# Teleparallel Minkowski Spacetime with Perturbative Approach for Teleparallel
Gravity on a Proper Frame

A. Landry<EMAIL_ADDRESS>Department of Mathematics and Statistics, Dalhousie
University, P.O. Box 15 000, Halifax, Nova Scotia, Canada, B3H 4R2  R. J. van
den Hoogen<EMAIL_ADDRESS>Department of Mathematics and Statistics, St.
Francis Xavier University, Antigonish, Nova Scotia, Canada, B2G 2W5

###### Abstract

A complete perturbation theory suitable for teleparallel gravity is developed.
The proposed perturbation scheme takes into account perturbations of the
coframe, the metric, and the spin-connection, while ensuring that the
resulting perturbed system continues to describe a teleparallel gravity
situation. The resulting perturbation scheme can be transformed to one in
which perturbations all take place within the co-frame. A covariant definition
of a teleparallel Minkowski geometry is proposed. We compute the perturbed
field equations for $f(T)$ teleparallel gravity and discuss the stability of
the teleparallel Minkowski geometry within $f(T)$ teleparallel gravity.

††preprint: arXiv:2303.16089

###### Contents

  1. I Introduction
  2. II Teleparallel Theories of Gravity
    1. II.1 Notation
    2. II.2 Torsion-Based Theories
    3. II.3 Geometrical Framework for Teleparallel Gravity
    4. II.4 Linear Transformations and Gauge Choices
    5. II.5 Action for $f(T)$ Teleparallel Gravity
    6. II.6 Field Equations for $f(T)$ Teleparallel Gravity
  3. III Constant Torsion Spacetimes
    1. III.0.1 Definition: Minkowski Geometry and Minkowski Spacetime
  4. IV Perturbations in Teleparallel Geometries
    1. IV.1 Proper Orthonormal Perturbation of the Co-Frame
    2. IV.2 Perturbed $f(T)$ Teleparallel Field Equations: General
    3. IV.3 Perturbed $f(T)$ Teleparallel Field Equations: Constant Torsion Scalar
    4. IV.4 Perturbed $f(T)$ Teleparallel Field Equations: Zero Torsion Scalar
    5. IV.5 Perturbed $f(T)$ Teleparallel Field Equations: The Zero Torsion Scalar Perturbation Limit
    6. IV.6 Perturbed $f(T)$ Teleparallel Field Equations: Minkowski
  5. V Effects of Perturbations and the Minkowski Spacetime Symmetries Conditions for Stability
    1. V.1 Rotation/Boost Perturbation in a Minkowski Background
    2. V.2 General Linear Perturbation in a Minkowski Background
    3. V.3 Perturbations on Trivial Coframes by Each Part of the Perturbation
      1. V.3.1 Trace
      2. V.3.2 Full Symmetric Perturbation
      3. V.3.3 Full Antisymmetric Perturbation
      4. V.3.4 A Mixed Situation and Minkowski Spacetime
  6. VI Discussion and Conclusions
  7. A Perturbed Physical Quantities in Teleparallel Theories
  8. B General Perturbed Torsion-Based Field Equation via Linearization
  9. C The Derivation of Minkowski Spacetime Symmetries: Conditions for Stability
    1. C.1 Rotation/Boost Perturbation
    2. C.2 General Linear Perturbation

##  I Introduction

There are two major classes of theories for physical phenomena: gravitational
theories and quantized theories [1, 2, 3, 4]. The first class of theories are
used to explain phenomena at the astrophysical scale; for example, General
Relativity (GR) has been very successful in explaining astrophysical phenomena
[5, 6, 7, 8]. However, the second class of theories concerns phenomena
occurring at the microscopic scale involving fundamental quantum particles.
Attempts have been made to reconcile the two classes of theories in order to
have a general, all-encompassing theory. A theory that is capable of dealing
with very low-amplitude physical and geometrical quantities, as is the case
for theories based on quantization, is desirable.

Indeed, Quantum Mechanics (QM) as well as Quantum Field Theory (QFT) have
well-established perturbative theories: a potential is perturbed, generating a
correction of the eigenvalues of the energies, as well as corrections to the
wave functions [1, 2, 3, 4]. QM and QFT are well established and have been
used to describe the gravitational corrections of curved spacetimes of
physical phenomena that can occur at the microscopic scale [9, 10, 11, 12].
Unfortunately, this perturbative approach to GR is problematic, primarily
because one requires an identifiable background on which to perform the
perturbations [13]. One can, of course, use gauge invariant variables to
address this challenge.

Recently, there has been a growing interest in the development of teleparallel
gravity as an alternative theory to GR [14, 15, 16, 17, 18, 19, 20, 21].
Teleparallel gravity needs to be better understood and developed in order to
address foundational, physical, and geometrical problems. Here, we will
illuminate some of the challenges and nuances that are present within
perturbative approaches to teleparallel gravity.

Golovnev and Guzman [22] studied a class of perturbations within a geometry
having a Minkowski metric. They applied perturbations to a particular boosted
coframe in which the metric has the Minkowski form and the torsion scalar is
zero, but where the torsion tensor is non-zero. One may argue that any
geometry in which the torsion tensor is non-zero is inherently not a Minkowski
geometry, but this is a matter of definition. In another paper, Jimenez et al.
performed perturbations of Minkowski spacetime in $f(T)$ teleparallel gravity
by using a trivial tetrad and having the perturbations encoded in
infinitesimal Lorentz transformations [23]. Their approach, while correct, is
restrictive when working towards a general perturbation theory within
teleparallel gravity. In ref [24], the authors develop a complete perturbation
theory that can be employed for perturbation analysis in Minkowski and flat
Robertson-Walker-type cosmological spacetimes. Our analysis provides a
different perspective and can be used as a general framework, and therefore,
it complements the work in ref [24].

Recently, within a cosmological setting, Bahamonde et al. [25] investigated
perturbations occurring on a FLRW-type background. They defined a very
specific form for the perturbation compatible with this background. They then
obtain the perturbed field equations. In addition, they investigated the
consequent effects of perturbations on the torsion and on different physical
quantities. Most of the types of perturbations studied lead to the flat FLRW
background case under some precise limits. On the other hand, some
perturbation modes do not propagate, which maintains the strong coupling. This
is the case of the scalar and the pseudo-scalar parts of the perturbations.
Here, we still have work with a limited scope; hence, the need for a more
general theory of perturbations in teleparallel gravity.

Bamba and Cai’s papers focus on Gravitational Waves (GWs) in teleparallel
gravity [26, 27]. GWs are a class of wave-like perturbations of Minkowski
spacetime. They are still dealing here with a specific case of perturbation.
In Bamba [26], they place themselves in the Minkowski background to process
the GWs in teleparallel gravity. In Cai [27], they place themselves in the
FLRW background. They therefore have a generalization of Bamba’s work for GWs
that are compatible with the cosmological models. In addition, in [27], they
add the effects of scalar fields in their perturbations. Not only are they
still dealing with specific cases of perturbations, but they are moving from
the Minkowski background to the FLRW background. However, they still do not
have a general theory for the Minkowski background. Therefore, a more general
and fundamental theory that is applicable for any perturbation and any co-
frame in Minkowski spacetime in teleparallel gravity is needed.

We begin this paper with a definition of Minkowski geometry and Minkowski
spacetime within teleparallel gravity. Then, we will investigate the effects
of perturbations in teleparallel gravity. After, we will study the stability
of Minkowski spacetime by using the perturbed quantities and field equations.

In teleparallel gravity, co-frames encode both the gravitational and inertial
effects. Our goal is to explore the perturbations of gravity, and therefore,
we shall carefully construct a perturbative theory that achieves this goal. If
we transform initially to “proper” frames which encode only the gravitational
effects and then perform perturbations on all physical quantities,
consequently ensuring that the resulting perturbed theory is still within the
class of teleparallel theories of gravity will yield the general allowable
form for perturbations within teleparallel gravity. We will perturb the
physical quantities which maintain the “proper frames”, thus avoiding the
challenge of interpreting the spurious inertial effects that may appear in
“non-proper frames” [28, 14, 29, 16, 15].

We want to highlight the effects of perturbations in teleparallel gravity. For
example, in an absolute vacuum, one can highlight the effects of perturbations
modifying this same vacuum. For example, we will determine the gravitational
Energy-Momentum associated with a perturbation. We will apply this theory of
perturbations in teleparallel gravity to some examples and problems of Physics
[30, 31, 16]. Particularly, we will study through these coframe perturbations
the stability of the Minkowski background, and determine the required symmetry
conditions to satisfy.

This paper is divided as follows. In Section II, we present a summary of
teleparallel gravity and propose a definition of Minkowski geometry within
teleparallel gravity. In Section IV, we will define the perturbations
maintaining the “proper frames”, the orthonormal framework, and we will also
provide the perturbed Field Equations (FEs). In Section V, we will explore
some coframe perturbations to determine the stability criterions for Minkowski
spacetime. We can also generalize these criterions to null and constant
torsion spacetimes.

##  II Teleparallel Theories of Gravity

###  II.1 Notation

Greek indices $(\mu,\nu,\dots)$ are employed to represent the spacetime
coordinate indices, while Latin indices $(a,b,\dots)$, are employed to
represent frame or tangent-space indices. As is standard notation, round
parentheses surrounding indices represent symmetrization, while square
brackets represent anti-symmetrization. Any quantity that is computed using a
Levi-Civita connection ${\overset{\circ}{\omega}}{}^{a}_{\phantom{a}b\mu}$
will have a circle above the symbol. A comma will denote a partial derivative.
The metric signature is assumed to be $(-,+,+,+)$.

###  II.2 Torsion-Based Theories

Torsion-based theories of gravity are a subclass of Einstein-Cartan theories
[32, 15, 16]. This superclass of theories contains theories based solely on
the curvature, for example, General Relativity, or $f\left(R\right)$ theories
where $R$ is Ricci curvature scalar. Einstein-Cartan theories of gravity also
contain theories of gravity that are based solely on the torsion, for example,
teleparallel theories of gravity, including New General Relativity [33] and
$f\left(T\right)$ theories where $T$ is the torsion scalar. In addition,
theories of gravity based on both the curvature and torsion scalars
($f\left(R,T\right)$-type) are also subclasses of the Einstein-Cartan theories
of gravity. Recently, there has been an emergence of theories based on non-
metricity ($f\left(Q\right)$-type), although they are less well known [34, 35,
16]. In this paper, we are interested in teleparallel gravity, and in
particular, $f(T)$ teleparallel gravity [14, 15, 17, 18, 19, 20, 16, 29].

###  II.3 Geometrical Framework for Teleparallel Gravity

Let $M$ be a $4$-dimensional differentiable manifold with coordinates
$x^{\mu}$. Then, the geometry of the manifold is characterized by the three
geometrical objects.

  * •

The Co-frame: $h^{a}=h^{a}_{\;\;\mu}dx^{\mu}$. This quantity generally encodes
both the gravitational and inertial effects in a gravitational system. The
dual of the co-frame is defined as the vector field
$h_{a}=h_{a}^{~{}\mu}\frac{\partial}{\partial x^{\mu}}$, such that
$h^{a}_{~{}\mu}h_{b}^{~{}\mu}=\delta^{a}_{b}$.

  * •

The Gauge Metric: $g_{ab}$. This object expresses the “metric” of the tangent
space, such that $g_{ab}=g(h_{a},h_{b})$. Having a metric allows one to define
the lengths and angles.

  * •

The Spin-connection: $\omega^{a}_{\;\;b}=\omega^{a}_{\;\;b\mu}dx^{\mu}$.
Having a connection allows one to “parallel transport’,’ or equivalently, it
allows one to define a covariant differentiation.

In teleparallel gravity, the co-frame, gauge metric, and spin connection are
restricted and interdependent, characterized by the following two postulates
[14, 15, 16]:

  * •

Null Curvature:

$R^{a}_{\;\;b\nu\mu}\equiv\omega^{a}_{~{}b\mu,\nu}-\omega^{a}_{~{}b\nu,\mu}+\omega^{a}_{~{}c\nu}\omega^{c}_{~{}b\mu}-\omega^{a}_{~{}c\mu}\omega^{c}_{~{}b\nu}=0$
(1)

  * •

Null Non-Metricity:

$Q_{ab\mu}\equiv-
g_{ab,\mu}+\omega^{c}_{~{}a\mu}g_{cb}+\omega^{c}_{~{}b\mu}g_{ac}=0$ (2)

In teleparallel gravity, the only remaining non-null field strength is the
torsion defined as

$T^{a}_{\phantom{a}\mu\nu}=h^{a}_{\phantom{a}\nu,\mu}-h^{a}_{\phantom{a}\mu,\nu}+\omega^{a}_{\phantom{a}b\mu}h^{b}_{\phantom{a}\nu}-\omega^{a}_{\phantom{a}b\nu}h^{b}_{\phantom{a}\mu}$
(3)

It is now possible to construct a gravitational theory that depends only on
the torsion. However, before proceeding, we illustrate the effects of gauge
transformations on the geometry, and how we can judiciously choose a gauge to
simplify our computations.

###  II.4 Linear Transformations and Gauge Choices

From the Principle of Relativity, we impose the requirement that the physical
gravitational system under consideration be invariant under $GL(4,\mathbb{R})$
local linear transformations of the frame. These types of transformations
allow one to pass from one frame of reference to another frame of reference.
For the fundamental geometrical quantities
$\\{h^{a},g_{ab},\omega^{a}_{~{}bc}\\}$, we have the following transformation
rules under a general linear transformation $M^{a}_{~{}b}\in
GL(4,\mathbb{R})$:

$\displaystyle h^{\prime a}_{~{}\mu}$ $\displaystyle=$ $\displaystyle
M^{a}_{~{}b}\,h^{b}_{~{}\mu},$ (4) $\displaystyle g^{\prime}_{ab}$
$\displaystyle=$ $\displaystyle M_{a}^{~{}e}\,M_{b}^{~{}f}\,g_{ef},$ (5)
$\displaystyle\omega^{\prime a}_{\,~{}b\mu}$ $\displaystyle=$ $\displaystyle
M^{a}_{~{}e}\,\omega^{e}_{~{}f\mu}\,M_{b}^{~{}f}+M^{a}_{~{}e}\,\partial_{\mu}\,M_{b}^{~{}e}.$
(6)

where $M_{b}^{~{}a}=(M^{-1})^{a}_{~{}b}$ represents the inverse matrix.
Equation (6) shows that the Spin-connection transforms non-homogeneously under
a general linear transformation.

#### Gauge Choices and Teleparallel Gravity

Physical phenomena must respect the principle of Gauge Invariance. The
physical phenomenon must be explainable and valid, regardless of the gauge and
its possible transformations. If this general principle is important for
quantized theories, then this same principle is also important for
teleparallel gravity. Generally, we have a tremendous choice of gauge,
depending on the assumed symmetries of the physical system. However, once we
have made a gauge choice, the consequent field equations describing the theory
must transform covariantly (i.e., they are invariant) under any remaining
gauge freedom.

##### Proper Orthonormal Frame

The Null Curvature postulate guarantees that there exists an element
$M^{a}_{~{}b}\in GL(4,\mathbb{R})$, such that

$\omega^{a}_{~{}b\mu}\equiv(M^{-1})^{a}_{~{}b}\partial_{\mu}(M^{b}_{~{}c})$
(7)

Since the connection transforms non-homogeneously under local linear
transformations, we can always apply the linear transformation $M^{a}_{~{}b}$
to transform to a proper frame in which $\omega^{a}_{~{}b\mu}=0$. Further,
within this proper frame, given the Null Non-Metricity postulate, it is then
possible to apply a second constant linear transformation to bring the gauge
metric to some desired form. For example, we can transform to a gauge in which
the spin connection is null and the gauge metric is
$g_{ab}=\mathrm{Diag}[-1,1,1,1]$, which we will call a “proper orthonormal
frame”. The only remaining gauge freedom in this case are global (constant)
Lorentz transformations.

