u^{\prime\mu}}{\partial
u^{\lambda}}\right|_{A}=\left.\frac{\partial u^{\prime\prime\nu}}{\partial
u^{\lambda}}\right|_{A}\,.$ (31)

The induced action of the coordinate transformations on the tangent space at
$A$ therefore constitutes a homomorphic mapping of the covariance group of
$\mathcal{M}$ into $J_{A}$, with the map given by (25). Note that this map is
spacetime-dependent.

This being a pseudo-Riemannian manifold, we can also define a symmetric inner
product for each tangent space:

$(\mathbf{V},\mathbf{W})_{A}=(\mathbf{W},\mathbf{V})_{A}\in\mathbb{R}\,.$ (32)

The image of this map on the coordinate basis is the metric at $A$:

$g_{\mu\nu}|_{A}=(\mathbf{e}_{\mu},\mathbf{e}_{\nu})_{A}\,.$ (33)

and the inner product acts linearly over the tangent space. We can use this to
find the transformation of the metric under a change of coordinates.

We can always define a set of coordinates $x^{\xi}$ for which the basis is
pseudo-orthonormal at our chosen point (with respect to the inner product). We
will call this ‘frame basis’ $\hat{\mathbf{n}}_{\xi}$:

$(\hat{\mathbf{n}}_{\xi},\hat{\mathbf{n}}_{\rho})_{A}=\eta_{\xi\rho}\,.$ (34)

We will denote the transformation between the chosen frame basis and the
chosen (unprimed) coordinate basis $j_{0}|_{A}$:

$(j_{0})_{\mu}{}^{\xi}|_{A}=\left.\frac{\partial x^{\xi}}{\partial
u^{\mu}}\right|_{A}\in
J_{A}:\hat{\mathbf{n}}_{\mu}\mapsto\mathbf{e}_{\mu}=(j_{0})_{\mu}{}^{\xi}\hat{\mathbf{n}}_{\xi}\,,$
(35)

while $j|_{A}$ will be used for a generic change of basis - for example,

$j|_{A}\in
J_{A}:\mathbf{e}_{\mu}\mapsto\mathbf{e^{\prime}}_{\mu}=j_{\mu}{}^{\nu}\mathbf{e}_{\nu}\,.$
(36)

Note that in this formalism, $V^{\mu}$ consequently transforms according to:

$j|_{A}:V^{\mu}_{(u)}|_{A}\mapsto
V^{\mu}_{(u^{\prime})}|_{A}=V^{\nu}_{(u)}|_{A}(j^{-1})_{\nu}{}^{\mu}\,.$ (37)

As mentioned in the introduction, $j_{0}$ can be decomposed using a pseudo-
orthogonal subgroup. The Minkowski metric444It has emerged that some authors
refer to this as the ‘Cartesian Minkowski metric’ - we follow the terminology
of, for example, D’Inverno[44]. (34) is invariant under spacetime rotations
(including boosts) and spacetime inversions (such as reflections) and
combinations of these, which make up a group $I_{A}$ isomorphic to $O(1,3)$.
$J_{A}$ can be partitioned into cosets of the form $l_{0}I_{A}$, so we can
always write

$j_{0}|_{A}=l_{0}|_{A}i_{0}|_{A}$ (38)

where $i_{0}\in I_{A}$. If we then define

$\hat{\mathbf{k}}_{\kappa}|_{A}=(i_{0})_{\kappa}{}^{\xi}|_{A}\;\hat{\mathbf{n}}_{\xi}|_{A}\,,$
(39)

we find that

$(\hat{\mathbf{k}}_{\kappa},\hat{\mathbf{k}}_{\lambda})_{A}=\eta_{\kappa\lambda}\,,$
(40)
$\mathbf{e}_{\mu}|_{A}=(l_{0})_{\mu}{}^{\kappa}|_{A}\;\hat{\mathbf{k}}_{\kappa}$
(41)

and

$g_{\mu\nu}|_{A}=(l_{0})_{\mu}{}^{\kappa}|_{A}\;(l_{0})_{\nu}{}^{\lambda}|_{A}\;\eta_{\kappa\lambda}\,.$
(42)

##  3 Connections and covariant derivatives along a curve

Having examined the tangent space at a given point $A$, we now want to look at
comparing the tangent spaces at different points. To do this, we need to use a
connection.

General relativity uses a particular connection, the Levi-Civita connection,
or Christoffel symbol. This has the advantages of being symmetric and being
uniquely defined - on a given manifold in a given coordinate system, its
components are single-valued at each point. However, when considering frame
bases as we are here, it makes more sense to introduce the concepts by
starting with connections on a curve, which can be generalized either to the
Levi-Civita connection and its associated spin connection, or to those of
teleparallelism.

Consider a curve $c(s)$ through $\Omega$ parametrized by the single variable
$s$. This may or may not be a geodesic. We take $s$ to be invariant under
changes of coordinate. Pick two points on it $A$ and $B$. We define any map
between the tangent spaces $T_{A}\mathcal{M}$ and $T_{B}\mathcal{M}$ which
preserves linearity and the inner product as a ‘parallel map’. There are an
infinite number of these.

Now choose frame bases at both points, $\hat{\mathbf{n}}_{\xi}|_{A}$ and
$\hat{\mathbf{n}}_{\xi}|_{B}$. Denote the parallel map $\tilde{}$ for which
the image of $\hat{\mathbf{n}}_{\xi}|_{A}$ is $\hat{\mathbf{n}}_{\xi}|_{B}$:

$\displaystyle\tilde{}:T_{A}\mathcal{M}$ $\displaystyle\rightarrow$
$\displaystyle T_{B}\mathcal{M}$ (43)
$\displaystyle\tilde{}:\hat{\mathbf{n}}_{\xi}|_{A}$ $\displaystyle\mapsto$
$\displaystyle\hat{\mathbf{n}}_{\xi}|_{B}\,.$ (44)

Then as $\tilde{}$ is a linear map,

$\tilde{}:\mathbf{e}_{\mu}|_{A}\mapsto\tilde{\mathbf{e}}_{\mu}=(j_{0}|_{A}\;j_{0}{}^{-1}|_{B})_{\mu}{}^{\nu}\mathbf{e}_{\nu}|_{B}\,.$
(45)

In the teleparallism formalism, this is valid regardless of how close or far
apart $A$ and $B$ are. However, we are looking to define a connection. We
therefore take $A$ and $B$ to be close to each other (the interval between
these events is small). We then note that we can also define parallel maps to
and from all the points on $c(s)$ between these points - this set of parallel
maps along this section of the curve constitutes a ‘parallelism’. We choose
this such that the transformation $j_{0}$ from the frame basis to the
coordinate basis varies continuously with $s$. (This means that not only must
the coordinate basis and the frame basis be related by the same group $J$ all
along the curve, but $j_{0}$ must be in the same connected component of $J$ at
all points.) This allows us to carry out a Taylor expansion of $j_{0}{}^{-1}$
in $s$, giving us

$\tilde{\mathbf{e}}_{\mu}=\left(\mathbf{1}+\delta
s\left.\left(j_{0}\partial_{s}j_{0}{}^{-1}\right)_{\mu}{}^{\nu}\right|_{A}\right)\mathbf{e}_{\nu}|_{B}+\mathcal{O}^{2}(s)\,,$
(46)

where $\partial_{s}$ denotes the differential $\frac{\partial}{\partial s}$.

