# Teleparallel bigravity

Daniel Blixt<EMAIL_ADDRESS>Scuola Superiore Merdionale, Largo S.
Marcellino 10, I-80138, Napoli, Italy  Manuel Hohmann<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Institute of Physics, University of Tartu,
W. Ostwaldi 1, 50411 Tartu, Estonia  Tomi Koivisto<EMAIL_ADDRESS>Laboratory of Theoretical Physics, Institute of Physics, University of Tartu,
W. Ostwaldi 1, 50411 Tartu, Estonia  National Institute of Chemical Physics
and Biophysics, Rävala pst. 10, 10143 Tallinn, Estonia  Luca Marzola
<EMAIL_ADDRESS>National Institute of Chemical Physics and Biophysics,
Rävala pst. 10, 10143 Tallinn, Estonia

###### Abstract

We write down the teleparallel equivalent to Hassan-Rosen bigravity, which is
written using a torsionful but curvature-free connection. The theories only
differ by a boundary term. The equivalence was proven, both by using
perturbation theory and Hamiltonian analysis. It is further shown how one can
construct novel bigravity theories within the teleparallel framework. Some of
those are analyzed through perturbation theory, and it is found that all of
the considered novel bigravity theories suffer from pathologies. In
particular, it is found that a construction with two copies of new general
relativity leads to ghostly degrees of freedom which are not present in the
single tetrad teleparallel corresponding theory. We demonstrate how the
teleparallel framework allows to easily create theories with derivative
interaction. However, it is shown through perturbation theory that the
simplest model is not viable. Furthermore, we demonstrate some steps in the
Hamiltonian analysis of teleparallel bigravity with two copies of new general
relativity and some toy models. The results rule out some of the novel
teleparallel bigravity theories, but also demonstrate techniques in
perturbation theory and Hamiltonian analysis which could be further used for
more profound theories in the future.

##  I Introduction

Quantum field theory has successfully led to a consistent description of all
fundamental forces of nature within the standard model of particle physics,
except gravity. The standard model in its present form has passed numerous
experimental tests, including most prominently the discovery of the Higgs
boson almost half a century after its prediction. Modifying such a robust
theory, to make it consistent with general relativity seems very difficult. A
more natural approach would be to look at modifications of general relativity.
The gauge structure encountered in the standard model and unexplained
observations in Nature guide us towards the most motivated ways to explore
theories beyond general relativity.
General relativity has also been very successful in passing experimental
tests. It has successfully predicted the deflection of light from the sun and
the precession of Mercury. One may ask why we should consider theories beyond
general relativity. Beside a philosophically appealing idea, that there would
exist a theory for everything, we need a quantum theory of gravity in order to
understand the early universe and black holes. However, general relativity is
a notoriously non-renormalizable theory, as is very nicely explained in
Zee:2003mt , so that the usual quantization methods used in the standard model
cannot be applied.
From observations at scales of galaxies and larger, we conclusively find
evidence for the so-called dark sector of the universe. The dark sector
consists of dark matter and dark energy, which make up about 95% of our
observable universe. Dark energy gives rise to an expanding universe at an
accelerating speed. The existence of dark matter can be inferred indirectly by
its gravitational effects on the motion of stars within galaxies,
gravitational lensing, and structure formation. However, neither of these
components has been observed in any laboratory experiment searching for a
suitable extension of the standard model. Since gravity is the dominant force
at the scales where dark matter and dark energy are observed, it is motivating
to look at extensions of general relativity which could give us an
understanding of the dark sector of the universe.
The first question to ask before investigating modifications of general
relativity is “which formulation of general relativity should be modified?”.
In BeltranJimenez:2019tjy  the different formulations of general relativity
are explained. Einstein already proposed an alternative formulation of general
relativity, which is nowadays called “teleparallel equivalent to general
relativity”doi:10.1002/3527608958.ch36 ; Maluf:2013gaa , or “TEGR”. However,
the third equivalent formulation called “symmetric teleparallel equivalent to
general relativity”, or “STEGR”, was not found until 1998 in Nester:1998mp
and only more recently unified into the general teleparallel theory in
BeltranJimenez:2019odq . In this article we will work in the framework of
“teleparallel gravity”, where our starting point will be the teleparallel
equivalent to general relativity. For reviews about teleparallel gravity, see
AP ; Bahamonde:2018rgz ; Bahamonde:2021zah ; Bahamonde:2021gfp ; Blixt:2020ekl
; Krssak:2018ywd . For literature about the viability of teleparallel gravity
theories, see Bahamonde:2022ohm ; BeltranJimenez:2020fvy ; Blagojevic:2000qs ;
Blagojevic:2020dyq ; Blixt:2018znp ; Blixt:2019ene ; Blixt:2019mkt ;
Capozziello:2023foy ; Chen:2014qtl ; Cheng:1988zg ; Ferraro:2016wht ;
Ferraro:2018tpu ; Ferraro:2020tqk ; Golovnev:2018wbh ; Golovnev:2019kcf ;
Golovnev:2020aon ; Golovnev:2017dox ; Golovnev:2020nln ; Hohmann:2019sys ;
Hohmann:2019gmt ; BeltranJimenez:2019nns ; Koivisto:2018loq ; Krssak:2015oua ;
Kuhfuss:1986rb ; Li:2011rn ; Maluf:2000ag ; Maluf:1994ji ; Mitric:2019rop ;
Nester:2017wau ; Okolow:2011np ; Okolow:2011nq ; Okolow:2013lwa ; Ong:2013qja
; Ortin:2015hya ; PhysRevD.19.3524 ; VanNieuwenhuizen:1973fi . Teleparallel
gravity, in contrast to the conventional curvature-based formulation of
gravity in terms of the Einstein-Hilbert Lagrangian, is a gauge theory of
gravity with a Yang-Mills-like structure111The alleged similarity with Yang-
Mills is though compromised by the degeneracy of the gauge group, the
inconsistency of the minimal coupling, the need for a non-canonical
Lagrangian, the violation of Yang-Mills Bianchi identity etc. There is,
however, a new approach to gauging translations which may alleviate these
issues Koivisto:2019ejt ; Koivisto:2022uvd . Blagojevic:2013xpa ; Cho:1975dh ,
similar to that of the standard model. Thus, teleparallel gravity theories are
more reminiscent of the standard model of particle physics.
The second question to ask is “how to modify gravity?”. In this article we
want to investigate the possibility of adding a massive spin-2 field. Already
in 1939 Fierz and Pauli addressed the question of a possible formulation of a
massive gravity theory Pauli:1939xp . There were some further historical
attempts, such as Boulware:1973my ; VanNieuwenhuizen:1973fi , but not until
2010 a consistent theory of massive gravity was formulated which avoids the
previously encountered problems of ghost instabilities deRham:2010kj . Shortly
after Hassan and Rosen wrote down a consistent gravitational theory for a
massless and a massive spin-2 field Hassan:2011zd . This theory was named
“bimetric gravity” but is sometimes referred as “bigravity” or “bimetric
theory”. For a review about bimetric gravity, see Schmidt-May:2015vnx ;
Heisenberg:2021dqm . For work on the consistency of bigravity theories, see
Hassan:2013pca ; Hinterbichler:2012cn ; Li:2015iwc ; Tukhashvili:2017eqc ;
Hassan:2014gta ; Hassan:2017ugh ; Kocic:2018yvr ; Kocic:2019ahm ;
Kocic:2019axa ; Schmidt-May:2016hsx ; Hassan:2012rq ; Hassan:2012gz ;
Hassan:2012wr ; Hassan:2012qv ; Hassan:2018mbl ; Molaee:2018brt ;
Higuchi:1986py ; Hassan:2018mcw ; Soloviev:2022qlh . Bimetric gravity has the
potential of describing dark matter Babichev:2016bxi ; Manita:2022tkl  and
viable cosmological solutions Akrami:2012vf ; Akrami:2013pna ; Bassi:2023ymf ;
Hogas:2019ywm ; Hogas:2021saw ; Hogas:2022owf ; Solomon:2014dua ;
Luben:2018ekw ; Luben:2019yyx . The aim of this article is to investigate if
we can write down viable teleparallel bigravity theories, that is, to
formulate a self-consistent theory of two dynamical tetrads using a flat,
metric compatible, but torsionful connection.
A promising feature coming from the structure of teleparallel bigravity is the
possibility to construct terms with derivative interactions which are much
more symmetric than those constructed in curvature based theories of bigravity
(see for example Li:2015iwc ; Li:2015izu ; Tukhashvili:2017eqc ). Derivative
interaction has gained interest in bigravity since they are related to
theories with conformal symmetries Apolo:2016ort ; Apolo:2016vkn . These
theories exhibit a symmetry which manifests with the existence of a partially
massless spin-2 field instead of the massive spin-2 field. There is, however,
a no-go theorem222This is rather a conjecture, since the argument only holds
up to cubic order, whereas the interacting theory might only start at quartic
order, as the authors stress in their conclusions. forbidding self-
interactions for a single partially massless spin-2 field Garcia-Saenz:2015mqi
. This no-go theorem is here avoided, by the fact that this is a theory of
several interacting fields as in Boulanger:2019zic .
We provide a new teleparallel equivalent formulation for Hassan-Rosen
bigravity (TEHR) where curvature is replaced by torsion analogously to the
teleparallel formulation of general relativity. This provides a theory of a
massless and a massive spin-2 field, which, in addition, more evidently
exhibits a Yang-Mills-like gauge structure. These properties make the theory
more reminiscent of the standard model of particle physics compared to the
formulation of general relativity the curvature formulation of the theory of
general relativity (see though footnote 1 in this paper and the first
paragraph in the paper Koivisto:2019ejt ).
We show that the Hamiltonian of TEHR has the same structure as in
Hinterbichler:2012cn  so we expect the number of degrees of freedom to be
consistent. Furthermore, the linear analysis around a Minkowski background
describes massless and massive spin-2 fields just as in Hassan-Rosen bimetric
gravity. This formulation will work as a conceptually different view of
Hassan-Rosen bimetric gravity, as well as a new base for possible extensions
of the theory. Furthermore, we present a very general class of novel theories
within the TEHR framework and look at the linear level of some interesting
cases and use the Dirac-Bergmann algorithm to express the Hamiltonians for
some examples.
In section II, we describe the theories of bigravity and teleparallel gravity
to finally introduce our novel theory of teleparallel bigravity. In section
III we make the linear analysis for a large subclass of teleparallel
bigravity, and in section IV we derive the primary constraints for these
theories and the Hamiltonian for two specific examples.
We adopt to the following conventions: metrics follow the
$\eta_{AB}=\mathrm{diag}(-1,1,1,1)$ sign convention. Greek indices denote
global coordinate indices, capital Latin indices represent Lorentz indices,
and small Latin indices denote spatial coordinate indices. We use the notation
for symmetrizing indices of a tensor by $2A_{(\mu\nu)}:=A_{\mu\nu}+A_{\nu\mu}$
and, similarly, for antisymmetrizing indices of a tensor we have
$2A_{[\mu\nu]}:=A_{\mu\nu}-A_{\nu\mu}$.

##  II Teleparallel bigravity

In this section we formulate a novel class of theories of gravity which we
choose to call teleparallel bigravity. The basic idea of this theory is quite
simple. By taking the Hassan-Rosen bimetric gravity theory as our starting
point, which is based on a formulation with a metric compatible and torsion-
free connection, one can view this action in the teleparallel framework. In
this formulation we are able to construct novel theories of gravity, as well
as recovering equivalent theories, such as the teleparallel equivalent to
Hassan-Rosen bimetric gravity. In order to reach a construction of
teleparallel bigravity we start by reviewing Hassan-Rosen bimetric gravity in
subsection II.1. Then, in order to clarify the distinction between the idea of
teleparallel bigravity and other novel theories with names which suggest very
similar ideas, we devote section II.2 to shortly review these theories. In
section II.3 we give a short review for teleparallel equivalent to general
relativity, and its extension to new general relativity is reviewed in section
II.4. Section II.5 is devoted to the introduction of teleparallel bigravity.

###  II.1 Hassan-Rosen bimetric gravity

Bimetric gravity (or bigravity, bimetric relativity, or bimetric theory) is an
extension to general relativity where one metric is added to the theory and is
given dynamics. It is a theory of two interacting spin-2 fields. One of the
fields is massless (as in general relativity) and the other is massive. In
Hassan:2011zd  the unique ghost-free bimetric gravity theory with the kinetic
terms being two copies of the Einstein-Hilbert action is presented. We refer
to this theory as Hassan-Rosen bimetric gravity, and its corresponding action
is

$\displaystyle\begin{split}S_{\mathrm{HR}}=&m^{2}_{e}\int\mathrm{d}^{4}x\sqrt{-\accentset{\mathfrak{e}}{g}}R\left(\accentset{\mathfrak{e}}{g}\right)+m^{2}_{f}\int\mathrm{d}^{4}x\sqrt{-\accentset{\mathfrak{f}}{g}}R\left(\accentset{\mathfrak{f}}{g}\right)-2m^{4}\int\mathrm{d}^{4}x\sqrt{-\accentset{\mathfrak{e}}{g}}\sum_{n=0}^{4}\beta_{n}e_{n}\left(\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}\right),\end{split}$
(1)

where, $\accentset{\mathfrak{e}}{g}$ and $\accentset{\mathfrak{f}}{g}$ are two
dynamical metrics (with corresponding tetrads $\mathfrak{e},\mathfrak{f}$ to
be defined later), and $\beta_{n}$ are arbitrary parameters in front of the
elementary symmetric polynomials of the eigenvalues $\lambda_{i}$: of the
matrix $\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}$,
given by

$\displaystyle
e_{0}\left(\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}\right)=1,$
(2) $\displaystyle
e_{1}\left(\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}\right)=\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4},$
(3) $\displaystyle
e_{2}\left(\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}\right)=\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3}+\lambda_{2}\lambda_{4}+\lambda_{3}\lambda_{4},$
(4) $\displaystyle
e_{3}\left(\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}\right)=\lambda_{1}\lambda_{2}\lambda_{3}+\lambda_{1}\lambda_{2}\lambda_{4}+\lambda_{1}\lambda_{3}\lambda_{4}+\lambda_{2}\lambda_{3}\lambda_{4},$
(5) $\displaystyle
e_{4}\left(\sqrt{\accentset{\mathfrak{e}}{g}^{-1}\accentset{\mathfrak{f}}{g}}\right)=\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}.$
(6)

As in general relativity

$\displaystyle\sqrt{-\accentset{\mathfrak{e}}{g}}:=\sqrt{-\mathrm{det}(\accentset{\mathfrak{e}}{g})},$
(7)

and $R(\accentset{\mathfrak{e}}{g})$ is the Ricci scalar of the Levi-Civita
connection, which essentially consists of up to second order derivatives of
the metric $\accentset{\mathfrak{e}}{g}_{\mu\nu}$. The parameters $m,\
m_{\mathfrak{e}},\ m_{\mathfrak{f}}$ have a dimension of mass and are needed
to make the action dimensionless.
Hassan-Rosen bimetric gravity can also be formulated in the vielbein (or
tetrad) formalism Hinterbichler:2012cn . In this formalism the action becomes:

$\displaystyle\begin{split}S_{\mathrm{HR}}=&m^{2}_{\mathfrak{e}}\int\mathrm{d}x^{4}\epsilon_{ABCD}E^{A}\wedge
E^{B}\wedge
R^{CD}(E)+m^{2}_{\mathfrak{f}}\int\mathrm{d}^{4}x\epsilon_{ABCD}F^{A}\wedge
F^{B}\wedge R^{CD}(F)-\\\
&-2m^{4}\int\epsilon_{ABCD}\left[\beta_{0}E^{A}\wedge E^{B}\wedge E^{C}\wedge
E^{D}+\beta_{1}E^{A}\wedge E^{B}\wedge E^{C}\wedge F^{D}+\right.\\\
&\left.+\beta_{2}E^{A}\wedge E^{B}\wedge F^{C}\wedge
F^{D}+\beta_{3}E^{A}\wedge F^{B}\wedge F^{C}\wedge F^{D}+\beta_{4}F^{A}\wedge
F^{B}\wedge F^{C}\wedge F^{D}\right],\end{split}$ (8)

where $R^{CD}(E)$ and $R^{CD}(F)$ are the curvature 2-forms of the Levi-Civita
connection constructed from the cotetrads $E^{A}$ and $F^{A}$ respectively,
which relates to the metrics $\accentset{\mathfrak{e}}{g}$ and
$\accentset{\mathfrak{f}}{g}$ in the following way:

$\displaystyle E^{A}=\mathfrak{e}^{A}{}_{\mu}\mathrm{d}x^{\mu},$ (9)
$\displaystyle F^{A}=\mathfrak{f}^{A}{}_{\mu}\mathrm{d}x^{\mu},$ (10)
$\displaystyle\accentset{\mathfrak{e}}{g}_{\mu\nu}=\mathfrak{e}^{A}{}_{\mu}\eta_{AB}\mathfrak{e}^{B}{}_{\nu},$
(11)
$\displaystyle\accentset{\mathfrak{f}}{g}_{\mu\nu}=\mathfrak{f}^{A}{}_{\mu}\eta_{AB}\mathfrak{f}^{B}{}_{\nu}.$
(12)

We also define the tetrads
$\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\mu}$, and
$\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{A}{}^{\mu}$, dual to the
cotetrads $\mathfrak{e}^{A}{}_{\mu}$, and $\mathfrak{f}^{A}{}_{\mu}$
respectively. For objects which are not metrics, we may suppress metrics and
place indices in the wrong position in order to make expressions look more
compact. For instance, if we have a cotetrad $\mathfrak{e}^{A}{}_{\mu}$, we
may, instead of
$\mathfrak{e}^{B}{}_{\nu}\eta_{AB}\accentset{\mathfrak{e}}{g}^{\mu\nu}$, write
$\mathfrak{e}_{A}{}^{\mu}$. This takes the correct index positions as the
tetrad, but it should not be confused with the tetrad which we, in this case,
denote by $\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\mu}$. Note
that the action is written slightly differently from Hinterbichler:2012cn  and
rather as in Li:2015iwc . This will then only be equivalent to (1) up to a
reparametrisation of the coefficients $\beta_{n}$.