##### Orthonormal Frame

If one prefers not to be restricted to a proper frame, then there is more
flexibility. Since the gauge metric is symmetric, we can still always choose
an “orthonormal frame” in which the gauge metric becomes
$g_{ab}=\mathrm{Diag}[-1,1,1,1]$, but where the spin connection may be non-
trivial. Assuming an orthonormal frame, the remaining gauge freedom is
represented by proper orthochronous Lorentz transformations in the
$SO^{+}(1,3)$ subgroup of $GL(4,\mathbb{R})$. Other gauge choices might
include Complex-Null, Half-Null, Angular-Null, and others [17, 18, 19]. In the
orthonormal frame, given the Null Curvature postulate, there exists a
$\Lambda^{a}_{~{}b}\in SO^{+}(1,3)$, such that the spin connection is  [36,
37]:

$\omega^{a}_{~{}b\mu}\equiv(\Lambda^{-1})^{a}_{~{}b}\partial_{\mu}(\Lambda^{b}_{~{}c})$
(8)

and given the Null Non-Metricity postulate, we have the restriction
$\omega_{(ab)\mu}=0$.

However, in either choice of gauge, we note that the spin connection,
$\omega^{a}_{~{}b\mu}$, is not a true dynamical variable and that it only
encodes inertial effects present in the choice of frame [14, 15, 17, 18, 19,
20, 16, 29, 28].

###  II.5 Action for $f(T)$ Teleparallel Gravity

In principle, one can construct a Lagrangian density from any of the scalars
built from the torsion tensor. One such scalar is [14, 15, 17, 18, 19, 20, 16,
29]:

$\displaystyle
T=\frac{1}{4}T^{a}_{~{}bc}T_{a}^{~{}bc}+\frac{1}{2}T^{a}_{~{}bc}T^{cb}_{~{}~{}a}-T^{a}_{~{}ca}T^{bc}_{~{}~{}b},$
(9)

which we will call “the” torsion scalar $T$ . Another related scalar, for
example, used in New General Relativity [33], is

$\displaystyle\widetilde{T}=c_{1}T^{a}_{~{}bc}T_{a}^{~{}bc}+c_{2}T^{a}_{~{}bc}T^{cb}_{~{}~{}a}+c_{3}T^{a}_{~{}ca}T^{bc}_{~{}~{}b}$
(10)

Other torsion scalars could be included, but these scalars are not invariant
under $SO^{+}(1,3)$, and they include parity violating terms [33].

Here, we are interested in a particular class of teleparallel gravity
theories, $f(T)$ teleparallel gravity. The action describing the $f(T)$
teleparallel theory of gravity containing matter is [14, 15, 17, 18, 19, 20,
16, 29]:

$S_{f\left(T\right)}=\int\,d^{4}\,x\,\left[\frac{h}{2\,\kappa}\,f\left(T\right)+\mathcal{L}_{Matter}\right].$
(11)

where $h=\mbox{Det}\left(h^{a}_{~{}\mu}\right)$ is the determinant of the
veilbein, the parameter $\kappa$ is the gravitational coupling constant which
contains the physical constants, and $f\left(T\right)$ is an arbitrary
function of the torsion scalar $T$, given by Equation (9).

###  II.6 Field Equations for $f(T)$ Teleparallel Gravity

From the action integral expressed by Equation (11), we determine the field
equations by varying with respect to the coframe $h^{a}_{~{}\mu}$ [14, 15, 17,
18, 19, 20, 16, 29]:

$\displaystyle\kappa\,\Theta_{a}^{~{}~{}\mu}=\frac{f_{T}(T)}{h}\,\partial_{\nu}\,\left(h\,S_{a}^{~{}~{}\mu\nu}\right)+f_{TT}(T)\,S_{a}^{~{}~{}\mu\nu}\,\partial_{\nu}T+\frac{f(T)}{2}\,h_{a}^{~{}~{}\mu}-f_{T}(T)\,\left(\omega^{b}_{~{}~{}a\nu}+T^{b}_{~{}~{}a\nu}\right)\,S_{b}^{~{}~{}\mu\nu}.$

The superpotential is defined as [14, 15, 18, 17]:

$S_{a}^{~{}\mu\nu}=\frac{1}{2}\left(T_{a}^{~{}\mu\nu}+T_{~{}~{}a}^{\nu\mu}-T_{~{}~{}a}^{\mu\nu}\right)-h_{a}^{~{}\nu}\,T^{\rho\mu}_{~{}~{}\rho}+h_{a}^{~{}\mu}\,T^{\rho\nu}_{~{}~{}\rho}.$
(13)

The canonical Energy-Momentum is defined as [18]:

$h\,\Theta_{a}^{~{}\mu}\equiv\frac{\delta\mathcal{L}_{Matter}}{\delta
h^{a}_{~{}\mu}}.$ (14)

Now, expressing the field equations (II.6) in terms of the tangent-space
components allows one to split the field equations into symmetric and
antisymmetric parts. The symmetric and antisymmetric parts of the $f(T)$
teleparallel gravity FEs are respectively [17, 18, 19]:

$\displaystyle\kappa\Theta_{\left(ab\right)}\,$ $\displaystyle=$
$\displaystyle\,f_{TT}\left(T\right)\,S_{\left(ab\right)}^{~{}~{}~{}\mu}\,\partial_{\mu}T+f_{T}\left(T\right)\,\overset{\
\circ}{G}_{ab}+\frac{g_{ab}}{2}\,\left[f\left(T\right)-T\,f_{T}\left(T\right)\right],$
$\displaystyle 0\,$ $\displaystyle=$
$\displaystyle\,f_{TT}\left(T\right)\,S_{\left[ab\right]}^{~{}~{}~{}\mu}\,\partial_{\mu}T,$
(15)

where $\overset{\ \circ}{G}_{ab}$ is the Einstein tensor computed from the
Levi-Civita connection of the metric.

We note with an orthonormal gauge choice, and consequent invariance under
$SO^{+}(1,3)$ transformations, it can be shown that

$\Theta_{[ab]}=0,$ (16)

and that the metrical energy-momentum $T_{ab}$ and the symmetric part of the
canonical energy-momentum satisfy

$\Theta_{(ab)}=T_{ab}\equiv\frac{1}{2}\frac{\delta L_{Matt}}{\delta g_{ab}}.$
(17)

##  III Constant Torsion Spacetimes

A class of interesting spacetimes are those leading to a constant torsion
scalar, i.e., $T=T_{0}=\text{Const}$. This class of spacetimes includes the
Minkowski spacetime, amongst others. In this case, the equations (II.6) will
simplify with $\partial_{\mu}T=0$ as follows, leaving only the symmetric part
of the field equations:

$\displaystyle\kappa\Theta_{\left(ab\right)}\,$ $\displaystyle=$
$\displaystyle f_{T}\left(T_{0}\right)\,\overset{\
\circ}{G}_{ab}+\frac{g_{ab}}{2}\,\left[f\left(T_{0}\right)-T_{0}\,f_{T}\left(T_{0}\right)\right].$
(18)

The antisymmetric part of the field equations becomes identically satisfied.
We can now divide Equation (18) by $f_{T}\left(T_{0}\right)$ to obtain:

$\displaystyle\kappa_{eff}\Theta_{\left(ab\right)}\,$ $\displaystyle=$
$\displaystyle\overset{\
\circ}{G}_{ab}+g_{ab}\,\left[\frac{f\left(T_{0}\right)}{2\,f_{T}\left(T_{0}\right)}-\frac{T_{0}}{2}\right]$
(19) $\displaystyle=$ $\displaystyle\overset{\
\circ}{G}_{ab}+g_{ab}\,\Lambda\left(T_{0}\right).$

where we define the re-scaled gravitational coupling constant
$\kappa_{eff}=\frac{\kappa}{f_{T}\left(T_{0}\right)}$ and an effective
cosmological constant $\Lambda\left(T_{0}\right)$, both dependent on the value
of $T=T_{0}$. We observe that if $T=T_{0}=\text{Const}$, then the $f(T)$
teleparallel field equations reduce to those of GR, having a re-scaled
gravitational coupling and a cosmological constant.

Due to its importance in characterizing the Minkowski geometry, we carefully
consider the case of $T_{0}=0$ for further consideration.

### Null Torsion Scalar Spacetimes

When $T_{0}=0$, the field equations reduce to:

$\displaystyle\kappa_{eff}\Theta_{\left(ab\right)}\,$ $\displaystyle=$
$\displaystyle\overset{\
\circ}{G}_{ab}+g_{ab}\,\left[\frac{f\left(0\right)}{2\,f_{T}\left(0\right)}\right],$
(20) $\displaystyle=$ $\displaystyle\overset{\
\circ}{G}_{ab}+g_{ab}\,\Lambda\left(0\right).$

where $\kappa_{eff}=\frac{\kappa}{f_{T}\left(0\right)}$ and
$\Lambda\left(0\right)=\frac{f(0)}{2\,f_{T}\left(0\right)}$. If $f(0)\neq 0$,
then the Cosmological Constant $\Lambda(0)\neq 0$.

####  III.0.1 Definition: Minkowski Geometry and Minkowski Spacetime

Before obtaining the field equations and introducing the perturbations on
such, one must clearly define the true nature of the Minkowski spacetime in
teleparallel gravity in a covariant way. This will make it possible to better
understand the nature and origin of the equations involving the dominant
quantities with respect to the perturbed quantities. This geometry is
characterized as follows:

  * •

Maximally symmetric: The Minkowski geometry is invariant under a $G_{10}$
group of transformations [18].

  * •

Null Curvature: $R_{~{}b\mu\nu}^{a}=0$

  * •

Null Torsion: $T^{a}_{~{}\mu\nu}=0$

  * •

Null Non-Metricity: $Q_{ab\mu}=0$

One of the consequences is that Minkowski geometry is everywhere a smooth
geometry without singularity. This covariant definition of teleparallel
Minkowski geometry has been proposed also by Beltran et al. [38].

We distinguish between Minkowski geometry and Minkowski spacetime in
teleparallel gravity as follows. Minkowski geometry is defined independently
of any field equations, while Minkowski spacetime is a Minkowski geometry that
is a solution to the teleparallel gravity field equations where the matter
source is a vacuum, $\Theta_{ab}=0$.

If the geometry is Minkowski, then the torsion scalar is identically zero.
Note that the converse is not necessarily true. The Einstein tensor
$\overset{\ \circ}{G}_{ab}=0$, and since the matter source is a vacuum,
$\Theta_{ab}=0$, the field equations (20) reduce to

$0=\frac{f\left(0\right)}{2}\,g_{ab}.$ (21)

From the field equations (21), if the geometry is Minkowski and
$\Theta_{ab}=0$, then $f(0)=0$. In this case, the solution is a Minkowski
spacetime, a Minkowski geometry that satisfies the field equations in vacuum.
Alternatively, if $f(0)\not=0$, then a solution to the field equations (21)
necessarily requires a non-null $\Theta_{ab}$, and consequently, this
spacetime is not a Minkowski spacetime, even though the geometry is Minkowski.
Of course, the non-trivial $\Theta_{ab}$ can be interpreted as the energy
density of the vacuum. Expressing the statement clearly, Minkowski geometry is
a solution to the vacuum $f(T)$ teleparallel gravity field equations only if
$f(0)=0$.

##  IV Perturbations in Teleparallel Geometries

###  IV.1 Proper Orthonormal Perturbation of the Co-Frame

As described earlier, a teleparallel geometry is characterized in general via
the triplet of quantities, the co-frame one form $h^{a}$, the spin connection
one-form $\omega^{a}_{~{}b}$, and the metric tensor field $g_{ab}$, with two
constraints, Null Curvature and Null Non-Metricity. As argued earlier,
assuming that the physical system is invariant under the $GL(4,\mathbb{R})$
linear transformations (see also ref. [38]), this means that even before
constructing a perturbative theory, one can always choose to begin in a
“proper orthonormal frame” as our background without a loss of generality:

${h}^{a}={h}^{a}_{~{}\mu}dx^{\mu},\qquad{\omega}^{a}_{~{}b}=0,\qquad{g}_{ab}=\eta_{ab}=\mathrm{Diag}[-1,1,1,1].$
(22)

Now, we apply a perturbation to all three quantities, as follows:

$h^{\prime a}={h}^{a}+\delta h^{a},\qquad\omega^{\prime
a}_{~{}b}=\delta\omega^{a}_{~{}b},\qquad g^{\prime}_{ab}=\eta_{ab}+\delta
g_{ab}$ (23)

The perturbed geometry is no longer expressed in a proper orthonormal frame.
The perturbed system is only proper if $\delta\omega^{a}_{~{}b}=0$, and
orthonormal if $\delta g_{ab}=0$. However, we shall show that we can always
transform to a proper orthonormal perturbation scheme.

We note that the perturbed geometry given by the triplet $\\{h^{\prime
a},\omega^{\prime a}_{~{}b},g^{\prime}_{ab}\\}$ must still satisfy the Null
Curvature and Null Non-Metricity constraints or else one is moving outside of
the theory of teleparallel gravity. In general, the perturbations $\delta
h^{a}$, $\delta\omega^{a}_{~{}b}$, and $\delta g_{ab}$ are not all
independent. The Null Curvature constraint for the perturbed connection
$\omega^{\prime a}_{~{}b}$ implies that there exists some local linear
transformation $L^{a}_{~{}b}\in GL(4,\mathbb{R})$, such that

$\delta\omega^{a}_{~{}b}=(L^{-1})^{a}_{~{}c}dL^{c}_{~{}b}$ (24)

where $d$ indicates the exterior derivative. This means that we can apply this
general linear transformation to the perturbed system to express it in a
perturbed proper frame

$\bar{h}^{\prime a}=L^{a}_{~{}b}({h}^{b}+\delta
h^{b}),\qquad\bar{\omega}^{\prime
a}_{~{}b}=0,\qquad\bar{g}^{\prime}_{ab}=(L^{-1})^{c}_{~{}a}(L^{-1})^{d}_{~{}b}(\eta_{cd}+\delta
g_{cd})$ (25)

where we have used a bar to indicate that we are now in a proper frame.

The Null Non-Metricity condition applied to this “perturbed proper frame” (25)
means that $\bar{g}^{\prime}_{ab}$ is a symmetric matrix of the constants
which can diagonalized. That is, there exists a matrix $P^{a}_{~{}b}\in
GL(4,\mathbb{R})$ of constants such that
$\bar{g}^{\prime}_{ab}=(P^{-1})^{c}_{~{}a}(P^{-1})^{d}_{~{}b}\eta_{cd}$. So,
we can apply this constant transformation $P^{a}_{~{}b}$ to the “perturbed
proper frame” (25) to obtain a “perturbed proper orthonormal frame” without a
loss of generality.