From the linear nature of the parallel map, we then find the image of any
vector $\mathbf{V}$:

$\tilde{}:\mathbf{V}|_{A}\mapsto\tilde{\mathbf{V}}=V^{\nu}|_{A}\mathbf{e}_{\nu}|_{B}+\delta
s\;V^{\mu}|_{A}\left.\left(j_{0}\partial_{s}j_{0}{}^{-1}\right)_{\mu}{}^{\nu}\right|_{A}\mathbf{e}_{\nu}|_{B}+\mathcal{O}^{2}(s)\,.$
(47)

The quantity in brackets is our archetypal connection (up to a change in
sign):

$\left.\left(\Gamma^{(u)}_{s}\right)_{\mu}{}^{\nu}\right|_{A}\equiv-\left.\left(j_{0}\partial_{s}j_{0}^{-1}\right)_{\mu}{}^{\nu}\right|_{A}=\left.\left(\partial_{s}(j_{0})j_{0}^{-1}\right)_{\mu}{}^{\nu}\right|_{A}\,.$
(48)

This is closely related to the Maurer-Cartan form $j_{0}\mathrm{d}j_{0}^{-1}$.
We can follow the method of non-linear realizations and use (38) to decompose
it into subgroup and coset space terms:

$\left.\left(\Gamma^{(u)}_{s}\right)_{\mu}{}^{\nu}\right|_{A}=-\left.\left(l_{0}\,\partial_{s}l_{0}{}^{-1}\right)_{\nu}{}^{\mu}\right|_{A}-\left.\left(l_{0}\,\omega_{s}l_{0}{}^{-1}\right)_{\nu}{}^{\mu}\right|_{A}$
(49)

where $\omega_{s}$ is defined by

$(\omega_{s})_{\nu}{}^{\mu}(u)=-\left(i_{0}\,\partial_{s}i_{0}{}^{-1}\right)_{\nu}{}^{\mu}$
(50)

This takes values in the Lie algebra of the Lorentz group and is known as the
Lorentz connection or spin connection associated with $\Gamma$. We will see in
Section (5) how this is helpful in understanding inertial forces.

Under a change of curvilinear coordinates, from $u^{\kappa}$ to
$u^{\prime\kappa}$, we simply replace $j_{0}$ in these expressions by
$jj_{0}$, where

$j_{\mu}{}^{\nu}=\frac{\partial u^{\nu}}{\partial u^{\prime\mu}}\,,$ (51)

giving us

$j:\left.\left(\Gamma^{(u)}_{s}\right)_{\mu}{}^{\nu}\right|_{A}\mapsto\left.\left(\Gamma^{(u^{\prime})}_{s}\right)_{\mu}{}^{\nu}\right|_{A}=\left.\left(j\Gamma_{s}j^{-1}\right)_{\mu}{}^{\nu}\right|_{A}-\left.\left(j\partial_{s}j^{-1}\right)_{\mu}{}^{\nu}\right|_{A}\,.$
(52)

One possible change of coordinates is to the set $x^{\xi}$ mentioned above,
with pseudo-orthonormal basis at $A$. Then $j=j_{0}^{-1}$, so

$\left.\left(\Gamma^{(x)}_{s}\right)_{\mu}{}^{\nu}\right|_{A}=0\,.$ (53)

If $c(s)$ is a geodesic, then $x$ can have pseudo-orthonormal basis, and
$\Gamma^{(x)}_{s}=0$, along the entire curve. This, as we shall see when we
look at the geodesic equation, is the world line for a freely falling observer
in the absence of non-gravitational forces.

More generally, the action of $j$ on a coset $l_{0}I$ is

$j:l_{0}I\mapsto l^{\prime}I=jl_{0}I$ (54)

(see (4)), so that

$jl_{0}=l^{\prime}i$ (55)

and hence

$jl_{0}i_{0}=l^{\prime}ii_{0}\,.$ (56)

This means that the connection (49) is transformed to

$\left.\left(\Gamma^{\prime(u)}_{s}\right)_{\mu}{}^{\nu}\right|_{A}=-\left.\left(l^{\prime}\,\partial_{s}l^{\prime}{}^{-1}\right)_{\nu}{}^{\mu}\right|_{A}-\left.\left(l^{\prime}\,\omega^{\prime}_{s}l^{\prime}{}^{-1}\right)_{\nu}{}^{\mu}\right|_{A}$
(57)

where

$(\omega^{\prime}_{s})_{\nu}{}^{\mu}(u)=-\left(i^{\prime}\,\partial_{s}i^{\prime}{}^{-1}\right)_{\nu}{}^{\mu}$
(58)

and

$i^{\prime}=ii_{0}\,.$ (59)

We can also look at changing parallelism. Consider a new parallelism
$\bar{}\;$, which again preserves orthonormality, so that

$\bar{}\;:\left.\hat{\mathbf{n}}_{\xi}\right|_{A}\mapsto\left.\bar{\mathbf{n}}_{\xi}\right|_{B}=i_{\xi}{}^{\rho}\left.\hat{\mathbf{n}}_{\rho}\right|_{B}\,.$
(60)

If $i$ is constant along $c(s)$, $\Gamma_{s}$ is unaffected. But if $i$ varies
with $s$ (we take it to be in the same connected component of $I$ at every
point),

$\bar{}\;:\left.\left(\Gamma_{s}\right)_{\mu}{}^{\nu}\right|_{A}\mapsto\left.\left(\Gamma^{\prime}_{s}\right)_{\mu}{}^{\nu}\right|_{A}-\left.\left(j_{0}i\partial_{s}(i^{-1})j_{0}^{-1}\right)_{\mu}{}^{\nu}\right|_{A}\,.$
(61)

We can use (47) to define a covariant derivative:

$D_{s}V^{\nu}=\partial_{s}V^{\nu}+V^{\mu}\left(\Gamma_{s}\right)_{\mu}{}^{\nu}\,.$
(62)

It is easy to show that this transforms covariantly:

$j:D^{(u)}_{s}V^{\mu}_{(u)}\mapsto
D^{(u^{\prime})}_{s}V^{\mu}_{(u^{\prime})}=D^{(u)}_{s}V^{\nu}_{(u)}\left(j^{-1}\right)_{\nu}{}^{\mu}\,.$
(63)

In the $x$ coordinates, this simply becomes

$D^{(x)}_{s}V^{\mu}_{(x)}=\partial_{s}V^{\mu}_{(x)}\,.$ (64)

##  4 Connections and covariant derivatives across the coordinate
neighbourhood

###  4.1 The Weitzenböck and Levi-Civita connections

It is possible to extend the way we defined $\Gamma$ above to the whole of
$\Omega$. Rather than just defining a parallelism - a set of parallel maps -
along a curve, we define a parallelism across the whole of $\Omega$. Note that
we are not defining it for the whole manifold. This cannot be done for most
Riemannian and pseudo-Riemannian manifolds (they are not ‘parallelisable’).
For example, a parallelism can be defined on the two-sphere, which fails at
least one point but is valid at all other points. However, this is more than
adequate for a coordinate neighbourhood, which may cover, for example, one
hemisphere, or even part of a hemisphere. For our manifold, we assume that we
can choose $\Omega$ to be small enough that a single parallelism can be used
for all of it.

This results in $j_{0}$ becoming a field over $u^{\xi}$. We can then define a
connection field using the same approach as in (46), except we now Taylor
expand in each of the curvilinear coordinates; this is known as the
Weitzenböck connection:

$\dot{\Gamma}_{\lambda\nu}{}^{\mu}(u)\equiv-\left(j_{0}\partial_{\lambda}j_{0}{}^{-1}\right)_{\nu}{}^{\mu}\equiv\left(\partial_{\lambda}(j_{0})j_{0}{}^{-1}\right)_{\nu}{}^{\mu}\,.$
(65)

This is not the most general connection. Other rules for parallel transporting
a vector exist, which do not take this form. More generally,

$\tilde{}:\mathbf{V}|_{A}\mapsto\tilde{\mathbf{V}}=V^{\nu}|_{A}\mathbf{e}_{\nu}|_{B}-\delta
u^{\lambda}\;V^{\mu}|_{A}\Gamma_{\lambda\mu}{}^{\nu}\mathbf{e}_{\nu}|_{B}+\mathcal{O}(\delta
u)^{2}\,.$ (66)

The transformation of $\Gamma_{\lambda\mu}{}^{\nu}$ under a local change of
basis is similar to the transformation for $\Gamma_{s}$, except that we now
need to act on the index $\lambda$:

$j(u):\Gamma^{(u)}_{\lambda\mu}{}^{\nu}\mapsto\Gamma^{(u^{\prime})}_{\lambda\mu}{}^{\nu}=j_{\lambda}{}^{\kappa}\left(j\Gamma^{(u)}_{\kappa}j^{-1}\right)_{\mu}{}^{\nu}-j_{\lambda}{}^{\kappa}\left(j\partial_{\kappa}j^{-1}\right)_{\mu}{}^{\nu}$
(67)

where $(\Gamma_{\lambda})_{\mu}{}^{\nu}\equiv\Gamma_{\lambda\mu}{}^{\nu}$.

