Non-Relativistic and Ultra-Relativistic Scaling Limits of Multimetric Gravity

Ertuğrul Ekiz${}^{~{}1}$aaaEmail<EMAIL_ADDRESS>Oguzhan
Kasikci${}^{~{}1}$bbbEmail<EMAIL_ADDRESS>Mehmet
Ozkan${}^{~{}1}$cccEmail<EMAIL_ADDRESS>Cemal Berfu
Senisik${}^{~{}1}$dddEmail<EMAIL_ADDRESS>
and Utku Zorba${}^{~{}2}$eeeEmail<EMAIL_ADDRESS>
1 Department of Physics, Istanbul Technical University,
Maslak 34469 Istanbul, Turkey

2 Physics Department, Boğaziçi University,
34342 Bebek, Istanbul, Turkey

Abstract

> We present a method of contraction that can be applied to re-construct the
> recent extended non-relativistic and ultra-relativistic algebras as well as
> corresponding action principles. The methodology involves the use of
> multiple copies of Poincaré algebra. Consequently, the contraction defines
> non-relativistic or ultra-relativistic limits of multimetric theories of
> gravity. In particular, we show that the non-relativistic scaling limit of
> bi-metric gravity corresponds to the recent formulation of an action
> principle for Newtonian gravity with a constant background mass density.

###### Contents

  1. 1 Introduction
  2. 2 Algebras and Actions
    1. 2.1 Lie Algebra Expansion and the Poincaré Algebra
    2. 2.2 Non-Relativistic Algebras and Actions
    3. 2.3 Ultra-Relativistic Algebras and Actions
  3. 3 Non-Relativistic Scaling Limit of Multimetric Gravity
    1. 3.1 $D=4$
    2. 3.2 $D=3$
    3. 3.3 Non-Relativistic Algebras as a Contraction of Multiple Poincaré Algebras
  4. 4 Ultra-Relativistic Scaling Limit of Multimetric Gravity
  5. 5 Discussion
  6. A Equivalence of the First Order and the Second Order Formulation of the Newtonian Gravity

##  1 Introduction

####

Lie algebra expansion [1, 2, 3, 4] is a powerful tool to generate interesting
gravitational theories starting from the first order formulation of the
(cosmological) Einstein-Hilbert action. On the one hand, massive gravity
theories that are consistent with the holographic c-theorem have been shown to
arise from the truncation of an infinite-dimensional Lie algebra that is
closely connected to the Lie algebra expansion of the $\rm AdS$ algebra and
the cosmological Einstein-Hilbert action [5]. On the other hand, this
procedure was the main tool to construct the action principle for Newtonian
gravity in first-order formulation (see [6, 7] for its second order
formulation and the relevant $1/c^{2}$ expansion) as well as establishing new,
extended, two and three-dimensional non/ultra-relativistic gravity models [8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34]. In fact, the massive gravity models of [5] arise as
scaling limits of ghost-free bi-gravity models [37, 38] which later lead to
the discovery that there exist trajectories in the parameter space of bi-
gravity theories that connect the central charges of bulk/boundary unitary
three-dimensional bi-gravity models to non-unitary massive gravity theories by
a continuous change of scaling parameter [39, 40, 41]. It is, thus, a natural
question whether one can unify the Lie algebra expansion and the scaling limit
together to define a non/ultra-relativistic limit for bimetric and multimetric
models of gravity to establish similar connections between physical
quantities. In this paper, we will show that this is indeed the case by
presenting a systematic procedure that relates the space-time decomposed
multimetric gravity to extended non-relativistic and ultra-relativistic models
of gravity.

In building extended non/ultra-relativistic gravity, the main motivation comes
from the formulation of an action principle for Newtonian gravity. This
construction requires one to go beyond the standard Bargmann algebra by an
extension with additional three new generators. This extension, which was
originally formulated by $1/c^{2}$ expansion of Einstein-Hilbert action [7],
separates the strong gravitational effects from the relativistic effects and
it has been shown that in the presence of a twistless torsion, the Newtonian
gravity action can successfully explain the three classical tests of general
relativity [42, 44, 43, 45]. Thus, although the starting point was the
$1/c^{2}$-expansion of the Einstein gravity, the models that arise in each
order in the expansion are novel both theoretically and phenomenologically,
and can be studied in their own right.

The main procedure to find extended non/ultra-relativistic algebras and
corresponding gravity models is the Lie algebra expansion. It is based on the
splitting of the generators of a Lie algebra into even and odd classes
followed their series expansion with respect to the class that they belong to.
If the algebra under consideration is chosen to be the space-time split
Poincaré algebra, then the expansion yields either extended non-relativistic
or extended ultra-relativistic algebras [4]. The corresponding gravity models
can also be found in the same spirit, that is, one can start with the first-
order formulation of the Einstein-Hilbert action, perform the space-time
splitting and expand the fields in accordance with their corresponding
generator [4]. This procedure successfully gives rise to the action principle
for the Newtonian gravity (see Appendix A for the equivalence of the first and
the second order formulations) and various two and three-dimensional models
have been constructed/reconstructed in the recent literature. The
transformation rules for the matter fields can also be found by this
methodology [46] (see [45] for the corresponding $1/c^{2}$-expansion)
including the rigid supermultiplets of extended superalgebras. The local
transformation rules for supersymmetric models is an open problem to date. In
particular, in three-dimensions, the nature of expansion does not even allow
for the rigid supermultiplets for the algebras where the supergravity actions
can be written [46].