###  II.2 Other bigravity theories

Our aim is to look at bigravity theories in the teleparallel framework, which
assumes vanishing curvature. This includes novel theories of gravity.
Recently, other novel bigravity theories have been proposed in articles where
either the title is reminiscent to the concept of teleparallel gravity, or the
structure of the theory is similar. Before going into the theory of
teleparallel gravity, we highlight the main differences between these
bigravity theories and teleparallel bigravity, in order to avoid any
confusion. In Damour:2019oru ; Nikiforova:2020fbz ; Nikiforova:2022pqy  a
novel theory named torsion bigravity has been considered. Here, a theory of
both curvature and torsion is formulated, where a massless spin-2 field is
mediated by the curvature, whereas the massive spin-2 field is mediated by the
torsion, which avoids the notion of two metrics (or tetrads) in the
construction of a bigravity theory.
In Markou:2018mhq  a bigravity theory is constructed with vierbeins as the
fundamental field in order to yield an antisymmetric part of the field
equations. It was motivated by the fact that antisymmetric field equations
played a big role in the quantization of string theory, which suggests that it
would make sense to include antisymmetric parts in a gravity theory. Their
construction differs from ours by being curvature-based. However, we cannot
exclude cases where the field equations might be equivalent. This theory was
extended from the single vierbein massive gravity theory proposed in
Markou:2018wxo . In Alexandrov:2019dxm  the structure of the action seems to
yield a physics similar to that of our theory, and even closer to that of
Li:2015iwc . The crucial difference with respect to our approach is that our
construction uses more than one connection. In teleparallel bigravity the
action contains two connections (corresponding to the two tetrad fields), but
only one spin connection. The construction of a bigravity theory with only one
connection was found to yield one graviton, instead of two as in other
bigravity theories, making this scheme significantly different from
teleparllel bigravityAlexandrov:2019dxm .

###  II.3 Teleparallel equivalent to general relativity

General relativity is formulated with the Ricci scalar which depends on the
spin connection, which is taken to be torsion-free. However, an equivalent
formulation can be made by setting the spin connection to be curvature-free
and replacing curvature with torsion. The resulting action reads:

$S_{\mathrm{TEGR}}=\int\mathrm{d}^{4}x\mathcal{L}_{\mathrm{TEGR}}=-m_{e}^{2}\int\mathrm{d}^{4}x\mathfrak{e}\accentset{\mathfrak{e}}{T}^{\rho\mu\nu}\left(\frac{1}{4}\accentset{\mathfrak{e}}{T}_{\rho\mu\nu}+\frac{1}{2}\accentset{\mathfrak{e}}{T}_{\nu\mu\rho}-\accentset{\mathfrak{e}}{g}_{\rho\nu}\accentset{\mathfrak{e}}{T}_{\mu}\right),$
(13)

where $\mathfrak{e}=\det(\mathfrak{e}^{A}{}_{\mu})$,
$\accentset{\mathfrak{e}}{T}_{\mu}=\accentset{\mathfrak{e}}{T}^{\nu}{}_{\mu\nu}$,
$\accentset{\mathfrak{e}}{T}^{\alpha}{}_{\mu\nu}=\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\accentset{\mathfrak{e}}{T}^{A}{}_{\mu\nu}$
and $\accentset{\mathfrak{e}}{T}^{A}{}_{\mu\nu}$ are the coefficients of the
two-form

$\displaystyle\mathbb{\accentset{\mathfrak{e}}{T}}^{A}=\frac{1}{2}\accentset{\mathfrak{e}}{T}^{A}{}_{\mu\nu}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu};\quad\accentset{\mathfrak{e}}{T}^{A}{}_{\mu\nu}=2\partial_{[\mu}\mathfrak{e}^{A}{}_{\nu]}=\partial_{\mu}\mathfrak{e}^{A}{}_{\nu}-\partial_{\nu}\mathfrak{e}^{A}{}_{\mu},$
(14)

and where, for our purposes, we chose to set a priori the so-called spin
connection to zero. In particular, it is known that this works for the
teleparallel equivalent of general relativity and its most popular
modifications Blixt:2022rpl ; Blixt:2018znp ; Golovnev:2021omn ;
Golovnev:2023yla .
This action is equivalent to the Einstein-Hilbert action modulo a boundary
term AP ; Maluf:2013gaa  (which does not appear in this theory). This can be
seen by realizing that the Ricci scalar of the teleparallel connection which
vanishes due to the flatness condition can be written as

$\displaystyle
0\equiv\accentset{\mathfrak{e}}{\mathcal{R}}=\accentset{\mathfrak{e}}{R}+\accentset{\mathfrak{e}}{T}-\frac{2}{\mathfrak{e}}\partial_{\mu}\left(\mathfrak{e}\accentset{\mathfrak{e}}{T}^{\mu}\right),$
(15)

where $\accentset{\mathfrak{e}}{R}$ denotes the curvature of the torsion-free
Levi-Civita connection and
$\mathcal{L}_{\mathrm{TEGR}}=-m_{g}^{2}\accentset{\mathfrak{e}}{T}$. One
realize immediately from this relation that

$\displaystyle
S_{\mathrm{EH}}:=m_{e}^{2}\int\mathrm{d}^{4}x\mathfrak{e}\accentset{\mathfrak{e}}{R}=\int\mathrm{d}^{4}x\mathcal{L}_{\mathrm{TEGR}}+2m_{e}^{2}\int\mathrm{d}^{4}x\partial_{\mu}\left(\mathfrak{e}\accentset{\mathfrak{e}}{T}^{\mu}\right)=S_{\mathrm{TEGR}}+B.T.,$
(16)

where $B.T.$ denotes the apparent boundary term. From this equation we
identify the curvature of the Levi-Civita connection and realize that in
teleparallel gravity it is equivalent to (13) up to a boundary term.

###  II.4 New general relativity

Considering (13) it is apparent that one easily can formulate more theories
quadratic in the torsion than the special case equivalent to general
relativity. By relaxing the coefficients in (13) to be arbitrary constants, we
get a generalized theory of gravity. It appears that any other term quadratic
in the torsion tensor and without any derivatives acting on the torsion would
violate parity PhysRevD.19.3524 . Our starting point is then

$S_{\mathrm{NGR}}=\int\mathrm{d}^{4}x\mathcal{L}_{\mathrm{NGR}}=-m_{e}^{2}\int\mathrm{d}^{4}x\mathfrak{e}\accentset{\mathfrak{e}}{T}^{\rho\mu\nu}\left(\accentset{\mathfrak{e}}{C}_{1}\accentset{\mathfrak{e}}{T}_{\rho\mu\nu}+\accentset{\mathfrak{e}}{C}_{2}\accentset{\mathfrak{e}}{T}_{\nu\mu\rho}+\accentset{\mathfrak{e}}{C}_{3}g_{\rho\nu}\accentset{\mathfrak{e}}{T}_{\mu}\right).$
(17)

This class of theories have extensively been studied in Blixt:2018znp ;
Blagojevic:2000qs ; Okolow:2011np ; PhysRevD.19.3524 ; Koivisto:2018loq ;
Blixt:2019ene ; BeltranJimenez:2019nns ; Hohmann:2019sys ; Hohmann:2018jso ;
Ualikhanova:2019ygl ; Cheng:1988zg . Sometimes this theory is named new
general relativity, however, the original name was reserved for only a
restricted class by forcing
$2\accentset{\mathfrak{e}}{C}_{1}+\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}=0$
PhysRevD.19.3524 . It is known that if one does not force this restriction the
parameterized post Newtonian parameters will deviate from those of general
relativity PhysRevD.19.3524 ; Ualikhanova:2019ygl  and ghost instabilities are
bound to arise BeltranJimenez:2019nns ; Koivisto:2018loq ; Kuhfuss:1986rb ;
Ortin:2015hya . The ghost instabilities will not get removed in the extension
to teleparallel bigravity. Since our goal is to make some prior investigation
for what novel teleparallel bigravity theories can be viable, this restriction
makes sense. However, to canonically write the action for this restricted case
one should add a Lagrange multiplier corresponding to the primary constraints
imposed by this restriction to (17)333And as well one should not put the spin
connection to vanish beforehand, however, this is shown in appendix A to not
affect the counting of the number of degrees of freedom.. Despite this naming,
it makes sense to not fix the coefficients $\accentset{\mathfrak{e}}{C}_{1},\
\accentset{\mathfrak{e}}{C}_{2}$, and $\accentset{\mathfrak{e}}{C}_{3}$
beforehand since we can then realize how the choice of parameters has to be
chosen to remove ghost-instabilities.

###  II.5 Teleparallel bigravity

Hassan-Rosen bimetric gravity is given by the action of either (1) or (8). It
has two Einstein-Hilbert terms, one for each metric, and an interaction
potential. Many equations are similar to general relativity, with the addition
of an analogous equation depending on the second metric.
In this section we will consider a new framework of gravity theories which we
call “teleparallel bigravity”. We write down a general action quadratic in the
torsion and that does not violate parity. Then we count degrees of freedom for
the special case of “teleparallel equivalent to Hassan-Rosen bigravity”,
showing that the theory is a ghost-free theory of massless and massive spin
two fields.

####  II.5.1 Novel terms with derivative interactions

Bigravity theories with derivative interactions have been considered in
Apolo:2016ort ; Tukhashvili:2017eqc ; Li:2015iwc . The naïve actions provided
in Li:2015iwc ; Li:2015izu ; deRham:2015rxa  generically have ghost
instabilities due to a relative sign and the full Hamiltonian analysis of
Apolo:2016ort  have not been carried out. The teleparallel gravity framework,
however, gives much more freedom to create bigravity theories with derivative
interactions. In this section we explore if examples that are potentially
ghost-free can be created. The most general action which is quadratic in
torsion in this framework reads

$\displaystyle
S_{\mathrm{TBG}}=-\int\mathrm{d}^{4}x\left[m_{e}^{2}\accentset{\mathfrak{e}}{T}^{A}{}_{\mu\nu}\accentset{\mathfrak{e}}{T}^{B}{}_{\rho\sigma}\overset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}+m_{ef}^{2}\accentset{\mathfrak{e}}{T}^{A}{}_{\mu\nu}\accentset{\mathfrak{f}}{T}^{B}{}_{\rho\sigma}\overset{kin}{G}{}_{AB}{}^{\mu\nu\rho\sigma}+m_{f}^{2}\accentset{\mathfrak{f}}{T}^{A}{}_{\mu\nu}\accentset{\mathfrak{f}}{T}^{B}{}_{\rho\sigma}\overset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}\right]+V(E,F),$
(18)

where $V(E,F)$ is an interaction potential, and
$\overset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma},\
\overset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma},$ and
$\overset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}$ are some combinations of
tetrads and metrics not containing any derivatives. Analogously to new general
relativity, we introduced

$\displaystyle\mathbb{\accentset{\mathfrak{f}}{T}}^{A}=\frac{1}{2}\accentset{\mathfrak{f}}{T}^{A}{}_{\mu\nu}\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu};\quad\accentset{\mathfrak{f}}{T}^{A}{}_{\mu\nu}=2\partial_{[\mu}\mathfrak{f}^{A}{}_{\nu]}=\partial_{\mu}\mathfrak{f}^{A}{}_{\nu}-\partial_{\nu}\mathfrak{f}^{A}{}_{\mu},$
(19)

where we again set the spin connection to zero, which we have shown to not
affect the number of degrees of freedom in appendix A. Below, we display a
figure for teleperallel bigravity and its limits.

Teleparallel Bigravity New General Relativity Bimetric Gravity Other Bigravity
Theories Bimetric Gravity Viable Novel Theories? Teleparallel equivalent to
general relativity General Relativity Add a tensor fieldTeleparallel framework
Generalized terms
---
quadratic in torsion
Add a second metric$\mathbf{\sim}$

The figure should be understood in the following way. In the lower part in
green color we display two equivalent formulations of general relativity. That
is, the standard formulation with a metric compatible and torsion free
connection and the teleparallel equivalent to general relativity. With the
standard formulation of general relativity, it is known from the literature
that one can add a second metric and make it dynamical to create a bimetric
gravity theory (following the red arrow in the figure). We note that the
concept bigravity can be used for more theories than just the standard Hassan-
Rosen bimetric gravity, with some of them briefly discussed in section II.2.
From bimetric gravity, one can first look into the tetrad formulation of the
theory, and then take inspiration from the known teleparallel gravity theories
to construct a teleparallel bigravity theory by changing to the teleparallel
framework (blue arrow).
Another logical path for constructing teleparallel bigravity is to start with
the teleparallel equivalent to general relativity. By just looking at this
action, one can realize that it can be generalized to new general relativity.
With new general relativity we can then add a second tetrad which we make
dynamical in a very similar way to what was done in the construction of
bimetric gravity. This as well leads to the formulation of teleparallel
bigravity. Teleparallel bigravity, hence, have the limits of Hassan-Rosen
bimetric gravity, new general relativity, and general relativity. In
teleparallel bigravity there are novel theories of gravity which may, or may
not be viable.

####  II.5.2 Teleparallel equivalent to Hassan-Rosen bigravity

We will work in the tetrad formalism since this has been useful in both
teleparallel and bimetric gravity. We make an ansatz that we simply replace
the Einstein-Hilbert terms in (8) with two copies of (13). We will in section
III show that it reproduces the linear equations of Hassan-Rosen bimetric
gravity, and in section II.5.3 that the counting gives the expected degrees of
freedom. The action then is:

$\displaystyle\begin{split}S_{\mathrm{TEHR}}&=-m^{2}_{e}\int\mathrm{d}^{4}x\mathfrak{e}\accentset{\mathfrak{e}}{T}-m^{2}_{f}\int\mathrm{d}^{4}x\mathfrak{f}\accentset{\mathfrak{f}}{T}-\\\
&-2m^{4}\int\epsilon_{ABCD}\left[\beta_{0}E^{A}\wedge E^{B}\wedge E^{C}\wedge
E^{D}+\beta_{1}E^{A}\wedge E^{B}\wedge E^{C}\wedge F^{D}+\right.\\\
&\left.+\beta_{2}E^{A}\wedge E^{B}\wedge F^{C}\wedge
F^{D}+\beta_{3}E^{A}\wedge F^{B}\wedge F^{C}\wedge F^{D}+\beta_{4}F^{A}\wedge
F^{B}\wedge F^{C}\wedge F^{D}\right].\end{split}$ (20)

Here $\mathfrak{f}=\det(\mathfrak{f}^{A}{}_{\mu})$, and, $E^{A}$, $F^{B}$,
$\accentset{\mathfrak{e}}{g}$, and $\accentset{\mathfrak{f}}{g}$ are defined
in (9), (10), (11), (12) respectively, $\accentset{\mathfrak{e}}{T}$ is
defined in equation (13) and defines $\mathcal{L}_{\mathrm{TEGR}}$ together
with the volume element $\mathfrak{e}$. This is formulated in a gauge where
the spin connection vanishes (which is not the canonical formulation). We show
in appendix A that the counting of degrees of freedom at the nonlinear level
(using Hamiltonian analysis) is independent of our gauge choice. Since this
greatly simplifies the Hamiltonian analysis, we choose to set the spin
connection to zero.
It is well known that the action given by (13) is equivalent to the Einstein-
Hilbert action up to a surface term given by (16). Hence, the kinetic part of
(20) is equivalent to the kinetic part of Hassan-Rosen bimetric gravity up to
a couple of surface terms.
To get a true equivalence we need to make sure that the theory contains the
same number of degrees of freedom as in Hassan-Rosen bimetric gravity. We
should also investigate whether the consistency-tests performed in Hassan-
Rosen bimetric gravity still holds in this new formulation of the theory.

####  II.5.3 Counting degrees of freedom

In Maluf:1994ji ; Blagojevic:2000qs ; Nester:2017wau  it is shown that the
degrees of freedom count for (13) coincides with that of general relativity.
Hence the teleparallel formulation of general relativity is a ghost free
theory for a spin-2 field as in general relativity.
We now describe how degrees of freedom are counted in Hassan-Rosen bimetric
gravity, since we are looking at an extension to this theory. Hassan-Rosen
bimetric gravity is formulated in terms of two dynamical metrics. Note that
first class constraints remove 2 degrees of freedom each in phase-space,
whereas second class constraints remove only 1. Starting with 20 degrees of
freedom for each metric (including conjugate momenta). 8 degrees of freedom
can be removed from first class constraints associated with a common
diffeomorphism invariance for the two metrics. 4 additional degrees of freedom
can be removed from gauge fixing one set of lapse and shift. 4 more are
removed by Hamiltonian and momentum constraints, normally realized from the
field equations of analogous to (25). One set of lapse and shift appears as
Lagrange multipliers which gives us 4 first class primary constraints which
removes 8 additional degrees of freedom. Finally, 2 more degrees of freedom
are removed by an additional second class constraint with an associated
secondary second class constraint. This leaves us with $40-26=14=(2+5)\times
2$ degrees of freedom corresponding to a massless and massive spin-2 field and
its corresponding conjugate momenta. Note that the 2 last second class
constraints follow from the specific form of the interaction potential
Hassan:2011zd ; Hassan:2018mbl  and removes the pathological Boulware-Deser
ghost444Note that in the absence of the interaction potential, there are no
connection between the kinetic terms and there will be two sets of
diffeomorphism invariances and this will bring us down to two massless spin-2
fields.. We should, hence, carefully check whether this constraint still
remains in our new theory.
In order to make the Hamiltonian analysis, we need the conjugate momenta,
which comes from the fields which have time-derivatives. Hence, we need to
distinguish between space and time and make a so-called 3+1
decomposition.555Note that a valid simultaneous 3+1 decomposition of the two
metrics must be done such that they both share a common spacelike hypersurface
Kocic:2018ddp ; Kocic:2018yvr ; Kocic:2019ahm . To avoid cumbersome repetition
only the objects associated with one metric ($\accentset{\mathfrak{e}}{g}$)
will be displayed, and the analogous parts follows. In the case of TEGR there
are 6 additional primary constraints compared the metric GR and they have been
shown to be of first class, thus removing the dynamics of all the extra
components introduced when treating tetrads as canonical fields instead of the
metric. These are only slightly different from the usual tetrad formulation of
bigravity Hinterbichler:2012cn  due to the presence of torsion terms.