Just as for $\Gamma_{s}$, we can apply a transformation $j_{0}^{-1}$ to reduce
the Weitzenböck connection to zero - except that we can now do it over the
whole of $\Omega$. However, on a curved manifold, the frame bases defined by

$\hat{\mathbf{n}}_{\xi}=\left(j_{0}^{-1}\right)_{\xi}{}^{\mu}\mathbf{e}_{\mu}$
(68)

at each point _do not represent the basis for any coordinate system_. This
must be the case: if these bases are the basis for a coordinate system, the
metric in these coordinates is the Minkowski metric. This is only possible if
the manifold is flat and the coordinates are Minkowski coordinates. Thus the
price we have to pay for adopting a parallelism across a coordinate
neighbourhood is giving up our interpretation of $j_{0}$ as a Jacobian. (The
technical definition of the parallelism is that the map $\tilde{}$ associates
these bases through (44) for any two points $A$ and $B$ on $\Omega$,
regardless of distance and independent of path between them.)

However, we can still take $j_{0}$ to be a Jacobian along a specific geodesic,
by using so called ‘Riemann normal coordinates’. If we consider a point
particle moving along a geodesic, we can always base a set of coordinates
$x^{\xi}$ on its rest frame. The geodesic is parametrized by $\tau$, the
particle’s proper time, which is proportional to $x^{0}$:

$x^{0}=\mathrm{c}\tau\,.$ (69)

These coordinates have pseudo-orthonormal basis along the entire geodesic, and
indeed the first derivatives of the metric are zero. By comparison with (53),
we therefore have

$\left.\dot{\Gamma}^{(x)}_{0\mu}{}^{\nu}\right|_{c(\tau)}=0\,.$ (70)

We look more closely at the physical interpretation of Riemann normal
coordinates and the issue of adapting a parallelism to a geodesic in Section
5.

For any connection $\Gamma^{(u)}_{\lambda\mu}{}^{\nu}$, we may define the
covariant derivative of a vector, with components

$D_{\lambda}V^{\nu}=\partial_{\lambda}V^{\nu}+V^{\mu}\Gamma_{\lambda\mu}{}^{\nu}\,.$
(71)

The covariant derivative at a point $A$ is an element of
$T_{A}\mathcal{M}\otimes T^{*}_{A}\mathcal{M}$. Under a local change of basis,
the inhomogeneous term in the transformation of $\Gamma$ is canceled by the
inhomogeneous term in the transformation of $\partial_{\lambda}V^{\mu}$.
Consequently, $D_{\lambda}V^{\nu}$ transforms covariantly:

$j:D^{(u)}_{\lambda}V^{\mu}_{(u)}\mapsto
D^{(u^{\prime})}_{\lambda}V^{\mu}_{(u^{\prime})}=j_{\lambda}{}^{\kappa}D^{(u)}_{\kappa}V^{\nu}_{(u)}\left(j^{-1}\right)_{\nu}{}^{\mu}\,.$
(72)

This can be extended in the normal way to tensors of other ranks.

It is easy to show that any connection for which (66) preserves the inner
product of vectors is metric compatible, that is

$D_{\lambda}g^{\mu\nu}=0\,.$ (73)

However, it is not necessarily symmetric. For example, the Weitzenböck
connection is metric compatible, but has a torsion:

$\dot{T}_{\lambda\mu}{}^{\nu}=\dot{\Gamma}_{\lambda\mu}{}^{\nu}-\dot{\Gamma}_{\mu\lambda}{}^{\nu}\neq
0\,.$ (74)

The only symmetric, metric-compatible connection is the Levi-Civita
connection:

$\mathring{\Gamma}_{\lambda\mu}{}^{\nu}=\mathring{\Gamma}_{\mu\lambda}{}^{\nu}=\frac{1}{2}g^{\nu\kappa}\left(\partial_{\kappa}g_{\lambda\mu}-\partial_{\lambda}g_{\kappa\mu}-\partial_{\mu}g_{\kappa\lambda}\right)\,.$
(75)

As shown by Pereira[42] and others, any non-symmetric, metric-compatible
connection, including the Weitzenböck connection, can be written as the sum of
its contorsion and the Levi-Civita connection:

$\Gamma_{\lambda\mu}{}^{\nu}=\mathring{\Gamma}_{\lambda\mu}{}^{\nu}+K_{\lambda\mu}{}^{\nu}$
(76)

where

$K_{\lambda\mu}{}^{\nu}=\frac{1}{2}\left(T_{\mu}{}^{\nu}{}_{\lambda}+T_{\lambda}{}^{\nu}{}_{\mu}-T_{\lambda\mu}{}^{\nu}\right)\,.$
(77)

Now for any geodesic $c(s)$, in the Riemann normal coordinates $x^{\xi}$, the
derivatives of the metric are zero, so

$\left.\mathring{\Gamma}^{(x)}_{\lambda\mu}{}^{\nu}\right|_{c(\tau)}=0\,.$
(78)

However, away from the geodesic the Levi-Civita connection is non-zero on a
curved manifold, even in this coordinate system. Note that incorporating (70),
we have

$\left.\dot{\Gamma}^{(x)}_{0\mu}{}^{\nu}\right|_{c(\tau)}=\left.\mathring{\Gamma}^{(x)}_{0\mu}{}^{\nu}\right|_{c(\tau)}=0\,.$
(79)

Thus we see that the timelike part of the Weitzenböck contorsion along a
geodesic is zero in Riemann normal coordinates. We look at the physical
interpretation of this in Section 5.

Note that the decomposition (76) is consistent with what we know about flat
regions of spacetime. In this case, $\hat{\mathbf{n}}_{\xi}$ is the coordinate
basis for the Minkowski coordinates $x^{\xi}$. This means that $j_{0}$ would
be a Jacobian matrix across such a region. Consequently, the Weitzenböck
connection could be written

$\dot{\Gamma}_{\lambda\nu}{}^{\mu}(u)=\partial_{\lambda}\left(\frac{\partial
x^{\xi}}{\partial u^{\nu}}\right)(j_{0}{}^{-1})_{\xi}{}^{\mu}\,.$ (80)

This has no torsion, so from its definition, the Weitzenböck contorsion tensor
would vanish. This, according to (76), would leave the Weitzenböck connection
equal to the Levi-Civita connection. Finally, this would mean that the Levi-
Civita connection is pure gauge and could be eliminated by a change of
coordinates (to the $x$ coordinates).

###  4.2 Field strengths

The Weitzenböck connection has zero field strength[42, 18]:

$\partial_{\lambda}\dot{\Gamma}_{\nu\kappa}{}^{\mu}-\partial_{\nu}\dot{\Gamma}_{\lambda\kappa}{}^{\mu}+\dot{\Gamma}_{\nu\kappa}{}^{\rho}\dot{\Gamma}_{\lambda\rho}{}^{\mu}-\dot{\Gamma}_{\lambda\kappa}{}^{\rho}\dot{\Gamma}_{\nu\rho}{}^{\mu}=0$
(81)

and, as noted above, it can be reduced to zero across $\Omega$ by a local
change of basis. The scalar curvature (the Ricci scalar) may be constructed
from its torsion tensor[42, 29]. For a given coordinate system on a given
manifold, this connection is not unique - its definition depends on the
parallelism chosen.