In a separate development, it has been found that the massive gravity models
that admit the holographic c-theorem arise from the truncation of an infinite
dimensional Lie algebra and its corresponding gauge theory of gravity [5].
This infinite-dimensional Lie algebra is a cosmological algebra and if the
cosmological parameter is set to zero, it simply becomes the Lie algebra
expansion of the $D$-dimensional Poincaré algebra. The inclusion of the
cosmological constant is equivalent to the infinite-dimensional expansion of
the $\rm(A)dS$ algebra given the fact that the cosmological constant is scaled
with the expansion parameter, i.e. $\Lambda\to\Lambda/\lambda^{2}$ where
$\Lambda$ is the cosmological constant and $\lambda$ represents the expansion
parameter. The truncation of the resulting infinite-dimensional Lie algebra
gives rise to gravity models with a set of auxiliary fields, which, when
solved and substituted back into the action, becomes massive gravity models
that are compatible with the holographic c-theorem. These models include the
new massive gravity [47] and its various extensions [48, 49], all of which
have been shown to be related to multimetric gravity by means of a scaling
limit [37, 38].

In this paper, we investigate the connection between these two seemingly
unrelated subjects. In particular, we have shown that the action principles
for non/ultra-relativistic gravity models arise as a scaling limit of
multimetric gravity. This can be thought as an important stepping stone
towards an understanding of the solutions and the phenomenological aspects of
non/ultra-relativistic gravity theories by performing the scaling limit that
we discuss in this paper. We begin in Section 2 by reminding the reader about
the basics of the Lie algebra expansion and present the general formulation of
extended non-relativistic and ultra-relativistic actions, giving particular
attention to three and four dimensions. In Section 3, we show that the non-
relativistic gravity models with larger symmetries are scaling limits of
multimetric theories with a Lorentzian signature. This point is one of our key
results, so let us be more precise with our statement. As mentioned, there is
a direct connection between the Lie algebra expanded (A)dS algebra and the
massive theories of gravity. When the cosmological constant is set to zero, a
model that comes from a consistent truncation of the infinite-dimensional
algebra does not describe massive gravity but it is a theory of gravity that
is coupled to a set of gauge fields [5]. These models arise from the scaling
limit of a multimetric theory in the absence of potential terms for the
vielbein. Nevertheless, as they are directly relevant to the expansion of the
Poincaré algebra, we first relate the multimetric models with no potential to
non-relativistic and ultra-relativistic gravity models. For example, the
scaling limit of a bimetric gravity without potential terms is the gauge
theory formulation of Newtonian gravity with no source. Based on our result
for how to take the scaling limit, we then turn on the potential terms and
establish their contribution to the non-relativistic and ultra-relativistic
models. In the case of bi-gravity, the potential terms give rise to a constant
background mass density for non-relativistic gravity. In Section 4, we show
that there is an analog construction for the ultra-relativistic gravity
models. We show that ultra-relativistic gravity with extended symmetries also
arises as a different limit of the same multi-gravity models, which resembles
the Galilei / Carroll limits of General Relativity. We give our comments and
conclusions in Section 5.

##  2 Algebras and Actions

####

The Lie algebra expansion is a method to generate higher-dimensional Lie
algebras starting from a lower-dimensional core Lie algebra. As we will
discuss the details momentarily, it is based on a series expansion of Maurer-
Cartan one-forms of the dual algebra. Thus, the expansion that generates
larger Lie algebras also generates action principles by expanding a core
action that is invariant under the core Lie algebra. In particular, for the
space-time decomposed Poincaré (or (A)dS) algebra and the corresponding first-
order formulation of the (cosmological) Einstein-Hilbert action, the expansion
yields non/ultra-relativistic gravity models at each order in expansion
parameter [4]. For example,

$\displaystyle\mathcal{L}_{\rm GR}$ $\displaystyle=$
$\displaystyle\lambda\mathcal{L}_{1}+\lambda^{3}\mathcal{L}_{3}+\lambda^{5}\mathcal{L}_{5}+\ldots\,,$
(2.1)

is the structure of the non-relativistic expansion where $\lambda$ is the
expansion parameter and $\mathcal{L}_{n}$ represents the Lagrangian at the
relevant $\lambda^{n}$ order [1, 2, 3, 4]. Note that each of these actions is
invariant under the corresponding order of the expanded core Lie algebra. As
can be seen from the structure of the expanded Lagrangian, the expansion and
the scaling limit $(\lambda\to 0)$ yield the same result at the lowest order.
For instance, in the case of (2.1), the lowest order Lagrangian in the
expansion, $\mathcal{L}_{1}$, can also be found by the same expansion of the
gauge fields, then rescaling the core Lagrangian $\mathcal{L}_{\rm core}$ by a
factor of $\lambda^{-1}$ and finally taking the limit $\lambda\to 0$ in which
case the coefficients of all $\mathcal{L}_{n}$ with $n>1$ vanishes. However,
it is not possible to single out a Lagrangian $\mathcal{L}_{n}$ with $n>1$ in
this way as rescaling the core Lagrangian with $\lambda^{-n}$ would yield
divergences in the coefficients of lower order terms. One way to isolate a
higher-order Lagrangian to perform the scaling limit is to consider multiple
copies of the same core algebra and combine the core Lagrangians to cancel out
any lower-order terms that would cause divergences. In the case of Poincaré
algebra, this means that we must consider multiple copies of Einstein-Hilbert
action to obtain a proper scaling limit. This is the leading technical notion
of this present paper, which then describes the non/ultra-relativistic scaling
limits of multi-gravity models. Thus, this section is aimed to discuss the
non-relativistic and ultra-relativistic expansions of the Poincaré algebra and
Einstein-Hilbert action in first-order formulation to set the stage for
multimetric models and their non/ultra-relativistic scaling limits.