By making a 3+1 decomposition of spacetime in ADM variables we can express the
Hamiltonian for teleparallel equivalent to general relativity Maluf:1994ji ;
Blixt:2018znp ; Blagojevic:2000qs ; Nester:2017wau . Following the notation of
Blixt:2018znp , but adding an overset label $\mathfrak{e}$ which will be
important when extending the result to the teleparallel equivalent to Hassan-
Rosen bigravity. The ADM-variables used are $\accentset{\mathfrak{e}}{N}$
(lapse), $\accentset{\mathfrak{e}}{N}^{i}$ (shift), spatial tetrad
$\mathfrak{e}^{A}{}_{i}$ and its conjugate momenta
$\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}$, induced metric
$\accentset{\mathfrak{e}}{\gamma}_{ij}$ and its Levi-Civita covariant
derivative $D_{i}$ and spatial volume element
$\sqrt{\accentset{\mathfrak{e}}{\gamma}}$, a normal vector to constant
hypersurfaces
$\accentset{\mathfrak{e}}{\xi}^{A}=-\frac{1}{6}\epsilon^{A}{}_{BCD}\mathfrak{e}^{B}{}_{i}\mathfrak{e}^{C}{}_{j}\mathfrak{e}^{D}{}_{k}\epsilon^{ijk}$,
and the resulting Hamiltonian reads

$\displaystyle\begin{split}\mathcal{H}&=\accentset{\mathfrak{e}}{N}\sqrt{\accentset{\mathfrak{e}}{\gamma}}\Bigg{[}\frac{1}{8m^{2}_{e}}\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\frac{\accentset{\mathfrak{e}}{\pi}_{B}{}^{l}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\accentset{\mathfrak{e}}{\gamma}^{jk}\accentset{\mathfrak{e}}{\gamma}_{i(l}\mathfrak{e}^{A}{}_{k)}\mathfrak{e}^{B}{}_{j}-\frac{1}{16m^{2}_{e}}\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\frac{\accentset{\mathfrak{e}}{\pi}_{B}{}^{j}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\mathfrak{e}^{A}{}_{i}\mathfrak{e}^{B}{}_{j}-\accentset{\mathfrak{e}}{\xi}^{A}D_{i}\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}-m^{2}_{e}\cdot{}^{3}\overset{\mathfrak{e}}{\mathbb{T}}\Bigg{]}\\\
&+\accentset{\mathfrak{e}}{N}^{k}\sqrt{\accentset{\mathfrak{e}}{\gamma}}\Big{[}\frac{\accentset{\mathfrak{e}}{\pi}_{B}{}^{j}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\accentset{\mathfrak{e}}{T}^{B}{}_{jk}-\mathfrak{e}^{A}{}_{k}\overset{\mathfrak{e}}{D}_{i}\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\Big{]}+\sqrt{\accentset{\mathfrak{e}}{\gamma}}\overset{\mathfrak{e}}{D}_{i}\left[\left(\accentset{\mathfrak{e}}{N}\accentset{\mathfrak{e}}{\xi}^{A}+\accentset{\mathfrak{e}}{N}^{j}\mathfrak{e}^{A}{}_{j}\right)\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\right]+\lambda^{ij}\accentset{\mathfrak{e}}{\mathcal{P}}_{ij}\\\
&\equiv\accentset{\mathfrak{e}}{N}\accentset{\mathfrak{e}}{\mathfrak{C}}+\accentset{\mathfrak{e}}{N}^{i}\accentset{\mathfrak{e}}{\mathfrak{C}}_{i}+\lambda^{ij}\accentset{\mathfrak{e}}{\mathcal{P}}_{ij},\end{split}$
(21)

where $\equiv$ signifies that a boundary term have been dropped as discussed
in Pati:2022nwi  and

$\displaystyle\accentset{\mathfrak{e}}{\mathcal{P}}_{ij}:=\begin{bmatrix}{}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{1}&{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{12}&{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{13}\\\
{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{21}&{}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{2}&{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{23}\\\
{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{31}&{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{32}&{}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{3}\end{bmatrix},$
(22)

with the 6 primary first class constraints

$\displaystyle{}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathfrak{C}}^{i}=-\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\accentset{\mathfrak{e}}{\xi}^{A}-2m^{2}_{e}\accentset{\mathfrak{e}}{T}^{B}{}_{jk}\accentset{\mathfrak{e}}{\gamma}^{ij}\mathfrak{e}_{B}{}^{k}\approx
0,$ (23)

and

$\displaystyle{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{mp}=\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}\accentset{\mathfrak{e}}{\gamma}_{i[p}\mathfrak{e}^{A}{}_{m]}+m^{2}_{e}\accentset{\mathfrak{e}}{T}^{B}{}_{pm}\accentset{\mathfrak{e}}{\xi}_{B}\approx
0,$ (24)

where $\mathcal{V}$ stands for “vector part” and $\mathcal{A}$ for
“antisymmetric” part of the irreducible decomposition under the rotation group
as in Blixt:2018znp ; Blagojevic:2000qs . Note that Hinterbichler:2012cn  did
not find the important
${}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathfrak{C}}^{i}$ constraints and
resorted instead to the field equations to demonstrate Lorentz invariance. The
constraint algebra satisfy the Lorentz algebra Blagojevic:2000qs ;
Ferraro:2016wht ; Maluf:2013gaa  which is essential for realizing they are of
first class.

From (20) it is straightforward to write down the Lagrangian as the back-
transformation of the primary Hamiltonian, for teleparallel equivalent to
Hassan-Rosen bigravity in a form like the one used in Hinterbichler:2012cn :

$\displaystyle
L_{TEHR}=\int\mathrm{d}x^{3}\left(\accentset{\mathfrak{e}}{\pi}\dot{\mathfrak{e}}+\accentset{\mathfrak{f}}{\pi}\dot{\mathfrak{f}}-\accentset{\mathfrak{e}}{N}\left(\accentset{\mathfrak{e}}{\mathfrak{C}}+{}^{\mathcal{M}}\accentset{\mathfrak{e}}{\mathfrak{C}}\right)-\accentset{\mathfrak{f}}{N}\left(\accentset{\mathfrak{f}}{\mathfrak{C}}+{}^{\mathcal{M}}\accentset{\mathfrak{f}}{\mathfrak{C}}\right)-\accentset{\mathfrak{e}}{N}^{i}\left(\accentset{\mathfrak{e}}{\mathfrak{C}}_{i}+{}^{\mathcal{M}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{i}\right)-\accentset{\mathfrak{f}}{N}^{i}\left(\accentset{\mathfrak{f}}{\mathfrak{C}}_{i}+{}^{\mathcal{M}}\accentset{\mathfrak{f}}{\mathfrak{C}}_{i}\right)-\lambda^{ij}\accentset{\mathfrak{e}}{\mathcal{P}}_{ij}-\tilde{\lambda}^{ij}\accentset{\mathfrak{f}}{\mathcal{P}}_{ij}\right),$
(25)

where
${}^{\mathcal{M}}\accentset{\mathfrak{e}}{\mathfrak{C}},{}^{\mathcal{M}}\accentset{\mathfrak{f}}{\mathfrak{C}},{}^{\mathcal{M}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{i},{}^{\mathcal{M}}\accentset{\mathfrak{f}}{\mathfrak{C}}_{i}$
comes, and are defined, from the interaction potential which is linear in
lapse and shifts due to the fact that they only appear (and are linear) in
$\mathfrak{e}^{A}{}_{0}$ and $\mathfrak{f}^{A}{}_{0}$ terms and the
antisymmetric property of the wedge product ensures that they appear exactly
one time in each term as discussed in Hinterbichler:2012cn . Now we have
reproduced the tetrad formulation of bigravity and we, hence, do not expect
any difference in the results of Hassan-Rosen bigravity. In section II.5.2 we
find that at the linear level we recover the linearized modes of Hassan-Rosen
bigravity as expected.

However, one should be cautious since in the special case of choosing the
kinetic action to be two copies of TEGR (and similarly by taking two copies of
the tetrad formulation of Einstein-Hilbert terms) the antisymmetric components
of the tetrads become non-dynamical Hassan:2018mcw . 666This is manifest in
section III where the antisymmetric modes disappear for the TEGR coefficients.
Variation with respect to the antisymmetric components generate the following
field equations Hinterbichler:2012cn ; Hassan:2018mcw  which turns out to be
constraints. Thus, the antisymmetric components are subject to the following
12 constraints:

$\displaystyle\frac{\delta
V(E,F)}{\delta\mathfrak{e}^{A}{}_{\mu}}\eta^{AB}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{\nu}-\frac{\delta
V(E,F)}{\delta\mathfrak{e}^{A}{}_{\nu}}\eta^{AB}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{\mu}=0,$
(26) $\displaystyle\frac{\delta
V(E,F)}{\delta\mathfrak{f}^{A}{}_{\mu}}\eta^{AB}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\nu}-\frac{\delta
V(E,F)}{\delta\mathfrak{f}^{A}{}_{\nu}}\eta^{AB}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\mu}=0.$
(27)

Note that this is special for TEHR (and for the tetrad HR formulation of HR
bigravity and multigravity extensions Hassan:2018mcw ) and in general
teleparallel bigravity theories the antisymmetric part of the tetrads become
dynamical, and the above constraints will be spoiled in the absence of
$\accentset{\mathfrak{e}}{\mathcal{P}}_{ij},\accentset{\mathfrak{f}}{\mathcal{P}}_{ij}$,
which constrains the antisymmetric kinetic part. Absence of primary
constraints for the the antisymmetric sector will reduce the number of
constraints, also in the case where at least one of
${}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathfrak{C}}^{i},{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathfrak{C}}_{mp}$,
or their corresponding $\mathfrak{f}$ constraints, is absent.
We will now turn to the linear analysis of general quadratic teleparallel
bigravity theories to confirm the results of Hassan-Rosen bigravity and new
general relativity, and further explore what novel theories might be viable.
In section IV we will do some preliminary investigation of these novel
theories.

####  II.5.4 Explicit examples worth considering

In this section we argue for what kind of choice of supermetrics provided in
teleparallel bigravity (18) are worth taking a deeper look into. We assume
that the interaction potential $V(E,F)$ is as in Hassan Rosen bigravity. That
is,

$\displaystyle\begin{split}V(E,F)=-2m^{4}\int\epsilon_{ABCD}\left[\beta_{0}E^{A}\wedge
E^{B}\wedge E^{C}\wedge E^{D}+\beta_{1}E^{A}\wedge E^{B}\wedge E^{C}\wedge
F^{D}+\right.\\\ \left.+\beta_{2}E^{A}\wedge E^{B}\wedge F^{C}\wedge
F^{D}+\beta_{3}E^{A}\wedge F^{B}\wedge F^{C}\wedge F^{D}+\beta_{4}F^{A}\wedge
F^{B}\wedge F^{C}\wedge F^{D}\right].\end{split}$ (28)

This interaction potential has a clear interpretation when expressed in
tetrads. Namely, the terms are the volume forms one can create by general
combinations of two tetrads in four dimensions. Furthermore, we have a special
case
$\accentset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{e})$
where the first kinetic term is that of new general relativity. We get another
copy of new general relativity by assuming
$\accentset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{f})$.
This is already interesting since these include novel theories for which we
already have some insights from the literature of new general relativity and
Hassan-Rosen bigravity. For the rest of the article we assume that the
interaction potential takes the form as in Hassan-Rosen bigravity, that
$\accentset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{e})$,
and
$\accentset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{f})$.
By including a non-zero $\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}$ we will
get novel terms which can be motivated from its conformal symmetry. This term
alone does, however, suffer from ghost-instabilities. However, if one later
would consider a similar theory with complex tetrads similar to the
constructions in Apolo:2016ort ; Apolo:2016vkn  the theory might turn out to
be viable. For the linearized level the explicit expressions of the non-
vanishing supermetrics does not play a role in the perturbations we define
around a Minkowski background. For the non-linear level we will assume that
$\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}(\text{\ae})$,
where we define the cotetrad $\text{\ae}^{A}{}_{\mu}$ to satisfy

$\displaystyle\text{\ae}^{A}{}_{\mu}\eta_{AB}\text{\ae}^{B}{}_{\nu}=\accentset{\text{\ae}}{g}_{\mu\nu},$
(29)

with its corresponding tetrad denoted by
$\rotatebox[origin={c}]{180.0}{\text{\ae}}_{A}{}^{\mu}$, and
$\accentset{\text{\ae}}{g}_{\mu\nu}$ being some new Lorentzian metric.

For the Hamiltonian analysis we derive the primary constraints for the theory
corresponding to this action. However, we restrict further analysis to the
special case
$\accentset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}=0$,
which should be viewed as a toy-model since it contains ghosts. The cotetrad
$\text{\ae}^{A}{}_{\mu}$ for a bigravity theory should depend on the two other
cotetrads $\mathfrak{e}^{A}{}_{\mu}$ and $\mathfrak{f}^{A}{}_{\mu}$, since the
theory describes only two dynamical tensor fields. However, the Hamiltonian
analysis can be made without much complications to the point where velocities
have been Legendre transformed to conjugate momenta. For this reason we leave
$\text{\ae}^{A}{}_{\mu}$ undetermined until this stage of the analysis. We
could already make some speculations of what $\text{\ae}^{A}{}_{\mu}$ could
be. In order to make the term with derivative interactions completely
symmetric in $\mathfrak{e}$ and $\mathfrak{f}$ there are in particular two
choices. First one is to assume
$\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{e})+\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{f})$.
The second choice is to take it to depend on the so-called geometric mean
defined via the relation

$\displaystyle\accentset{\text{\ae}}{g}_{\mu\nu}=\mathfrak{e}^{A}{}_{(\mu}\mathfrak{f}^{B}{}_{\nu)}\eta_{AB}=\text{\ae}^{A}{}_{\mu}\text{\ae}^{B}{}_{\nu}\eta_{AB},$
(30)

where $\accentset{\text{\ae}}{g}_{\mu\nu}$ is an intermediate metric between
$\accentset{\mathfrak{e}}{g}_{\mu\nu}$ and
$\accentset{\mathfrak{f}}{g}_{\mu\nu}$ Kocic:2018ddp . The explicit
expressions for the supermetrics are,

$\displaystyle\begin{split}\accentset{\mathfrak{e}}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{e})&=\accentset{\mathfrak{e}}{C}_{1}\eta_{AB}\eta^{CD}\eta^{EF}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{C}{}^{\rho}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{D}{}^{[\mu}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{F}{}^{\nu]}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{F}{}^{\sigma}-\accentset{\mathfrak{e}}{C}_{2}\eta^{CD}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{[\mu}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{C}{}^{\nu]}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{D}{}^{[\rho}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\sigma]}-\accentset{\mathfrak{e}}{C}_{3}\eta^{CD}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{[\mu}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{C}{}^{\nu]}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{D}{}^{[\rho}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{\sigma]}\end{split},$
(31)
$\displaystyle\begin{split}\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}(\text{\ae})&=\accentset{\textrm{\ae}}{C}_{1}\eta_{AB}\eta^{CD}\eta^{EF}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{C}{}^{\rho}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{D}{}^{[\mu}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{F}{}^{\nu]}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{F}{}^{\sigma}-\accentset{\textrm{\ae}}{C}_{2}\eta^{CD}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{B}{}^{[\mu}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{C}{}^{\nu]}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{D}{}^{[\rho}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{A}{}^{\sigma]}\\\
&-\accentset{\textrm{\ae}}{C}_{3}\eta^{CD}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{A}{}^{[\mu}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{C}{}^{\nu]}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{D}{}^{[\rho}\rotatebox[origin={c}]{180.0}{\text{\ae}}_{B}{}^{\sigma]}\end{split},$
(32)
$\displaystyle\begin{split}\accentset{\mathfrak{f}}{G}_{AB}{}^{\mu\nu\rho\sigma}(\mathfrak{f})&=\accentset{\mathfrak{f}}{C}_{1}\eta_{AB}\eta^{CD}\eta^{EF}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{C}{}^{\rho}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{D}{}^{[\mu}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{F}{}^{\nu]}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{F}{}^{\sigma}-\accentset{\mathfrak{f}}{C}_{2}\eta^{CD}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{[\mu}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{C}{}^{\nu]}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{D}{}^{[\rho}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{A}{}^{\sigma]}-\accentset{\mathfrak{f}}{C}_{3}\eta^{CD}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{A}{}^{[\mu}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{C}{}^{\nu]}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{D}{}^{[\rho}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\sigma]}\end{split},$
(33)

where we introduced, a priory, arbitrary coefficients
$\accentset{\mathfrak{f}}{C}_{1},\accentset{\mathfrak{f}}{C}_{2},\accentset{\mathfrak{f}}{C}_{3}$,
for the part which only depends on $\mathfrak{f}$ and
$\accentset{\textrm{\ae}}{C}_{1},\accentset{\textrm{\ae}}{C}_{2},\accentset{\textrm{\ae}}{C}_{3}$
for the terms with derivative interactions. For the linear analysis, we will
use a different supermetric

$\displaystyle\accentset{kin}{G}_{AB}{}^{\mu\nu\rho\sigma}=\accentset{\textrm{\ae}}{C}_{1}\accentset{\mathfrak{e}}{g}_{\alpha\delta}\,\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\accentset{\mathfrak{f}}{g}^{\lambda\gamma}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\delta}\,+\accentset{\textrm{\ae}}{C}_{2}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\delta}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\gamma}\,+\accentset{\textrm{\ae}}{C}_{3}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\gamma}\,,$
(34)

pointing out that its use in our perturbation theory still gives compatible
results since we only look at lower order expansions around a Minkowski
background, where the contractions are reduced to contractions with the
Minkowski metric.