The field strength of the Levi-Civita connection is the Riemann curvature
tensor:

$\mathring{R}^{\mu}{}_{\kappa\lambda\nu}=\partial_{\lambda}\mathring{\Gamma}_{\nu\kappa}{}^{\mu}-\partial_{\nu}\mathring{\Gamma}_{\lambda\kappa}{}^{\mu}+\mathring{\Gamma}_{\nu\kappa}{}^{\rho}\mathring{\Gamma}_{\lambda\rho}{}^{\mu}-\mathring{\Gamma}_{\lambda\kappa}{}^{\rho}\mathring{\Gamma}_{\nu\rho}{}^{\mu}$
(82)

and the connection cannot be reduced to zero across $\Omega$ by a local change
of basis, except on a flat spacetime. For a given coordinate system on a given
manifold, it is unique. The Riemann tensor can also be viewed in terms of the
action of the covariant derivatives on a vector field:

$\left[\mathring{D}_{\kappa},\mathring{D}_{\rho}\right]V^{\xi}=\mathring{R}^{\xi}{}_{\lambda\kappa\rho}V^{\lambda}\,.$
(83)

###  4.3 Lorentz connections

Each metric-compatible connection $\Gamma_{\lambda\nu}{}^{\mu}$ has an
associated Lorentz connection or spin connection, taking values in the Lie
algebra of the Lorentz group[42]. This means that at least two of its indices
must be frame indices. It therefore has two forms, one of which has all three
indices as frame indices, while the other has two frame indices and one
coordinate index. In the formalism of this paper, the Lorentz connection with
three frame indices is considered to be the usual connection in the frame
basis. The frame basis at a point $A$ is the basis at that point for some set
of Riemann normal coordinates $x^{\xi}$, so we can write this connection at
this point as $\left.\Gamma^{(x)}_{\lambda\kappa}{}^{\mu}\right|_{A}$. The
form with two frame indices and one coordinate index is considered to be in a
mix of two different bases. We shall write this as follows:

$\left.\omega_{\mu}{}^{\xi\rho}\right|_{A}M_{\xi\rho}\in\mathcal{I_{A}}\,,$
(84)

where the first index is taken to be a coordinate index and the last two are
frame indices.

If we choose a frame $\hat{\mathbf{n}}_{\mu}$ at $A$ related to the coordinate
basis by (35) where $j_{0}$ can be decomposed using (38), any metric-
compatible connection in the coordinate basis can be related to a Lorentz
connection as follows:

$\Gamma^{(u)}_{\mu\lambda}{}^{\nu}=(l_{0})_{\lambda}{}^{\kappa}\omega_{\mu\kappa}{}^{\rho}(l_{0}^{-1})_{\rho}{}^{\nu}+(l_{0})_{\lambda}{}^{\kappa}\partial_{\mu}(l_{0}^{-1})_{\kappa}{}^{\nu}\,,$
(85)

where the first index of $\omega_{\mu\kappa}{}^{\rho}$ is taken to be in the
coordinate basis, with frame indices raised and lowered using
$\eta^{\xi\kappa}$ and $\eta_{\xi\kappa}$. This equation can be inverted to
give:

$\omega_{\mu\xi}{}^{\rho}=(l_{0}^{-1})_{\xi}{}^{\lambda}\Gamma^{(u)}_{\mu\lambda}{}^{\nu}(l_{0})_{\nu}{}^{\rho}+(l_{0}^{-1})_{\xi}{}^{\lambda}\partial_{\mu}(l_{0})_{\lambda}{}^{\rho}\,.$
(86)

However, even for a given connection $\Gamma$, this Lorentz connection is not
unique: any local change of frame $i(u)$ (including $i_{0}(u)$) results in
another Lorentz connection. $\omega_{\mu\xi}{}^{\rho}$ transforms under a
local change of frame according to:

$i(u):\omega_{\mu\xi}{}^{\rho}\mapsto\omega^{\prime}_{\mu\xi}{}^{\rho}=\left(i\omega_{\mu}i^{-1}\right)_{\xi}{}^{\rho}+\left(i\partial_{\mu}i^{-1}\right)_{\xi}{}^{\rho}\,.$
(87)

For the Weitzenböck connection, (85) amounts to a Cartan decomposition using
(38):

$\dot{\Gamma}^{(u)}_{\mu\lambda}{}^{\nu}=(l_{0})_{\lambda}{}^{\xi}\dot{\omega}_{\mu\xi}{}^{\rho}(l_{0}^{-1})_{\rho}{}^{\nu}+\dot{\rho}_{\mu\lambda}{}^{\nu}$
(88)

where

$\dot{\omega}_{\mu\xi}{}^{\rho}=(i_{0}\partial_{\mu}i_{0}^{-1})_{\xi}{}^{\rho}=-[(\partial_{\mu}i_{0})i_{0}^{-1}]_{\xi}{}^{\rho}$
(89)

and

$\dot{\rho}_{\mu\xi}{}^{\rho}=(l_{0}\partial_{\mu}l_{0}^{-1})_{\xi}{}^{\rho}=-[(\partial_{\mu}l_{0})l_{0}^{-1}]_{\xi}{}^{\rho}\,.$
(90)

(This is the extension of (49) to the coordinate neighbourhood $\Omega$.) (89)
means that the Weitzenböck spin connection can be reduced to zero everywhere
by a local change of frame, whereas the Levi-Civita spin connection
cannot[42]. The choice of gauge which reduces it to zero is often called the
Weitzenböck gauge and $\dot{\rho}_{\mu\xi}{}^{\rho}$ is generally viewed in
the literature as the Weitzenböck connection in this Weitzenböck gauge.

This Cartan decomposition can be applied to the transformation of the
Weitzenböck connection under $j$, given by (67). The results are again similar
to those for $\Gamma_{s}$, but with an extra transformation of the third
index:

$j:\dot{\Gamma}^{(u)}_{\lambda\mu}{}^{\nu}\mapsto\dot{\Gamma^{\prime}}^{(u)}_{\kappa\mu}{}^{\nu}=-j_{\kappa}{}^{\lambda}[\left(l^{\prime}\,\partial_{\lambda}l^{\prime}{}^{-1}\right)+\left(l^{\prime}\,\dot{\omega}^{\prime}_{\lambda}l^{\prime}{}^{-1}\right)]_{\nu}{}^{\mu}$
(91)

where

$(\dot{\omega}^{\prime}_{\lambda})_{\nu}{}^{\mu}(u)\equiv\dot{\omega}^{\prime}_{\lambda\nu}{}^{\mu}(u)=-\left(i^{\prime}\,\partial_{\lambda}i^{\prime}{}^{-1}\right)_{\nu}{}^{\mu}\,.$
(92)

and $l^{\prime}$ and $i^{\prime}$ are again given by (55) and (59).

It will be noted that $\dot{\rho}_{\mu\xi}{}^{\rho}$ depends only on $l_{0}$
and its derivatives. Given that the metric only carries the degrees of freedom
of $l_{0}$, it may be wondered whether this term is related to the Levi-Civita
connection. Actually, it can be shown that it is. From (42) (when applied
across the chart), we have

$\displaystyle\partial_{\kappa}g_{\mu\nu}$ $\displaystyle=$
$\displaystyle(\partial_{\kappa}(l_{0})_{\mu}{}^{\xi}(l_{0})_{\nu}{}^{\rho}+(l_{0})_{\mu}{}^{\xi}\partial_{\kappa}(l_{0})_{\nu}{}^{\rho})\;\eta_{\xi\rho}$
(93) $\displaystyle=$
$\displaystyle[-(l_{0}\partial_{\kappa}l_{0}^{-1})_{\mu}{}^{\lambda}(l_{0})_{\lambda}{}^{\xi}(l_{0})_{\nu}{}^{\rho}-(l_{0}\partial_{\kappa}l_{0}^{-1})_{\nu}{}^{\lambda}(l_{0})_{\mu}{}^{\xi}(l_{0})_{\lambda}{}^{\rho}]\;\eta_{\xi\rho}$
(94) $\displaystyle=$
$\displaystyle-(\dot{\rho}_{\kappa})_{\left\\{\mu\right.}{}^{\lambda}g_{\lambda\left.\nu\right\\}}\,,$
(95)

where to get the second line, we have inserted $l_{0}l_{0}^{-1}$ into both
terms then used
$l_{0}^{-1}\partial_{\kappa}l_{0}=-\partial_{\kappa}(l_{0}^{-1})l_{0}$. The
braces round the indices in the last line represent symmetrization on those
indices. From this we find that the Levi-Civita connection is

$\displaystyle\mathring{\Gamma}_{\kappa\nu}{}^{\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{\rho\mu}(\partial_{\kappa}g_{\mu\nu}+\partial_{\nu}g_{\mu\kappa}-\partial_{\mu}g_{\kappa\nu})$
(96) $\displaystyle=$
$\displaystyle\frac{1}{2}g^{\rho\mu}(-\dot{\rho}_{[\kappa\mu]}{}^{\lambda}g_{\lambda\nu}-\dot{\rho}_{[\kappa\nu]}{}^{\lambda}g_{\mu\lambda}+\dot{\rho}_{\left\\{\mu\nu\right\\}}{}^{\lambda}g_{\kappa\lambda})$
(97)

where the square brackets round the indices represent antisymmetrization on
those indices. Thus the Levi-Civita connection depends only on
$\dot{\rho}_{\mu\xi}{}^{\rho}$ and the metric - we can thus see why it has
become so popular to carry out calculations in the Weitzenböck gauge.