###  2.1 Lie Algebra Expansion and the Poincaré Algebra

####

The Lie algebra expansion is a method that takes a core Lie algebra
$\mathfrak{g}$ and produces new, higher dimensional algebras as long as
$\mathfrak{g}$ can be written as a direct sum of two subspaces $V_{0}$ and
$V_{1}$ that satisfies the following relations

$\displaystyle\left[V_{0},V_{0}\right]$ $\displaystyle\subset V_{0}\,,$
$\displaystyle\left[V_{0},V_{1}\right]$ $\displaystyle\subset V_{1}\,,$
$\displaystyle\left[V_{1},V_{1}\right]$ $\displaystyle\subset V_{0}\,.$ (2.2)

Based on these relations, $V_{0}$ represent the even class of generators while
$V_{1}$ represents the odd class. The direct sum structure of the core Lie
algebra suggests that we may also assign a gauge field to each of the
generators

$\displaystyle A_{\mu}=A_{\mu}^{i}X_{i}+A_{\mu}^{\alpha}Y_{\alpha}\,,$ (2.3)

where $X^{i}$ represents the even subset of generators while $Y^{\alpha}$
represents the odd ones. In the next step, we expand the gauge fields with an
expansion parameter $\lambda$ with respect to the class that they belong to

$\displaystyle A^{i}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}A^{i}_{(2n)}\,,$ $\displaystyle
A^{\alpha}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}A^{\alpha}_{(2n+1)}\,.$ (2.4)

Here, the sum can be extended to infinity to produce and infinite dimensional
Lie algebra. The consistent truncation at order $g=(N_{0},N_{1})$ requires
that either $N_{0}=N_{1}$ or $N_{0}=N_{1}+1$ is satisfied. With this expansion
in hand, one can start with the Maurer-Cartan equations of the core Lie
algebra $\mathfrak{g}$, expand the gauge fields with respect to (2.4) and read
off the structure constants of the expanded algebra from the expanded Maurer-
Cartan equations at each order. Equivalently, based on their even/odd
character, we may expand the generators $\\{X^{i},Y^{\alpha}\\}$ with
$X_{i}\in V_{0}$ and $Y_{\alpha}\in V_{1}$ as follows

$\displaystyle X_{i}^{(2n)}$ $\displaystyle=\lambda^{2n}\otimes X_{i}\,,$
$\displaystyle Y_{\alpha}^{(2n+1)}$ $\displaystyle=\lambda^{2n+1}\otimes
Y_{\alpha}\,.$ (2.5)

Then, using the commutation relations of the core algebra

$\displaystyle\left[X_{i},X_{j}\right]$ $\displaystyle=f_{ij}{}^{k}X_{k}\,,$
$\displaystyle\left[X_{i},Y_{\alpha}\right]$
$\displaystyle=f_{i\alpha}{}^{\beta}Y_{\beta}\,,$
$\displaystyle\left[Y_{\alpha},Y_{\beta}\right]$
$\displaystyle=f_{\alpha\beta}{}^{i}X_{i}\,,$ (2.6)

we may give the commutation relations for the expanded algebra as [19]

$\displaystyle\left[X_{i}^{(2m)},X_{j}^{(2n)}\right]$
$\displaystyle=f_{ij}{}^{k}X_{k}^{(2m+2n)}\,,$
$\displaystyle\left[X_{i}^{(2m)},Y_{\alpha}^{(2n+1)}\right]$
$\displaystyle=f_{i\alpha}{}^{\beta}Y_{\beta}^{(2m+2n+1)}\,,$ $\displaystyle\
\left[Y_{\alpha}^{(2m+1)},Y_{\beta}^{(2n+1)}\right]$
$\displaystyle=f_{\alpha\beta}{}^{i}X_{i}^{(2m+2n+2)}\,.$ (2.7)

Note that the Jacobi identities for algebras are closed at each order without
referring to higher-order commutators. Consequently, the group theoretical
curvatures of the resulting truncated algebras satisfy the Bianchi identity
without referring to higher-order terms [1, 2, 3, 4]. With this result in
hand, let us now turn our attention to the space-time split Poincaré algebra
and its non/ultra-relativistic Lie algebra expansions. The $D$-dimensional
Poincaré algebra consists of translations $(P_{A})$ and Lorentz
transformations $M_{AB}$ with the following non-vanishing commutation
relations

$\displaystyle\left[M_{AB},P_{C}\right]$ $\displaystyle=2\eta_{C[B}P_{A]}\,,$
$\displaystyle\left[M_{AB},M_{CD}\right]$
$\displaystyle=4\eta_{[A[C}M_{D]B]}\,.$ (2.8)

The space-time decomposition can be achieved by decomposing the
$D$-dimensional index $A$ as $A=(0,a)$ in which case the generators are split
as

$\displaystyle M_{AB}$ $\displaystyle=\\{M_{0a}\equiv G_{a},J_{ab}\\}\,,$
$\displaystyle P_{a}$ $\displaystyle=\\{P_{0}\equiv H,P_{a}\\}\,.$ (2.9)