$\displaystyle\hat{h}^{\prime a}$ $\displaystyle=$ $\displaystyle
P^{a}_{~{}b}\bar{h}^{\prime b}=P^{a}_{~{}b}L^{b}_{~{}c}({h}^{c}+\delta
h^{c}),$ (26a) $\displaystyle\hat{\omega}^{\prime a}_{~{}b}$ $\displaystyle=$
$\displaystyle 0,$ (26b) $\displaystyle\hat{g}_{ab}^{\prime}$ $\displaystyle=$
$\displaystyle\eta_{ab}.$ (26c)

We observe that we can investigate perturbations in teleparallel geometries by
simply looking at the perturbations in a co-frame, using proper orthonormal
frames. Doing so ensures that the Null Curvature and Null Non-Metricity
constraints are respected. If we define the compositions of the two linear
transformations as matrix $M^{a}_{~{}b}=P^{a}_{~{}c}L^{c}_{~{}b}\in
GL(4,\mathbb{R})$, then the “perturbed proper orthonormal frame” becomes

$\hat{h}^{\prime a}=M^{a}_{~{}b}\left({h}^{b}+\delta h^{b}\right).$ (27)

which encodes all possible perturbations within a proper orthonormal
framework. If $M^{a}_{~{}b}=\delta^{a}_{b}$, then the only perturbations are
perturbations in the original proper orthonormal frame. The matrix
$M^{a}_{~{}b}$ encodes the perturbations that took place originally in the
spin connection and metric, but it ensures that the resulting perturbed system
is teleparallel in nature. For completeness, the original perturbations can be
expressed in terms of $M^{a}_{~{}b}$, as

$\delta\omega^{a}_{~{}b}=(M^{-1})^{a}_{~{}c}dM^{c}_{~{}b},\qquad\delta
g_{ab}=(M^{-1})^{c}_{~{}a}(M^{-1})^{d}_{~{}b}\eta_{cd}-\eta_{ab}$ (28)

Now, in a perturbative approach, to the first order, we have that

$\displaystyle M^{a}_{~{}b}$ $\displaystyle\approx$
$\displaystyle\delta^{a}_{b}+\mu^{a}_{~{}b}$ (29) $\displaystyle\delta h^{a}$
$\displaystyle\approx$ $\displaystyle\nu^{a}_{~{}b}h^{b}$ (30)

for some $\mu^{a}_{~{}b}$ and $\nu^{a}_{~{}b}\in\mathfrak{gl}(4,\mathbb{R})$.
Therefore, putting it all together, we have to first order

$\displaystyle\hat{h}^{\prime a}$ $\displaystyle=$ $\displaystyle
h^{a}+(\mu^{a}_{~{}b}+\nu^{a}_{~{}b})h^{b}=h^{a}+\lambda^{a}_{~{}b}h^{b},$
(31a) $\displaystyle\hat{\omega}^{\prime a}_{~{}b}$ $\displaystyle=$
$\displaystyle 0,$ (31b) $\displaystyle\hat{g}_{ab}^{\prime}$ $\displaystyle=$
$\displaystyle\eta_{ab},$ (31c)

where $\lambda^{a}_{~{}b}\in M(4,\mathbb{R})$, the set of $4\times 4$ real-
valued matrices. Perturbations of the independent quantities in teleparallel
geometry can always be transformed to the form (31). The matrix $\lambda$ can
be invariantly decomposed into trace, symmetric trace-free, and anti-symmetric
parts.

For the next section and in the appendix, we will apply the perturbations

$\delta
h^{a}=\lambda^{a}_{~{}b}h^{b},\qquad\delta\omega^{a}_{~{}b}=0,\qquad\delta
g_{ab}=0,$ (32)

to the $f(T)$ teleparallel field equations in a proper orthonormal frame. In
particular, we will look at perturbations of constant scalar torsion
spacetimes.

###  IV.2 Perturbed $f(T)$ Teleparallel Field Equations: General

Considering the perturbations of the field equations (II.6), we obtain

$\displaystyle\kappa\left[\Theta_{\left(ab\right)}+\delta\Theta_{(ab)}\right]$
$\displaystyle=$ $\displaystyle f_{TT}\left(T+\delta
T\right)\,\left[S_{(ab)}^{~{}~{}~{}\mu}+\delta
S_{(ab)}^{~{}~{}~{}\mu}\right]\left[\partial_{\mu}T+\partial_{\mu}\left(\delta
T\right)\right]$ (33a) $\displaystyle\quad+f_{T}\left(T+\delta
T\right)\,\left[\overset{\ \circ}{G}_{ab}+\delta\overset{\
\circ}{G}_{ab}\right]$
$\displaystyle\quad+\frac{g_{ab}}{2}\left[f\left(T+\delta
T\right)-\left(T+\delta T\right)\,f_{T}\left(T+\delta T\right)\right],$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle f_{TT}\left(T+\delta
T\right)\,\left[S_{[ab]}^{~{}~{}~{}\mu}+\delta
S_{[ab]}^{~{}~{}~{}\mu}\right]\partial_{\mu}\left(T+\delta T\right),$ (33b)

which to the first order in the perturbations yields

$\displaystyle\kappa\,\delta\Theta_{\left(ab\right)}$ $\displaystyle\approx$
$\displaystyle\left[f_{TTT}\,S_{(ab)}^{~{}~{}~{}\mu}\partial_{\mu}T+f_{TT}\,\left(\overset{\
\circ}{G}_{ab}-\frac{T}{2}\,g_{ab}\right)\right]\,\delta
T+f_{T}\,\delta\overset{\ \circ}{G}_{ab}$ (34a)
$\displaystyle\quad+f_{TT}\left[\delta
S_{(ab)}^{~{}~{}~{}\mu}\,\partial_{\mu}T+S_{(ab)}^{~{}~{}~{}\mu}\,\partial_{\mu}\left(\delta
T\right)\right]+O\left(|\delta h|^{2}\right),$ $\displaystyle 0$
$\displaystyle\approx$ $\displaystyle
f_{TTT}\,\left[S_{[ab]}^{~{}~{}~{}\mu}\partial_{\mu}T\right]\,\delta
T+f_{TT}\,\left[S_{[ab]}^{~{}~{}~{}\mu}\partial_{\mu}\left(\delta
T\right)+\delta S_{[ab]}^{~{}~{}~{}\mu}\partial_{\mu}T\right]+O\left(|\delta
h|^{2}\right),$ (34b)

where we no longer explicitly show functional dependence in $F$.

In Appendix A, perturbations of different dependent quantities are explicitly
computed in terms of the perturbations (32), for example $\delta T,\delta
S_{[ab]}^{~{}~{}~{}\mu}$, etc. Here, $\delta T$ is given by Equation (5) and
$\delta S_{ab}^{~{}~{}~{}\mu}$ is given by Equation (8). Equation (34) gives
us expressions for the perturbations to the matter resulting from the
perturbations in the co-frame, and constraints on the perturbations to the
antisymmetric part of the super-potential.

###  IV.3 Perturbed $f(T)$ Teleparallel Field Equations: Constant Torsion
Scalar

To study the effects of perturbations of the co-frame in constant torsion
scalar spacetimes, one substitutes $T=T_{0}=\text{Const}$ into Equation (34).
This means $\partial_{\nu}T=0$. If we divide by $f_{T}\left(T_{0}\right)$,
Equation (34) becomes:

$\displaystyle\kappa_{eff}\,\delta\Theta_{(ab)}$ $\displaystyle\approx$
$\displaystyle\delta\overset{\
\circ}{G}_{ab}+\frac{f_{TT}\left(T_{0}\right)}{f_{T}\left(T_{0}\right)}\left[S_{(ab)}^{~{}~{}~{}\mu}\,\partial_{\mu}\left(\delta
T\right)+\delta T\,\left(\overset{\
\circ}{G}_{ab}-\frac{T_{0}}{2}\,g_{ab}\right)\right]+O\left(|\delta
h|^{2}\right),$ (35a) $\displaystyle 0$ $\displaystyle\approx$
$\displaystyle\left(\frac{f_{TT}\left(T_{0}\right)}{f_{T}\left(T_{0}\right)}\right)S_{[ab]}^{~{}~{}~{}\mu}\partial_{\mu}(\delta
T)+O\left(|\delta h|^{2}\right),$ (35b)

where $\kappa_{eff}=\frac{\kappa}{f_{T}\left(T_{0}\right)}$. In general,
$S_{[ab]}^{~{}~{}\mu}\neq 0$, and therefore, the perturbations in the torsion
scalar are constant. Of course, in situations in which some component of
$S_{[ab]}^{~{}~{}\mu}=0$, and then the corresponding $\partial_{\mu}(\delta
T)\not=0$.

###  IV.4 Perturbed $f(T)$ Teleparallel Field Equations: Zero Torsion Scalar

For spacetimes that have a zero torsion scalar, $T=0$, and Equations (35a) and
(35b) become:

$\displaystyle\kappa_{eff}\,\delta\Theta_{\left(ab\right)}$
$\displaystyle\approx$ $\displaystyle\delta\overset{\
\circ}{G}_{ab}+\frac{f_{TT}\left(0\right)}{f_{T}\left(0\right)}\left[S_{(ab)}^{~{}~{}~{}\mu}\,\partial_{\mu}\left(\delta
T\right)+\delta T\,\overset{\ \circ}{G}_{ab}\right]+O\left(|\delta
h|^{2}\right),$ (36a) $\displaystyle 0$ $\displaystyle\approx$
$\displaystyle\left(\frac{f_{TT}\left(0\right)}{f_{T}\left(0\right)}\right)S_{[ab]}^{~{}~{}~{}\mu}\partial_{\mu}(\delta
T)+O\left(|\delta h|^{2}\right),$ (36b)

where $\kappa_{eff}=\frac{\kappa}{f_{T}\left(0\right)}$. As before, in
general, $S_{ab}^{~{}~{}\mu}\neq 0$, and therefore, the perturbations in the
torsion scalar are constant. These equations represent perturbations in non-
Minkowski but zero torsion scalar spacetimes. However, they can reduce to
perturbations of the $f(T)$ teleparallel field equations with a teleparallel
Minkowksi geometry when $S_{ab}^{~{}~{}\mu}=0$ and $\overset{\
\circ}{G}_{ab}=0$, which are the conditions that are compatible with a
teleparallel Minkowski spacetime, as defined in Section III.0.1.

###  IV.5 Perturbed $f(T)$ Teleparallel Field Equations: The Zero Torsion
Scalar Perturbation Limit

We are curious to know what happens in the restricted perturbation scheme in
which $\delta T\rightarrow 0$ only. Starting with Equation (34), we take the
limit $\delta T\rightarrow 0$, and these perturbed field equations become:

$\displaystyle\kappa\,\delta\Theta_{\left(ab\right)}$ $\displaystyle\approx$
$\displaystyle f_{T}\,\delta\overset{\ \circ}{G}_{ab}+f_{TT}\left[\delta
S_{(ab)}^{~{}~{}~{}\mu}\,\partial_{\mu}T+S_{(ab)}^{~{}~{}~{}\mu}\,\partial_{\mu}\left(\delta
T\right)\right]+O\left(|\delta h|^{2}\right),$ (37a) $\displaystyle 0$
$\displaystyle\approx$ $\displaystyle f_{TT}\,\left[\delta
S_{[ab]}^{~{}~{}~{}\mu}\partial_{\mu}T+S_{[ab]}^{~{}~{}~{}\mu}\,\partial_{\mu}\left(\delta
T\right)\right]+O\left(|\delta h|^{2}\right).$ (37b)

Looking at Equation (37b), given that in general,
$S_{[ab]}^{~{}~{}~{}\mu}\not=0$ and $\delta S_{[ab]}^{~{}~{}~{}\mu}\not=0$ (or
equivalently, the torsion tensor and perturbations of the torsion tensor are
non-trivial, respectively), we observe that if the torsion scalar is not
constant, $\partial_{\mu}T\not=0$, and then the perturbations of the torsion
scalar are also not constant, that is, $\partial_{\mu}(\delta T)\not=0$.
Conversely, if $\partial_{\mu}T=0$, then $\partial_{\mu}(\delta T)=0$.

###  IV.6 Perturbed $f(T)$ Teleparallel Field Equations: Minkowski

For the Minkowski spacetimes, as defined in Section III.0.1, since the torsion
tensor is zero by definition, the superpotential terms
$S_{(ab)}^{~{}~{}~{}\mu}=S_{[ab]}^{~{}~{}~{}\mu}=0$. Further, the Einstein
tensor $\overset{\ \circ}{G}_{ab}=0$, and as argued before, $f(0)=0$, so that
Equations (36a) and (36b) reduce as follows:

$\displaystyle\kappa_{eff}\,\delta\Theta_{(ab)}$ $\displaystyle\approx$
$\displaystyle\delta\overset{\ \circ}{G}_{ab}+O\left(|\delta h|^{2}\right),$
(38a) $\displaystyle 0$ $\displaystyle\approx$ $\displaystyle O\left(|\delta
h|^{2}\right).$ (38b)

Equation (38b) for the antisymmetric part of the field equations is
identically satisfied, while Equation (38a) shows that a variation
$\delta\overset{\ \circ}{G}_{ab}$ associated with a perturbation is directly
related to a variation of the energy-momentum tensor
$\delta\Theta_{\left(ab\right)}$. This shows that the perturbations of
Minkowski spacetime as defined in Section III.0.1 for $f(T)$ teleparallel
gravity follow the perturbative treatments of Minkowski spacetime in GR.

##  V Effects of Perturbations and the Minkowski Spacetime Symmetries
Conditions for Stability

###  V.1 Rotation/Boost Perturbation in a Minkowski Background

We would like to know if orthonormal coframe perturbations as expressed by
Equation (32) lead to the stability of a pure Minkowski spacetime background.
To achieve this goal, we will first test the stability for the rotation/boost
perturbations as described in Equation (32). Secondly, we will also test the
stability and its impact for a translated form of this Equation (32). We will
finish by studying the effects of the trace, symmetric, and antisymmetric
parts of perturbation, and their respective impacts on torsion and
superpotential perturbations.

In fact, Equation (32) for the orthonormal gauge is exactly the rotation/boost
perturbation in Minkowski spacetime. The perturbation is described as follows:

$\delta h^{a}_{\;\;\mu}=\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\mu}.$ (39)

By substituting Equation (9) inside Equation (38a), the field equation with
the Equation (39) perturbation inside is exactly:

$\displaystyle\kappa_{eff}\,\delta\Theta_{(ab)}$ $\displaystyle\approx$
$\displaystyle\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\Bigg{[}h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}\,\delta{\overset{\
\circ}{R}}_{~{}m\alpha\nu}^{k}-\frac{\eta^{cd}\,\eta_{ef}}{2}\,\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}\,\delta\overset{\
\circ}{R}_{~{}m\alpha\rho}^{k}\Bigg{]}$ $\displaystyle\quad+O\left(|\delta
h|^{2}\right),$ $\displaystyle 0$ $\displaystyle\approx$ $\displaystyle
O\left(|\delta h|^{2}\right).$ (40)

Here, we obtain the perturbed FEs in terms of $\delta\overset{\
\circ}{R}_{~{}m\alpha\rho}^{k}$ and $h^{a}_{\;\;\mu}$. If we have that
$\delta\overset{\ \circ}{R}_{~{}m\alpha\nu}^{k}\rightarrow 0$, then we obtain
$\delta\Theta_{(ab)}\rightarrow 0$ for Equation (39), as is also required by
GR and TEGR. We might also express Equation (V.1) in terms of
$\lambda^{a}_{\;\;b}$, and we have shown that pure Minkowski spacetime is
stable from the zero curvature criteria, as required by the teleparallel
postulates.

From Equation (5), and by substituting Equation (39), the torsion scalar
perturbation $\delta T$ is expressed by Equation (1) in Appendix C. This last
equation can be summarized as:

$\displaystyle\delta T\rightarrow
0\quad\quad\quad\text{for}\;T^{a}_{~{}\mu\nu}=\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\rightarrow
0.$ (41)

From here, we obtain that the condition for $\delta T\rightarrow 0$ is
described by the zero torsion tensor criteria $T^{a}_{~{}~{}\mu\nu}=0$
relation as:

$\displaystyle\partial_{\mu}\left(h^{a}_{\;\;\nu}\right)\approx\partial_{\nu}\left(h^{a}_{\;\;\mu}\right)$
(42)

From Equation (8), and by substituting Equation (39), the superpotential
perturbation $\delta S_{ab}^{~{}~{}~{}\mu}$ is expressed by Equation (2) in
Appendix C. This equation can be summarized as:

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}\rightarrow
0\quad\quad\quad\text{for}\;\delta
T^{a}_{~{}\mu\nu}=\partial_{\mu}\,\left(\lambda^{a}_{~{}c}\,h^{c}_{\;\;\nu}\right)-\partial_{\nu}\,\left(\lambda^{a}_{~{}c}\,h^{c}_{\;\;\mu}\right)\rightarrow
0.$ (43)

From this result, we obtain that the condition for $\delta
S_{ab}^{~{}~{}~{}\mu}\rightarrow 0$ is also described by the zero perturbed
torsion tensor criteria $\delta T^{a}_{~{}~{}\mu\nu}=0$ relation as:

$\displaystyle\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)\approx\partial_{\nu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\mu}\right).$
(44)

Equation (44) (the zero perturbed torsion criteria) is complementary to
Equation (42) (zero torsion criteria) for obtaining the limit $\delta
S_{ab}^{~{}~{}~{}\mu}\rightarrow 0$. We apply Equation (42) before applying
Equation (44). From here, the Equations (42) and (44) are the two fundamental
symmetry conditions for Minkowski spacetime stability.