Note that when $\dot{\rho}_{\mu\xi}{}^{\rho}$ is symmetric on its lower
indices, it reduces to the Levi-Civita connection (with one index raised and
another one lowered). However, there does not appear to be anything requiring
this to be the case. In general, both terms in (88) carry torsion, so this
decomposition is a different one to that given in (76). More precisely, all
three terms in (76) for the Weitzenböck connection are dependent on
$\dot{\rho}$, but only the first and last terms carry an
$\dot{\omega}$-dependence.

##  5 Using the coset decomposition to study Lorentz gauge transformations and
inertial forces on an observer

###  5.1 Inertial forces are not induced by Lorentz transformations

As mentioned in Section 1.5, there has recently been considerable debate about
the role of the Lorentz gauge transformation. As we will show below, the coset
decomposition is the appropriate framework in which to address this question.

However, it is first worth correcting an issue which seems to be causing some
confusion. It often seems to be assumed that a change of reference frame which
induces inertial forces, such as centrifugal forces, can be represented by a
local Lorentz transformation. It seems a natural assumption to make, following
the logic that a rotation of coordinates is a Lorentz transformation, so a
rotation of coordinates that changes over time is a local Lorentz
transformation. However, this is not actually the case; furthermore, we do not
need the apparatus of teleparallelism to show this. The flaw in the above
logic is that time itself is one of the coordinates and this results in a non-
pseudo-orthogonal transformation, as we now show.

Consider two coordinate systems, $x^{\mu}$ and $x^{\prime\mu}$. For
convenience, we will place them on a flat spacetime and make $x^{\prime\mu}$
the Minkowski coordinates. If $x^{\prime\mu}$ is rotated with respect to
$x^{\mu}$ through an angle $\theta$ in the $x$-$y$ plane, then

$x^{\prime}=x\cos\theta+y\sin\theta,\hskip
22.76219pty^{\prime}=-x\sin\theta+y\cos\theta,\hskip
22.76219ptz^{\prime}=z,\hskip 22.76219ptt^{\prime}=t\,.$ (98)

If we now let $\theta$ change over time, so that $x^{\prime\mu}$ rotates with
respect to $x^{\mu}$ at a rate of $k$ radians per second,

$x^{\prime}=x\cos kt+y\sin kt,\hskip 19.91692pty^{\prime}=-x\sin kt+y\cos
kt,\hskip 19.91692ptz^{\prime}=z,\hskip 19.91692ptt^{\prime}=t\,;$ (99)

it is then a simple calculation to show that

$\displaystyle ds^{\prime 2}$ $\displaystyle=$ $\displaystyle c^{2}dt^{\prime
2}-dx^{\prime 2}-dy^{\prime 2}-dz^{\prime 2}$ $\displaystyle=$
$\displaystyle(c^{2}-k^{2}x^{2}-k^{2}y^{2})dt^{2}-dx^{2}-dy^{2}-2ky\,dx\,dt-2kx\,dy\,dt-
dz^{2}\,.$

Given that $ds^{2}$ is an invariant under changes of coordinates, it is clear
that the metric is not preserved (it is no longer the Minkowski metric
$\eta_{\mu\nu}$), so the transformation cannot be pseudo-orthogonal.

From the GR perspective, it makes sense that the introduction of an inertial
force requires a change of metric. Without this, the Levi-Civita connection
remains zero. The experience of an observer following a path $u(\tau)$ is
determined by the geodesic equation

$\frac{\partial^{2}u^{\mu}}{\partial\tau^{2}}+\mathring{\Gamma}_{\kappa\lambda}{}^{\mu}\frac{\partial
u^{\kappa}}{\partial\tau}\frac{\partial u^{\lambda}}{\partial\tau}=0$ (102)

and a zero Levi-Civita connection means that the observer feels no force.

###  5.2 Choices of parallelism and coordinates, and the forces experienced by
an observer

Let us consider two situations, familiar to students of general relativity.

Situation 1 is that of a free observer situated in a flat spacetime (this
could, for example, represent deep space). A natural choice of coordinates for
this spacetime would be Minkowski coordinates, which can be used over the
entire spacetime (at least, for the extent in each direction in which it
remains flat). The coordinate basis is then also a frame basis at every point.

By associating this basis at a point $A$ with the basis at every other point,
(so that they are images of each other under parallel transport), we can
define a parallelism. We could say that this parallelism is ‘adapted to’ the
Minkowski coordinates on this spacetime. With these choices, $j_{0}$ reduces
to the identity: $\mathbf{e}_{\mu}$ coincides with $\hat{\mathbf{n}}_{\mu}$.

However, other choices are available. While retaining the Minkowski
coordinates, we could change to a different parallelism, which is not adapted
to these coordinates, using (87). This may seem a perverse thing to do, but
the paralellism is just a choice. $l_{0}$ remains the identity, so
$\mathbf{e}_{\nu}=\hat{\mathbf{k}}_{\nu}=i_{\nu}{}^{\mu}\hat{\mathbf{n^{\prime}}}_{\mu}$.

Alternatively, we could change to a set of curvilinear coordinates. This
changes the coordinate basis according to (35).

Situation 2 is that of an observer following a geodesic on a curved spacetime
- that is, falling freely in a gravitational field. A natural choice of
coordinates for this situation are those of the observer’s local inertial
frame - these are Riemann normal coordinates whose basis is orthonormal along
the geodesic.

We can choose a parallelism which is adapted to these coordinates on this
geodesic, so that again $j_{0}$ is the identity element along the geodesic.
_However, on a curved spacetime, there is no parallelism which is adapted to
all the geodesics across an extended multi-dimensional region of the
spacetime._ Thus $j_{0}$ cannot be the identity everywhere. (Indeed, $l_{0}$
cannot be the identity everywhere, as this would make the metric the Minkowski
one everywhere.)

Again, other choices are available. While retaining the Riemann normal
coordinates, we could choose a different parallelism, for example one adapted
to a world line which crosses that of our observer.

Alternatively, we could change to a different set of curvilinear coordinates -
for example, the rest frame for an observer whose path crosses that of our
observer, or the rest frame of the mass distribution causing the curvature.

The question is then what impact these changes have on the experience of the
observer. In GR, this is determined by the geodesic equation (102). The
crucial point is that this only depends upon the Levi-Civita connection; it is
independent of the choice of parallelism. (We could use (76) to rewrite the
connection term above, but then it would depend upon both the Weitzenböck
connection _and_ its contorsion.) This must also be the case for any theory in
which the geodesic equation holds - including TEGR, as it is equivalent to GR
at the level of physical observations.

This can be understood physically as follows. In general relativity, and hence
TEGR, the observer ‘lives’ in the frame of reference defined by the
coordinates $u^{\mu}$. This is independent of the choice of frame bases along
their world line - that is, the parallelism used - which is simply a choice
for the researcher carrying out calculations. This should also hold for
modified theories. The parallelism is a choice of Lorentz gauge chosen by the
analyst to make calculations easier. The experience of the observer should not
change if this gauge choice changes, as long as the coordinate basis
$\mathbf{e}_{\mu}$ remains the same555It should be stressed that this is for a
classical, pointlike observer, as in the Einstein lift experiments - if the
analysis were extended to incorporate field configurations which take values
in a Hilbert space, then it could be that measurements of quantum numbers
could distinguish between parallelisms. Such analysis lies outside the scope
of this paper..

A change of coordinates, however, represents a change of reference frame -
from one observer to another - and this changes the Levi-Civita connection.
Hence one observer will not necessarily feel the same force as another.

In situation 1, in Minkowski coordinates, the Levi-Civita connection vanishes.
The observer therefore feels no force. A change of coordinate system from
Minkowski coordinates to a curvilinear system induces a non-zero Levi-Civita
connection, which is felt as inertial forces.