In this case, the Poincaré algebra decomposes as

$\displaystyle\left[G_{a},P_{b}\right]$ $\displaystyle=\delta_{ab}H\,,$
$\displaystyle\left[G_{a},H\right]$ $\displaystyle=P_{a}\,,$
$\displaystyle\left[J_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,$
$\displaystyle\left[J_{ab},G_{c}\right]$
$\displaystyle=\delta_{bc}G_{a}-\delta_{ac}G_{b}\,,$
$\displaystyle\left[J_{ab},J_{cd}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}\,,$
$\displaystyle\left[G_{a},G_{b}\right]$ $\displaystyle=J_{ab}\,.$ (2.10)

Based on the spacetime decomposed Poincaré algebra, we may discuss the non-
relativistic and ultra-relativistic Lie algebra expansions and action
principles.

###  2.2 Non-Relativistic Algebras and Actions

####

The non-relativistic higher-dimensional algebras are achieved with the
following choice for the generators [4]

$\displaystyle V_{0}$ $\displaystyle=\\{J_{ab},H\\}\,,$ $\displaystyle V_{1}$
$\displaystyle=\\{P_{a},G_{a}\\}\,.$ (2.11)

With this choice of generators, we can follow the prescription that we
presented in (2.7). Thus, the Lie algebra expansion of the spacetime
decomposed Poincaré algebra is given by

$\displaystyle\left[G_{a}^{(2m+1)},P_{b}^{(2n+1)}\right]$
$\displaystyle=\delta_{ab}H^{(2m+2n+2)}\,,$
$\displaystyle\left[G_{a}^{(2m+1)},H^{(2n)}\right]$
$\displaystyle=P_{a}^{(2m+2n+1)}\,,$
$\displaystyle\left[J_{ab}^{(2m)},P_{c}^{(2n+1)}\right]$
$\displaystyle=2\delta_{c[b}P_{a]}^{(2m+2n+1)}\,,$
$\displaystyle\left[J_{ab}^{(2m)},G_{c}^{(2n+1)}\right]$
$\displaystyle=2\delta_{c[b}G_{a]}^{(2m+2n+1)}\,,$
$\displaystyle\left[J_{ab}^{(2m)},J_{cd}^{(2n)}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}^{(2m+2n)}\,,$
$\displaystyle\left[G_{a}^{(2m+1)},G_{b}^{(2n+1)}\right]$
$\displaystyle=J_{ab}^{(2m+2n+2)}\,.$ (2.12)

To provide the well-known non-relativistic algebras that arise as a consistent
truncation of this infinite-dimensional algebra, let’s first focus on the
simplest case where we have two even and two odd generators,
$P_{a}^{(1)},G_{a}^{(1)},H^{(0)},J_{ab}^{(0)}$. These generators satisfy the
Galilei algebra

$\displaystyle\left[G_{a},H\right]$ $\displaystyle=P_{a}\,,$
$\displaystyle\left[J_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,$
$\displaystyle\left[J_{ab},G_{c}\right]$
$\displaystyle=\delta_{bc}G_{a}-\delta_{ac}G_{b}\,,$
$\displaystyle\left[J_{ab},J_{cd}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}\,,$ (2.13)

where we relabeled the generators as
$P_{a}^{(1)}=P_{a},G_{a}^{(1)}=G_{a},H^{(0)}=H$ and $J_{ab}^{(0)}=J_{ab}$.
According to the consistent truncation conditions, we may now add two more
generators that belong to $V_{0}$, i.e., $H^{(2)}$ and $J_{ab}^{(2)}$ 111The
algebra with these extra two generators is known as the extended Bargmann
algebra, which extends the Bargmann algebra with a new generator
$J_{ab}^{(2)}=S_{ab}$ if $H^{(2)}$ is identified as the mass generator
$H^{(2)}\equiv M$. The Lie algebra expansion skips these two cases and
directly goes from the Galilei algebra to the Newtonian algebra in four and
higher dimensions [4].. However, as we will discuss momentarily, this
truncation does not have a corresponding invariant action principle that can
be achieved by the expansion of Einstein-Hilbert action for in four and higher
dimensions. The next consistent truncation requires two more additional
generators of odd character, $P_{a}^{(3)}$ and $G_{a}^{(3)}$, in which case
the algebra becomes identical to the one that underlies the first-order
formulation of Newtonian gravity [43]

$\displaystyle\left[G_{a},H\right]$ $\displaystyle=P_{a}\,,$
$\displaystyle\left[J_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,$
$\displaystyle\left[J_{ab},G_{c}\right]$
$\displaystyle=\delta_{bc}G_{a}-\delta_{ac}G_{b}\,,$
$\displaystyle\left[J_{ab},T_{c}\right]$
$\displaystyle=\delta_{bc}T_{a}-\delta_{ac}T_{b}\,,$
$\displaystyle\left[J_{ab},B_{c}\right]$
$\displaystyle=\delta_{bc}B_{a}-\delta_{ac}B_{b}\,,$
$\displaystyle\left[J_{ab},J_{cd}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}\,,$
$\displaystyle\left[J_{ab},S_{cd}\right]$
$\displaystyle=4\delta_{[a[c}S_{d]b]}\,,$
$\displaystyle\left[G_{a},P_{b}\right]$ $\displaystyle=\delta_{ab}M\,,$
$\displaystyle\left[B_{a},H\right]$ $\displaystyle=T_{a}\,,$
$\displaystyle\left[G_{a},M\right]$ $\displaystyle=T_{a}\,,$
$\displaystyle\left[G_{a},G_{b}\right]$ $\displaystyle=S_{ab}\,,$
$\displaystyle\left[S_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}T_{a}-\delta_{ac}T_{b}\,,$
$\displaystyle\left[S_{ab},G_{c}\right]$
$\displaystyle=\delta_{bc}B_{a}-\delta_{ac}B_{b}\,,$ (2.14)