If we set $\delta T\rightarrow 0$ and $\delta S_{ab}^{~{}~{}~{}\mu}\rightarrow
0$ for Equations (36a) and (36b) for all zero torsion spacetimes, we still
respect Equations (42) and (44), as for pure Minkowski spacetimes. Hence, the
zero torsion tensor and zero perturbed torsion tensor criterions are still
valid for all zero torsion spacetimes, Minkowski or not.

Even for the constant torsion spacetimes, by always setting $\delta
T\rightarrow 0$ and $\delta S_{ab}^{~{}~{}~{}\mu}\rightarrow 0$ inside
Equations (35a) and (35b), we respect again Equations (42) and (44), as for
the zero torsion scalar spacetimes. This is another generalization of the
Minkowski spacetime result to a most general class of spacetimes as the
constant torsion ones.

There are some other consequences for Minkowski spacetime on a proper frame.
By applying the null covariant derivative criteria to Equation (39), we use
Equation (3) in the Appendix C result to obtain as a relation:

$\displaystyle\delta\Gamma^{\rho}_{\;\;\nu\mu}=h_{a}^{\;\;\rho}\left[\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)-\left(h_{c}^{\;\;\sigma}\,\partial_{\mu}\,h^{c}_{\;\;\nu}\right)\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\sigma}\right)\right],$
(45)

where
$\Gamma^{\rho}_{\;\;\nu\mu}=h_{c}^{\;\;\rho}\,\partial_{\mu}\,h^{c}_{\;\;\nu}$
is the Weitzenbock connection for a proper frame. For trivial coframes as
$h^{a}_{\;\;\mu}=\delta^{a}_{\;\;\mu}=Diag[1,1,1,1]$, Equation (45) becomes:

$\displaystyle\delta\Gamma^{\rho}_{\;\;\nu\mu}=h_{a}^{\;\;\rho}\left[\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\mu}\right)\right]=\delta_{a}^{\;\;\rho}\,\partial_{\mu}\left(\lambda^{a}_{\;\;b}\right)\,\delta^{b}_{\;\;\mu}.$
(46)

In the next subsection, we will study the effect of a translation applied to
the perturbation described by Equation (39) on Equations (45) and (46). The
goal is to know the effects of the perturbations on the Weitzenbock connection
and its perturbation.

We can now see by the Equations (V.1)–(46) the effect of the perturbation
described by Equation (39), maintaining the proper frame and respecting the
$GL(4,\mathbb{R})$ invariance transformation. In addition, Equations (42) and
(44) give the Minkowski spacetime stability conditions on proper frames for
the perturbation described by Equation (39) [39, 40, 41, 42].

###  V.2 General Linear Perturbation in a Minkowski Background

A more general perturbation scheme requires one to deal with the following
general linear perturbation:

$\delta
h^{a}_{\;\;\mu}=\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\mu}+\epsilon^{a}_{\;\;\mu},$
(47)

where $|\lambda^{a}_{\;\;b}|,|\epsilon^{a}_{\;\;\mu}|\ll 1$. We have here the
transformation described by Equation (39), superposed with a translation in
Minkowski tangent space.

For the Equation (47) perturbation, Equation (V.1) becomes as follows:

$\displaystyle\kappa_{eff}\,\delta\Theta_{(ab)}$ $\displaystyle\approx$
$\displaystyle\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\Bigg{[}h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}\,\delta{\overset{\
\circ}{R}}^{k}_{~{}m\alpha\nu}-\frac{\eta^{cd}\,\eta_{ef}}{2}\,\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}\,\delta{\overset{\
\circ}{R}}^{k}_{~{}m\alpha\rho}\Bigg{]}$ $\displaystyle\quad+O\left(|\delta
h|^{2}\right),$ $\displaystyle 0$ $\displaystyle\approx$ $\displaystyle
O\left(|\delta h|^{2}\right).$ (48)

Here, again we obtain the perturbed FEs in terms of $\delta{\overset{\
\circ}{R}}^{k}_{~{}m\alpha\rho}$ and $h^{a}_{\;\;\mu}$. As for Equation (V.1),
$\delta{\overset{\ \circ}{R}}^{k}_{~{}m\alpha\nu}\rightarrow 0$, we still then
obtain $\delta\Theta_{(ab)}\rightarrow 0$ for Equation (47), as is also
required by GR and TEGR [39, 40, 41, 42]. We might express Equation (V.2) in
terms of $\lambda^{a}_{\;\;b}$ and $\epsilon^{a}_{\;\;\mu}$. Here again, we
have shown that pure Minkowski spacetime is still stable from the zero
curvature criteria, as required by teleparallel postulates.

From Equation (5), and by substituting Equation (47), the torsion scalar
perturbation $\delta T$ is expressed by Equation (1) in Appendix C and can be
summarized as:

$\displaystyle\delta T\rightarrow
0\quad\quad\quad\,\text{for}\;T^{a}_{~{}\mu\nu}=\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\rightarrow
0.$ (49)

The condition for $\delta T\rightarrow 0$ is still described by Equation (42)
for the zero torsion tensor criteria $T^{a}_{~{}~{}\mu\nu}=0$.

From Equation (8), and by substituting Equation (47), the superpotential
perturbation $\delta S_{ab}^{~{}~{}~{}\mu}$ is expressed by Equation (2) in
Appendix C and can also be summarized as:

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}\rightarrow 0.$ (50)

Equation (50) is satisfied if we respect
$\partial_{a}\epsilon_{b}^{\;\;\mu}=\partial_{b}\epsilon_{a}^{\;\;\mu}=0$ (a
constant translation condition for Equation (47)) and after applying the
Equation (42) criteria. The condition for $\delta
S_{ab}^{~{}~{}~{}\mu}\rightarrow 0$ is still described by Equation (44) for
the zero perturbed torsion tensor criteria $\delta T^{a}_{~{}~{}\mu\nu}=0$,
only if the constant translation criteria are respected as:

$\displaystyle\partial_{\mu}\epsilon^{a}_{\;\;\nu}=\partial_{\nu}\epsilon^{a}_{\;\;\mu}=0.$
(51)

Hence, for the Equation (47) perturbation, we still respect Equations (42) and
(44) as the two first symmetry conditions for Minkowski spacetime stability,
but we must also respect Equation (51) before Equation (44). A simple
translation does not affect these Equations (42) and (44) only if we respect
Equation (51), and the translation term $\epsilon^{a}_{\;\;\nu}$ must be
constant inside Equation (47). This constant translation criteria as expressed
by Equation (51) is a third symmetry condition for Minkowski spacetime
stability.

As for Equations (45) and (46), we apply the null covariant derivative
criteria to Equation (47) and we obtain as a relation:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}+\epsilon^{a}_{\;\;\nu}\right)-\left(h_{c}^{\;\;\rho}\,\partial_{\mu}\,h^{c}_{\;\;\nu}\right)\,\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\rho}+\epsilon^{a}_{\;\;\rho}\right)-\delta\Gamma^{\rho}_{\;\;\nu\mu}h^{a}_{\;\;\rho}$
(52)
$\displaystyle\Rightarrow\delta\Gamma^{\rho}_{\;\;\nu\mu}=h_{a}^{\;\;\rho}\left[\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)-\left(h_{c}^{\;\;\sigma}\,\partial_{\mu}\,h^{c}_{\;\;\nu}\right)\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\sigma}+\epsilon^{a}_{\;\;\sigma}\right)\right].$

where
$\Gamma^{\sigma}_{\;\;\nu\mu}=h_{c}^{\;\;\sigma}\,\partial_{\mu}\,h^{c}_{\;\;\nu}$
is the Weitzenbock connection for a proper frame and
$\partial_{\mu}\,\epsilon^{a}_{\;\;\nu}=0$ because of constant translation.
Equation (52) is slightly different from Equation (45) according to the term
$-\left(h_{c}^{\;\;\sigma}\,\partial_{\mu}\,h^{c}_{\;\;\nu}\right)\,\epsilon^{a}_{\;\;\sigma}$.
For non-trivial coframes (i.e., $\partial_{\mu}\,h^{c}_{\;\;\nu}\neq 0$),
Equation (52) is not invariant under Equation (51). For trivial coframes
(i.e., $\partial_{\mu}\,h^{c}_{\;\;\nu}=0$), Equation (52) becomes exactly
Equation (46), as for the perturbation described by Equation (39). From this
result, we now respect the constant coframe criteria as (or null Weitzenbock
connection $\Gamma^{\rho}_{\;\;\nu\mu}=0$ criteria):

$\displaystyle\partial_{\mu}\,h^{c}_{\;\;\nu}=0.$ (53)

With Equation (53), we also satisfy the invariance under Equation (51), the
constant translation criteria for the Weitzenbock connection perturbation.
Hence, Equations (45) and (52) show that the Weitzenbock connection
perturbation $\delta\Gamma^{\rho}_{\;\;\nu\mu}$ is invariant only if we
respect Equation (53), the constant coframe criteria. This criteria, as
expressed by Equation (53), is a fourth symmetry condition for Minkowski
spacetime stability.

Now, Equations (V.2)–(53) generalize Equations (V.1)–(46) by applying a
constant translation $\epsilon^{a}_{\;\;\nu}$ to the linear transformation
described by Equation (39), which maintains the proper frame and the
invariance under the $GL(4,\mathbb{R})$ transformation. By respecting Equation
(51), the constant translation criteria, we still respect Equations (42) and
(44) for Equation (47), and this generalization shows that Minkowski spacetime
and all zero torsion spacetimes are stable everytime [39, 40, 41, 42].
However, Equations (45) and (52), both giving Equation (46), show that the
Weitzenbock connection perturbation $\delta\Gamma^{\rho}_{\;\;\nu\mu}$ is
invariant only if we work with constant or trivial coframes respecting
Equation (53).

###  V.3 Perturbations on Trivial Coframes by Each Part of the Perturbation

Before properly dealing with more complex cases of coframes, it is imperative
to deal with perturbations on the trivial coframe. This coframe is defined as
follows:

$h^{a}_{\;\;\mu}=\delta^{a}_{\;\;\mu}=Diag\left[1,\,1,\,1,\,1\right].$ (54)

The coframe described by Equation (54) is defined in the orthonormal gauge.
This equation (54) respects Equation (53), the fourth symmetry condition for
Minkowski spacetime stability. From there, we will study the following general
perturbations which will be applied to Equation (54) in terms of
$\lambda^{a}_{~{}b}$ and respecting Equations (42) and (44), and if necessary,
Equation (51). In addition, we will compare with another recent similar study
on so-called “cosmological” perturbations in order to better situate the
results for Minkowski spacetime for a scale factor of $1$ [24]. Their
$\lambda^{a}_{~{}b}$ equivalent matrix is expressed as:

$\displaystyle\left(\lambda^{a}_{~{}b}\right)_{Golov}=\left[\begin{array}[]{cc}\phi&\partial_{a}\,\xi+v_{a}\\\
\partial_{i}\,\beta+u_{i}&\left[-\psi\,\delta^{a}_{j}+\partial^{2}_{a\,j}\sigma+\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right]\end{array}\right],$
(57)

where we must respect the constraints $\partial^{a}\,v_{a}=0$,
$\partial^{k}\,w_{k}=0$, $\partial^{i}\,u_{i}=0$, and $\partial^{a}\,c_{a}=0$,
and the tensorial part is also traceless.

####  V.3.1 Trace

We have first as $\lambda^{a}_{~{}b}$ for a full trace perturbation:

$\left(\lambda^{a}_{~{}b}\right)_{Trace}=\lambda=Trace\left[Diag\left[a_{00},\,a_{11},\,a_{22},\,a_{33}\right]\right]=a_{00}+a_{11}+a_{22}+a_{33}.$
(58)

Equation (39) will be exactly $\left(\delta
h^{a}_{\;\;\mu}\right)_{Trace}=\frac{\lambda}{4}\delta^{a}_{\;\;\mu}$, and by
setting $h^{a}_{\;\;\mu}=\delta^{a}_{\;\;\mu}$, Equations (41) and (43) are:

$\displaystyle\delta T\approx O\left(|\delta h|^{2}\right)\rightarrow 0,$ (59)

which respects Equation (42) and

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}$
$\displaystyle\rightarrow\Bigg{[}\frac{1}{8}\left(\partial_{a}\left(\lambda\,\delta_{b}^{~{}\mu}\right)-\partial_{b}\left(\lambda\,\delta_{a}^{~{}\mu}\right)\right)+\frac{1}{4}\left(\partial_{b}\left(\lambda\,\delta_{~{}a}^{\mu}\right)-\partial_{a}\left(\lambda\,\delta_{~{}b}^{c}\,h_{~{}c}^{\mu}\right)\right)$
$\displaystyle\quad\quad-\frac{1}{4}\delta_{~{}b}^{\mu}\,\delta_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda\,\delta_{a}^{~{}\rho}\right)-\partial_{a}\,\left(\lambda\,\delta_{c}^{~{}\rho}\right)\right]+\frac{1}{4}\,\delta_{~{}a}^{\mu}\,\delta_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda\,\delta_{b}^{~{}\rho}\right)-\partial_{b}\,\left(\lambda\,\delta_{c}^{~{}\rho}\right)\right]\Bigg{]}$
$\displaystyle\quad\quad+O\left(|\delta h|^{2}\right)\quad\quad\text{by
applying Equation \eqref{436a} (the zero torsion criteria).}$
$\displaystyle\rightarrow
0\quad\quad\quad\quad\quad\quad\quad\,\text{for}\;\delta
T^{a}_{~{}\mu\nu}=\partial_{\mu}\,\left(\lambda\right)\,\delta^{a}_{\;\;\nu}-\partial_{\nu}\,\left(\lambda\right)\,\delta^{a}_{\;\;\mu}\rightarrow
0.$ (60)

Equation (44) will be expressed as:

$\displaystyle\partial_{\mu}\,\left(\lambda\right)\,\delta^{a}_{\;\;\nu}\approx\partial_{\nu}\,\left(\lambda\right)\,\delta^{a}_{\;\;\mu}.$
(61)

By comparing with Equation (57), we obtain the following equations for the
rectangular coordinates [24]:

  * •

Equation (58) becomes:

$\displaystyle\left(\lambda^{a}_{~{}b}\right)_{Trace\,Golov}=\lambda_{Golov}=\phi-\psi+\partial^{2}\,\sigma+\frac{h}{2},$
(62)

where $\epsilon_{ajk}=0$ because $a=j$ and $h=Trace(h_{aj})$.