##  III Linear analysis of teleparallel bigravity

In this section we examine teleparallel bigravity at the linear level, where
we retain only linear perturbations of the field equations or, equivalently,
second order perturbations of the Lagrangian. The results motivate a further
examination of these theories at the nonlinear level, which we carry out in
section IV.

###  III.1 The spectrum of teleparallel bigravity

The full action and Lagrangian density of the theory are specified as

$\displaystyle
S=\int\mathrm{d}^{4}x\,\left(\overset{\mathfrak{e}}{\mathcal{L}}+\overset{\mathfrak{f}}{\mathcal{L}}+\overset{kin}{\mathcal{L}}\right)-\overset{int}{\mathcal{L}}\,,$
(35)

where

$\displaystyle\overset{\mathfrak{e}}{\mathcal{L}}:$
$\displaystyle=-\mathfrak{e}\,m^{2}_{\mathfrak{e}}\Big{(}\accentset{\mathfrak{e}}{C}_{1}\accentset{\mathfrak{e}}{g}_{\alpha\delta}\,\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\accentset{\mathfrak{e}}{g}^{\lambda\gamma}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{\delta}\,\accentset{\mathfrak{e}}{T}^{A}{}_{\kappa\lambda}\,\accentset{\mathfrak{e}}{T}^{B}{}_{\beta\gamma}+\accentset{\mathfrak{e}}{C}_{2}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\delta}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{\gamma}\,\accentset{\mathfrak{e}}{T}^{A}{}_{\kappa\gamma}\,\accentset{\mathfrak{e}}{T}^{B}{}_{\beta\delta}+\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{B}{}^{\gamma}\,\accentset{\mathfrak{e}}{T}^{A}{}_{\kappa\alpha}\,\accentset{\mathfrak{e}}{T}^{B}{}_{\beta\gamma}\Big{)}$
(36) $\displaystyle\overset{\mathfrak{f}}{\mathcal{L}}:$
$\displaystyle=-\mathfrak{f}\,m^{2}_{\mathfrak{f}}\Big{(}\accentset{\mathfrak{f}}{C}_{1}\accentset{\mathfrak{f}}{g}_{\alpha\delta}\,\accentset{\mathfrak{f}}{g}^{\kappa\beta}\,\accentset{\mathfrak{f}}{g}^{\lambda\gamma}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\delta}\,\accentset{\mathfrak{f}}{T}^{A}{}_{\kappa\lambda}\,\accentset{\mathfrak{f}}{T}^{B}{}_{\beta\gamma}+\accentset{\mathfrak{f}}{C}_{2}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{A}{}^{\delta}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\gamma}\,\accentset{\mathfrak{f}}{T}^{A}{}_{\kappa\gamma}\,\accentset{\mathfrak{f}}{T}^{B}{}_{\beta\delta}+\accentset{\mathfrak{f}}{C}_{3}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\gamma}\,\accentset{\mathfrak{f}}{T}^{A}{}_{\kappa\alpha}\,\accentset{\mathfrak{f}}{T}^{B}{}_{\beta\gamma}\Big{)}$
(37) $\displaystyle\overset{kin}{\mathcal{L}}$
$\displaystyle:=-\mathfrak{e}\,m^{2}_{\mathfrak{e}\mathfrak{f}}\Big{(}\accentset{\textrm{\ae}}{C}_{1}\accentset{\mathfrak{e}}{g}_{\alpha\delta}\,\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\accentset{\mathfrak{f}}{g}^{\lambda\gamma}\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\delta}\,\accentset{\mathfrak{e}}{T}^{A}{}_{\kappa\lambda}\,\accentset{\mathfrak{f}}{T}^{B}{}_{\beta\gamma}+\accentset{\textrm{\ae}}{C}_{2}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\delta}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\gamma}\,\accentset{\mathfrak{e}}{T}^{A}{}_{\kappa\gamma}\,\accentset{\mathfrak{f}}{T}^{B}{}_{\beta\delta}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\mathfrak{e}}{g}^{\kappa\beta}\,\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}{}^{\alpha}\rotatebox[origin={c}]{180.0}{$\mathfrak{f}$}_{B}{}^{\gamma}\,\accentset{\mathfrak{e}}{T}^{A}{}_{\kappa\alpha}\,\accentset{\mathfrak{f}}{T}^{B}{}_{\beta\gamma}\Big{)}$
(38)
$\displaystyle\begin{split}\overset{int}{\mathcal{L}}&:=m^{4}_{int}\epsilon_{ABCD}\Big{(}\beta_{0}E^{A}\wedge
E^{B}\wedge E^{C}\wedge E^{D}+\beta_{1}E^{A}\wedge E^{B}\wedge E^{C}\wedge
F^{D}+\beta_{2}E^{A}\wedge E^{B}\wedge F^{C}\wedge F^{D}\\\
&+\beta_{3}E^{A}\wedge F^{B}\wedge F^{C}\wedge F^{D}+\beta_{4}F^{A}\wedge
F^{B}\wedge F^{C}\wedge F^{D}\Big{)}\end{split}$ (39)

The expressions used for the perturbations of the tetrad, cotetrad and volume
form are

$\displaystyle{\mathfrak{e}^{A}}_{\delta}$
$\displaystyle\to{\delta^{A}}_{\delta}+\frac{\epsilon}{2}\,\eta^{AB}{\delta_{B}}^{\alpha}\left(\accentset{\mathfrak{e}}{h}_{\alpha\delta}+\accentset{\mathfrak{e}}{b}_{\alpha\delta}\right)={\delta^{A}}_{\delta}+\epsilon\,\eta^{AB}{\delta_{B}}^{\alpha}\accentset{\mathfrak{e}}{B}_{\alpha\delta};\qquad\accentset{\mathfrak{e}}{B}_{\alpha\delta}:=\frac{\accentset{\mathfrak{e}}{h}_{\alpha\delta}+\accentset{\mathfrak{e}}{b}_{\alpha\delta}}{2}$
(40)
$\displaystyle{\rotatebox[origin={c}]{180.0}{$\mathfrak{e}$}_{A}}^{\gamma}$
$\displaystyle\to{\delta_{A}}^{\gamma}-\accentset{\mathfrak{e}}{\overline{g}}^{\gamma\beta}{\delta_{A}}^{\alpha}\left(\frac{\epsilon}{2}\left(\accentset{\mathfrak{e}}{h}_{\beta\alpha}+\accentset{\mathfrak{e}}{b}_{\beta\alpha}\right)+\frac{\epsilon^{2}}{4}\left(\accentset{\mathfrak{e}}{b}_{\alpha\nu}{\accentset{\mathfrak{e}}{b}_{\beta}}^{\nu}-\accentset{\mathfrak{e}}{h}_{\alpha\nu}{\accentset{\mathfrak{e}}{b}_{\beta}}^{\nu}+\accentset{\mathfrak{e}}{b}_{\alpha\nu}{\accentset{\mathfrak{e}}{h}_{\beta}}^{\nu}-\accentset{\mathfrak{e}}{h}_{\alpha\nu}{\accentset{\mathfrak{e}}{h}_{\beta}}^{\nu}\right)\right)=$
(41)
$\displaystyle\quad={\delta_{A}}^{\gamma}-\accentset{\mathfrak{e}}{\overline{g}}^{\gamma\beta}{\delta_{A}}^{\alpha}\left(\epsilon\,\frac{\accentset{\mathfrak{e}}{h}_{\beta\alpha}+\accentset{\mathfrak{e}}{b}_{\beta\alpha}}{2}+\epsilon^{2}\,\frac{-{\accentset{\mathfrak{e}}{b}_{\beta}}^{\nu}\accentset{\mathfrak{e}}{b}_{\nu\alpha}-{\accentset{\mathfrak{e}}{b}_{\beta}}^{\nu}\accentset{\mathfrak{e}}{h}_{\nu\alpha}-{\accentset{\mathfrak{e}}{h}_{\beta}}^{\nu}\accentset{\mathfrak{e}}{b}_{\nu\alpha}-{\accentset{\mathfrak{e}}{h}_{\beta}}^{\nu}\accentset{\mathfrak{e}}{h}_{\nu\alpha}}{4}\right)=$
$\displaystyle\quad={\delta_{A}}^{\gamma}+\accentset{\mathfrak{e}}{\overline{g}}^{\gamma\beta}{\delta_{A}}^{\alpha}\left(-\epsilon\,\accentset{\mathfrak{e}}{B}_{\beta\alpha}+\epsilon^{2}\,\accentset{\mathfrak{e}}{B}_{\beta}{}^{\nu}\accentset{\mathfrak{e}}{B}_{\nu\alpha}\right),$
$\displaystyle\mathfrak{e}\equiv\frac{1}{4!}\epsilon_{ABCD}\epsilon^{\alpha\beta\gamma\delta}{\mathfrak{e}^{A}}_{\alpha}{\mathfrak{e}^{B}}_{\beta}{\mathfrak{e}^{C}}_{\gamma}{\mathfrak{e}^{D}}_{\delta}\to\bar{\mathfrak{e}}\left(1+\frac{1}{2}{\accentset{\mathfrak{e}}{h}^{A}}_{A}+\frac{1}{8}\accentset{\mathfrak{e}}{b}_{AB}\accentset{\mathfrak{e}}{b}^{AB}-\frac{1}{8}\accentset{\mathfrak{e}}{h}_{AB}\accentset{\mathfrak{e}}{h}^{AB}+\frac{1}{8}({\accentset{\mathfrak{e}}{h}^{A}}_{A})^{2}\right)$
(42)

where we used the permutation symbol $\epsilon_{ABCD}$. Analogous relations
hold for the  $\mathfrak{f}$ tetrad. Notice the presence of the background
metric $\accentset{\mathfrak{e}}{\overline{g}}^{\gamma\beta}$ and that
$\bar{\mathfrak{e}}=\sqrt{\left|{\accentset{\mathfrak{e}}{\overline{g}}}\right|}$.

The resulting linearized Lagrangian then is

$\displaystyle
S=\int\mathrm{d}^{4}x\left(\overset{\mathfrak{e}}{\mathcal{L}}_{hh}+\overset{\mathfrak{e}}{\mathcal{L}}_{hb}+\overset{\mathfrak{e}}{\mathcal{L}}_{bb}+\overset{\mathfrak{f}}{\mathcal{L}}_{hh}+\overset{\mathfrak{f}}{\mathcal{L}}_{hb}+\overset{\mathfrak{f}}{\mathcal{L}}_{bb}+\overset{kin}{\mathcal{L}}_{hh}+\overset{kin}{\mathcal{L}}_{hb}+\overset{kin}{\mathcal{L}}_{bb}+\overset{int}{\mathcal{L}}_{hh}+\overset{int}{\mathcal{L}}_{hb}+\overset{int}{\mathcal{L}}_{bb}+\overset{int}{\mathcal{L}}_{1}+\overset{int}{\mathcal{L}}_{0}\right)$
(43)

with the second order expansions given by:

$\displaystyle\overset{\mathfrak{e}}{\mathcal{L}}_{hh}$
$\displaystyle=\frac{m_{\mathfrak{e}}^{2}}{4}\accentset{\mathfrak{e}}{h}^{AB}\left(-2\,\accentset{\mathfrak{e}}{C}_{3}\partial_{B}\partial_{A}\accentset{\mathfrak{e}}{h}+(-2\accentset{\mathfrak{e}}{C}_{1}-\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\partial_{C}\partial_{B}\accentset{\mathfrak{e}}{h}_{A}{}^{C}+(2\accentset{\mathfrak{e}}{C}_{1}+\accentset{\mathfrak{e}}{C}_{2})\square\accentset{\mathfrak{e}}{h}_{AB}+\accentset{\mathfrak{e}}{C}_{3}\eta_{AB}\,\square\accentset{\mathfrak{e}}{h}\right)$
(44) $\displaystyle\overset{\mathfrak{e}}{\mathcal{L}}_{hb}$
$\displaystyle=-\frac{m_{\mathfrak{e}}^{2}}{2}\accentset{\mathfrak{e}}{h}^{AB}\left(2\accentset{\mathfrak{e}}{C}_{1}+\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\right)\partial_{C}\partial_{B}\accentset{\mathfrak{e}}{b}_{A}{}^{C}$
(45) $\displaystyle\overset{\mathfrak{e}}{\mathcal{L}}_{bb}$
$\displaystyle=\frac{m_{\mathfrak{e}}^{2}}{4}\accentset{\mathfrak{e}}{b}^{AB}\left((-2\accentset{\mathfrak{e}}{C}_{1}+3\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\partial_{C}\partial_{B}\accentset{\mathfrak{e}}{b}_{A}{}^{C}+(2\accentset{\mathfrak{e}}{C}_{1}-\accentset{\mathfrak{e}}{C}_{2})\square\accentset{\mathfrak{e}}{b}_{AB}\right)$
(46) $\displaystyle\overset{\mathfrak{f}}{\mathcal{L}}_{hh}$
$\displaystyle=\frac{m_{\mathfrak{f}}^{2}}{4}\accentset{\mathfrak{f}}{h}^{AB}\left(-2\accentset{\mathfrak{f}}{C}_{3}\partial_{B}\partial_{A}\accentset{\mathfrak{f}}{h}+(-2\accentset{\mathfrak{f}}{C}_{1}-\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{h}_{A}{}^{C}+(2\accentset{\mathfrak{f}}{C}_{1}+\accentset{\mathfrak{f}}{C}_{2})\square\accentset{\mathfrak{f}}{h}_{AB}+\accentset{\mathfrak{f}}{C}_{3}\eta_{AB}\,\square\accentset{\mathfrak{f}}{h}\right)$
(47) $\displaystyle\overset{\mathfrak{f}}{\mathcal{L}}_{hb}$
$\displaystyle=-\frac{m_{\mathfrak{f}}^{2}}{2}\accentset{\mathfrak{f}}{h}^{AB}\left(2\accentset{\mathfrak{f}}{C}_{1}+\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}\right)\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{b}_{A}{}^{C}$
(48) $\displaystyle\overset{\mathfrak{f}}{\mathcal{L}}_{bb}$
$\displaystyle=\frac{m_{\mathfrak{f}}^{2}}{4}\accentset{\mathfrak{f}}{b}^{AB}\left((-2\accentset{\mathfrak{f}}{C}_{1}+3\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{b}_{A}{}^{C}+(2\accentset{\mathfrak{f}}{C}_{1}-\accentset{\mathfrak{f}}{C}_{2})\square\accentset{\mathfrak{f}}{b}_{AB}\right)$
(49) $\displaystyle\overset{kin}{\mathcal{L}}_{hh}$
$\displaystyle=\frac{m_{\mathfrak{e}\mathfrak{f}}^{2}}{4}\accentset{\mathfrak{e}}{h}^{AB}\left(-\accentset{\textrm{\ae}}{C}_{3}\partial_{B}\partial_{A}\accentset{\mathfrak{f}}{h}+(-2\accentset{\textrm{\ae}}{C}_{1}-\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{h}_{A}{}^{C}+(2\accentset{\textrm{\ae}}{C}_{1}+\accentset{\textrm{\ae}}{C}_{2})\square\accentset{\mathfrak{f}}{h}_{AB}+\accentset{\textrm{\ae}}{C}_{3}\eta_{AB}\,(-\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{h}^{BC}+\square\accentset{\mathfrak{f}}{h})\right)$
(50) $\displaystyle\overset{kin}{\mathcal{L}}_{hb}$
$\displaystyle=-\frac{m_{\mathfrak{e}\mathfrak{f}}^{2}}{4}\left(2\accentset{\textrm{\ae}}{C}_{1}+\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3}\right)\left(\accentset{\mathfrak{e}}{h}^{AB}\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{b}_{A}{}^{C}+\accentset{\mathfrak{e}}{b}^{AB}\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{h}_{A}{}^{C}\right)$
(51) $\displaystyle\overset{kin}{\mathcal{L}}_{bb}$
$\displaystyle=\frac{m_{\mathfrak{e}\mathfrak{f}}^{2}}{4}\accentset{\mathfrak{e}}{b}^{AB}\left((-2\accentset{\textrm{\ae}}{C}_{1}+3\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\partial_{C}\partial_{B}\accentset{\mathfrak{f}}{b}_{A}{}^{C}+(2\accentset{\textrm{\ae}}{C}_{1}-\accentset{\textrm{\ae}}{C}_{2})\square\accentset{\mathfrak{f}}{b}_{AB}\right)$
(52) $\displaystyle\overset{int}{\mathcal{L}}_{0}$
$\displaystyle=-(\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4})m^{4}_{int}$
(53) $\displaystyle\overset{int}{\mathcal{L}}_{1}$
$\displaystyle=-\frac{m^{4}_{int}}{8}\left((4\beta_{0}+3\beta_{1}+2\beta_{2}+\beta_{3})\accentset{\mathfrak{e}}{h}+(\beta_{1}+2\beta_{2}+3\beta_{3}+4\beta_{4})\accentset{\mathfrak{f}}{h}\right)$
(54) $\displaystyle\overset{int}{\mathcal{L}}_{hh}$
$\displaystyle=\frac{m^{4}_{int}}{48}\left((6\beta_{0}+3\beta_{1}+\beta_{2})\accentset{\mathfrak{e}}{h}_{AB}\accentset{\mathfrak{e}}{h}^{AB}+(\beta_{2}+3\beta_{3}+6\beta_{4})\accentset{\mathfrak{f}}{h}^{AB}\accentset{\mathfrak{f}}{h}_{AB}+(3\beta_{1}+4\beta_{2}+3\beta_{3})\accentset{\mathfrak{e}}{h}^{AB}\accentset{\mathfrak{f}}{h}_{AB}+\right.$
(55)
$\displaystyle\quad\left.-(6\beta_{0}+3\beta_{1}+\beta_{2})\accentset{\mathfrak{e}}{h}\accentset{\mathfrak{e}}{h}-(\beta_{2}+3\beta_{3}+6\beta_{4})\accentset{\mathfrak{f}}{h}\accentset{\mathfrak{f}}{h}-(3\beta_{1}+4\beta_{2}+3\beta_{3})\accentset{\mathfrak{e}}{h}\accentset{\mathfrak{f}}{h}\right)$
$\displaystyle\overset{int}{\mathcal{L}}_{hb}$ $\displaystyle=0$ (56)
$\displaystyle\overset{int}{\mathcal{L}}_{bb}$
$\displaystyle=-\frac{m^{4}_{int}}{48}\left((6\beta_{0}+3\beta_{1}+\beta_{2})\accentset{\mathfrak{e}}{b}_{AB}\accentset{\mathfrak{e}}{b}^{AB}+(\beta_{2}+3\beta_{3}+6\beta_{4})\accentset{\mathfrak{f}}{b}_{AB}\accentset{\mathfrak{f}}{b}^{AB}+(3\beta_{1}+4\beta_{2}+3\beta_{3})\accentset{\mathfrak{f}}{b}_{AB}\accentset{\mathfrak{e}}{b}^{AB}\right)\,,$
(57)

where we defined
$\accentset{\mathfrak{e}}{h}:=\accentset{\mathfrak{e}}{h}^{A}{}_{A}$ and
$\accentset{\mathfrak{f}}{h}:=\accentset{\mathfrak{f}}{h}^{A}{}_{A}$.