where we labeled $H^{(2)}=M$, $J_{ab}^{(2)}=S_{ab},P_{a}^{(3)}=T_{a}$ and
$G_{a}^{(3)}=B_{a}$. For the construction of an action principle, we first
need to space-time decompose the gauge fields of the Poincaré algebra, namely
vielbein $E_{\mu}{}^{A}$ and the spin connection $\Omega_{\mu}{}^{AB}$, i.e.

$\displaystyle E^{A}$ $\displaystyle=\\{E^{0}=T,E^{a}\\}\,,$
$\displaystyle\Omega^{AB}$
$\displaystyle=\\{\Omega^{0a}=\Omega^{a},\Omega^{ab}\\}\,.$ (2.15)

This step can then be followed by their expansion in line with the expansion
of their corresponding generator (2.9).

$\displaystyle T$ $\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\tau_{(2n)}\,,$
$\displaystyle E^{a}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}e^{a}_{(2n+1)}\,,$
$\displaystyle\Omega^{ab}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\omega^{ab}_{(2n)}\,,$
$\displaystyle\Omega^{a}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\omega^{a}_{(2n+1)}\,.$ (2.16)

These expressions can be used in the space-time decomposed Einstein-Hilbert
action in the first order formulation to generate invariant non-relativistic
gravity models. To perform the expansion explicitly, let us focus on three and
four dimensions, however our arguments are dimension independent. In four
dimensions, the space-time decomposed action is given by

$\displaystyle\mathcal{L}_{EH}$ $\displaystyle=$
$\displaystyle\epsilon_{ABCD}R^{AB}\wedge E^{C}\wedge
E^{D}=2\epsilon_{abc}\left(R(\Omega^{a})\wedge E^{b}\wedge
E^{c}-R(\Omega^{ab})\wedge E^{c}\wedge T\right)\,.$ (2.17)

Here $R^{AB}$ refers to the group-theoretical curvature of the spin connection
and $R(\Omega^{a})$ and $R(\Omega^{ab})$ refer to its spacetime decomposition
with respect to (2.15). While these curvatures can simply be read off from the
Poincaré algebra, it is useful to present their expression explicitly for
future purposes

$\displaystyle R(\Omega^{a})$
$\displaystyle=d\Omega^{a}+\Omega^{ab}\wedge\Omega_{b}\,,$ $\displaystyle
R(\Omega^{ab})$
$\displaystyle=d\Omega^{ab}-\Omega^{ac}\wedge\Omega^{b}{}_{c}-\Omega^{a}\wedge\Omega^{b}\,.$
(2.18)

The expanded action is then given by

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-2\epsilon_{abc}\sum_{m=0}^{N_{0}}\sum_{n=0}^{N_{1}}\sum_{\ell=0}^{N_{0}}\lambda^{2m+2n+2\ell+1}R(\omega^{ab}_{(2m)})\wedge
e^{c}_{(2n+1)}\wedge\tau_{(2\ell)}$ (2.19)
$\displaystyle+2\epsilon_{abc}\sum_{m=0}^{N_{1}}\sum_{n=0}^{N_{1}}\sum_{\ell=0}^{N_{1}}\lambda^{2m+2n+2\ell+3}R(\omega^{a}_{(2m+1)})\wedge
e^{b}_{(2n+1)}\wedge e^{c}_{(2\ell+1)}\,.$

As mentioned, the consistent truncation of the algebra as well as the action
requires a certain relation between $N_{0}$ and $N_{1}$. In the case of the
algebra, the necessary condition was that either $N_{0}=N_{1}$ or
$N_{0}=N_{1}+1$ must be satisfied. In the case of the action, the situation is
more subtle. For example, let us consider the expansion to order
$(N_{0},N_{1})=(1,1)$

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\lambda\left(-2\epsilon_{abc}R^{ab}(\omega)\wedge
e^{a}\wedge\tau\right)$ (2.20)
$\displaystyle+\lambda^{3}\Big{(}-2\epsilon_{abc}\left(R^{ab}(\omega)\wedge
e^{c}\wedge m+R^{ab}(s)\wedge e^{c}\wedge\tau+R^{ab}(\omega)\wedge
t^{c}\wedge\tau\right)$
$\displaystyle\qquad\,\,\,+2\epsilon_{abc}R^{a}(\Omega)\wedge e^{b}\wedge
e^{c}\Big{)}\,,$

where we set

$\displaystyle\tau_{(0)}$ $\displaystyle=\tau\,,$ $\displaystyle\tau_{(2)}$
$\displaystyle=m\,,$ $\displaystyle\omega^{ab}_{(0)}$
$\displaystyle=\omega^{ab}\,,$ $\displaystyle\omega^{ab}_{(2)}$
$\displaystyle=s^{ab}\,,$ $\displaystyle e^{a}_{(1)}$ $\displaystyle=e^{a}\,,$
$\displaystyle e^{a}_{(3)}$ $\displaystyle=t^{a}\,,$
$\displaystyle\omega^{a}_{(1)}$ $\displaystyle=\Omega^{a}$
$\displaystyle\omega^{a}_{(3)}$ $\displaystyle=b^{a}\,.$ (2.21)