In situation 2, in Riemann normal coordinates, the Levi-Civita connection
vanishes along the observer’s world line, so the observer feels no force.
However, it is non-zero away from this, representing the tidal effects of
gravity. On changing coordinates, the Levi-Civita connection becomes non-zero.
This could, for example, be changing to the reference frame of an observer on
the surface of a solid gravitating body, at rest with respect to that body. In
this case, the Levi-Civita connection would represent the gravitational force
as felt by the observer. Alternatively, we could choose a more exotic
reference frame, such as one representing someone spinning with respect to the
freely-falling observer. The Levi-Civita connection along the world line would
then represent the centrifugal force felt by the new observer; beyond their
world line it would combine this force with the gravitational tidal effects.

The Levi-Civita connection, therefore, recognizes both gravity and inertial
forces and does not distinguish between them at the local level, as observed
by Aldrovandi _et al_[36]. One has to consider how it varies over spacetime to
distinguish between them. This is done through the Riemann tensor, which is
zero when spacetime is flat and nonzero when it is curved. This means that
when gravity is reduced to zero, the Levi-Civita connection becomes pure gauge
- it is zero in Minkowski coordinates, and in any other coordinate system it
reduces to the form (65), where $j_{0}$ is the Jacobian matrix from the
Minkowski coordinates to the curvilinear ones. As the deviation between the
Weitzenböck and Levi-Civita connections is the Weitzenböck contorsion tensor,
this must reduce to zero for Riemann-flat spacetime (in any coordinates).
Conversely, as the field strength of the Weitzenböck connection is always
zero, a spacetime with Riemann curvature will always have non-zero Weitzenböck
contorsion tensor. Thus the Weitzenböck contorsion tensor also has the
property that it can be used to determine whether a gravitational field is
present.

Researchers have found that in some situations, it is easier to work with the
Weitzenböck torsion or contorsion tensors than the Riemann tensor, as these
depend only on the first derivatives of the ‘tetrad fields’ - that is, of the
matrices relating different bases. The price of this is that it becomes
difficult to maintain general covariance unless one keeps track of not just
the $l$ degrees of freedom but also the $i$ degrees of freedom - despite the
fact that these extra degrees of freedom do not affect physical observables.

###  5.3 How changes of coordinate and parallelism both induce Lorentz gauge
transformations

Having looked at the impact of these changes on the Levi-Civita connection, we
can also look at how they affect the two terms in the Cartan decomposition of
the Weitzenböck connection. A change of parallelism is given by a local
Lorentz transformation of the frame basis, $i(u)$. This means that the
coordinate basis $\mathbf{e}_{\mu}$ is related to the new frame basis
$\hat{\mathbf{n^{\prime}}}_{\xi}$ by

$\mathbf{e}_{\mu}=(l_{0}i_{0}i)_{\mu}{}^{\xi}\hat{\mathbf{n^{\prime}}}_{\xi}\,.$
(103)

This changes the Lorentz connection according to (87) but has no impact on
$\dot{\rho}_{\mu\nu}{}^{\xi}$.

A change of coordinates, on the other hand, in general666There will be some
specific changes of coordinates which do not affect either or both of these
terms. If the action of a Jacobian matrix $j$ leaves $l_{0}$ invariant, it
must leave both the components of the metric and $\dot{\rho}_{\mu\nu}{}^{\xi}$
invariant. This is not exactly the same as an isometry, which leaves the
_functional form_ of the metric invariant - this distinction is explained in
Section 6.2. Transformations which leave the functional forms of these
quantities invariant have been studied by Hohmann _et al_[41], albeit in the
tetrad formulation, without the insights contained in this section. The
relation between these two types of invariance is unclear, at least within the
context of the coset formulation presented here, and is likely to be worthy of
further examination. affects both of these terms, as shown in (91) and (92).
However, it is only the change of $\dot{\rho}_{\mu\nu}{}^{\xi}$ which is felt
by the observer, as this is the only term which contributes to the Levi-Civita
connection. The Lorentz connection remains a gauge choice that has no physical
impact on the observer. However, this gauge choice does change with the
coordinates. This means, in particular, that the Weitzenböck gauge is specific
to the choice of coordinates - one may choose to carry out calculations in the
Weitzenböck gauge, but when the coordinates are changed, the system will no
longer be in this gauge.

Naturally, the preceding paragraph includes coordinate transformations for
which the Jacobian matrix is pseudo-orthogonal. In this case, we can replace
$j$ with $i$. Then in place of (56) we have

$il_{0}i_{0}=il_{0}i^{-1}ii_{0}=l^{\prime}ii_{0}$ (104)

so that the $l^{\prime}$ in (91) is given by

$l^{\prime}=il_{0}i^{-1}\,,$ (105)

while the $i$ in (59), which contributes to the $i^{\prime}$ in (92), is
precisely the Jacobian matrix for the change of curvilinear basis.

Finally, we can consider the case where

$\left.l_{0}\right|_{c(\tau)}=0$ (106)

\- that is, the metric is the Minkowski one along the observer’s world line.
This means that the observer is in free fall. (Note that we are not
restricting $l_{0}$ away from the world line - it may be non-zero away from
it, which means that we may have gravitational forces present.) We do not have
to have a parallelism adapted to this world line, so we can still have

$\left.i_{0}\right|_{c(\tau)}\neq 0\,.$ (107)

This means that the Lorentz connection can be non-zero, but
$\dot{\rho}_{\mu\xi}{}^{\nu}$ must be zero. In this case, the Jacobian matrix
for an inertial coordinate transformation acts directly on $i_{0}$, with the
action (59). It therefore changes the Lorentz connection, but the metric
remains the Minkowski one along the world line.

##  6 Translations

###  6.1 Translations as coordinate transformations

As mentioned in Section 1, there has recently been a vigorous debate about
whether teleparallel gravity constitutes a gauge theory of translations. Here,
we see how the methodology of this paper allows us to study this issue. Our
starting point is the formula for coordinate transformations, (15). Consider
the subset for which all coefficients after the first two terms are zero -
that is, the (global) inhomogeneous linear transformations:

$u^{\prime\xi}=f^{\xi}+f^{\xi}{}_{\rho}u^{\rho}\,.$ (108)

For these, the Jacobian matrix is

$\frac{\partial u^{\prime\xi}}{\partial u^{\kappa}}=f^{\xi}{}_{\kappa}\,.$
(109)

From this, we can immediately see that such a transformation preserves the
Minkowski metric if and only if $f^{\xi}{}_{\kappa}$ is pseudo-orthogonal.
Such transformations, with the general form

$u^{\prime\xi}=f^{\xi}+i^{\xi}{}_{\rho}u^{\rho}$ (110)

comprise the Poincaré group. Amongst these are the global translations

$u^{\prime\xi}=f^{\xi}+u^{\xi}$ (111)

for which the Jacobian matrix is a Kronecker delta. This means that under the
homomorphism (25) from the covariance group into $J_{A}$, the translations lie
in the kernel, so bases and vector components are untransformed. (Note that
for vector _fields_ , here we are talking about the values of the vector
components at a given point, not their functional forms - their expressions in
terms of the coordinates, which will obviously change.)

We now want to consider what happens when the translation parameters are made
spacetime-dependent. First, we consider the transformation of the basis on the
tangent space. This transforms by contraction with the inverse Jacobian
matrix, which is

$\left.\frac{\partial u^{\xi}}{\partial
u^{\prime\kappa}}\right|_{A}=\delta^{\xi}_{\kappa}-\left.\frac{\partial
f^{\xi}}{\partial u^{\prime\kappa}}\right|_{A}\,,$ (112)

so

$f(u^{\rho}):\mathbf{e}_{\kappa}|_{A}\mapsto\mathbf{e}^{\prime}_{\kappa}|_{A}=\mathbf{e}_{\kappa}|_{A}-\left.\frac{\partial
f^{\xi}}{\partial u^{\prime\kappa}}\right|_{A}\mathbf{e}_{\xi}|_{A}\,.$ (113)

The partial derivative operator transforms in the same way (indeed, it can be
seen as a representation of the basis):

$f(u^{\rho}):\partial_{\kappa}|_{A}\mapsto\partial^{\prime}_{\kappa}|_{A}=\partial_{\kappa}|_{A}-(\partial_{\kappa}f^{\xi})|_{A}\,\partial_{\xi}|_{A}\,.$
(114)

Thus for a scalar field $\phi$,

$f(u^{\rho}):\partial_{\kappa}\phi|_{A}\mapsto\partial^{\prime}_{\kappa}\phi|_{A}=\partial_{\kappa}\phi|_{A}-(\partial_{\kappa}f^{\xi})|_{A}\,\partial_{\xi}\phi|_{A}$
(115)

\- that is, we have a minimal coupling to the 16 variables of
$\partial_{\kappa}f^{\xi}|_{A}$.