###  III.2 Linear analysis of the teleparallel equivalent to Hassan-Rosen
bimetric gravity

As a first application of the general theory, we recover the standard bimetric
theory in tetrad formulation. To this purpose we eliminate the new terms
brought by the kinetic mixing in the Lagrangian setting
$\accentset{\textrm{\ae}}{C}_{i}=0$. For the sake of visualization, we also
rewrite the above Lagrangian in matrix formalism, using a basis where the
field vector has the components $\Phi:=$($\accentset{\mathfrak{e}}{h}_{AB}$,
$\accentset{\mathfrak{f}}{h}_{AB}$, $\accentset{\mathfrak{e}}{b}_{AB}$,
$\accentset{\mathfrak{f}}{b}_{AB}$)T and by integrating by parts if required

$\displaystyle\mathcal{L}=\Phi^{T}(\mathbb{K}+\mathbb{M}^{2})\Phi+\overset{int}{\mathcal{L}}_{1}+\overset{int}{\mathcal{L}}_{0}.$
(58)

The matrices $\mathbb{K}$ and $\mathbb{M}^{2}$ for the general case can be
straightforwardly read from the Lagrangian in Eq. (43). Below we give their
explicit forms only for the cases under consideration.

Requiring the absence of linear terms for $\accentset{\mathfrak{e}}{h}$ and
$\accentset{\mathfrak{f}}{h}$ in the bimetric Lagrangian yields the following
conditions on the coefficients entering the interaction term

$\begin{cases}4\beta_{0}=-3\beta_{1}-2\beta_{2}-\beta_{3}\\\
4\beta_{4}=-3\beta_{3}-2\beta_{2}-\beta_{1}\end{cases}$

and, as a result, we also obtain that

$\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}\equiv 0,$

in the same fashion, for instance, as the condition (3.2) and (3.3) of
Hinterbichler:2012cn . After the cancellation of these tadpole terms,
enforcing $\overset{int}{\mathcal{L}}_{1}=\overset{int}{\mathcal{L}}_{0}=0$,
the mass term reduces to

$\displaystyle\mathbb{M}^{2}$
$\displaystyle:=\dfrac{m^{4}_{int}\left(3\beta_{1}+4\beta_{2}+3\beta_{3}\right)}{96}\begin{pmatrix}-1&1&0&0\\\
1&-1&0&0\\\ 0&0&1&-1\\\
0&0&-1&1\end{pmatrix}\left(\eta^{AD}\eta^{BE}-\eta^{AB}\eta^{DE}\right),$ (59)

where, in the last step, we used
$\accentset{\mathfrak{f}}{b}_{A}{}^{A}=\accentset{\mathfrak{e}}{b}_{A}{}^{A}\equiv
0$.

As for the kinetic terms, in order to recover standard bimetric theory, we
enforce the cancellation of the $h-b$ mode mixing by setting
$-2\accentset{\mathfrak{e}}{C}_{1}=\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}$,
as well as
$-2\accentset{\mathfrak{f}}{C}_{1}=\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}$
yielding two copies of the one parameter family of new general relativity
introduced in PhysRevD.19.3524 . The resulting kinetic term presents then the
following structure

$\displaystyle\mathbb{K}$
$\displaystyle:=-\frac{1}{4}\begin{pmatrix}m_{\mathfrak{e}}^{2}\,\accentset{\mathfrak{e}}{C}_{3}&0&0&0\\\
0&m_{\mathfrak{f}}^{2}\,\accentset{\mathfrak{f}}{C}_{3}&0&0\\\
0&0&m_{\mathfrak{e}}^{2}(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})&0\\\
0&0&0&m_{\mathfrak{f}}^{2}(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\end{pmatrix}\mathcal{E}^{ABDE}$
(60)

where the Lichnerowicz operator is defined as

$\displaystyle\mathcal{E}^{ABDE}:=\eta^{AD}\eta^{BE}\square-2\eta^{AD}\partial^{E}\partial^{B}+(\eta^{DE}\partial^{A}\partial^{B}+\eta^{AB}\partial^{D}\partial^{E})-\eta^{AB}\eta^{DE}\square$
(61)

and we have again used the fact that
$\accentset{\mathfrak{f}}{b}_{A}{}^{A}=\accentset{\mathfrak{e}}{b}_{A}{}^{A}\equiv
0$.

With the kinetic term being already in a diagonal form, we need only to
diagonalize the remaining mass term. As a first step, we give the fields the
canonical mass dimension through the field redefinition

$\displaystyle\accentset{\mathfrak{e}}{h}_{AB}\to\frac{\accentset{\mathfrak{e}}{h}_{AB}}{m_{\mathfrak{e}}},\quad\accentset{\mathfrak{f}}{h}_{AB}\to\frac{\accentset{\mathfrak{f}}{h}_{AB}}{m_{\mathfrak{f}}},\quad\accentset{\mathfrak{e}}{b}_{AB}\to\frac{\accentset{\mathfrak{e}}{b}_{AB}}{m_{\mathfrak{e}}},\quad\accentset{\mathfrak{f}}{b}_{AB}\to\frac{\accentset{\mathfrak{f}}{b}_{AB}}{m_{\mathfrak{f}}}$
(62)

which puts the Lagrangian in the form:

$\displaystyle\begin{split}\mathcal{L}_{bm}&=\frac{1}{4}\begin{pmatrix}\accentset{\mathfrak{e}}{h}_{AB}&\accentset{\mathfrak{f}}{h}_{AB}&\accentset{\mathfrak{e}}{b}_{AB}&\accentset{\mathfrak{f}}{b}_{AB}\end{pmatrix}\left(\mathbb{K}_{bm}+\mathbb{M}^{2}_{bm}\right)\begin{pmatrix}\accentset{\mathfrak{e}}{h}_{DE}\\\
\accentset{\mathfrak{f}}{h}_{DE}\\\ \accentset{\mathfrak{e}}{b}_{DE}\\\
\accentset{\mathfrak{f}}{b}_{DE}\end{pmatrix}\,.\end{split}$ (63)

with

$\displaystyle\mathbb{K}_{bm}=-\begin{pmatrix}\,\accentset{\mathfrak{e}}{C}_{3}&0&0&0\\\
0&\,\accentset{\mathfrak{f}}{C}_{3}&0&0\\\
0&0&2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}&0\\\
0&0&0&2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}\end{pmatrix}\mathcal{E}^{ABDE},$
(64)

and

$\displaystyle\mathbb{M}^{2}_{bm}=\dfrac{m^{4}_{int}\left(3\beta_{1}+4\beta_{2}+3\beta_{3}\right)}{24}\begin{pmatrix}-\dfrac{1}{m_{\mathfrak{e}}^{2}}&\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}&0&0\\\
\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}&-\dfrac{1}{m_{\mathfrak{f}}^{2}}&0&0\\\
0&0&\dfrac{1}{m_{\mathfrak{e}}^{2}}&-\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}\\\
0&0&-\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}&\dfrac{1}{m_{\mathfrak{f}}^{2}}\end{pmatrix}\left(\eta^{AD}\eta^{BE}-\eta^{AB}\eta^{DE}\right).$
(65)

Before analyzing the two emerging blocks in isolation, we further simplify the
Lagrangian by writing, without loss of generality,

$\displaystyle\dfrac{m^{4}_{int}\left(3\beta_{1}+4\beta_{2}+3\beta_{3}\right)}{24}=:m_{FP}^{2}\frac{m_{\mathfrak{e}}^{2}m_{\mathfrak{f}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,.$
(66)

As usual in bimetric theories, we have re-absorbed the dependence of the
Lagrangian on $\beta_{i=1,2,3}$ into the definition of the mass scale
$m_{FP}$.

####  III.2.1 Symmetric sector

Focusing now on the symmetric perturbations, we have

$\displaystyle\begin{split}\mathcal{L}_{bm}^{hh}&=-\frac{1}{4}\accentset{\mathfrak{e}}{h}_{AB}\,\accentset{\mathfrak{e}}{C}_{3}\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{e}}{h}_{DE}-\frac{1}{4}\accentset{\mathfrak{f}}{h}_{AB}\,\accentset{\mathfrak{f}}{C}_{3}\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{f}}{h}_{DE}\\\
&-\frac{m_{FP}^{2}}{4}\frac{m_{\mathfrak{e}}^{2}m_{\mathfrak{f}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\left[\left(\frac{\accentset{\mathfrak{e}}{h}_{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}_{AB}}{m_{\mathfrak{f}}}\right)\left(\frac{\accentset{\mathfrak{e}}{h}^{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}^{AB}}{m_{\mathfrak{f}}}\right)-\left(\frac{\accentset{\mathfrak{e}}{h}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}}{m_{\mathfrak{f}}}\right)^{2}\right]\end{split}$
(67)

which is diagonalized through the usual field redefinition

$\displaystyle
M_{AB}:=\frac{m_{\mathfrak{e}}m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\left(\frac{\accentset{\mathfrak{e}}{h}_{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}_{AB}}{m_{\mathfrak{f}}}\right)=\frac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\accentset{\mathfrak{e}}{h}_{AB}-\frac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\accentset{\mathfrak{f}}{h}_{AB}$
(68)

ensuring that the norm of $M_{AB}$ in field space is the unity.

The orthogonal combination
$G_{AB}=\alpha\accentset{\mathfrak{e}}{h}_{AB}+\beta\accentset{\mathfrak{f}}{h}_{AB}$
is determined through

$\displaystyle\begin{cases}G_{AB}\cdot M_{AB}=0\\\ G_{AB}\cdot
G_{AB}=M_{AB}\cdot
M_{AB}\end{cases}\implies\begin{cases}\dfrac{\alpha}{m_{\mathfrak{e}}}-\dfrac{\beta}{m_{\mathfrak{f}}}=0\\\
\alpha^{2}+\beta^{2}=1\end{cases}\implies\begin{cases}\alpha=\dfrac{m_{\mathfrak{e}}\beta}{m_{\mathfrak{f}}}\\\
\beta^{2}\left(1+\dfrac{m_{\mathfrak{e}}^{2}}{m_{\mathfrak{f}}^{2}}\right)=1\end{cases}\implies\begin{cases}\alpha=\dfrac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\\\
\beta=\dfrac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\end{cases}$
(69)

where the $\cdot$ operator stands for the scalar product in field space. As a
result, we obtain

$\displaystyle
G_{AB}:=\frac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\accentset{\mathfrak{e}}{h}_{AB}+\frac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\accentset{\mathfrak{f}}{h}_{AB}\,,$
(70)

showing that the field transformation linking the mass eigenstates to the
original fields is simply a rotation in field space:

$\displaystyle\begin{pmatrix}M_{AB}\\\
G_{AB}\end{pmatrix}=\begin{pmatrix}\dfrac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}&-\dfrac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\\\
\dfrac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}&\dfrac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\end{pmatrix}\begin{pmatrix}\accentset{\mathfrak{e}}{h}_{AB}\\\
\accentset{\mathfrak{f}}{h}_{AB}\end{pmatrix}\,.$ (71)

In terms of the mass eigenstates, the Lagrangian of the symmetric sector is
then given by

$\displaystyle\mathcal{L}_{bm}^{hh}$
$\displaystyle=\frac{1}{4}G_{AB}\,\frac{\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,\tilde{\mathcal{E}}^{ABDE}\,G_{DE}+\frac{1}{4}M_{AB}\,\frac{\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{f}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,\tilde{\mathcal{E}}^{ABDE}\,M_{DE}$
(72)
$\displaystyle+\frac{1}{2}M_{AB}\,\frac{\left(\accentset{\mathfrak{e}}{C}_{3}-\accentset{\mathfrak{f}}{C}_{3}\right)m_{\mathfrak{e}}m_{\mathfrak{f}}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,\tilde{\mathcal{E}}^{ABDE}\,G_{DE}-\frac{m_{FP}^{2}}{4}\left(M_{AB}M^{AB}-M^{2}\right)\,,$

showing that the condition
$\accentset{\mathfrak{e}}{C}_{3}=\accentset{\mathfrak{f}}{C}_{3}$ is necessary
for a complete diagonalization of the symmetric sector. In fact, because the
performed field redefinition is simply a rotation, the kinetic term matrix of
the original field must be proportional to the unit matrix in order to be
unaffected by the field redefinition.

####  III.2.2 Antisymmetric sector

As for the antisymmetric sector, we have

$\displaystyle\begin{split}\mathcal{L}_{bm}^{bb}&=-\frac{1}{4}\accentset{\mathfrak{e}}{b}_{AB}\,\left(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\right)\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{e}}{b}_{DE}-\frac{1}{4}\accentset{\mathfrak{f}}{b}_{AB}\,\left(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}\right)\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{f}}{b}_{DE}\\\
&+\frac{m_{FP}^{2}}{4}\frac{m_{\mathfrak{e}}^{2}m_{\mathfrak{f}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\left(\frac{\accentset{\mathfrak{e}}{b}_{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{b}_{AB}}{m_{\mathfrak{f}}}\right)\left(\frac{\accentset{\mathfrak{e}}{b}^{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{b}^{AB}}{m_{\mathfrak{f}}}\right)\end{split}$
(73)

which is diagonalized through a rotation analogous to that of the symmetric
sector. In particular, by defining

$\displaystyle\begin{pmatrix}E_{AB}\\\
F_{AB}\end{pmatrix}=\begin{pmatrix}\dfrac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}&-\dfrac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\\\
\dfrac{m_{\mathfrak{e}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}&\dfrac{m_{\mathfrak{f}}}{\sqrt{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}}\end{pmatrix}\begin{pmatrix}\accentset{\mathfrak{e}}{b}_{AB}\\\
\accentset{\mathfrak{f}}{b}_{AB}\end{pmatrix}$ (74)

we have

$\displaystyle\mathcal{L}_{bm}^{bb}$
$\displaystyle=-\frac{1}{4}F_{AB}\,\frac{\left(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\right)m_{\mathfrak{e}}^{2}+\left(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}\right)m_{\mathfrak{f}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,\mathcal{E}^{ABDE}\,F_{DE}$
(75)
$\displaystyle-\frac{1}{4}E_{AB}\,\frac{\left(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\right)m_{\mathfrak{f}}^{2}+\left(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}\right)m_{\mathfrak{e}}^{2}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,\mathcal{E}^{ABDE}\,E_{DE}$
$\displaystyle-\frac{1}{2}F_{AB}\,\frac{\left(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}-2\accentset{\mathfrak{f}}{C}_{2}-\accentset{\mathfrak{f}}{C}_{3}\right)m_{\mathfrak{e}}m_{\mathfrak{f}}}{m_{\mathfrak{e}}^{2}+m_{\mathfrak{f}}^{2}}\,\mathcal{E}^{ABDE}\,E_{DE}+\frac{m_{FP}^{2}}{4}E_{AB}E^{AB}\,.$

In analogy with the case of the symmetric sector, the condition
$2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}=2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}$
ensures the decoupling of the modes.