Furthermore, the group theoretical curvatures $R^{ab}(\omega),R^{ab}(s)$ and
$R^{a}(\Omega)$ can easily be read off by expanding the curvatures (2.18) to
the $\lambda^{3}$ order

$\displaystyle R^{ab}(\omega)$
$\displaystyle=d\omega^{ab}-\omega^{ac}\wedge\omega^{b}{}_{c}\,,$
$\displaystyle R^{ab}(s)$ $\displaystyle=ds^{ab}-2\omega^{ac}\wedge
s^{b}{}_{c}-\omega^{a}\wedge\omega^{b}\,,$ $\displaystyle R^{a}(\omega)$
$\displaystyle=d\omega^{a}+\omega^{ab}\wedge\omega_{b}\,.$ (2.22)

In this Lagrangian, the $\lambda^{1}$-order model represents the Galilei
gravity which is invariant under the Galilei algebra (2.13). The
$\lambda^{3}$-order Lagrangian is the first-order formulation of the Newtonian
gravity which is invariant under (2.14). While we truncated the algebra and
the expansion at order $(N_{0},N_{1})=(1,1)$, we could have stopped at order
$(N_{0},N_{1})=(1,0)$. When this happens, the $\lambda$-order Lagrangian
remains unaltered but the $\lambda^{3}$-order does not have the
$R^{ab}(\omega)\wedge t^{a}\wedge\tau$ term in Lagrangian which spoils its
exact gauge invariance without referring to next-order terms. Although we
provide a four-dimensional example here, the argument holds in general. It is
better to think about the expansion as a truncation of the infinite-
dimensional algebra and the action. Unless all terms that contribute to a
certain $\lambda$-order are taken into account, the action is not gauge-
invariant [4]. In four-dimensions, the expanded Lagrangian (2.19) indicates
that the consistent truncation requires $N_{0}=N_{1}$.

In three-dimensions, for instance, the argument we provide above means that
the consistent truncation occurs if $N_{0}=N_{1}+1$. To see that, let us take
a look at the spacetime decomposed Einstein-Hilbert Lagrangian [4]

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle 2R(\Omega)\wedge
T+2\epsilon_{ab}R(\Omega^{a})\wedge E^{b}\,,$ (2.23)

where

$\displaystyle R(J)$
$\displaystyle=d\Omega-\epsilon_{ab}\Omega^{a}\wedge\Omega^{b}\,,$
$\displaystyle R^{a}(\Omega)$
$\displaystyle=d\Omega^{a}-\epsilon^{a}{}_{b}\Omega\wedge\Omega^{b}\,,$ (2.24)

where we used the fact that in two dimensions $\Omega^{ab}$ can be written as
$\Omega^{ab}=\epsilon^{ab}\Omega$. The expansion of the gauge fields (2.16)
gives rise to the following Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
2\sum_{m=0}^{N_{0}}\sum_{n=0}^{N_{0}}\lambda^{2m+2n}R(\omega_{(2m)})\wedge\tau_{(2n)}+2\epsilon_{ab}\sum_{m=0}^{N_{1}}\sum_{n=0}^{N_{1}}\lambda^{2m+2n+2}R(\omega^{a}_{(2m+1)})\wedge
e^{b}_{(2n+1)}\,.\qquad$ (2.25)

This form of the Lagrangian suggests that unless $N_{0}=N_{1}+1$ we cannot
obtain all contribution to any given $\lambda^{2n}$-order. For instance, let
us consider the expansion of the Lagrangian to $\lambda^{2}$-order

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\left(2R(\omega)\wedge\tau\right)+\lambda^{2}\left(2\left(R(\omega)\wedge
m+R(s)\wedge\tau+\epsilon_{ab}R^{a}(\omega)\wedge e^{b}\right)\right)\,,$
(2.26)

where

$\displaystyle R(\omega)$ $\displaystyle=d\omega\,,$ $\displaystyle R(s)$
$\displaystyle=ds-\epsilon_{ab}\omega^{a}\wedge\omega^{b}\,,$ $\displaystyle
R^{a}(\omega)$
$\displaystyle=d\omega^{a}-\epsilon^{a}{}_{b}\,\omega\wedge\omega^{b}\,.$
(2.27)

The zeroth-order action is the three-dimensional Galilei gravity. The
$\lambda^{2}$-order theory is known as the extended Bargmann gravity [8]. The
complete $\lambda^{2}$ contribution to the Lagrangian occurs as long as
$N_{0}=1$ and $N_{1}=0$. Consequently, the necessary transformation rules for
the gauge fields can be found by truncating the infinite-dimensional algebra
(2.12) by keeping the number of even generators two more than the number of
odd generators, namely $N_{0}=N_{1}+1$. In the case of extended Bargmann
gravity, the corresponding truncation is known as the extended Bargmann
algebra [8]

$\displaystyle\left[G_{a},H\right]$ $\displaystyle=P_{a}\,,$
$\displaystyle\left[J,P_{a}\right]$ $\displaystyle=-\epsilon_{ab}P^{b}\,,$
$\displaystyle\left[J,G_{a}\right]$ $\displaystyle=-\epsilon_{ab}G^{b}$
$\displaystyle\left[G_{a},P_{b}\right]$ $\displaystyle=\delta_{ab}M\,,$
$\displaystyle\left[G_{a},G_{b}\right]$ $\displaystyle=-\epsilon_{ab}S\,,$
(2.28)

where we set $J_{ab}=\epsilon_{ab}J$ and $S_{ab}=\epsilon_{ab}S$. In the next
order for the consistent truncation, one obtains the Extended Newtonian
Gravity [12], which we defer its details to Section 3.2.