  * •

From Equation (58), we obtain as the supplementary constraints:

$\displaystyle\partial_{a}\,\xi+v_{a}=0\quad\text{and}\quad\partial_{i}\,\beta+u_{i}=0$
(63)

  * •

Equation (61) will be expressed in terms of Equations (62) and (63):

$\displaystyle\partial_{\mu}\,\left(\lambda_{Golov}\right)\,\delta^{a}_{\;\;\nu}$
$\displaystyle\approx$
$\displaystyle\partial_{\nu}\,\left(\lambda_{Golov}\right)\,\delta^{a}_{\;\;\mu}$
$\displaystyle\partial_{\mu}\,\left(\phi-\psi+\partial^{2}\,\sigma+\frac{h}{2}\right)\,\delta^{a}_{\;\;\nu}$
$\displaystyle\approx$
$\displaystyle\partial_{\nu}\,\left(\phi-\psi+\partial^{2}\,\sigma+\frac{h}{2}\right)\,\delta^{a}_{\;\;\mu}.$
(64)

####  V.3.2 Full Symmetric Perturbation

For the perfect symmetric perturbation, we have as the $\lambda^{a}_{~{}b}$
perturbation with null diagonal components:

$\left(\lambda^{a}_{~{}b}\right)_{Sym}=\tilde{\lambda}^{a}_{~{}b}=\left[\begin{array}[]{cccc}0&b_{10}&b_{20}&b_{30}\\\
b_{10}&0&b_{12}&b_{13}\\\ b_{20}&b_{12}&0&b_{23}\\\
b_{30}&b_{13}&b_{23}&0\end{array}\right].$ (65)

Equation (39) will be exactly $\left(\delta
h^{a}_{\;\;\mu}\right)_{Sym}=\tilde{\lambda}^{a}_{\;\;b}\,\delta^{b}_{\;\;\mu}$,
and by setting $h^{a}_{\;\;\mu}=\delta^{a}_{\;\;\mu}$, Equation (41) is still
expressed by Equation (59), respecting the Equations (42) and (43):

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}=$
$\displaystyle\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\tilde{\lambda}_{b}^{~{}c}\,\delta_{c}^{~{}\mu}\right)-\partial_{b}\left(\tilde{\lambda}_{a}^{~{}c}\,\delta_{c}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\tilde{\lambda}_{~{}a}^{c}\,\delta_{~{}c}^{\mu}\right)-\partial_{a}\left(\tilde{\lambda}_{~{}b}^{c}\,\delta_{~{}c}^{\mu}\right)\right)$
$\displaystyle\quad\quad-\delta_{~{}b}^{\mu}\,\delta_{~{}\rho}^{c}\left[\partial_{c}\,\left(\tilde{\lambda}_{a}^{~{}f}\delta_{f}^{~{}\rho}\right)-\partial_{a}\,\left(\tilde{\lambda}_{c}^{~{}f}\delta_{f}^{~{}\rho}\right)\right]+\delta_{~{}a}^{\mu}\,\delta_{~{}\rho}^{c}\left[\partial_{c}\,\left(\tilde{\lambda}_{b}^{~{}f}\delta_{f}^{~{}\rho}\right)-\partial_{b}\,\left(\tilde{\lambda}_{c}^{~{}f}\delta_{f}^{~{}\rho}\right)\right]\Bigg{]}$
$\displaystyle\quad\quad+O\left(|\delta h|^{2}\right)\quad\quad\text{by
applying Equation \eqref{436a} (the zero torsion criteria).}$
$\displaystyle\rightarrow
0\quad\quad\quad\quad\quad\quad\quad\,\text{for}\;\delta
T^{a}_{~{}\mu\nu}=\partial_{\mu}\,\left(\lambda^{a}_{~{}c}\right)\delta^{c}_{\;\;\nu}-\partial_{\nu}\,\left(\lambda^{a}_{~{}c}\right),\delta^{c}_{\;\;\mu}\rightarrow
0.$ (66)

Equation (44) will be expressed as:

$\displaystyle\partial_{\mu}\,\left(\tilde{\lambda}^{a}_{~{}c}\right)\delta^{c}_{\;\;\nu}\approx\partial_{\nu}\,\left(\tilde{\lambda}^{a}_{~{}c}\right),\delta^{c}_{\;\;\mu}.$
(67)

By comparing with Equation (57) again, we obtain the following equations for
the rectangular coordinates [24]:

  * •

Equation (65) becomes:

$\displaystyle\left(\lambda^{a}_{~{}b}\right)_{Sym\,Golov}$ $\displaystyle=$
$\displaystyle\left(\tilde{\lambda}^{a}_{~{}b}\right)_{Golov}$ (70)
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}0&\partial_{a}\,\xi+v_{a}\\\
\partial_{a}\,\xi+v_{a}&\left[\partial^{2}_{a\,j}\sigma+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right]\end{array}\right],$

where $a\neq j\neq k$, $\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)=0$
and $\partial_{a}\,\xi+v_{a}=\partial_{i}\,\beta+u_{i}$ because we have a
symmetric perturbation. As a supplement, we deduce that $\phi=0$ and $\psi=0$
for Equation (65), because of the null diagonal components.

  * •

The Equation (67) components will be expressed in terms of Equation (70):

$\displaystyle\partial_{\mu}\,\left(\partial_{a}\,\xi+v_{a}\right)\,\delta^{a}_{\;\;\nu}$
$\displaystyle\approx$
$\displaystyle\partial_{\nu}\,\left(\partial_{a}\,\xi+v_{a}\right)\,\delta^{a}_{\;\;\mu}$
$\displaystyle\partial_{\mu}\,\left(\partial^{2}_{a\,j}\sigma+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right)\,\delta^{a}_{\;\;\nu}$
$\displaystyle\approx$
$\displaystyle\partial_{\nu}\,\left(\partial^{2}_{a\,j}\sigma+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right)\,\delta^{a}_{\;\;\mu}$
(71)

####  V.3.3 Full Antisymmetric Perturbation

For the full antisymmetric perturbation, we have as the $\lambda^{a}_{~{}b}$
perturbation with null diagonal components:

$\left(\lambda^{a}_{~{}b}\right)_{AntiSym}=\bar{\lambda}^{a}_{~{}b}=\left[\begin{array}[]{cccc}0&b_{10}&b_{20}&b_{30}\\\
-b_{10}&0&b_{12}&b_{13}\\\ -b_{20}&-b_{12}&0&b_{23}\\\
-b_{30}&-b_{13}&-b_{23}&0\end{array}\right].$ (72)

Equation (39) will be exactly $\left(\delta
h^{a}_{\;\;\mu}\right)_{AntiSym}=\bar{\lambda}^{a}_{\;\;b}\,\delta^{b}_{\;\;\mu}$,
and by setting $h^{a}_{\;\;\mu}=\delta^{a}_{\;\;\mu}$, Equation (41) is still
expressed by Equation (59), respecting Equations (42) and (43):

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}=$
$\displaystyle\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\bar{\lambda}_{b}^{~{}c}\,\delta_{c}^{~{}\mu}\right)-\partial_{b}\left(\bar{\lambda}_{a}^{~{}c}\,\delta_{c}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\bar{\lambda}_{~{}a}^{c}\,\delta_{~{}c}^{\mu}\right)-\partial_{a}\left(\bar{\lambda}_{~{}b}^{c}\,\delta_{~{}c}^{\mu}\right)\right)$
$\displaystyle\quad\quad-\delta_{~{}b}^{\mu}\,\delta_{~{}\rho}^{c}\left[\partial_{c}\,\left(\bar{\lambda}_{a}^{~{}f}\delta_{f}^{~{}\rho}\right)-\partial_{a}\,\left(\bar{\lambda}_{c}^{~{}f}\delta_{f}^{~{}\rho}\right)\right]+\delta_{~{}a}^{\mu}\,\delta_{~{}\rho}^{c}\left[\partial_{c}\,\left(\bar{\lambda}_{b}^{~{}f}\delta_{f}^{~{}\rho}\right)-\partial_{b}\,\left(\bar{\lambda}_{c}^{~{}f}\delta_{f}^{~{}\rho}\right)\right]\Bigg{]}$
$\displaystyle\quad\quad+O\left(|\delta h|^{2}\right)\quad\quad\text{by
applying Equation \eqref{436a} (the zero torsion criteria).}$
$\displaystyle\rightarrow
0\quad\quad\quad\quad\quad\quad\quad\,\text{for}\;\delta
T^{a}_{~{}\mu\nu}=\partial_{\mu}\,\left(\bar{\lambda}^{a}_{~{}c}\right)\,\delta^{c}_{\;\;\nu}-\partial_{\nu}\,\left(\bar{\lambda}^{a}_{~{}c}\right)\,\delta^{c}_{\;\;\mu}\rightarrow
0.$ (73)

Equation (44) will be expressed as:

$\displaystyle\partial_{\mu}\,\left(\bar{\lambda}^{a}_{~{}c}\right)\,\delta^{c}_{\;\;\nu}\approx\partial_{\nu}\,\left(\bar{\lambda}^{a}_{~{}c}\right)\,\delta^{c}_{\;\;\mu}.$
(74)

By still comparing with Equation (57), we obtain the following equations for
the rectangular coordinates [24]:

  * •

Equation (72) becomes:

$\displaystyle\left(\lambda^{a}_{~{}b}\right)_{AntiSym\,Golov}$
$\displaystyle=$ $\displaystyle\left(\bar{\lambda}^{a}_{~{}b}\right)_{Golov}$
(77) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}0&\partial_{a}\,\xi+v_{a}\\\
-\left(\partial_{a}\,\xi+v_{a}\right)&\left[\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right]\end{array}\right]$

where $a\neq j\neq k$,
$\partial^{2}_{a\,j}\sigma=-\partial^{2}_{j\,a}\sigma=0$,
$\partial_{j}\,c_{a}=-\partial_{a}\,c_{j}$, $h_{aj}=-h_{ja}$, and
$\partial_{a}\,\xi+v_{a}=-\left(\partial_{i}\,\beta+u_{i}\right)$ because we
have an antisymmetric perturbation. We deduce again that $\phi=0$ and $\psi=0$
for Equation (72), because of the null diagonal components.

  * •

The Equation (74) components will be expressed in terms of Equation (77):

$\displaystyle\partial_{\mu}\,\left(\partial_{a}\,\xi+v_{a}\right)\,\delta^{a}_{\;\;\nu}$
$\displaystyle\approx$
$\displaystyle\partial_{\nu}\,\left(\partial_{a}\,\xi+v_{a}\right)\,\delta^{a}_{\;\;\mu}$
$\displaystyle\partial_{\mu}\,\left(\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right)\,\delta^{a}_{\;\;\nu}$
$\displaystyle\approx$
$\displaystyle\partial_{\nu}\,\left(\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right)\,\delta^{a}_{\;\;\mu}$

####  V.3.4 A Mixed Situation and Minkowski Spacetime

Here, we will treat the most general case. It is the combination of the three
previous sorts, as:

$\displaystyle\left(\lambda^{a}_{~{}b}\right)_{Mixed}=\lambda^{a}_{~{}b}$
$\displaystyle=$
$\displaystyle\frac{\delta^{a}_{~{}b}}{4}\,\left(\lambda^{a}_{~{}b}\right)_{Trace}+\left(\lambda^{a}_{~{}b}\right)_{Sym}+\left(\lambda^{a}_{~{}b}\right)_{AntiSym},$
(79) $\displaystyle=$
$\displaystyle\frac{\lambda}{4}\delta^{a}_{~{}b}+\tilde{\lambda}^{a}_{~{}b}+\bar{\lambda}^{a}_{~{}b}.$

In general, we always obtain for Equation (79) that
$\left(\lambda^{a}_{~{}b}\right)_{Mixed}$ is exactly Equation (57) when we
compare it to the linear parametrization of ref [24]. Then, we obtain as the
components of Equation (44) the most general relations for perturbation in the
Minkowski background as:

$\displaystyle\partial_{\mu}\,\phi\delta^{c}_{\;\;\nu}$
$\displaystyle\approx\;\partial_{\nu}\,\phi,\delta^{c}_{\;\;\mu}$ (80a)
$\displaystyle\partial_{\mu}\,\left(\partial_{a}\,\xi+v_{a}\right)\delta^{c}_{\;\;\nu}$
$\displaystyle\approx\;\partial_{\nu}\,\left(\partial_{a}\,\xi+v_{a}\right),\delta^{c}_{\;\;\mu}$
(80b)
$\displaystyle\partial_{\mu}\,\left(\partial_{i}\,\beta+u_{i}\right)\delta^{c}_{\;\;\nu}$
$\displaystyle\approx\;\partial_{\nu}\,\left(\partial_{i}\,\beta+u_{i}\right),\delta^{c}_{\;\;\mu}$
(80c)
$\displaystyle\partial_{\mu}\,\left[-\psi\,\delta^{a}_{j}+\partial^{2}_{a\,j}\sigma+\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right]\delta^{c}_{\;\;\nu}$
$\displaystyle\approx\;\partial_{\nu}\,\left[-\psi\,\delta^{a}_{j}+\partial^{2}_{a\,j}\sigma+\epsilon_{ajk}\,\left(\partial_{k}\,s+w_{k}\right)+\partial_{j}\,c_{a}+\frac{h_{aj}}{2}\right]\delta^{c}_{\;\;\mu}.$
(80d)

Equation (39) will be exactly $\left(\delta
h^{a}_{\;\;\mu}\right)_{Mixed}=\lambda^{a}_{\;\;b}\,\delta^{b}_{\;\;\mu}$, and
we exactly obtain Equations (41) and (43) by respecting Equations (42) and
(44) via superposition. In Equation (79), the first two terms (Trace and
Symmetric terms) represent the symmetric part of
$\left(\lambda^{a}_{~{}b}\right)_{Mixed}$, and the last term (Antisymmetric
term) represents the Antisymmetric part of
$\left(\lambda^{a}_{~{}b}\right)_{Mixed}$. For every case, we satisfy the
Equations (42), (44), (51), and (53) in the supplement of the energy-momentum
stability from $\delta\overset{\ \circ}{R}^{k}_{~{}m\alpha\nu}\rightarrow 0$,
leading to $\delta\Theta_{(ab)}\rightarrow 0$ [39, 40, 41, 42].

For the trivial coframe cases expressed by Equation (54), we verify the
energy-momentum stability by Equation (V.1); and the four other symmetries
conditions stated by Equations (42), (44), (51), and (53) are all satisfied.
The Minkowski spacetime is stable with these four symmetry conditions. From
these considerations for pure Minkowski spacetime, we have shown that
$\delta\Theta_{(ab)}\rightarrow 0$ by Equations (V.1) and (V.2) when all of
the perturbed quantities proceed to zero. From this, we must absolutely have
$\Theta_{(ab)}=0$ when we are in a pure vacuum: the full absence of a
gravitational source.

##  VI Discussion and Conclusions

The purpose of this paper is to clarify the meaning of Minkowski and constant
scalar torsion geometries within a teleparallel gravity framework. A
perturbation scheme is developed, which is general and applicable to all
possible teleparallel spacetimes that respect the Null Curvature and Null Non-
Metricity postulates. The perturbation scheme is then applied to different
constant torsion scalar scenarios, with a particular emphasis on perturbations
of the teleparallel Minkowksi spacetimes.

We obtained in Section IV the perturbed field equations (perturbed FEs) in
terms of the perturbed torsion scalar $\delta T$ and perturbed superpotential
$\delta S_{ab}^{~{}~{}~{}\mu}$. These two quantities are themselves dependent
on the coframe perturbation $\delta h^{a}_{\;\;\mu}$. The perturbed field
equations make it possible to relate these perturbed quantities to the
perturbation of the energy-momentum $\delta\Theta_{\left(ab\right)}$. This is
analogous to the field equations for the non-perturbed quantities and how they
relate to the physical quantities in the Energy-Momentum
$\Theta_{\left(ab\right)}$.

In Section V, we look at the field Equations (V.1) and (V.2) when the
curvature perturbation criteria $\delta{\overset{\
\circ}{R}}^{k}_{~{}m\alpha\nu}$ proceeds to zero, and we observe that the
energy-momentum perturbation $\delta\Theta_{\left(ab\right)}$ also goes to
zero, as in GR. In GR, it is known that a curvature perturbation leads to an
energy-momentum perturbation. We show that the same thing occurs for the
teleparallel Minkowski spacetime with Equations (V.1) and (V.2).

Then, we obtain via the null torsion tensor and the null perturbed torsion
tensor criteria as defined by Equations (42) and (44) that the torsion scalar
perturbation $\delta T$ and superpotential perturbation $\delta
S_{ab}^{~{}~{}~{}\mu}$ go to zero for pure Minkowski spacetime when we use the
Equation (39) perturbation (boost/rotation perturbation). These Equations (42)
and (44) are the two first fundamental Minkowski spacetime stability
conditions on proper frames. However, if we use the more general linear
perturbation as defined by Equation (47), we need to respect the constant
translation criteria as defined by Equation (51) in order for the
superpotential perturbation $\delta S_{ab}^{~{}~{}~{}\mu}$ to proceed to zero.
This is a third Minkowski spacetime stability condition for the proper frames
to respect for the Equation (47) perturbation. In this way, by respecting
Equation (51), we then respect the Equations (42) and (44), as for the
Equation (39) perturbation.