####  III.2.3 Teleparallel Equivalent of Bimetric Gravity

In terms of the mass eigenstates we have identified and through the above
conditions, the full Lagrangian then becomes:

$\displaystyle\mathcal{L}_{bm}$
$\displaystyle=-\frac{1}{4}\accentset{\mathfrak{e}}{C}_{3}\,G_{AB}\,\mathcal{E}^{ABDE}\,G_{DE}-\frac{1}{4}\accentset{\mathfrak{e}}{C}_{3}\,M_{AB}\,\mathcal{E}^{ABDE}\,M_{DE}-\frac{m_{FP}^{2}}{4}\left(M_{AB}M^{AB}-M^{2}\right)$
(76)
$\displaystyle\quad-\frac{1}{4}\left(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\right)\,F_{AB}\,\mathcal{E}^{ABDE}\,F_{DE}-\frac{1}{4}\left(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\right)\,E_{AB}\,\mathcal{E}^{ABDE}\,E_{DE}+\frac{m_{FP}^{2}}{4}E_{AB}E^{AB}\,.$

Setting now the remaining coefficients to their TEGR values, we notice that
$\accentset{\mathfrak{e}}{C}_{3}=\accentset{\mathfrak{f}}{C}_{3}=-1$ ensures a
proper normalizes the Lichnerowicz operator $\mathcal{E}^{ABDE}$ and casts the
Lagrangian of the symmetric sector into the usual bimetric form:

$\displaystyle\mathcal{L}_{bm}^{hh}=\frac{1}{4}G_{AB}\,\mathcal{E}^{ABDE}\,G_{DE}+\frac{1}{4}M_{AB}\,\mathcal{E}^{ABDE}\,M_{DE}-\frac{m_{FP}^{2}}{4}\left(M_{AB}M^{AB}-M^{2}\right)\,.$
(77)

Differently, the choice
$\accentset{\mathfrak{e}}{C}_{2}=\accentset{\mathfrak{f}}{C}_{2}=C_{2}=1/2$
prevents the propagation of all antisymmetric modes, reducing the
corresponding Lagrangian to

$\displaystyle\mathcal{L}_{bm}^{bb}=\frac{m_{FP}^{2}}{4}E_{AB}E^{AB}\,.$ (78)

The resulting equation of motion then forces constraint $E_{AB}=0$, which
through eq.$\eqref{eq:EFsub}$ recovers

$\displaystyle\frac{\accentset{\mathfrak{e}}{b}_{AB}}{m_{\mathfrak{e}}}=\frac{\accentset{\mathfrak{f}}{b}_{AB}}{m_{\mathfrak{f}}}\,.$
(79)

At the first order of the perturbation expansion, this condition coincides
with the requirement
$\eta_{AB}{\mathfrak{e}^{A}}_{[\mu}{\mathfrak{f}^{B}}_{\nu]}\stackrel{{\scriptstyle!}}{{=}}0$,
often discussed in the literature of bigravity Kocic:2018ddp ; Kocic:2018yvr ;
Kocic:2019ahm ; Hinterbichler:2012cn .

We remark that the final form of the Lagrangian was obtained by using a
rotation in field space, rather than a generic linear transformation, to
diagonalize the mass term. Whether this might seem as an overly restrictive
choice, we remind that in the teleparallel equivalent of bimetric gravity, the
coefficients $C_{i}$ of the two tetrads are set from the very beginning to
their TEGR values and, as a consequence, the identified rotation is the only
transformation that recasts the symmetric sector in the standard form.

We also notice that prior to setting the coefficients $C_{i}$ to their TEGR
values, the Lagrangian of the antisymmetric sector mirrors that of the
symmetric one, albeit an important distinction. The mass terms of the massive
modes, in fact, have opposite signs. Whether a generic theory containing
propagating symmetric and antisymmetric modes would probably contain ghosts,
at this stage it seems still possible to recover a regime of the theory
characterized by the propagation of a massive and a massless antisymmetric
mode and no symmetric perturbations.

###  III.3 Linear analysis of the full Lagrangian

We now face the full Lagrangian, allowing for the kinetic mixing terms that we
neglected in the previous section. Note that in the limit $m_{FP}\to 0$ of
vanishing Fierz-Pauli mass, a kinetic coupling in the symmetric sector, which
contains two massless spin-2 fields in this case, inevitably leads to the
existence of a scalar ghost mode Boulanger:2000rq . For this reason, such
kinetic coupling is not considered in the Hassan-Rosen bigravity Hassan:2011zd
.

The treatment of the linear and 0-th order terms proceeds as in the standard
bimetric case: enforcing the absence of tadpole term by setting

$\displaystyle\begin{cases}4\beta_{0}=-3\beta_{1}-2\beta_{2}-\beta_{3}\\\
4\beta_{4}=-3\beta_{3}-2\beta_{2}-\beta_{1}\end{cases}\,,$ (80)

we obtain $\beta_{0}+\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}\equiv 0$, and the
mass term consequently acquires again the form

$\displaystyle\mathbb{M}^{2}=\dfrac{m^{4}_{int}\left(3\beta_{1}+4\beta_{2}+3\beta_{3}\right)}{96}\begin{pmatrix}-\dfrac{1}{m_{\mathfrak{e}}^{2}}&\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}&0&0\\\
\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}&-\dfrac{1}{m_{\mathfrak{f}}^{2}}&0&0\\\
0&0&\dfrac{1}{m_{\mathfrak{e}}^{2}}&-\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}\\\
0&0&-\dfrac{1}{m_{\mathfrak{e}}m_{\mathfrak{f}}}&\dfrac{1}{m_{\mathfrak{f}}^{2}}\end{pmatrix}\left(\eta^{AD}\eta^{BE}-\eta^{AB}\eta^{DE}\right)$
(81)

in terms of the properly normalized fields given by

$\displaystyle\accentset{\mathfrak{e}}{h}_{AB}\to\frac{\accentset{\mathfrak{e}}{h}_{AB}}{m_{\mathfrak{e}}},\quad\accentset{\mathfrak{f}}{h}_{AB}\to\frac{\accentset{\mathfrak{f}}{h}_{AB}}{m_{\mathfrak{f}}},\quad\accentset{\mathfrak{e}}{b}_{AB}\to\frac{\accentset{\mathfrak{e}}{b}_{AB}}{m_{\mathfrak{e}}},\quad\accentset{\mathfrak{f}}{b}_{AB}\to\frac{\accentset{\mathfrak{f}}{b}_{AB}}{m_{\mathfrak{f}}}\,.$
(82)

As for the kinetic term of the normalized fields, in this first analysis we
choose to set

$\displaystyle\begin{cases}-2\accentset{\mathfrak{e}}{C}_{1}=\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}\\\
-2\accentset{\mathfrak{f}}{C}_{1}=\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}\\\
-2\accentset{\textrm{\ae}}{C}_{1}=\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3}\end{cases}$
(83)

in order to obtain the same block diagonal structure of the mass term. Whereas
this condition is in principle unnecessary, it allows us to tackle the
symmetric and antisymmetric modes separately, with a considerable
simplification of the diagonalization procedure. In order to further simplify
the Lagrangian, we also opt to set
$m^{2}_{\mathfrak{e}\mathfrak{f}}\stackrel{{\scriptstyle!}}{{=}}m_{\mathfrak{e}}m_{\mathfrak{f}}$,
obtaining:

$\displaystyle\mathbb{K}$
$\displaystyle=-\frac{1}{4}\begin{pmatrix}\accentset{\mathfrak{e}}{C}_{3}&\dfrac{\accentset{\textrm{\ae}}{C}_{3}}{2}&0&0\\\
\dfrac{\accentset{\textrm{\ae}}{C}_{3}}{2}&\accentset{\mathfrak{f}}{C}_{3}&0&0\\\
0&0&(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})&\dfrac{1}{2}(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\\\
0&0&\dfrac{1}{2}(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})&(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\end{pmatrix}\mathcal{E}^{ABDE}\,,$
(84)

with the Lichnerowicz operator $\mathcal{E}^{ABDE}$ defined as in the standard
case. Because the interaction term is the same as in the standard bimetric
gravity, the field rotations in eq.$\eqref{eq:eigrot}$ and $\eqref{eq:EFsub}$
certainly diagonalize it. However, if we restrict ourselves to these
transformations, the coefficients in the kinetic term will have to be set in a
way that the kinetic term becomes proportional to the unit matrix in field
space. In particular, this choice will force the absence of kinetic mixing,
leading inevitably to scenarios close to the standard bimetric case or that,
at most, allow for the propagation of a massless and a negative mass
antisymmetric mode. For this reason, in the following we consider instead
generic superpositions of symmetric and antisymmetric perturbations rather
than rotations in the corresponding field spaces.

####  III.3.1 Symmetric sector

Upon partial integration of the off-diagonal kinetic terms, the Lagrangian for
the symmetric perturbation is written as:

$\displaystyle\mathcal{L}^{hh}$
$\displaystyle=-\frac{1}{4}\,\accentset{\mathfrak{e}}{h}_{AB}\accentset{\mathfrak{e}}{C}_{3}\mathcal{E}^{ABDE}\,\accentset{\mathfrak{e}}{h}_{DE}-\frac{1}{4}\,\accentset{\mathfrak{f}}{h}_{AB}\accentset{\mathfrak{f}}{C}_{3}\mathcal{E}^{ABDE}\,\accentset{\mathfrak{f}}{h}_{DE}-\frac{1}{4}\,\accentset{\mathfrak{e}}{h}_{AB}\accentset{\textrm{\ae}}{C}_{3}\mathcal{E}^{ABDE}\,\accentset{\mathfrak{f}}{h}_{DE}$
(85)
$\displaystyle-\dfrac{m^{4}_{int}}{4}\left[\left(\frac{\accentset{\mathfrak{e}}{h}_{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}_{AB}}{m_{\mathfrak{f}}}\right)\left(\frac{\accentset{\mathfrak{e}}{h}^{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}^{AB}}{m_{\mathfrak{f}}}\right)-\left(\frac{\accentset{\mathfrak{e}}{h}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}}{m_{\mathfrak{f}}}\right)^{2}\right]$

where, without loss of generality, we have set
$\left(3\beta_{1}+4\beta_{2}+3\beta_{3}\right)=24$. In order to diagonalize
the Lagrangian, we now consider the generic linear transformation

$\displaystyle\begin{pmatrix}\accentset{\mathfrak{e}}{h}_{AB}\\\
\accentset{\mathfrak{f}}{h}_{AB}\end{pmatrix}=:\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}\begin{pmatrix}M_{AB}\\\ G_{AB}\end{pmatrix}$ (86)

where $G_{AB}$ and $M_{AB}$ are respectively the graviton and massive
bigraviton excitation. However generic the transformation be, we still require
that an inverse exists by imposing that the determinant of the transformation
never vanishes. We will verify that the solutions we identify indeed satisfy
this criterion.

Analyzing the mass term in isolation, we see that

$\displaystyle\left(\frac{\accentset{\mathfrak{e}}{h}_{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{h}_{AB}}{m_{\mathfrak{f}}}\right)=\left[\left(\frac{\alpha}{m_{\mathfrak{e}}}-\frac{\gamma}{m_{\mathfrak{f}}}\right)\,M_{AB}+\left(\frac{\beta}{m_{\mathfrak{e}}}-\frac{\delta}{m_{\mathfrak{f}}}\right)\,G_{AB}\right]$
(87)

and analogous expressions hold for the remaining terms. It is then clear that
setting

$\displaystyle\delta=\frac{m_{\mathfrak{f}}}{m_{\mathfrak{e}}}\beta$ (88)

forbids a mass term for the $G$ perturbation. Similarly, requiring that the
kinetic terms for $G$ and $M$ have the canonical form, as well as the presence
of no mixing, results in three further conditions that fully determine the
parameters in eq.$\eqref{eq:gen_t}$. Explicitly, we have

$\displaystyle\begin{split}\mathcal{L}^{hh}&\supset-\frac{1}{4}\left(\accentset{\mathfrak{e}}{C}_{3}\alpha^{2}+\accentset{\mathfrak{f}}{C}_{3}\gamma^{2}+\accentset{\textrm{\ae}}{C}_{3}\alpha\gamma\right)\,M_{AB}\,\mathcal{E}^{ABDE}\,M_{DE}-\frac{1}{4}\left(\accentset{\mathfrak{e}}{C}_{3}\beta^{2}+\accentset{\mathfrak{f}}{C}_{3}\delta^{2}+\accentset{\textrm{\ae}}{C}_{3}\beta\delta\right)\,G_{AB}\,\mathcal{E}^{ABDE}\,G_{DE}\\\
&-\frac{1}{4}\left(2\accentset{\mathfrak{e}}{C}_{3}\alpha\,\beta+2\accentset{\mathfrak{f}}{C}_{3}\gamma\,\delta+\accentset{\textrm{\ae}}{C}_{3}(\alpha\delta+\beta\gamma)\right)\,M_{AB}\,\mathcal{E}^{ABDE}\,G_{DE},\end{split}$
(89)

where $\supset$ denotes that some terms in the Lagrangian are omitted, and
imposing the above condition, as well as

$\displaystyle\begin{cases}\accentset{\mathfrak{e}}{C}_{3}\alpha^{2}+\accentset{\mathfrak{f}}{C}_{3}\gamma^{2}+\accentset{\textrm{\ae}}{C}_{3}\alpha\gamma=-1\\\
\accentset{\mathfrak{e}}{C}_{3}\beta^{2}+\accentset{\mathfrak{f}}{C}_{3}\delta^{2}+\accentset{\textrm{\ae}}{C}_{3}\beta\delta=-1\\\
2\accentset{\mathfrak{e}}{C}_{3}\alpha\,\beta+2\accentset{\mathfrak{f}}{C}_{3}\gamma\,\delta+\accentset{\textrm{\ae}}{C}_{3}(\alpha\delta+\beta\gamma)=0\end{cases}$
(90)

results in four possible solutions for the involved parameters which differ
only by the verse associated to the selected linear combinations in field
space. In particular, the choice

$\displaystyle\begin{cases}\alpha=-\dfrac{\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}+2\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}}{\sqrt{\left(\accentset{\textrm{\ae}}{C}_{3}^{2}-4\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{f}}{C}_{3}\right)\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\\\
\beta=\dfrac{m_{\mathfrak{e}}}{\sqrt{-\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\\\
\gamma=\dfrac{\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{f}}+2\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}}{\sqrt{\left(\accentset{\textrm{\ae}}{C}_{3}^{2}-4\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{f}}{C}_{3}\right)\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\\\
\delta=\dfrac{m_{\mathfrak{f}}}{\sqrt{-\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\end{cases}$
(91)

correctly reduces to the (inverse of the) rotation matrix in
eq.$\eqref{eq:eigrot}$. Real and finite coefficients are obtained when

$\displaystyle\begin{cases}\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}<0\\\
\accentset{\textrm{\ae}}{C}_{3}^{2}-4\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{f}}{C}_{3}<0\end{cases}$
(92)

which also ensure that the transformation is invertible:

$\displaystyle\det\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}\overset{\text{eq.}\eqref{eq:goodsub}}{=}\frac{2\sqrt{-\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}{\sqrt{\left(\accentset{\textrm{\ae}}{C}_{3}^{2}-4\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{f}}{C}_{3}\right)\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}$
(93)

We notice the following particular values of
$\accentset{\textrm{\ae}}{C}_{3}$:

  * •

$\accentset{\textrm{\ae}}{C}_{3}=\pm
2\sqrt{\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{f}}{C}_{3}}$. In
this case the kinetic term acquires the form
$\mathcal{L}\supset
A^{2}\,M_{AB}\,\tilde{\mathcal{E}}^{ABDE}\,M_{DE}+B^{2}\,G_{AB}\,\tilde{\mathcal{E}}^{ABDE}\,G_{DE}+2AB\,G_{AB}\,\tilde{\mathcal{E}}^{ABDE}\,M_{DE}$,
implying that a vanishing mixing term necessarily requires at least either of
the kinetic terms to vanish.

  * •

$\accentset{\textrm{\ae}}{C}_{3}=-(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2})/m_{e}m_{f}$.
Once eq.$\eqref{eq:noGmass}$ is imposed to prevent a mass term for the
$G_{AB}$ perturbation, this particular value of
$\accentset{\textrm{\ae}}{C}_{3}$ nullifies the kinetic term of the same
perturbation.

Excluding the above values of $\accentset{\textrm{\ae}}{C}_{3}$, the
Lagrangian of the symmetric sector simply becomes

$\displaystyle\mathcal{L}^{hh}=\frac{1}{4}\,M_{AB}\,\mathcal{E}^{ABDE}\,M_{DE}+\frac{1}{4}\,G_{AB}\,\mathcal{E}^{ABDE}\,G_{DE}-\frac{m_{FP}^{2}}{4}\left(M_{AB}M^{AB}-M^{2}\right)$
(94)

where the involved perturbations are determined from the original
$\accentset{\mathfrak{e}}{h}_{AB}$ and $\accentset{\mathfrak{f}}{h}_{AB}$
fields through the inverse of eq.$\eqref{eq:gen_t}$ and where we have defined

$\displaystyle
m_{FP}^{2}:=m_{int}^{4}\left(\frac{\alpha}{m_{\mathfrak{e}}}-\frac{\gamma}{m_{\mathfrak{f}}}\right)^{2}=m_{int}^{4}\frac{4\left(\accentset{\mathfrak{e}}{C}_{3}m_{\mathfrak{e}}^{2}+\accentset{\textrm{\ae}}{C}_{3}m_{\mathfrak{e}}m_{\mathfrak{f}}+\accentset{\mathfrak{f}}{C}_{3}m_{\mathfrak{f}}^{2}\right)}{m_{\mathfrak{e}}^{2}m_{\mathfrak{f}}^{2}\left(\accentset{\textrm{\ae}}{C}_{3}^{2}-4\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{f}}{C}_{3}\right)}\,.$
(95)

Notice that the Fierz-Pauli mass is positive defined if the conditions in
eq.$\eqref{eq:k3cond}$ are satisfied, and becomes vanishing or singular in
correspondence of the two values of $\accentset{\textrm{\ae}}{C}_{3}$ that
make the transformation singular. This seems to imply that if a fluctuation
does not propagate, the other must necessarily be massless.