###  2.3 Ultra-Relativistic Algebras and Actions

####

The ultra-relativistic higher-dimensional algebras are achieved with the
following choice for the generators

$\displaystyle V_{0}$ $\displaystyle=\\{J_{ab},P_{a}\\}\,,$ $\displaystyle
V_{1}$ $\displaystyle=\\{H,G_{a}\\}\,.$ (2.29)

Thus, the Lie algebra expansion of the spacetime decomposed Poincaré algebra
is given by

$\displaystyle\left[G_{a}^{(2m+1)},P_{b}^{(2n)}\right]$
$\displaystyle=\delta_{ab}H^{(2m+2n+1)}\,,$
$\displaystyle\left[G_{a}^{(2m+1)},H^{(2n+1)}\right]$
$\displaystyle=P_{a}^{(2m+2n+2)}\,,$
$\displaystyle\left[J_{ab}^{(2m)},P_{c}^{(2n)}\right]$
$\displaystyle=2\delta_{c[b}P_{a]}^{(2m+2n)}\,,$
$\displaystyle\left[J_{ab}^{(2m)},G_{c}^{(2n+1)}\right]$
$\displaystyle=2\delta_{c[b}G_{a]}^{(2m+2n+1)}\,,$
$\displaystyle\left[J_{ab}^{(2m)},J_{cd}^{(2n)}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}^{(2m+2n)}\,,$
$\displaystyle\left[G_{a}^{(2m+1)},G_{b}^{(2n+1)}\right]$
$\displaystyle=J_{ab}^{(2m+2n+2)}\,.$ (2.30)

The simplest truncation of this infinite-dimensional algebra is known as the
Carroll algebra [51, 52, 50, 53, 54, 55]

$\displaystyle[C_{a},P_{b}]$ $\displaystyle=\delta_{ab}H\,,$
$\displaystyle\left[J_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,$
$\displaystyle\left[J_{ab},C_{c}\right]$
$\displaystyle=\delta_{bc}C_{a}-\delta_{ac}C_{b}\,,$
$\displaystyle\left[J_{ab},J_{cd}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}\,,$ (2.31)

where we set $G_{a}^{(1)}=C_{a}$ to indicate that it is the generator of
Carrollian boosts. Due to the change of the expansion character of the
generators, the spacetime decomposed vielbein and the spin-connection are now
given by

$\displaystyle T$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\tau_{(2n+1)}\,,$
$\displaystyle E^{a}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}e^{a}_{(2n)}\,,$
$\displaystyle\Omega^{ab}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\omega^{ab}_{(2n)}\,,$
$\displaystyle\Omega^{a}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\omega^{a}_{(2n+1)}\,.$ (2.32)

As a result, the spacetime decomposed $D=4$ Einstein-Hilbert action is
expanded as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-2\epsilon_{abc}\sum_{m=0}^{N_{0}}\sum_{n=0}^{N_{0}}\sum_{\ell=0}^{N_{1}}\lambda^{2m+2n+2\ell+1}R(\omega^{ab}_{(2m)})\wedge
e^{c}_{(2n)}\wedge\tau_{(2\ell+1)}$ (2.33)
$\displaystyle+2\epsilon_{abc}\sum_{m=0}^{N_{1}}\sum_{n=0}^{N_{0}}\sum_{\ell=0}^{N_{0}}\lambda^{2m+2n+2\ell+1}R(\omega^{a}_{(2m+1)})\wedge
e^{b}_{(2n)}\wedge e^{c}_{(2\ell)}\,.$

Note that this case is much simpler than the non-relativistic case. All terms
in the Lagrangian are expanded to the same $\lambda$-order. Hence, it is
guaranteed to have all possible contributions to any given $\lambda$-order as
long as we have equal number of even and odd generators. At the lowest order
$(\lambda)$ we have the Carroll gravity [35, 36, 54, 55, 56]

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-2\epsilon_{abc}R^{ab}(\omega)\wedge
e^{a}\wedge\tau+2\epsilon_{abc}R^{a}(\omega)\wedge e^{b}\wedge e^{c}\,,$
(2.34)

where we set

$\displaystyle\tau_{(1)}$ $\displaystyle=\tau\,,$ $\displaystyle e^{a}_{(0)}$
$\displaystyle=e^{a}\,,$ $\displaystyle\omega^{ab}_{(0)}$
$\displaystyle=\omega^{ab}\,,$ $\displaystyle\omega^{a}_{(1)}$
$\displaystyle=\omega^{a}\,.$ (2.35)