Another consequence from the Equation (47) perturbation is about the
Weitzenbock connection perturbation $\delta\Gamma^{\rho}_{\;\;\nu\mu}$.
Equations (45) and (52) have shown that we need to respect the constant
coframe criteria as defined by Equation (53). Equation (53) is a fourth
Minkowski spacetime stability condition for proper frames to respect for the
Equation (47) perturbation, allowing for the invariance for the Weitzenbock
connection perturbation.

To generalize, these steps applied for the Minkowski spacetime, given these
stability criteria, can also be applied for null torsion scalar spacetimes, as
well as the constant torsion scalar spacetimes. Indeed, with the analysis made
in Sections V.1 and V.2, and the stability criteria obtained for the Minkowski
spacetime, Equations (36a) and (36b) for the null torsion scalar spacetimes
make it possible to generalize these treatments, and in the end to obtain the
same stability criteria, which are Equations (42) and (44), and if necessary,
Equations (51) and (53). This is also the case for the constant torsion scalar
spacetimes described by the Equations (35a) and (35b) if we take the limits
$\delta T\rightarrow 0$ and $\delta S_{ab}^{~{}~{}~{}\mu}\rightarrow 0$, as
for the Minkowski and null torsion scalar spacetimes.

One can expand upon the work here on perturbations in covariant teleparallel
gravity to more general teleparallel spacetimes and to broader classes of
teleparallel gravity theories. For example, in the case of static spherically
symmetric teleparallel spacetimes [43, 44] in which the torsion scalar is not
constant, what is the stability of the static spherically symmetric solution?
Further, this perturbation scheme can also be applied to cosmological
geometries in $f(T)$ teleparallel gravity [21], thereby enhancing the previous
work of [24]. Additionally, one can also look at perturbations in other
non-$f(T)$ teleparallel gravity theories.

The current analysis could also bring some light to a couple of unresolved
challenges in teleparallel gravity. The first challenge concerns the number of
degrees of freedom (DOF) in 4-dimensional $f(T)$ teleparallel gravity [45, 46,
47, 48]. In [45], the authors employ a Hamiltonian analysis to determine that
$f(T)$ teleparallel gravity has three extra DOF when compared to GR.
Unfortunately, it appears that the analysis is flawed, in that it is not
general, for they assumed a diagonal metric to reach some of their
conclusions. Later, Ferraro and Guzman [46] made an argument that the number
of extra DOF is 1. However, the analysis appears to be somewhat incomplete and
only applicable to teleparallel gravity geometries in which the torsion scalar
is constant [48]. More recently, the authors of [47] go through a Hamiltonian
analysis to conclude that the number of extra DOF is 3. A couple of challenges
in their results have been identified in [48]. Obviously, this is still an
unresolved problem which requires further investigation. Another unresolved
complex physical problem is the strong coupling of teleparallel perturbations.
This physical problem occurs as one approaches the Planck scale where the
quantum field effects become non-negligible, particularly for second-order
perturbations and higher. At these scales, the kinetic energy part will become
dominant when compared to the gravity and background parts. This strong
coupling issue with teleparallel perturbations needs further development and
understanding within the covariant $f(T)$ teleparallel-gravity framework.

Here, with the material developed in this present paper, we have a more
complete perturbation framework that is suitable for use in teleparallel
gravity, and the toolkit needed for studying several and more complex problems
in teleparallel gravity.

## Acknowledgments

R.J.v.d.H. is supported by the Natural Sciences and Engineering Research
Council of Canada, and by the W.F. James Chair of Studies in the Pure and
Applied Sciences at St.F.X. A.L. is supported by an AARMS fellowship.

## Abbreviations

The following abbreviations are used in this paper:

FE | Field Equation
---|---
GR | General Relativity
TEGR | Teleparallel Equivalent of General Relativity
DOF | Degrees of Freedom

##  Appendix A Perturbed Physical Quantities in Teleparallel Theories

To complete the analysis of Teleparallel theories and geometries, we want to
perturb various physical quantities that may be involved. As explained in
Section IV.1, we are able to always consider perturbations of the co-frame
only within a proper orthonormal gauge.

$\displaystyle\hat{h}^{\prime a}_{~{}\mu}$ $\displaystyle=$ $\displaystyle
h^{a}_{~{}\mu}+\delta h^{a}_{~{}\mu},$ (81a)
$\displaystyle\hat{\omega}^{\prime a}_{~{}\,b\mu}$ $\displaystyle=$
$\displaystyle 0,$ (81b) $\displaystyle\hat{g}^{\prime}_{ab}$ $\displaystyle=$
$\displaystyle\eta_{ab},$ (81c)

where $\delta h^{a}=\delta h^{a}_{~{}\mu}\,dx^{\mu}=\lambda^{a}_{~{}b}h^{b}$.
Here, we apply the coframe perturbations to the main physical and geometrical
quantities involved in Teleparallel Gravity.

  1. 1.

The inverse coframe perturbation $\delta h_{a}^{~{}\mu}$:

$\displaystyle h_{a}^{~{}\mu}+\delta h_{a}^{~{}\mu}$ $\displaystyle=$
$\displaystyle
h_{a}^{~{}\mu}+\left[\lambda_{~{}a}^{b}\right]^{-1}\,h_{a}^{~{}\mu},$
$\displaystyle=$ $\displaystyle
h_{a}^{~{}\mu}+\lambda_{a}^{~{}b}\,h_{a}^{~{}\mu},$
$\displaystyle\Rightarrow\delta h_{a}^{~{}\mu}$ $\displaystyle=$
$\displaystyle\lambda_{a}^{~{}b}\,h_{a}^{~{}\mu}$ (82)

  2. 2.

Determinant of the co-frame $h=\text{Det}(h^{a}_{~{}\mu})$:

$\displaystyle h+\delta h$ $\displaystyle=$
$\displaystyle\text{Det}(h^{a}_{~{}\mu}+\delta h^{a}_{~{}\mu})$
$\displaystyle\approx$ $\displaystyle
h+\text{Det}(\lambda^{a}_{~{}b}\,h^{b}_{~{}\mu})=h+\lambda\,h$
$\displaystyle\Rightarrow\delta h$ $\displaystyle\approx$
$\displaystyle\lambda\,h$ (83)

where $\lambda=\text{Det}(\lambda^{a}_{~{}b})\ll 1$ and $\text{Det}(\delta
h^{a}_{~{}\mu})=\text{Det}(\lambda^{a}_{~{}b}\,h^{b}_{~{}\mu})=\lambda\,h$.

  3. 3.

Metric tensor $g_{\mu\nu}$:

$\displaystyle g_{\mu\nu}+\delta g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta_{ab}\left[h^{a}_{\;\;\mu}+\delta
h^{a}_{\;\;\mu}\right]\left[h^{b}_{\;\;\nu}+\delta h^{b}_{\;\;\nu}\right],$
$\displaystyle\approx$ $\displaystyle g_{\mu\nu}+\eta_{ab}\left[\delta
h^{a}_{\;\;\mu}h^{b}_{\;\;\nu}+h^{a}_{\;\;\mu}\delta
h^{b}_{\;\;\nu}\right]+O\left(|\delta h|^{2}\right),$
$\displaystyle\Rightarrow\delta g_{\mu\nu}$ $\displaystyle\approx$
$\displaystyle\eta_{ab}\left[\delta
h^{a}_{\;\;\mu}h^{b}_{\;\;\nu}+h^{a}_{\;\;\mu}\delta
h^{b}_{\;\;\nu}\right]+O\left(|\delta h|^{2}\right).$ (84)

  4. 4.

Torsion tensor $T^{a}_{\;\;\mu\nu}$ and $T^{\rho}_{\;\;\mu\nu}$:

$\displaystyle T^{a}_{\;\;\mu\nu}+\delta T^{a}_{\;\;\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}h^{a}_{\;\;\nu}+\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}h^{a}_{\;\;\mu}+\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)$ $\displaystyle\approx$ $\displaystyle
T^{a}_{\;\;\mu\nu}+\left[\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)\right]+O\left(|\delta h|^{2}\right)$
$\displaystyle\Rightarrow\delta T^{a}_{\;\;\mu\nu}$ $\displaystyle\approx$
$\displaystyle\left[\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)\right]+O\left(|\delta h|^{2}\right)$ (85)

If we also have that
$T^{\rho}_{\;\;\mu\nu}=h^{~{}\rho}_{a}\,T^{a}_{\;\;\mu\nu}$, then:

$\displaystyle T^{\rho}_{\;\;\mu\nu}+\delta T^{\rho}_{\;\;\mu\nu}$
$\displaystyle=$ $\displaystyle\left(h^{~{}\rho}_{a}+\delta
h^{~{}\rho}_{a}\right)\left(T^{a}_{\;\;\mu\nu}+\delta
T^{a}_{\;\;\mu\nu}\right)$ $\displaystyle\approx$ $\displaystyle
T^{\rho}_{\;\;\mu\nu}+\delta
h^{~{}\rho}_{a}\,T^{a}_{\;\;\mu\nu}+h^{~{}\rho}_{a}\,\delta
T^{a}_{\;\;\mu\nu}+O\left(|\delta h|^{2}\right)$
$\displaystyle\Rightarrow\delta T^{\rho}_{\;\;\mu\nu}$ $\displaystyle\approx$
$\displaystyle\delta
h^{~{}\rho}_{a}\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]+h^{~{}\rho}_{a}\,\left[\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)\right]+O\left(|\delta h|^{2}\right)$

  5. 5.

Torsion scalar $T$:

$\displaystyle T+\delta T$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(T^{a}_{\;\;\mu\nu}+\delta
T^{a}_{\;\;\mu\nu}\right)\left(T_{a}^{\;\;\mu\nu}+\delta
T_{a}^{\;\;\mu\nu}\right)+\frac{1}{2}\left(T^{a}_{\;\;\mu\nu}+\delta
T^{a}_{\;\;\mu\nu}\right)\left(T^{\nu\mu}_{\;\;a}+\delta
T^{\nu\mu}_{\;\;a}\right)$
$\displaystyle\qquad-\left(T^{\nu}_{\;\;\mu\nu}+\delta
T^{\nu}_{\;\;\mu\nu}+\right)\left(T^{\rho\mu}_{\;\;\rho}+\delta
T^{\rho\mu}_{\;\;\rho}\right)$ $\displaystyle=$ $\displaystyle
T+\frac{1}{4}\left(\delta
T^{a}_{\;\;\mu\nu}T_{a}^{\;\;\mu\nu}+T^{a}_{\;\;\mu\nu}\delta
T_{a}^{\;\;\mu\nu}\right)+\frac{1}{2}\left(\delta
T^{a}_{\;\;\mu\nu}T^{\nu\mu}_{\;\;a}+T^{a}_{\;\;\mu\nu}\delta
T^{\nu\mu}_{\;\;a}\right)$ $\displaystyle\qquad-\left(\delta
T^{\nu}_{\;\;\mu\nu}T^{\rho\mu}_{\;\;\rho}+T^{\nu}_{\;\;\mu\nu}\delta
T^{\rho\mu}_{\;\;\rho}\right)+O\left(|\delta h|^{2}\right)$
$\displaystyle\Rightarrow\delta T$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\delta
T^{a}_{\;\;\mu\nu}T_{a}^{\;\;\mu\nu}+T^{a}_{\;\;\mu\nu}\delta
T_{a}^{\;\;\mu\nu}\right)+\frac{1}{2}\left(\delta
T^{a}_{\;\;\mu\nu}T^{\nu\mu}_{\;\;a}+T^{a}_{\;\;\mu\nu}\delta
T^{\nu\mu}_{\;\;a}\right)$ (87) $\displaystyle\qquad-\left(\delta
T^{\nu}_{\;\;\mu\nu}T^{\rho\mu}_{\;\;\rho}+T^{\nu}_{\;\;\mu\nu}\delta
T^{\rho\mu}_{\;\;\rho}\right)+O\left(|\delta h|^{2}\right)$

In terms of Equations (4) and (4), Equation (5) becomes as:

$\displaystyle\delta T=$
$\displaystyle\frac{1}{4}\Bigg{[}\left(\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)\right)\left(\partial^{\mu}\,h_{a}^{\;\;\nu}-\partial^{\nu}\,h_{a}^{\;\;\mu}\right)+\left(\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right)$
$\displaystyle\times\,\left(\partial^{\mu}\,\left(\delta
h_{a}^{\;\;\nu}\right)-\partial^{\nu}\,\left(\delta
h_{a}^{\;\;\mu}\right)\right)\Bigg{]}+\frac{1}{2}\Bigg{[}\left(\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)\right)\left(\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right)$
$\displaystyle+\left(\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right)\left(\partial^{\nu}\,\left(\delta
h^{\mu}_{~{}a}\right)-\partial^{\mu}\,\left(\delta
h^{\nu}_{~{}a}\right)\right)\Bigg{]}$ $\displaystyle-\Bigg{[}\left(\delta
h^{~{}\nu}_{a}\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]+h^{~{}\nu}_{a}\,\left[\partial_{\mu}\left(\delta
h^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\delta
h^{a}_{\;\;\mu}\right)\right]\right)\left(h^{a}_{~{}\rho}\left(\partial^{\rho}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\rho}_{~{}a}\right)\right)$
$\displaystyle+\left(h^{~{}\nu}_{a}\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]\right)\left(\delta
h^{a}_{~{}\rho}\left(\partial^{\rho}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\rho}_{~{}a}\right)+h^{a}_{~{}\rho}\left(\partial^{\rho}\,\left(\delta
h^{\mu}_{~{}a}\right)-\partial^{\mu}\,\left(\delta
h^{\rho}_{~{}a}\right)\right)\right)\Bigg{]}$ $\displaystyle+O\left(|\delta
h|^{2}\right).$ (88)

  6. 6.

Lagrangian density $\mathcal{L}_{Grav}$:

$\displaystyle\mathcal{L}_{Grav}+\delta\mathcal{L}_{Grav}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\left(h+\delta h\right)\,f\left(T+\delta
T\right),$ $\displaystyle\approx$
$\displaystyle\mathcal{L}_{Grav}+\frac{1}{2\kappa}\left[\delta
h\,f\left(T\right)+h\,f_{T}\left(T\right)\,\delta T\right]+O\left(|\delta
h|^{2}\right),$ $\displaystyle\Rightarrow\delta\mathcal{L}_{Grav}$
$\displaystyle\approx$ $\displaystyle\frac{1}{2\kappa}\left[\delta
h\,f\left(T\right)+h\,f_{T}\left(T\right)\,\delta T\right]+O\left(|\delta
h|^{2}\right).$ (89)

  7. 7.

Sum of the Torsion and Ricci Curvature scalar $\overset{\ \circ}{R}+T$: Here,
$\overset{\ \circ}{R}$ is the Ricci scalar computed from the Levi-Civita
connection.