What we are doing here is to view the action of the general linear group from
a new perspective. We previously considered the action of $j$ by contraction
on the frame basis, (36). We now consider the displacement of the basis under
this action:

$\delta\mathbf{e}_{\kappa}|_{A}=\mathbf{e}^{\prime}_{\kappa}|_{A}-\mathbf{e}_{\kappa}|_{A}=(j_{\kappa}{}^{\xi}|_{A}-\delta_{\kappa}{}^{\xi})\,\mathbf{e}_{\xi}|_{A}$
(116)

or in terms of the translation parameters $f^{\xi}$:

$\delta\mathbf{e}_{\kappa}|_{A}=-\partial_{\kappa}f^{\xi}|_{A}\mathbf{e}_{\xi}|_{A}\,.$
(117)

Thus we see that the local translations of the coordinates induce _the same
transformations of the tangent space_ as described by the action of the
general linear group. They are related by

$j_{\kappa}{}^{\xi}|_{A}-\delta_{\kappa}{}^{\xi}=-\partial_{\kappa}f^{\xi}|_{A}$
(118)

and therefore contain the same 16 degrees of freedom. As mentioned in Section
1, this differs from the conclusion of Hohmann _et al_[35] that translations
are generated by an _Abelian subgroup_ of the general linear group. Instead,
we find that _all_ general linear transformations _of the basis at $A$_ may be
expressed in terms of the local values of derivatives of translation
parameters. This implies that some of the changes of bases induced by local
translations correspond to local Lorentz transformations. By combining (118)
with the definition of a pseudo-orthogonal transformation in
$T_{A}\mathcal{M}$,

$j_{\xi}{}^{\kappa}|_{A}j_{\rho}{}^{\lambda}|_{A}\eta_{\kappa\lambda}=\eta_{\xi\rho}\,,$
(119)

it emerges that $f^{\xi}$ represents such a transformation if and only if

$\eta_{\kappa\lambda}(\partial_{\xi}f^{\kappa})|_{A}(\partial_{\rho}f^{\lambda})|_{A}=\eta_{\kappa\rho}(\partial_{\xi}f^{\kappa})|_{A}+\eta_{\xi\lambda}(\partial_{\rho}f^{\lambda})|_{A}\,.$
(120)

(Note that the very meaning of translations on a curved space is rather
different from the simple understanding of them on a flat space. $f^{\xi}$
represents a quantity which is added to a chosen curvilinear coordinate - this
could be, for example, an angular variable.)

While any change of basis $j$ _in a given tangent space_ may be represented in
terms of the translation parameters $f^{\xi}$ according to (118), it is not
true that any change of basis _across a coordinate neighbourhood_ , $j(u)$,
may be represented in this way. If a field-valued translation induces a change
of basis $j_{t}(u)$ across a coordinate neighbourhood, its non-trivial part
will take the form of a derivative:

$(j_{t})_{\kappa}{}^{\xi}-\delta_{\kappa}{}^{\xi}=-\partial_{\kappa}f^{\xi}$
(121)

This puts a constraint on its form as a function of the coordinates $u^{\mu}$.
(Therefore, rather than the subgroup structure described by Hohmann _et al_ ,
the functions $j_{t}(u)$ are actually a subset of all of the functions
$j(u)$.) In particular, we see that

$\partial_{\mu}(j_{t})_{\kappa}{}^{\xi}=-\partial_{\mu}\partial_{\kappa}f^{\xi}\,.$
(122)

This has a significant implication when $j_{t}$ is the Jacobian matrix
relating the coordinate basis to the frame basis, $j_{0}$, as this then
becomes

$\partial_{\mu}(j_{0})_{\kappa}{}^{\xi}=-\partial_{\mu}\partial_{\kappa}(f_{0})^{\xi}\,.$
(123)

As this second derivative is symmetric on its lower indices, the Weitzenböck
connection (65) has no torsion[45]. As we saw in the last section, this can
only happen in flat spacetime.

Our conclusion regarding $j_{0}$ is therefore (in general accord with Hohmann
_et al_): _the Weitzenböck connection can only be constructed from translation
parameters in a flat region of spacetime_. This can be understood also in
terms of the following argument. $j_{0}$ relates the curvilinear coordinate
basis $\mathbf{e}_{\kappa}$ to the frame basis $\hat{\mathbf{n}}_{\kappa}$,
while $f^{\xi}$ relates two coordinate systems. If $j_{0}$ can be expressed in
terms of such translation parameters $(f_{0})^{\xi}$,
$\hat{\mathbf{n}}_{\kappa}$ must be the basis corresponding to a coordinate
system. This can only happen for an extended region of spacetime (that is,
beyond just a geodesic) if that region is flat. In such a case, we may write

$(j_{0})_{\kappa}{}^{\xi}=\delta_{\kappa}{}^{\xi}-\partial_{\kappa}(f_{0})^{\xi}$
(124)

where

$(f_{0})^{\xi}=u^{\xi}-x^{\xi}\,.$ (125)

Having looked at the action of local translations on the basis, we now want to
turn to the action on the components of a vector field. Under a coordinate
transformation, $\mathbf{V}|_{A}\in T_{A}\mathcal{M}$ remains
$\mathbf{V}|_{A}\in T_{A}\mathcal{M}$; however, as we have seen above, it can
be decomposed in different bases on that tangent space. Let us say that in two
such bases, it has components $V^{\mu}|_{A}$ and
$V^{\prime\mu}|_{A}=V^{\mu}|_{A}+\delta V^{\mu}|_{A}$. As
$\mathbf{e}_{\mu}|_{A}$ is a complete linear basis for the tangent space, any
other basis vectors can be related to it by (36). Thus

$(V^{\mu}+\delta
V^{\mu})|_{A}\,\mathbf{e}^{\prime}_{\mu}|_{A}=V^{\lambda}|_{A}(j^{-1})_{\lambda}{}^{\mu}|_{A}\mathbf{e}^{\prime}_{\mu}|_{A}\,,$
(126)

from which we find

$\delta
V^{\mu}|_{A}=V^{\lambda}|_{A}[(j^{-1})_{\lambda}{}^{\mu}|_{A}-\delta_{\lambda}^{\mu}]\,.$
(127)

Thus any displacements of the vector components induced by coordinate
transformations have the original vector components as a factor.

Note that one cannot induce inhomogeneous transformations - displacements of
the vector components - from a coordinate transformation. These just change
the basis under which the value of a vector at a given point is decomposed
into components. However, translations can be applied as _point
transformations_ 777We are using the terminology of D’Inverno[44] here. to any
field to generate such transformations. We comment very briefly on this in the
next subsection.