####  III.3.2 Antisymmetric sector

Analogously, for the antisymmetric perturbations we have

$\displaystyle\mathcal{L}^{bb}$
$\displaystyle=-\frac{1}{4}\,\accentset{\mathfrak{e}}{b}_{AB}\,(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{e}}{b}_{DE}-\frac{1}{4}\,\accentset{\mathfrak{f}}{b}_{AB}\,(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{f}}{b}_{DE}-\frac{1}{4}\,\accentset{\mathfrak{e}}{b}_{AB}\,(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,\mathcal{E}^{ABDE}\,\accentset{\mathfrak{f}}{b}_{DE}$
(96)
$\displaystyle+\frac{m_{int}^{4}}{4}\left(\frac{\accentset{\mathfrak{e}}{b}_{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{b}_{AB}}{m_{\mathfrak{f}}}\right)\left(\frac{\accentset{\mathfrak{e}}{b}^{AB}}{m_{\mathfrak{e}}}-\frac{\accentset{\mathfrak{f}}{b}^{AB}}{m_{\mathfrak{f}}}\right),$

where we used $\left(3\beta_{1}+4\beta_{2}+3\beta_{3}\right)=24$. In order to
diagonalize the above Lagrangian we introduce the states $E_{AB}$ and $F_{AB}$
such that

$\displaystyle\begin{pmatrix}\accentset{\mathfrak{e}}{b}_{AB}\\\
\accentset{\mathfrak{f}}{b}_{AB}\end{pmatrix}=:\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}E_{AB}\\\ F_{AB}\end{pmatrix}\,,$ (97)

and express the mass term as

$\displaystyle\mathcal{L}^{bb}\supset\frac{m^{4}_{int}}{4}\left[\left(\frac{a}{m_{\mathfrak{e}}}-\frac{c}{m_{\mathfrak{f}}}\right)E_{AB}+\left(\frac{b}{m_{\mathfrak{e}}}-\frac{d}{m_{\mathfrak{f}}}\right)F_{AB}\right]\left[\left(\frac{a}{m_{\mathfrak{e}}}-\frac{c}{m_{\mathfrak{f}}}\right)E^{AB}+\left(\frac{b}{m_{\mathfrak{e}}}-\frac{d}{m_{\mathfrak{f}}}\right)F^{AB}\right]\,.$
(98)

Imposing the absence of mass mixing then forces, again, the presence of a
massless state. In analogy to the symmetric case, we choose here

$\displaystyle\frac{b}{m_{\mathfrak{e}}}\stackrel{{\scriptstyle!}}{{=}}\frac{d}{m_{\mathfrak{f}}}$
(99)

so that

$\displaystyle\mathcal{L}^{bb}\supset\frac{m^{4}_{int}}{4}\left(\frac{a}{m_{\mathfrak{e}}}-\frac{c}{m_{\mathfrak{f}}}\right)^{2}E_{AB}E^{AB}=:\frac{\mu_{FP}^{2}}{4}E_{AB}E^{AB}\,.$
(100)

Notice that $\mu^{2}_{FP}\geq 0$ and consequently the mass term appears in the
Lagrangian inevitably with the wrong sign. Leaving this issue aside for the
moment, we proceed to express the kinetic term in terms of the newly defined
perturbations

$\displaystyle\mathcal{L}^{bb}$
$\displaystyle\supset-\frac{1}{4}\left((2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,a^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,c^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,ac\right)\,E_{AB}\,{\mathcal{E}}^{ABDE}\,E_{DE}$
(101)
$\displaystyle-\frac{1}{4}\left((2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,b^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,d^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,bd\right)\,F_{AB}\,{\mathcal{E}}^{ABDE}\,F_{DE}+$
$\displaystyle-\frac{1}{4}\left(2(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,ab+2(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,cd+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,(ad+bc)\right)\,E_{AB}\,{\mathcal{E}}^{ABDE}\,F_{DE}$
(102)

. Requiring now

$\displaystyle\begin{cases}(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,a^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,c^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,ac=-1\\\
(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,b^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,d^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,bd=-1\\\
2(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\,ab+2(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\,cd+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\,(ad+bc)=0\end{cases}$
(103)

results in

$\displaystyle\begin{cases}a=-\dfrac{(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})m_{\mathfrak{e}}+2(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})m_{\mathfrak{f}}}{\sqrt{\left((2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})^{2}-4(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\right)\left((2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})m_{\mathfrak{e}}^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})m_{\mathfrak{f}}^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\\\
b=\dfrac{m_{\mathfrak{e}}}{\sqrt{-\left((2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})m_{\mathfrak{e}}^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})m_{\mathfrak{f}}^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\\\
c=\dfrac{(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})m_{\mathfrak{f}}+2(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})m_{\mathfrak{e}}}{\sqrt{\left((2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})^{2}-4(2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\right)\left((2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})m_{\mathfrak{e}}^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})m_{\mathfrak{f}}^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\\\
d=\dfrac{m_{\mathfrak{f}}}{\sqrt{-\left((2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})m_{\mathfrak{e}}^{2}+(2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})m_{\mathfrak{f}}^{2}+(2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})m_{\mathfrak{e}}m_{\mathfrak{f}}\right)}}\end{cases}$
(104)

and every other result can, likewise, be derived from those of the symmetric
sector via the substitution $\accentset{\textrm{\ae}}{C}_{3}\to
2\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3}$,
$\accentset{\mathfrak{e}}{C}_{3}\to
2\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3}$,
$\accentset{\mathfrak{f}}{C}_{3}\to
2\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3}$.

As in the case with vanishing kinetic mixing, the general theory seems to
admit the propagation of a ghost-like massive antisymmetric perturbation.
Although it is certainly possible to force the kinetic terms of this
problematic perturbation to vanish through a proper choice of the
coefficients, we argue below that this might not be necessary. As
antisymmetric perturbations correspond, in essence, to the propagation of
scalar fields, the presence of a negative mass term could simply indicate the
emergence of a spontaneously broken symmetry. In fact, for the sake of the
viability of the theory, it could be enough that the terms of higher order in
the perturbative expansion give rise to self interactions of the scalar field
which result in a potential bounded from below. The situation is analogous to
that of the Higgs boson in the standard model, where the negative mass term of
the field triggers the electroweak symmetry breaking.

##  IV Nonlinear analysis of teleparallel bigravity

We saw in the previous section that some teleparallel bigravity theories
contain ghosts. Still, it is of interest to understand their symmetries and
view them as toy models. For instance, the terms with derivative interactions
might be constructed in a viable way if we allow for a construction with
complex tetrads similar to the theory considered in Apolo:2016ort ;
Apolo:2016vkn . We dedicate this section to investigate teleparallel bigravity
at the nonlinear level. For this we make a 3+1 decomposition and find primary
constraints for the action considered in the linear analysis.

###  IV.1 Primary constraints

We remind ourselves that the linear analysis applies to the action (18) with
the supermetrics defined by (31),(32), and (33). Note that the interaction
potential is independent of velocities in the fields and, hence, do not
contribute to the conjugate momenta. Furthermore,
$\overset{\mathfrak{e}}{\mathcal{L}}$ and
$\overset{\mathfrak{f}}{\mathcal{L}}$ are just new general relativity terms
which are already known from the literature Blixt:2018znp ; Blixt:2019ene ;
Blixt:2020ekl ; Blagojevic:2000qs ; Mitric:2019rop . Hence, we can determine
the primary constraints by making the 3+1 decomposition of
$\overset{kin}{\mathcal{L}}$ and calculate its primary constraints and add
this information. The lapse and shift are denoted by
$\accentset{\text{\ae}}{N},\ \accentset{\text{\ae}}{N}^{i}$, respectively and
the induced metric is denoted by $\accentset{\text{\ae}}{\gamma}$, whereas the
normal vector components to the hypersurface of constant time slices is
denoted by $\accentset{\text{\ae}}{\xi}^{A}$. The Lagrangian 3+1 decomposition
of $\overset{kin}{\mathcal{L}}$ is

$\displaystyle\begin{split}\overset{kin}{\mathcal{L}}&=\frac{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\accentset{\mathfrak{e}}{T}^{A}{}_{i0}\accentset{\mathfrak{f}}{T}^{B}{}_{j0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\ B}\\\
&+\frac{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\left(\accentset{\mathfrak{e}}{T}^{A}{}_{i0}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}+\accentset{\mathfrak{f}}{T}^{A}{}_{i0}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\right)\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{il}\left(\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right)\right]\\\
&+\frac{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\left(\accentset{\mathfrak{e}}{T}^{A}{}_{ij}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}+\accentset{\mathfrak{f}}{T}^{A}{}_{ij}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\right)\left[\frac{1}{2}\accentset{\textrm{\ae}}{M}^{\
i\ k}_{\ A\
B}\accentset{\text{\ae}}{N}^{j}\accentset{\text{\ae}}{N}^{l}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{jl}\accentset{\text{\ae}}{\xi}_{A}\left(\accentset{\textrm{\ae}}{C}_{2}\text{\ae}_{B}{}^{i}\accentset{\text{\ae}}{N}^{k}+\accentset{\textrm{\ae}}{C}_{3}\text{\ae}_{B}{}^{k}\accentset{\text{\ae}}{N}^{i}\right)\right]\\\
&+\accentset{\text{\ae}}{N}m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}\cdot{}^{3}\overset{\text{\ae}}{\mathbb{T}},\end{split}$
(105)

where

$\displaystyle\begin{split}\accentset{\textrm{\ae}}{M}^{i\ j}_{\ A\
B}&=4\accentset{\text{\ae}}{N}^{2}\accentset{kin}{G}_{AB}^{\ \ \ \ i0j0}\\\
&=2\accentset{\textrm{\ae}}{C}_{1}\accentset{\text{\ae}}{\gamma}^{ij}\eta_{AB}-(\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3})\accentset{\text{\ae}}{\xi}_{A}\accentset{\text{\ae}}{\xi}_{B}\accentset{\text{\ae}}{\gamma}^{ij}+\accentset{\textrm{\ae}}{C}_{2}\text{\ae}_{A}{}^{j}\text{\ae}_{B}{}^{i}+\accentset{\textrm{\ae}}{C}_{3}\text{\ae}_{A}{}^{i}\text{\ae}_{B}{}^{j},\end{split}$
(106)

and

$\displaystyle\begin{split}{}^{3}\overset{\text{\ae}}{\mathbb{T}}\equiv-\accentset{\textrm{\ae}}{C}_{1}\eta_{AB}\accentset{\mathfrak{e}}{T}^{A}{}_{ij}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\accentset{\text{\ae}}{\gamma}^{ik}\accentset{\text{\ae}}{\gamma}^{jl}-\accentset{\textrm{\ae}}{C}_{2}\text{\ae}_{A}{}^{i}\text{\ae}_{B}{}^{j}\accentset{\mathfrak{e}}{T}^{A}{}_{kj}\accentset{\mathfrak{f}}{T}^{B}{}_{li}\accentset{\text{\ae}}{\gamma}^{kl}-\accentset{\textrm{\ae}}{C}_{3}\text{\ae}_{A}{}^{i}\text{\ae}_{B}{}^{j}\accentset{\text{\ae}}{\gamma}^{kl}\accentset{\mathfrak{e}}{T}^{A}{}_{ki}\accentset{\mathfrak{f}}{T}^{B}{}_{lj}.\end{split}$
(107)

The conjugate momenta hence becomes

$\displaystyle\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{m_{ef}^{2}}=\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\dot{f}^{B}{}_{j}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\overset{\mathfrak{f}}{D}_{j}f^{B}{}_{0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{il}\left(\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right)\right],$
(108)
$\displaystyle\frac{\accentset{\mathfrak{f}}{\pi}_{A}{}^{i}}{m_{ef}^{2}}=\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\dot{e}^{B}{}_{j}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\overset{\mathfrak{e}}{D}_{j}e^{B}{}_{0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{il}\left(\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right)\right].$
(109)

We decompose everything into irreducible parts, with respect to æ, as done in
Blagojevic:2000qs ; Blixt:2018znp .

$\displaystyle\dot{\mathfrak{e}}^{A}{}_{i}$
$\displaystyle={}^{\mathcal{V}}\dot{\mathfrak{e}}_{i}\accentset{\text{\ae}}{\xi}^{A}+{}^{\mathcal{A}}\dot{\mathfrak{e}}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{S}}\dot{\mathfrak{e}}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{T}}\dot{\mathfrak{e}}\text{\ae}^{A}{}_{i}$
(110) $\displaystyle\dot{\mathfrak{f}}^{A}{}_{i}$
$\displaystyle={}^{\mathcal{V}}\dot{\mathfrak{f}}_{i}\accentset{\text{\ae}}{\xi}^{A}+{}^{\mathcal{A}}\dot{\mathfrak{f}}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{S}}\dot{\mathfrak{f}}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{T}}\dot{\mathfrak{f}}\text{\ae}^{A}{}_{i}$
(111) $\displaystyle\accentset{\mathfrak{e}}{\pi}^{A}{}_{i}$
$\displaystyle={}^{\mathcal{V}}\accentset{\mathfrak{e}}{\pi}_{i}\accentset{\text{\ae}}{\xi}^{A}+{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\pi}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{S}}\accentset{\mathfrak{e}}{\pi}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{T}}\accentset{\mathfrak{e}}{\pi}\text{\ae}^{A}{}_{i}$
(112) $\displaystyle\accentset{\mathfrak{f}}{\pi}^{A}{}_{i}$
$\displaystyle={}^{\mathcal{V}}\accentset{\mathfrak{f}}{\pi}_{i}\accentset{\text{\ae}}{\xi}^{A}+{}^{\mathcal{A}}\accentset{\mathfrak{f}}{\pi}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{S}}\accentset{\mathfrak{f}}{\pi}_{ji}\accentset{\text{\ae}}{\gamma}^{kj}\text{\ae}^{A}{}_{k}+{}^{\mathcal{T}}\accentset{\mathfrak{f}}{\pi}\text{\ae}^{A}{}_{i}.$
(113)

The inversion for a general theory (18) is quite involved and lies beyond the
scope of this article. However, we can, as a toy example, derive the inversion
formula between the velocities and conjugate momenta for the subtheory
$\accentset{\mathfrak{e}}{C}_{1}=\accentset{\mathfrak{e}}{C}_{2}=\accentset{\mathfrak{e}}{C}_{3}=\accentset{\mathfrak{f}}{C}_{1}=\accentset{\mathfrak{f}}{C}_{2}=\accentset{\mathfrak{f}}{C}_{3}=0$,
for which the results are very similar to Blixt:2018znp . First we define the
sources which are independent of velocities:

$\displaystyle\overset{\mathfrak{e}}{S}_{A}{}^{i}=\accentset{\text{\ae}}{N}\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}+\overset{\mathfrak{f}}{D}_{j}\mathfrak{f}^{B}{}_{0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}+\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\gamma}^{il}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}-\accentset{\text{\ae}}{N}\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\gamma}^{il}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right]=\dot{\mathfrak{f}}^{B}{}_{j}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\ B},$ (114)
$\displaystyle\overset{\mathfrak{f}}{S}_{A}{}^{i}=\accentset{\text{\ae}}{N}\frac{\accentset{\mathfrak{f}}{\pi}_{A}{}^{i}}{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}+\overset{\mathfrak{e}}{D}_{j}\mathfrak{e}^{B}{}_{0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}+\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\gamma}^{il}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}-\accentset{\text{\ae}}{N}\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\gamma}^{il}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right]=\dot{\mathfrak{e}}^{B}{}_{j}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\ B},$ (115)

where $\accentset{\textrm{\ae}}{M}^{i\ j}_{\ A\ B}$ and its inverse can be
written in irreducible parts:

$\displaystyle\accentset{\textrm{\ae}}{M}^{i\ j}_{\ A\
B}={}^{\mathcal{V}}\accentset{\textrm{\ae}}{M}^{ij}\accentset{\text{\ae}}{\xi}_{A}\accentset{\text{\ae}}{\xi}_{B}+{}^{\mathcal{A}}\accentset{\textrm{\ae}}{M}^{[ir][js]}\text{\ae}^{C}{}_{r}\eta_{AC}\text{\ae}^{D}{}_{s}\eta_{BD}+{}^{\mathcal{S}}\accentset{\textrm{\ae}}{M}^{[ir][js]}\text{\ae}^{C}{}_{r}\eta_{AC}\text{\ae}^{D}{}_{s}\eta_{BD}+{}^{\mathcal{T}}\accentset{\textrm{\ae}}{M}\text{\ae}_{A}{}^{i}\text{\ae}_{B}{}^{j},$
(116)

which yields:

$\displaystyle\begin{split}\accentset{\textrm{\ae}}{M}^{i\ j}_{\ A\
B}&=-2\accentset{\textrm{\ae}}{A}_{\mathcal{V}}\accentset{\text{\ae}}{\xi}_{A}\accentset{\text{\ae}}{\xi}_{B}\accentset{\text{\ae}}{\gamma}^{ij}+2\accentset{\textrm{\ae}}{A}_{\mathcal{A}}\accentset{\text{\ae}}{\gamma}^{i[j}\accentset{\text{\ae}}{\gamma}^{s]r}\text{\ae}^{C}{}_{r}\eta_{AC}\text{\ae}^{D}{}_{s}\eta_{BD}+2\accentset{\textrm{\ae}}{A}_{\mathcal{S}}\left(\accentset{\text{\ae}}{\gamma}^{i(j}\accentset{\text{\ae}}{\gamma}^{s)r}+\frac{1}{3}\accentset{\text{\ae}}{\gamma}^{ir}\accentset{\text{\ae}}{\gamma}^{js}\right)\text{\ae}^{C}{}_{r}\eta_{AC}\text{\ae}^{D}{}_{s}\eta_{BD}\\\
&+\frac{2}{3}\accentset{\textrm{\ae}}{A}_{\mathcal{T}}\text{\ae}_{A}{}^{i}\text{\ae}_{B}{}^{j},\end{split}$
(117)

where

$\displaystyle\accentset{\textrm{\ae}}{A}_{\mathcal{V}}$
$\displaystyle=2\accentset{\textrm{\ae}}{C}_{1}+\accentset{\textrm{\ae}}{C}_{2}+\accentset{\textrm{\ae}}{C}_{3},$
(118) $\displaystyle\accentset{\textrm{\ae}}{A}_{\mathcal{A}}$
$\displaystyle=2\accentset{\textrm{\ae}}{C}_{1}-\accentset{\textrm{\ae}}{C}_{2}$
(119) $\displaystyle\accentset{\textrm{\ae}}{A}_{\mathcal{S}}$
$\displaystyle=2\accentset{\textrm{\ae}}{C}_{1}+\accentset{\textrm{\ae}}{C}_{2}$
(120) $\displaystyle\accentset{\textrm{\ae}}{A}_{\mathcal{T}}$
$\displaystyle=2\accentset{\textrm{\ae}}{C}_{1}+\accentset{\textrm{\ae}}{C}_{2}+3\accentset{\textrm{\ae}}{C}_{3}.$
(121)