$\displaystyle\delta(\overset{\ \circ}{R}+T)$ $\displaystyle=$
$\displaystyle\,\delta\left[\frac{2}{h}\,\delta_{\mu}\left(h\,T^{\nu\mu}_{\;\;\nu}\right)\right]=2\left[\delta\left(\frac{1}{h}\right)\,\delta_{\mu}\left(h\,T^{\nu\mu}_{\;\;\nu}\right)+\frac{1}{h}\delta_{\mu}\left[\delta\left(h\,T^{\nu\mu}_{\;\;\nu}\right)\right]\right]$
(90) $\displaystyle\approx$ $\displaystyle\frac{2}{h}\,\left[-\frac{\delta
h}{h}(\delta_{\mu}h)\,T^{\nu\mu}_{\;\;\nu}+\left(\delta_{\mu}(\delta
h)\right)\,T^{\nu\mu}_{\;\;\nu}+\left(\delta_{\mu}h\right)\,\delta
T^{\nu\mu}_{\;\;\nu}+h\,\delta_{\mu}\left(\delta
T^{\nu\mu}_{\;\;\nu}\right)\right]$ $\displaystyle+O\left(|\delta
h|^{2}\right)$

By using Equation (4), Equation (90) becomes as:

$\displaystyle\delta(\overset{\ \circ}{R}+T)\approx$
$\displaystyle\frac{2}{h}\,\Bigg{[}-\frac{\delta
h}{h}(\delta_{\mu}h)\,\left(h^{a}_{~{}\nu}\left[\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right]\right)+\left(\delta_{\mu}(\delta
h)\right)\,\left(h^{a}_{~{}\nu}\left[\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right]\right)$
$\displaystyle+\left(\delta_{\mu}h\right)\,\left(\delta
h^{a}_{~{}\nu}\left[\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right]+h^{a}_{~{}\nu}\left[\partial^{\nu}\,\left(\delta
h^{\mu}_{~{}a}\right)-\partial^{\mu}\,\left(\delta
h^{\nu}_{~{}a}\right)\right]\right)$
$\displaystyle+h\,\delta_{\mu}\left(\delta
h^{a}_{~{}\nu}\left[\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right]+h^{a}_{~{}\nu}\left[\partial^{\nu}\,\left(\delta
h^{\mu}_{~{}a}\right)-\partial^{\mu}\,\left(\delta
h^{\nu}_{~{}a}\right)\right]\right)\Bigg{]}+O\left(|\delta h|^{2}\right)$

  8. 8.

Superpotential $S_{ab}^{\;\;\mu}$:

$\displaystyle S_{ab}^{\;\;\;\mu}+\delta S_{ab}^{\;\;\;\mu}=$
$\displaystyle\,\frac{1}{2}\left(T_{ab}^{\;\;\mu}+\delta
T_{ab}^{\;\;\mu}+T^{\mu}_{\;\;ba}+\delta
T^{\mu}_{\;\;ba}-T^{\mu}_{\;\;ab}-\delta T^{\mu}_{\;\;ab}\right)$
$\displaystyle\qquad-\left(h_{~{}b}^{\mu}+\delta
h_{~{}b}^{\mu}\right)\left(T_{\rho a}^{\;\;\rho}+\delta T_{\rho
a}^{\;\;\rho}\right)+\left(h_{~{}a}^{\mu}+\delta
h_{~{}a}^{\mu}\right)\left(T_{\rho b}^{\;\;\rho}+\delta T_{\rho
b}^{\;\;\rho}\right)$ $\displaystyle\approx$
$\displaystyle\,S_{a}^{\;\;\mu\nu}+\Bigg{[}\frac{1}{2}\left(\delta
T_{ab}^{\;\;\mu}+\delta T^{\mu}_{\;\;ba}-\delta T^{\mu}_{\;\;ab}\right)-\delta
h_{~{}b}^{\mu}T_{\rho a}^{\;\;\rho}-h_{~{}b}^{\mu}\delta T_{\rho
a}^{\;\;\rho}+\delta h_{~{}a}^{\mu}T_{\rho b}^{\;\;\rho}$
$\displaystyle\quad+h_{~{}a}^{\mu}\delta T_{\rho
b}^{\;\;\rho}\Bigg{]}+O\left(|\delta h|^{2}\right)$
$\displaystyle\Rightarrow\delta S_{ab}^{\;\;\;\mu}\approx$
$\displaystyle\left[\frac{1}{2}\left(\delta T_{ab}^{\;\;\mu}+2\,\delta
T^{\mu}_{\;\;ba}\right)-\delta h_{~{}b}^{\mu}T_{\rho
a}^{\;\;\rho}-h_{~{}b}^{\mu}\delta T_{\rho a}^{\;\;\rho}+\delta
h_{~{}a}^{\mu}T_{\rho b}^{\;\;\rho}+h_{~{}a}^{\mu}\delta T_{\rho
b}^{\;\;\rho}\right]+O\left(|\delta h|^{2}\right)$

In terms of $\delta h^{a}_{~{}\mu}$, Equation (8) becomes:

$\displaystyle\delta S_{ab}^{\;\;\;\mu}\approx$
$\displaystyle\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\delta
h_{b}^{~{}\mu}\right)-\partial_{b}\left(\delta
h_{a}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\delta
h_{~{}a}^{\mu}\right)-\partial_{a}\left(\delta
h_{~{}b}^{\mu}\right)\right)-\delta
h_{~{}b}^{\mu}\left(h_{~{}\rho}^{c}\left[\partial_{c}\,h_{a}^{~{}\rho}-\partial_{a}\,h_{c}^{~{}\rho}\right]\right)$
$\displaystyle-h_{~{}b}^{\mu}\left(\delta
h_{~{}\rho}^{c}\left[\partial_{c}\,h_{a}^{~{}\rho}-\partial_{a}\,h_{c}^{~{}\rho}\right]+h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\delta
h_{a}^{~{}\rho}\right)-\partial_{a}\,\left(\delta
h_{c}^{~{}\rho}\right)\right]\right)+\delta
h_{~{}a}^{\mu}\left(h_{~{}\rho}^{c}\left[\partial_{c}\,h_{b}^{~{}\rho}-\partial_{b}\,h_{c}^{~{}\rho}\right]\right)$
$\displaystyle+h_{~{}a}^{\mu}\left(\delta
h_{~{}\rho}^{c}\left[\partial_{c}\,h_{b}^{~{}\rho}-\partial_{b}\,h_{c}^{~{}\rho}\right]+h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\delta
h_{b}^{~{}\rho}\right)-\partial_{b}\,\left(\delta
h_{c}^{~{}\rho}\right)\right]\right)\Bigg{]}+O\left(|\delta h|^{2}\right)$
(93)

  9. 9.

Einstein tensor $\overset{\ \circ}{G}_{\mu\nu}$:

$\displaystyle\overset{\ \circ}{G}_{ab}+\delta\overset{\ \circ}{G}_{ab}$
$\displaystyle=$ $\displaystyle\left(\overset{\
\circ}{G}_{\mu\nu}+\delta\overset{\
\circ}{G}_{\mu\nu}\right)\left(h_{a}^{\;\;\mu}+\delta
h_{a}^{\;\;\mu}\right)\left(h_{b}^{\;\;\nu}+\delta h_{b}^{\;\;\nu}\right)$
$\displaystyle\approx$ $\displaystyle\overset{\
\circ}{G}_{ab}+\left[\overset{\ \circ}{G}_{\mu\nu}\left(\delta
h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}+h_{a}^{\;\;\mu}\delta
h_{b}^{\;\;\nu}\right)+\delta\overset{\
\circ}{G}_{\mu\nu}\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\right]+O\left(|\delta
h|^{2}\right)$ $\displaystyle\Rightarrow\delta\overset{\ \circ}{G}_{ab}$
$\displaystyle\approx$ $\displaystyle\left[\overset{\
\circ}{G}_{\mu\nu}\left(\delta
h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}+h_{a}^{\;\;\mu}\delta
h_{b}^{\;\;\nu}\right)+\delta\overset{\
\circ}{G}_{\mu\nu}\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\right]+O\left(|\delta
h|^{2}\right).$ (94)

If $\overset{\ \circ}{G}_{\mu\nu}=\overset{\
\circ}{R}_{\mu\nu}-\frac{1}{2}\,g^{\sigma\rho}\,g_{\mu\nu}\,\overset{\
\circ}{R}_{\sigma\rho}=\overset{\
\circ}{R}_{\mu\nu}-\frac{\eta^{cd}\,\eta_{ab}}{2}\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{a}_{\;\;\mu}\,h^{b}_{\;\;\nu}\right]\,\overset{\
\circ}{R}_{\sigma\rho}$, then we obtain from Equation (3):

$\displaystyle\delta\overset{\ \circ}{G}_{\mu\nu}\approx$
$\displaystyle\,\delta\overset{\
\circ}{R}_{\mu\nu}-\frac{\eta^{cd}\,\eta_{ab}}{2}\,\Bigg{[}\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{a}_{\;\;\mu}\,h^{b}_{\;\;\nu}\right]\,\delta\overset{\
\circ}{R}_{\sigma\rho}+\Bigg{[}\delta
h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{a}_{\;\;\mu}\,h^{b}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,\delta
h_{d}^{\;\;\rho}\,h^{a}_{\;\;\mu}\,h^{b}_{\;\;\nu}$
$\displaystyle+h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,\delta
h^{a}_{\;\;\mu}\,h^{b}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{a}_{\;\;\mu}\,\delta
h^{b}_{\;\;\nu}\Bigg{]}\,\overset{\
\circ}{R}_{\sigma\rho}\Bigg{]}+O\left(|\delta h|^{2}\right)$ (95)

By substituting Equation (9) into Equation (9), we obtain that:

$\displaystyle\delta\overset{\ \circ}{G}_{ab}\approx$
$\displaystyle\Bigg{[}\overset{\
\circ}{R}_{\mu\nu}-\frac{\eta^{cd}\,\eta_{ef}}{2}\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,\overset{\
\circ}{R}_{\sigma\rho}\Bigg{]}\left(\delta
h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}+h_{a}^{\;\;\mu}\delta h_{b}^{\;\;\nu}\right)$
$\displaystyle+\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\Bigg{[}\delta\overset{\
\circ}{R}_{\mu\nu}-\frac{\eta^{cd}\,\eta_{ef}}{2}\,\Bigg{[}\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,\delta\overset{\
\circ}{R}_{\sigma\rho}$ $\displaystyle+\left[\delta
h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,\delta
h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,\delta
h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,\delta
h^{f}_{\;\;\nu}\right]\,\overset{\ \circ}{R}_{\sigma\rho}\Bigg{]}\Bigg{]}$
$\displaystyle+O\left(|\delta h|^{2}\right)$ (96)

Now, if we have that $\overset{\
\circ}{R}_{\mu\nu}=h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}\,\overset{\
\circ}{R}^{k}_{~{}m\alpha\nu}$, then Equation (9) becomes

$\displaystyle\delta\overset{\ \circ}{G}_{ab}\approx$
$\displaystyle\Bigg{[}h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}\,\overset{\
\circ}{R}^{k}_{~{}m\alpha\nu}-\frac{\eta^{cd}\,\eta_{ef}}{2}\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}\,\overset{\
\circ}{R}^{k}_{~{}m\alpha\rho}\Bigg{]}\left(\delta
h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}+h_{a}^{\;\;\mu}\delta h_{b}^{\;\;\nu}\right)$
$\displaystyle+\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\Bigg{[}\left[\left(\delta
h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}+h_{k}^{~{}\alpha}\,\delta
h^{m}_{~{}\mu}\right)\,\overset{\
\circ}{R}^{k}_{~{}m\alpha\nu}+h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}\,\delta\overset{\
\circ}{R}^{k}_{~{}m\alpha\nu}\right]$
$\displaystyle-\frac{\eta^{cd}\,\eta_{ef}}{2}\,\Bigg{[}\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,\left[\left(\delta
h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}+h_{k}^{~{}\alpha}\,\delta
h^{m}_{~{}\mu}\right)\,\overset{\
\circ}{R}^{k}_{~{}m\alpha\rho}+h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}\,\delta\overset{\
\circ}{R}^{k}_{~{}m\alpha\rho}\right]$ $\displaystyle+\left[\delta
h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,\delta
h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,\delta
h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}+h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,\delta
h^{f}_{\;\;\nu}\right]$
$\displaystyle\quad\times\,h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}\,\overset{\
\circ}{R}^{k}_{~{}m\alpha\rho}\Bigg{]}\Bigg{]}+O\left(|\delta h|^{2}\right)$
(97)

For pure Minkowski spacetime, we have that $\overset{\
\circ}{R}^{k}_{~{}m\alpha\rho}=0$ by default and Equation (9) reduces as:

$\displaystyle\delta\overset{\ \circ}{G}_{ab}\approx$
$\displaystyle\left(h_{a}^{\;\;\mu}h_{b}^{\;\;\nu}\right)\Bigg{[}h_{k}^{~{}\alpha}\,h^{m}_{~{}\mu}\,\delta\overset{\
\circ}{R}^{k}_{~{}m\alpha\nu}-\frac{\eta^{cd}\,\eta_{ef}}{2}\,\left[h_{c}^{\;\;\sigma}\,h_{d}^{\;\;\rho}\,h^{e}_{\;\;\mu}\,h^{f}_{\;\;\nu}\right]\,h_{k}^{~{}\alpha}\,h^{m}_{~{}\sigma}\,\delta\overset{\
\circ}{R}^{k}_{~{}m\alpha\rho}\Bigg{]}+O\left(|\delta h|^{2}\right).$

Equation (9) is useful for Equations (V.1) and (V.2) and the energy-momentum
stability test.

##  Appendix B General Perturbed Torsion-Based Field Equation via
Linearization

Here, we can also obtain the perturbed field equation (Equations (33) and
(34)) using Equation (6), with a matter contribution as follows:

$\displaystyle\delta\mathcal{L}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2\kappa}\left[\delta
h\,f\left(T\right)+h\,f_{T}\left(T\right)\,\delta
T\right]+\delta\mathcal{L}_{Matter}+O\left(|\delta h|^{2}\right)$ (99)

As for the non-perturbed FEs, we have here that $\delta\Theta_{(ab)}=\delta
T_{ab}\equiv\frac{1}{2}\frac{\delta\left(\delta L_{Matt}\right)}{\delta
g_{ab}}$.

For the term $\frac{1}{2\kappa}\,\delta h\,f\left(T\right)$, we obtain by
analogy with Equation (II.6) the following part (here, $\delta g_{ab}=0$ for
the orthonormal framework):

$\displaystyle\frac{\delta h\,f\left(T\right)}{2\kappa}\,$
$\displaystyle\rightarrow$ $\displaystyle f_{TT}\left[\delta
S_{\left(ab\right)}^{~{}~{}~{}\mu}\,\partial_{\mu}T+S_{\left(ab\right)}^{~{}~{}~{}\mu}\,\partial_{\mu}\left(\delta
T\right)\right]+f_{T}\,\delta\overset{\
\circ}{G}_{ab}-\frac{g_{ab}}{2}\,f_{T}\,\delta T\quad\quad\text{Symmetric}$
$\displaystyle\rightarrow$ $\displaystyle
f_{TT}\,\left[S_{\left[ab\right]}^{~{}~{}~{}\mu}\partial_{\mu}\left(\delta
T\right)+\delta
S_{\left[ab\right]}^{~{}~{}~{}\mu}\partial_{\mu}T\right]\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{Antisymmetric}$

At Equation (B), we only perturb the physical quantities linked by $\delta h$,
giving $\delta T$, $\delta\overset{\ \circ}{G}_{ab}$, and $\delta
S_{ab}^{\;\;\mu}$. We do not perturb $f(T)$ and its derivatives.

For the term $\frac{1}{2\kappa}\,h\,f_{T}\left(T\right)\,\delta T$, we still
obtain by analogy with Equation (II.6) the part (here again, $\delta
g_{ab}=0$):

$\displaystyle\frac{h\,f_{T}\left(T\right)\,\delta T}{2\kappa}\,$
$\displaystyle\rightarrow$
$\displaystyle\left[f_{TTT}\,S_{\left(ab\right)}^{~{}~{}~{}\mu}\partial_{\mu}T+f_{TT}\,\overset{\
\circ}{G}_{ab}+\frac{g_{ab}}{2}\left(f_{T}-T\,f_{TT}\right)\right]\,\delta
T\quad\quad\text{Symmetric}$ $\displaystyle\rightarrow$ $\displaystyle
f_{TTT}\,\left[S_{\left[ab\right]}^{~{}~{}~{}\mu}\partial_{\mu}T\right]\,\delta
T\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{Antisymmetric}$

At Equation (B), we only change $f(T)\rightarrow f_{T}(T)\,\delta T$,
$f_{T}(T)\rightarrow f_{TT}(T)\,\delta T$, and $f_{TT}(T)\rightarrow
f_{TTT}(T)\,\delta T$. We does not perturb the physical quantities themselves.