We now need to find an expression for $j_{t}{}^{-1}$ in terms of the
translation parameters. We do this using the local version of (111):

$(j_{t}{}^{-1})_{\kappa}{}^{\xi}=\frac{\partial u^{\prime\xi}}{\partial
u^{\kappa}}=\delta^{\xi}_{\kappa}+\frac{\partial f^{\xi}}{\partial
u^{\kappa}}\,.$ (128)

Similarly, for a flat region of spacetime or a geodesic in a curved spacetime,

$(j_{0}{}^{-1})_{\kappa}{}^{\xi}=\frac{\partial u^{\xi}}{\partial
x^{\kappa}}=\delta_{\xi}{}^{\kappa}+\frac{\partial f_{0}^{\xi}}{\partial
x^{\kappa}}$ (129)

where $x^{\kappa}$ are the Minkowski or Riemann normal coordinates. Note that
the non-trivial term on the right-hand side here is not that we have in (114),
as the derivative is with respect to $x^{\kappa}$. We can sort this out with
an iterative procedure:

$\displaystyle(j_{0}{}^{-1})_{\kappa}{}^{\xi}=\frac{\partial u^{\xi}}{\partial
x^{\kappa}}$ $\displaystyle=$
$\displaystyle\delta^{\xi}_{\kappa}+\frac{\partial f_{0}^{\xi}}{\partial
u^{\lambda}}\frac{\partial u^{\lambda}}{\partial x^{\kappa}}$ (130)
$\displaystyle=$ $\displaystyle\delta^{\xi}_{\kappa}+\frac{\partial
f_{0}^{\xi}}{\partial
u^{\lambda}}\left(\delta^{\lambda}_{\kappa}+\frac{\partial
f_{0}^{\lambda}}{\partial u^{\mu}}\frac{\partial u^{\mu}}{\partial
x^{\kappa}}\right)$ (131) $\displaystyle=$
$\displaystyle\delta^{\xi}_{\kappa}+\partial_{\kappa}f_{0}^{\xi}+(\partial_{\kappa}f_{0}^{\lambda})(\partial_{\lambda}f_{0}^{\xi})+\ldots$
(132)

and similarly

$((j_{t}){}^{-1})_{\xi}{}^{\kappa}=\delta^{\xi}_{\kappa}+\partial_{\kappa}f^{\xi}+(\partial_{\kappa}f^{\lambda})(\partial_{\lambda}f^{\xi})+\ldots\,.$
(133)

The implications of the convergence criteria for these series remains an open
question888It has come to the author’s attention at a late stage in the
preparation of this paper that this perturbative expansion has also been
carried out by Beltran Jiminez and Dialektopoulos[46]..

Equation (121) and (133) can then be substituted into any equations in the
earlier sections which contain $j$ and/or $j^{-1}$, to get equations
containing the translation parameters. Similarly, (124) and (132) can be
substituted into any equations for flat spacetime or a geodesic (_but only for
these_) which contain $j_{0}$ and/or $j_{0}^{-1}$. For example, the covariant
derivative (62) along a geodesic is

$D_{\tau}V^{\nu}=\partial_{\tau}V^{\mu}+V^{\mu}\left(\Gamma_{\tau}\right)_{\mu}{}^{\nu}\,,$
(134)

where $\tau$ denotes proper time and

$\left(\Gamma^{(u)}_{\tau}\right)_{\mu}{}^{\nu}=\left(\partial_{\tau}(j_{0})j_{0}^{-1}\right)_{\mu}{}^{\nu}=-\partial_{\tau}\partial_{\mu}f_{0}^{\xi}\left(\delta^{\nu}_{\xi}+\partial_{\xi}f_{0}^{\nu}+(\partial_{\xi}f_{0}^{\rho})(\partial_{\rho}f_{0}^{\nu})+\ldots\right)\,,$
(135)

while a covariant derivative with respect to the coordinates (involving any
connection) transforms according to

$\displaystyle f^{\xi}:D^{(u)}_{\lambda}V^{\mu}_{(u)}\mapsto
D^{(u^{\prime})}_{\lambda}V^{\mu}_{(u^{\prime})}=(\delta_{\lambda}{}^{\kappa}-\partial_{\lambda}f^{\kappa})D^{(u)}_{\kappa}V^{\nu}_{(u)}\left(\delta^{\mu}_{\nu}+\partial_{\nu}f^{\mu}\right.$
$\displaystyle\left.+(\partial_{\nu}f^{\rho})(\partial_{\rho}f^{\mu})+\ldots\right)\,.$
(136)

###  6.2 Translations as point transformations

In everything preceding this subsection, we have focused on coordinate
transformations.

Each component of a vector field is itself field-valued and its functional
form may be very different in two coordinate systems. For example, if the
$x$-component of a vector field is

$V^{1}=x^{2}+3y\,,$ (137)

then at a point $A$ with coordinates $(x=2,y=3)$ it has a value of 13. Under
the change of coordinates

$(x,y)\mapsto(x^{\prime},y^{\prime})=(x+3,y)\,,$ (138)

it takes the form

$V^{1}=x^{\prime 2}-6x^{\prime}+9+3y^{\prime}$ (139)

\- a very different functional form, but note that the basis has not changed.
The point $A$ now has coordinates $(x^{\prime}=5,y^{\prime}=3)$, but $V^{1}$
still has the value 13 at this point. This naturally goes for scalar, spinor
and tensor fields as well.

A ‘point translation’ is either an active transformation - a physical
displacement of the field across spacetime - or, equivalently, the change to
its functional form which results from applying the inverse transformation to
the underlying coordinates. (In the above case, either shifting the field
configuration three units to the left, or shifting the axes three units to the
right.) The resulting transformation of the field is found through a Taylor
expansion:

$V^{1}(x^{\prime})=V^{1}(x)+\delta x\,\partial_{x}V+\mathcal{O}^{2}(\delta
x)\,,$ (140)

which clearly allows an inhomogeneous transformation of the vector component.
The generator of such a transformation is the partial derivative operator,
which is proportional to the quantum mechanical momentum operator. These
transformations seem to be the ones used by Böhmer _et al_[39], who use a
Noether-type argument that teleparallel gravity may be described as a gauge
theory of translations. (It is not clear that these are the transformations
used by Pereira and Obukhov[32], who seem to postulate a translation action on
a vector at the outset, which is not derived as a point transformation or a
coordinate transformation.) These point transformations seem to represent the
action on field configurations taking values in a Hilbert space. The
transformations of quantum fields under the Poincaré group was examined by
Wigner[47], who identified the unitary representations of the group, including
vector representations.

It appears there could be merit in a closer study of the relation between
coordinate transformations and point transformations, as this could shed light
on the relation between the classical/Einsteinian and quantum perspectives.

##  7 Conclusions

This paper has shown the valuable insights that can be gained from linking
tangent space symmetries back to coordinate transformations on the base
manifold. These transformations act on the tangent space through elements of
the group $GL(4,R)$. We have seen how the coset space decomposition of this
group with respect to its Lorentz subgroup helps one to understand the nature
of the Lorentz gauge symmetry inherent in teleparallelism. Essentially, the
Weitzenböck connection can be constructed using an infinite number of
different parallelisms, all related by local Lorentz transformations, so the
Lorentz gauge freedom in teleparallelism is a reflection of this freedom of
choice of parallelism.

A change of parallelism can be represented by a Lorentz transformation, which
acts on the matrix relating the coordinate basis to the frame basis from the
right. However, in general, a change of coordinates will have a Lorentz
component to its action on the coordinate basis, and some changes of
coordinate can be represented purely as Lorentz transformations. These act on
the matrix relating the coordinate basis to the frame basis from the left.
These are not independent of the changes of parallelism - the action from the
left ‘feeds through’ to $i_{0}$, inducing a gauge transformation of the
Lorentz connection. This means that a choice of Lorentz gauge is not respected
under changes of coordinate.

This paper also makes it clear that frame bases can also be coordinate bases
along a geodesic, when Riemann normal coordinates are employed. This choice of
coordinates provides further insights into the relations between different
connections.

Finally, we noted that the homomorphic mapping from the covariance group into
the general linear group includes a mapping of the Poincaré group. For global
transformations, translations are in the kernel of this map. Local
translations, on the other hand, provide a complete cover of the general
linear group, through the derivatives of their parameters. Thus general linear
transformations on a given tangent space may be represented as local
translations and _vice versa_. However, general linear maps between coordinate
and frame basis fields across a coordinate neighbourhood can only be
represented as local translations on a Riemann flat spacetime. Inhomogeneous
transformations of the vector components cannot be represented in this way,
but can result from a ‘point translation’, as shown in Section 6.2.

While we have applied this theory in the case of four-dimensional spacetime,
all of it is valid any any number of dimensions, regardless of the action. We
assume only that the spacetime is pseudo-Riemannian, that all transformations
on the tangent bundle are induced by changes of coordinate, and that
observations are made by a classical, point-like observer.

## Acknowledgments

The author would like to thank Eric Huguet, Morgan Le Delliou, María José
Guzmán, Tomi Koivisto and Martin Krs̆s̆ak for thought-provoking conversations
and comments on the topics in this paper. I would also like to thank all those
who have translated the classic papers listed below, those who put helpful
postings on sites such as StackExchange and ResearchGate and those who have
edited Wikipedia pages on these topics. Particular thanks to Alexander
Unzicker for the key insights I gained from material on his website.

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