Putting $\accentset{\textrm{\ae}}{A}_{\mathcal{I}}=0\
(\mathcal{I}\in\\{\mathcal{V},\mathcal{A},\mathcal{S},\mathcal{T}\\}$) we get
the following primary constraints ${}^{\mathcal{I}}C\approx 0$ and
${}^{\mathcal{I}}\tilde{C}\approx 0$:

$\displaystyle\begin{split}{}^{\mathcal{V}}\accentset{\mathfrak{e}}{\mathcal{C}}^{i}=\frac{{}^{\mathcal{V}}\accentset{\mathfrak{e}}{\pi}^{i}}{\sqrt{\accentset{\text{\ae}}{\gamma}}}-\accentset{\textrm{\ae}}{C}_{3}m_{ef}^{2}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\accentset{\text{\ae}}{\gamma}^{il}\text{\ae}_{B}{}^{k}\approx
0,\\\
{}^{\mathcal{V}}\accentset{\mathfrak{f}}{\mathcal{C}}^{i}=\frac{{}^{\mathcal{V}}\accentset{\mathfrak{f}}{\pi}^{i}}{{\sqrt{\accentset{\text{\ae}}{\gamma}}}}-\accentset{\textrm{\ae}}{C}_{3}m_{ef}^{2}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\accentset{\text{\ae}}{\gamma}^{il}\text{\ae}_{B}{}^{k}\approx
0.\end{split}$ (122)
$\displaystyle\begin{split}{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\mathcal{C}}_{ij}=\frac{{}^{\mathcal{A}}\accentset{\mathfrak{e}}{\pi}_{ij}}{\accentset{\text{\ae}}{\gamma}}-\accentset{\textrm{\ae}}{C}_{2}m_{ef}^{2}\accentset{f}{T}^{B}{}_{ij}\accentset{\text{\ae}}{\xi}_{B}\approx
0,\\\
{}^{\mathcal{A}}\accentset{\mathfrak{f}}{\mathcal{C}}_{ij}=\frac{{}^{\mathcal{A}}\accentset{\mathfrak{f}}{\pi}_{ij}}{\sqrt{\accentset{\text{\ae}}{\gamma}}}-\accentset{\textrm{\ae}}{C}_{2}m_{ef}^{2}\accentset{\mathfrak{e}}{T}^{B}{}_{ij}\accentset{\text{\ae}}{\xi}_{B}\approx
0.\end{split}$ (123)
$\displaystyle\begin{split}{}^{\mathcal{S}}\accentset{\mathfrak{e}}{\mathcal{C}}_{ij}=\frac{{}^{\mathcal{S}}\accentset{\mathfrak{e}}{\pi}_{ij}}{\sqrt{\accentset{\text{\ae}}{\gamma}}}\approx
0,\\\
{}^{\mathcal{S}}\accentset{\mathfrak{f}}{\mathcal{C}}_{ij}=\frac{{}^{\mathcal{S}}\accentset{\mathfrak{f}}{\pi}_{ij}}{\sqrt{\accentset{\text{\ae}}{\gamma}}}\approx
0.\end{split}$ (124)
$\displaystyle\begin{split}{}^{\mathcal{T}}\accentset{\mathfrak{e}}{\mathcal{C}}=\frac{{}^{\mathcal{T}}\accentset{\mathfrak{e}}{\pi}}{\sqrt{\accentset{\text{\ae}}{\gamma}}}\approx
0,\\\
{}^{\mathcal{T}}{\accentset{\mathfrak{f}}{\mathcal{C}}}=\frac{{}^{\mathcal{T}}\accentset{\mathfrak{f}}{\pi}}{\sqrt{\accentset{\text{\ae}}{\gamma}}}\approx
0.\end{split}$ (125)

We now want to include the contribution from
$\overset{\mathfrak{e}}{\mathcal{L}}$ and
$\overset{\mathfrak{f}}{\mathcal{L}}$. The 3+1 decomposition of this
Lagrangian reads

$\displaystyle\begin{split}\overset{\mathfrak{e}}{\mathcal{L}}+\overset{kin}{\mathcal{L}}+\overset{\mathfrak{f}}{\mathcal{L}}&=\frac{m_{e}^{2}\sqrt{\accentset{\mathfrak{e}}{\gamma}}}{\accentset{\mathfrak{e}}{N}}\accentset{\mathfrak{e}}{T}^{A}{}_{i0}\accentset{\mathfrak{e}}{T}^{B}{}_{j0}\accentset{\mathfrak{e}}{M}^{i\
j}_{\ A\
B}+\frac{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\accentset{\mathfrak{e}}{T}^{A}{}_{i0}\accentset{\mathfrak{f}}{T}^{B}{}_{j0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}+\frac{m_{f}^{2}\sqrt{\accentset{\mathfrak{f}}{\gamma}}}{\accentset{\mathfrak{f}}{N}}\accentset{\mathfrak{f}}{T}^{A}{}_{i0}\accentset{\mathfrak{f}}{T}^{B}{}_{j0}\accentset{\mathfrak{f}}{M}^{i\
j}_{\ A\ B}\\\
&+\frac{m_{e}^{2}\sqrt{\accentset{\mathfrak{e}}{\gamma}}}{\accentset{\mathfrak{e}}{N}}\accentset{\mathfrak{e}}{T}^{A}{}_{i0}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\left[\accentset{\mathfrak{e}}{M}^{i\
l}_{\ A\
B}\accentset{\mathfrak{e}}{N}^{k}-2\accentset{\mathfrak{e}}{N}\accentset{\mathfrak{e}}{\gamma}^{il}\left(\accentset{\mathfrak{e}}{C}_{2}\accentset{\mathfrak{e}}{\xi}_{B}\mathfrak{e}_{A}{}^{k}+\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{e}}{\xi}_{A}\mathfrak{e}_{B}{}^{k}\right)\right]\\\
&+\frac{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\left(\accentset{\mathfrak{e}}{T}^{A}{}_{i0}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}+\accentset{\mathfrak{f}}{T}^{A}{}_{i0}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\right)\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{il}\left(\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right)\right]\\\
&+\frac{m_{f}^{2}\sqrt{\accentset{\mathfrak{f}}{\gamma}}}{\accentset{\mathfrak{f}}{N}}\accentset{\mathfrak{f}}{T}^{A}{}_{i0}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\left[\accentset{\mathfrak{f}}{M}^{i\
l}_{\ A\
B}\accentset{\mathfrak{f}}{N}^{k}-2\accentset{\mathfrak{f}}{N}\accentset{\mathfrak{f}}{\gamma}^{il}\left(\accentset{\mathfrak{f}}{C}_{2}\accentset{\mathfrak{f}}{\xi}_{B}\mathfrak{f}_{A}{}^{k}+\accentset{\mathfrak{f}}{C}_{3}\accentset{\mathfrak{f}}{\xi}_{A}\mathfrak{f}_{B}{}^{k}\right)\right]\\\
&+\frac{m_{e}^{2}\sqrt{\accentset{\mathfrak{e}}{\gamma}}}{\accentset{\mathfrak{e}}{N}}\accentset{\mathfrak{e}}{T}^{A}{}_{ij}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\accentset{\mathfrak{e}}{N}^{i}\left[\frac{1}{2}\accentset{\mathfrak{e}}{M}^{j\
l}_{\ A\
B}\accentset{\mathfrak{e}}{N}^{k}-2\accentset{\mathfrak{e}}{N}\accentset{\mathfrak{e}}{\gamma}^{jl}\left(\accentset{\mathfrak{e}}{C}_{2}\accentset{\mathfrak{e}}{\xi}_{B}\mathfrak{e}_{A}{}^{k}+\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{e}}{\xi}_{A}\mathfrak{e}_{B}{}^{k}\right)\right]\\\
&+\frac{m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\left(\accentset{\mathfrak{e}}{T}^{A}{}_{ij}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}+\accentset{\mathfrak{f}}{T}^{A}{}_{ij}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\right)\left[\frac{1}{2}\accentset{\textrm{\ae}}{M}^{\
i\ k}_{\ A\
B}\accentset{\text{\ae}}{N}^{j}\accentset{\text{\ae}}{N}^{l}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{jl}\accentset{\text{\ae}}{\xi}_{A}\left(\accentset{\textrm{\ae}}{C}_{2}\text{\ae}_{B}{}^{i}\accentset{\text{\ae}}{N}^{k}+\accentset{\textrm{\ae}}{C}_{3}\text{\ae}_{B}{}^{k}\accentset{\text{\ae}}{N}^{i}\right)\right]\\\
&+\frac{m_{f}^{2}\sqrt{\accentset{\mathfrak{f}}{\gamma}}}{\accentset{\mathfrak{f}}{N}}\accentset{\mathfrak{f}}{T}^{A}{}_{ij}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\accentset{\mathfrak{f}}{N}^{i}\left[\frac{1}{2}\accentset{\mathfrak{f}}{M}^{j\
l}_{\ A\
B}\accentset{\mathfrak{f}}{N}^{k}-2\accentset{\mathfrak{f}}{N}\accentset{\mathfrak{f}}{\gamma}^{jl}\left(\accentset{\mathfrak{f}}{C}_{2}\accentset{\mathfrak{f}}{\xi}_{B}\mathfrak{f}_{A}{}^{k}+\accentset{\mathfrak{f}}{C}_{3}\accentset{\mathfrak{f}}{\xi}_{A}\mathfrak{f}_{B}{}^{k}\right)\right]\\\
&+\accentset{\mathfrak{e}}{N}m_{e}^{2}\sqrt{\accentset{\mathfrak{e}}{\gamma}}\cdot{}^{3}\overset{\mathfrak{e}}{\mathbb{T}}+\accentset{\text{\ae}}{N}m_{ef}^{2}\sqrt{\accentset{\text{\ae}}{\gamma}}\cdot{}^{3}\overset{\text{\ae}}{\mathbb{T}}+\accentset{\mathfrak{f}}{N}m_{f}^{2}\sqrt{\accentset{\mathfrak{f}}{\gamma}}\cdot{}^{3}\overset{\mathfrak{f}}{\mathbb{T}},\end{split}$
(126)

where we introduce

$\displaystyle\begin{split}\accentset{\mathfrak{e}}{M}^{i\ j}_{\ A\
B}&=4\accentset{\mathfrak{e}}{N}^{2}\accentset{\mathfrak{e}}{G}_{AB}^{\ \ \ \
i0j0}\\\
&=2\accentset{\mathfrak{e}}{C}_{1}\accentset{\mathfrak{e}}{\gamma}^{ij}\eta_{AB}-(\accentset{\mathfrak{e}}{C}_{2}+\accentset{\mathfrak{e}}{C}_{3})\accentset{\mathfrak{e}}{\xi}_{A}\accentset{\mathfrak{e}}{\xi}_{B}\accentset{\mathfrak{e}}{\gamma}^{ij}+\accentset{\mathfrak{e}}{C}_{2}\mathfrak{e}_{A}{}^{j}\mathfrak{e}_{B}{}^{i}+\accentset{\mathfrak{e}}{C}_{3}\mathfrak{e}_{A}{}^{i}\mathfrak{e}_{B}{}^{j},\end{split}$
(127) $\displaystyle\begin{split}\accentset{\mathfrak{f}}{M}^{i\ j}_{\ A\
B}&=4\accentset{\mathfrak{f}}{N}^{2}\accentset{\mathfrak{f}}{G}_{AB}^{\ \ \ \
i0j0}\\\
&=2\accentset{\mathfrak{f}}{C}_{1}\accentset{\mathfrak{f}}{\gamma}^{ij}\eta_{AB}-(\accentset{\mathfrak{f}}{C}_{2}+\accentset{\mathfrak{f}}{C}_{3})\accentset{\mathfrak{f}}{\xi}_{A}\accentset{\mathfrak{f}}{\xi}_{B}\accentset{\mathfrak{f}}{\gamma}^{ij}+\accentset{\mathfrak{f}}{C}_{2}\mathfrak{f}_{A}{}^{j}\mathfrak{f}_{B}{}^{i}+\accentset{\mathfrak{f}}{C}_{3}\mathfrak{f}_{A}{}^{i}\mathfrak{f}_{B}{}^{j},\end{split}$
(128)
$\displaystyle\begin{split}{}^{3}\overset{\mathfrak{e}}{\mathbb{T}}\equiv-\accentset{\mathfrak{e}}{C}_{1}\eta_{AB}\accentset{\mathfrak{e}}{T}^{A}{}_{ij}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\accentset{\mathfrak{e}}{\gamma}^{ik}\accentset{\mathfrak{e}}{\gamma}^{jl}-\accentset{\mathfrak{e}}{C}_{2}\mathfrak{e}_{A}{}^{i}\mathfrak{e}_{B}{}^{j}\accentset{\mathfrak{e}}{T}^{A}{}_{kj}\accentset{\mathfrak{e}}{T}^{B}{}_{li}\accentset{\mathfrak{e}}{\gamma}^{kl}-\accentset{\mathfrak{e}}{C}_{3}\mathfrak{e}_{A}{}^{i}\mathfrak{e}_{B}{}^{j}\accentset{\mathfrak{e}}{\gamma}^{kl}\accentset{\mathfrak{e}}{T}^{A}{}_{ki}\accentset{\mathfrak{e}}{T}^{B}{}_{lj},\end{split}$
(129)
$\displaystyle\begin{split}{}^{3}\overset{\mathfrak{f}}{\mathbb{T}}\equiv-\accentset{\mathfrak{f}}{C}_{1}\eta_{AB}\accentset{\mathfrak{f}}{T}^{A}{}_{ij}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\accentset{\mathfrak{f}}{\gamma}^{ik}\accentset{\mathfrak{f}}{\gamma}^{jl}-\accentset{\mathfrak{f}}{C}_{2}\mathfrak{f}_{A}{}^{i}\mathfrak{f}_{B}{}^{j}\accentset{\mathfrak{f}}{T}^{A}{}_{kj}\accentset{\mathfrak{f}}{T}^{B}{}_{li}\accentset{\mathfrak{f}}{\gamma}^{kl}-\accentset{\mathfrak{f}}{C}_{3}\mathfrak{f}_{A}{}^{i}\mathfrak{f}_{B}{}^{j}\accentset{\mathfrak{f}}{\gamma}^{kl}\accentset{\mathfrak{f}}{T}^{A}{}_{ki}\accentset{\mathfrak{f}}{T}^{B}{}_{lj}.\end{split}$
(130)

The conjugate momenta for this action are

$\displaystyle\begin{split}\frac{\accentset{\mathfrak{e}}{\pi}_{A}{}^{i}}{m_{ef}^{2}}&=\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\dot{\mathfrak{f}}^{B}{}_{j}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\overset{\mathfrak{f}}{D}_{j}\mathfrak{f}^{B}{}_{0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\accentset{\mathfrak{f}}{T}^{B}{}_{kl}\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{il}\left(\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right)\right]\\\
&+\frac{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}{\accentset{\mathfrak{e}}{N}}\dot{\mathfrak{e}}^{B}{}_{j}\accentset{\mathfrak{e}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}{\accentset{\mathfrak{e}}{N}}\overset{\mathfrak{e}}{D}_{j}\mathfrak{e}^{B}{}_{0}\accentset{\mathfrak{e}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\mathfrak{e}}{\gamma}}}{\accentset{\mathfrak{e}}{N}}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\left[\accentset{\mathfrak{e}}{M}^{i\
l}_{\ A\
B}\accentset{\mathfrak{e}}{N}^{k}-\accentset{\mathfrak{e}}{N}\accentset{\mathfrak{e}}{\gamma}^{il}\left(\accentset{\mathfrak{e}}{C}_{2}\accentset{\mathfrak{e}}{\xi}_{B}\mathfrak{e}_{A}{}^{k}+\accentset{\mathfrak{e}}{C}_{3}\accentset{\mathfrak{e}}{\xi}_{A}\mathfrak{e}_{B}{}^{k}\right)\right],\end{split}$
(131)
$\displaystyle\begin{split}\frac{\accentset{\mathfrak{f}}{\pi}_{A}{}^{i}}{m_{ef}^{2}}&=\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\dot{\mathfrak{e}}^{B}{}_{j}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\overset{\mathfrak{e}}{D}_{j}\mathfrak{e}^{B}{}_{0}\accentset{\textrm{\ae}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\text{\ae}}{\gamma}}}{\accentset{\text{\ae}}{N}}\accentset{\mathfrak{e}}{T}^{B}{}_{kl}\left[\accentset{\textrm{\ae}}{M}^{i\
l}_{\ A\
B}\accentset{\text{\ae}}{N}^{k}-\accentset{\text{\ae}}{N}\accentset{\text{\ae}}{\gamma}^{il}\left(\accentset{\textrm{\ae}}{C}_{2}\accentset{\text{\ae}}{\xi}_{B}\text{\ae}_{A}{}^{k}+\accentset{\textrm{\ae}}{C}_{3}\accentset{\text{\ae}}{\xi}_{A}\text{\ae}_{B}{}^{k}\right)\right]\\\
&+\frac{\sqrt{\accentset{\mathfrak{f}}{\gamma}}}{\accentset{\mathfrak{f}}{N}}\dot{\mathfrak{f}}^{B}{}_{j}\accentset{\mathfrak{f}}{M}^{i\
j}_{\ A\
B}-\frac{\sqrt{\accentset{\mathfrak{f}}{\gamma}}}{\accentset{\mathfrak{f}}{N}}\overset{\mathfrak{f}}{D}_{j}\mathfrak{f}^{B}{}_{0}\accentset{\mathfrak{f}}{M}^{i\
j}_{\ A\