By adding the Equations (B) and (B), we obtain exactly at the first order the
Equations (33) and (34). This is the sign that the linearization of gravity
and the direct perturbation of the field equation described by Equation (II.6)
are both equivalent. Through these two methods, we obtain the field equation
described by Equations (33) and (34), which is in the order of things.

##  Appendix C The Derivation of Minkowski Spacetime Symmetries: Conditions
for Stability

In order to shorten the text, we put in this appendix some long calculations
that are necessary for the results of Sections V.1 and V.2.

###  C.1 Rotation/Boost Perturbation

  1. 1.

Torsion scalar perturbation $\delta T$: by using Equation (5) and by
substituting Equation (39) inside, we obtain the expression:

$\displaystyle\delta T=$
$\displaystyle\frac{1}{4}\Bigg{[}\left(\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)-\partial_{\nu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\mu}\right)\right)\left(\partial^{\mu}\,h_{a}^{\;\;\nu}-\partial^{\nu}\,h_{a}^{\;\;\mu}\right)+\left(\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right)$
$\displaystyle\times\left(\partial^{\mu}\,\left(\lambda_{a}^{\;\;b}\,h_{b}^{\;\;\nu}\right)-\partial^{\nu}\,\left(\lambda_{a}^{\;\;b}\,h_{b}^{\;\;\mu}\right)\right)\Bigg{]}+\frac{1}{2}\Bigg{[}\left(\partial_{\mu}\left(\lambda^{a}_{~{}b}\,h^{b}_{~{}\nu}\right)-\partial_{\nu}\left(\lambda^{a}_{~{}b}\,h^{b}_{~{}\mu}\right)\right)\left(\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right)$
$\displaystyle+\left(\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right)\left(\partial^{\nu}\,\left(\lambda^{b}_{~{}a}\,h^{\mu}_{~{}b}\right)-\partial^{\mu}\,\left(\lambda^{b}_{~{}a}\,h^{\nu}_{~{}b}\right)\right)\Bigg{]}$
$\displaystyle-\Bigg{[}\left(\lambda_{a}^{\;\;b}\,h^{~{}\nu}_{b}\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]+h^{~{}\nu}_{a}\,\left[\partial_{\mu}\left(\lambda^{a}_{~{}b}\,h^{b}_{\;\;\nu}\right)-\partial_{\nu}\left(\lambda^{a}_{~{}b}\,h^{b}_{\;\;\mu}\right)\right]\right)\left(h^{a}_{~{}\rho}\left(\partial^{\rho}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\rho}_{~{}a}\right)\right)$
$\displaystyle+\left(h^{~{}\nu}_{a}\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]\right)\left(\lambda^{a}_{~{}b}h^{b}_{~{}\rho}\left(\partial^{\rho}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\rho}_{~{}a}\right)+h^{a}_{~{}\rho}\left(\partial^{\rho}\,\left(\lambda^{b}_{~{}a}\,h^{\mu}_{~{}b}\right)-\partial^{\mu}\,\left(\lambda^{b}_{~{}a}\,h^{\rho}_{~{}b}\right)\right)\right)\Bigg{]}$
$\displaystyle+O\left(|\delta h|^{2}\right)$ $\displaystyle\rightarrow 0$
(102)

We need to impose
$T^{a}_{~{}\mu\nu}=\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\rightarrow
0$ to obtain the final result for Equation (1).

  2. 2.

Superpotential perturbation $\delta S_{ab}^{~{}~{}~{}\mu}$: by using Equation
(8) and by substituting Equation (39) inside, we obtain the expression:

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}$ $\displaystyle=$
$\displaystyle\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\lambda_{b}^{~{}c}\,h_{c}^{~{}\mu}\right)-\partial_{b}\left(\lambda_{a}^{~{}c}\,h_{c}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\lambda_{~{}a}^{c}\,h_{~{}c}^{\mu}\right)-\partial_{a}\left(\lambda_{~{}b}^{c}\,h_{~{}c}^{\mu}\right)\right)-\lambda_{~{}b}^{e}\,h_{~{}e}^{\mu}\left(h_{~{}\rho}^{c}\left[\partial_{c}\,h_{a}^{~{}\rho}-\partial_{a}\,h_{c}^{~{}\rho}\right]\right)$
$\displaystyle-
h_{~{}b}^{\mu}\left(\lambda^{c}_{~{}e}\,h_{~{}\rho}^{e}\left[\partial_{c}\,h_{a}^{~{}\rho}-\partial_{a}\,h_{c}^{~{}\rho}\right]+h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{a}^{~{}f}h_{f}^{~{}\rho}\right)-\partial_{a}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}\right)\right]\right)$
$\displaystyle+\lambda_{~{}a}^{e}h_{~{}e}^{\mu}\left(h_{~{}\rho}^{c}\left[\partial_{c}\,h_{b}^{~{}\rho}-\partial_{b}\,h_{c}^{~{}\rho}\right]\right)+h_{~{}a}^{\mu}\lambda^{c}_{~{}e}h_{~{}\rho}^{e}\left[\partial_{c}\,h_{b}^{~{}\rho}-\partial_{b}\,h_{c}^{~{}\rho}\right]$
$\displaystyle+h_{~{}a}^{\mu}\,h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{b}^{~{}f}h_{f}^{~{}\rho}\right)-\partial_{b}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}\right)\right]\Bigg{]}+O\left(|\delta
h|^{2}\right)$
$\displaystyle\rightarrow\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\lambda_{b}^{~{}c}\,h_{c}^{~{}\mu}\right)-\partial_{b}\left(\lambda_{a}^{~{}c}\,h_{c}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\lambda_{~{}a}^{c}\,h_{~{}c}^{\mu}\right)-\partial_{a}\left(\lambda_{~{}b}^{c}\,h_{~{}c}^{\mu}\right)\right)$
$\displaystyle\quad\quad-
h_{~{}b}^{\mu}\,h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{a}^{~{}f}h_{f}^{~{}\rho}\right)-\partial_{a}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}\right)\right]+h_{~{}a}^{\mu}\,h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{b}^{~{}f}h_{f}^{~{}\rho}\right)-\partial_{b}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}\right)\right]\Bigg{]}$
$\displaystyle\quad\quad+O\left(|\delta h|^{2}\right)\quad\quad\text{by
applying Equation \eqref{436a} (the zero torsion criteria).}$
$\displaystyle\rightarrow 0.$ (103)

We need to impose $\delta
T^{a}_{~{}\mu\nu}=\partial_{\mu}\,\left(\lambda^{a}_{~{}c}\,h^{c}_{\;\;\nu}\right)-\partial_{\nu}\,\left(\lambda^{a}_{~{}c}\,h^{c}_{\;\;\mu}\right)\rightarrow
0$ to obtain the final result for Equation (2).

  3. 3.

Weitzenbock connection perturbation $\delta\Gamma^{\rho}_{\;\;\nu\mu}$: from
the null covariant derivative criteria, we make the following derivation as:

$\displaystyle 0=\nabla_{\mu}\,\delta h^{a}_{\;\;\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\,\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)=\partial_{\mu}\,\delta
h^{a}_{\;\;\nu}-\Gamma^{\rho}_{\;\;\nu\mu}\,\delta
h^{a}_{\;\;\rho}-\delta\Gamma^{\rho}_{\;\;\nu\mu}h^{a}_{\;\;\rho}$
$\displaystyle=$
$\displaystyle\partial_{\mu}\,\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)-\left(h_{c}^{\;\;\rho}\,\partial_{\mu}\,h^{c}_{\;\;\nu}\right)\,\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\rho}\right)-\delta\Gamma^{\rho}_{\;\;\nu\mu}h^{a}_{\;\;\rho}$
$\displaystyle\Rightarrow\delta\Gamma^{\rho}_{\;\;\nu\mu}=h_{a}^{\;\;\rho}\left[\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}\right)-\left(h_{c}^{\;\;\sigma}\,\partial_{\mu}\,h^{c}_{\;\;\nu}\right)\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\sigma}\right)\right],$

where
$\Gamma^{\rho}_{\;\;\nu\mu}=h_{c}^{\;\;\rho}\,\partial_{\mu}\,h^{c}_{\;\;\nu}$
is the Weitzenbock connection for a proper frame.

###  C.2 General Linear Perturbation

  1. 1.

The torsion scalar perturbation $\delta T$:

$\displaystyle\delta T=$
$\displaystyle\frac{1}{4}\Bigg{[}\left(\partial_{\mu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\nu}+\epsilon^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\lambda^{a}_{\;\;b}\,h^{b}_{\;\;\mu}+\epsilon^{a}_{\;\;\mu}\right)\right)\left(\partial^{\mu}\,h_{a}^{\;\;\nu}-\partial^{\nu}\,h_{a}^{\;\;\mu}\right)+\left(\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right)$
$\displaystyle\times\left(\partial^{\mu}\,\left(\lambda_{a}^{\;\;b}\,h_{b}^{\;\;\nu}+\epsilon_{a}^{\;\;\nu}\right)-\partial^{\nu}\,\left(\lambda_{a}^{\;\;b}\,h_{b}^{\;\;\mu}+\epsilon_{a}^{\;\;\mu}\right)\right)\Bigg{]}$
$\displaystyle+\frac{1}{2}\Bigg{[}\left(\partial_{\mu}\left(\lambda^{a}_{~{}b}\,h^{b}_{~{}\nu}+\epsilon^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\lambda^{a}_{~{}b}\,h^{b}_{~{}\mu}+\epsilon^{a}_{\;\;\mu}\right)\right)\left(\partial^{\nu}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\nu}_{~{}a}\right)$
$\displaystyle+\left(\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right)\left(\partial^{\nu}\,\left(\lambda^{b}_{~{}a}\,h^{\mu}_{~{}b}+\epsilon^{\mu}_{~{}a}\right)-\partial^{\mu}\,\left(\lambda^{b}_{~{}a}\,h^{\nu}_{~{}b}+\epsilon^{\nu}_{~{}a}\right)\right)\Bigg{]}$
$\displaystyle-\Bigg{[}\left(\left(\lambda_{a}^{\;\;b}\,h^{~{}\nu}_{b}+\epsilon^{~{}\nu}_{a}\right)\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]+h^{~{}\nu}_{a}\,\left[\partial_{\mu}\left(\lambda^{a}_{~{}b}\,h^{b}_{\;\;\nu}+\epsilon^{a}_{\;\;\nu}\right)-\partial_{\nu}\left(\lambda^{a}_{~{}b}\,h^{b}_{\;\;\mu}+\epsilon^{a}_{\;\;\mu}\right)\right]\right)$
$\displaystyle\times\left(h^{a}_{~{}\rho}\left(\partial^{\rho}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\rho}_{~{}a}\right)\right)+\left(h^{~{}\nu}_{a}\,\left[\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\right]\right)$
$\displaystyle\times\left(\left(\lambda^{a}_{~{}b}h^{b}_{~{}\rho}+\epsilon^{a}_{~{}\rho}\right)\left(\partial^{\rho}\,h^{\mu}_{~{}a}-\partial^{\mu}\,h^{\rho}_{~{}a}\right)+h^{a}_{~{}\rho}\left(\partial^{\rho}\,\left(\lambda^{b}_{~{}a}\,h^{\mu}_{~{}b}+\epsilon^{\mu}_{~{}a}\right)-\partial^{\mu}\,\left(\lambda^{b}_{~{}a}\,h^{\rho}_{~{}b}+\epsilon^{\rho}_{~{}a}\right)\right)\right)\Bigg{]}$
$\displaystyle+O\left(|\delta h|^{2}\right)$ $\displaystyle\rightarrow 0.$
(105)

We again need to impose
$T^{a}_{~{}\mu\nu}=\partial_{\mu}\,h^{a}_{\;\;\nu}-\partial_{\nu}\,h^{a}_{\;\;\mu}\rightarrow
0$ as for Equation (1) to obtain Equation (1).

  2. 2.

The superpotential perturbation $\delta S_{ab}^{~{}~{}~{}\mu}$ is expressed
as:

$\displaystyle\delta S_{ab}^{~{}~{}~{}\mu}=$
$\displaystyle\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\lambda_{b}^{~{}c}\,h_{c}^{~{}\mu}+\epsilon_{b}^{~{}\mu}\right)-\partial_{b}\left(\lambda_{a}^{~{}c}\,h_{c}^{~{}\mu}+\epsilon_{a}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\lambda_{~{}a}^{c}\,h_{~{}c}^{\mu}+\epsilon_{~{}a}^{\mu}\right)-\partial_{a}\left(\lambda_{~{}b}^{c}\,h_{~{}c}^{\mu}+\epsilon_{~{}b}^{\mu}\right)\right)$
$\displaystyle-\left(\lambda_{~{}b}^{e}\,h_{~{}e}^{\mu}+\epsilon_{~{}b}^{\mu}\right)\left(h_{~{}\rho}^{c}\left[\partial_{c}\,h_{a}^{~{}\rho}-\partial_{a}\,h_{c}^{~{}\rho}\right]\right)$
$\displaystyle-
h_{~{}b}^{\mu}\left(\left(\lambda^{c}_{~{}e}\,h_{~{}\rho}^{e}+\epsilon_{~{}\rho}^{c}\right)\left[\partial_{c}\,h_{a}^{~{}\rho}-\partial_{a}\,h_{c}^{~{}\rho}\right]+h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{a}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{a}^{~{}\rho}\right)-\partial_{a}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{c}^{~{}\rho}\right)\right]\right)$
$\displaystyle+\left(\lambda_{~{}a}^{e}h_{~{}e}^{\mu}+\epsilon_{~{}a}^{\mu}\right)\left(h_{~{}\rho}^{c}\left[\partial_{c}\,h_{b}^{~{}\rho}-\partial_{b}\,h_{c}^{~{}\rho}\right]\right)+h_{~{}a}^{\mu}\left(\lambda^{c}_{~{}e}h_{~{}\rho}^{e}+\epsilon_{~{}\rho}^{c}\right)\left[\partial_{c}\,h_{b}^{~{}\rho}-\partial_{b}\,h_{c}^{~{}\rho}\right]$
$\displaystyle+h_{~{}a}^{\mu}\,h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{b}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{b}^{~{}\rho}\right)-\partial_{b}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{c}^{~{}\rho}\right)\right]\Bigg{]}+O\left(|\delta
h|^{2}\right)$
$\displaystyle\rightarrow\Bigg{[}\frac{1}{2}\left(\partial_{a}\left(\lambda_{b}^{~{}c}\,h_{c}^{~{}\mu}+\epsilon_{b}^{~{}\mu}\right)-\partial_{b}\left(\lambda_{a}^{~{}c}\,h_{c}^{~{}\mu}+\epsilon_{a}^{~{}\mu}\right)\right)+\left(\partial_{b}\left(\lambda_{~{}a}^{c}\,h_{~{}c}^{\mu}+\epsilon_{~{}a}^{\mu}\right)-\partial_{a}\left(\lambda_{~{}b}^{c}\,h_{~{}c}^{\mu}+\epsilon_{~{}b}^{\mu}\right)\right)$
$\displaystyle\quad\quad-
h_{~{}b}^{\mu}\,h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{a}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{a}^{~{}\rho}\right)-\partial_{a}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{c}^{~{}\rho}\right)\right]$
$\displaystyle\quad\quad+h_{~{}a}^{\mu}\,h_{~{}\rho}^{c}\left[\partial_{c}\,\left(\lambda_{b}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{b}^{~{}\rho}\right)-\partial_{b}\,\left(\lambda_{c}^{~{}f}h_{f}^{~{}\rho}+\epsilon_{c}^{~{}\rho}\right)\right]\Bigg{]}+O\left(|\delta
h|^{2}\right)$ $\displaystyle\rightarrow 0,$ (106)

where we apply Equation (42) and we respect the
$\partial_{a}\epsilon_{b}^{\;\;\mu}=\partial_{b}\epsilon_{a}^{\;\;\mu}=0$
condition.

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