For the next-to-leading order Lagrangian $(\lambda^{3})$, we have

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-2\epsilon_{abc}R^{ab}(s)\wedge
e^{a}\wedge\tau-2\epsilon_{abc}R^{ab}(\omega)\wedge
t^{a}\wedge\tau-2\epsilon_{abc}R^{ab}(\omega)\wedge e^{a}\wedge m$ (2.36)
$\displaystyle+2\epsilon_{abc}R^{a}(b)\wedge e^{b}\wedge
e^{c}+4\epsilon_{abc}R^{a}(\omega)\wedge e^{b}\wedge t^{c}\,.$

where we have

$\displaystyle\tau_{(1)}$ $\displaystyle=\tau\,,$ $\displaystyle\tau_{(3)}$
$\displaystyle=m\,,$ $\displaystyle\omega^{ab}_{(0)}$
$\displaystyle=\omega^{ab}\,,$ $\displaystyle\omega^{ab}_{(2)}$
$\displaystyle=s^{ab}\,,$ $\displaystyle e^{a}_{(0)}$ $\displaystyle=e^{a}\,,$
$\displaystyle e^{a}_{(2)}$ $\displaystyle=t^{a}\,,$
$\displaystyle\omega^{a}_{(1)}$ $\displaystyle=\omega^{a}$
$\displaystyle\omega^{a}_{(3)}$ $\displaystyle=b^{a}\,.$ (2.37)

Here, the group theoretical curvatures read

$\displaystyle R^{ab}(\omega)$
$\displaystyle=d\omega^{ab}-\omega^{ac}\wedge\omega^{b}{}_{c}\,,$
$\displaystyle R^{a}(\omega)$
$\displaystyle=d\omega^{a}+\omega^{ab}\wedge\omega_{b}\,,$ $\displaystyle
R^{ab}(s)$ $\displaystyle=ds^{ab}-2\omega^{ac}\wedge
s^{b}{}_{c}-\omega^{a}\wedge\omega^{b}\,,$ $\displaystyle R^{a}(b)$
$\displaystyle=db^{a}+\omega^{ab}\wedge b_{b}+s^{ab}\wedge\omega_{b}\,.$
(2.38)

This Lagrangian is invariant under the following extension of the Carroll
algebra

$\displaystyle[C_{a},P_{b}]$ $\displaystyle=\delta_{ab}H\,,$
$\displaystyle\left[J_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,$
$\displaystyle\left[J_{ab},C_{c}\right]$
$\displaystyle=\delta_{bc}C_{a}-\delta_{ac}C_{b}\,,$
$\displaystyle\left[J_{ab},J_{cd}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}\,,$ $\displaystyle[B_{a},P_{b}]$
$\displaystyle=\delta_{ab}M\,,$ $\displaystyle[C_{a},T_{b}]$
$\displaystyle=\delta_{ab}M\,,$ $\displaystyle[C_{a},H]$
$\displaystyle=T_{a}\,,$ $\displaystyle\left[J_{ab},T_{c}\right]$
$\displaystyle=\delta_{bc}T_{a}-\delta_{ac}T_{b}\,,$
$\displaystyle\left[S_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}T_{a}-\delta_{ac}T_{b}\,,$
$\displaystyle\left[J_{ab},B_{c}\right]$
$\displaystyle=\delta_{bc}B_{a}-\delta_{ac}B_{b}\,,$
$\displaystyle\left[S_{ab},C_{c}\right]$
$\displaystyle=\delta_{bc}B_{a}-\delta_{ac}B_{b}\,,$
$\displaystyle\left[J_{ab},S_{cd}\right]$
$\displaystyle=4\delta_{[a[c}S_{d]b]}\,,$
$\displaystyle\left[G_{a},G_{b}\right]$ $\displaystyle=S_{ab}\,,$ (2.39)

where we labeled $H^{(3)}=M$ and $P_{a}^{(2)}=T_{a}$. To our knowledge, it is
still an open problem to show the equivalence of the first-order action (2.37)
and the second order beyond-Carrollian action in [58]. Nevertheless, we refer
to this model as the beyond-Carrollian gravity. Note that all terms in the
spacetime decomposed three-dimensional Einstein-Hilbert Lagrangian are still
in the same order in the expansion, see (2.23). Thus, the argument that we
present for consistency of truncation also applies to three dimensions and we
will not present $D=3$ as an exceptional case.

##  3 Non-Relativistic Scaling Limit of Multimetric Gravity

####

In this section, we introduce a non/ultra-relativistic scaling limit for
multimetric gravity models based on the Lie algebra expansion that we
discussed in the previous section. As mentioned, the fundamental idea to have
a well-defined scaling limit is the necessity of using multiple copies of
Einstein-Hilbert action to get rid of divergent lower order Lagrangian(s). To
provide a concrete example, let us consider the non-relativistic expansion of
Einstein-Hilbert action to order $\lambda$, i.e.

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
M_{1}^{2}\lambda\left(-2\epsilon_{abc}R^{ab}(\omega)\wedge
e^{a}\wedge\tau\right)+\mathcal{O}(\lambda^{3})$ (3.1)

where $M_{1}$ is a mass parameter that multiplies the Einstein-Hilbert action.
Note that we expand the fields based on their expansion character (2.16)

$\displaystyle\tau_{(0)}$ $\displaystyle=\tau\,,$ $\displaystyle e^{a}_{(1)}$
$\displaystyle=e^{a}\,,$ $\displaystyle\omega^{ab}_{(0)}$
$\displaystyle=\omega^{ab}\,,$ $\displaystyle\omega^{a}_{(1)}=\omega^{a}\,.$
(3.2)

If we rescale the mass parameter $M_{1}^{2}$ as $M_{1}^{2}\to
M_{1}^{2}/\lambda$ and take the limit $\lambda\to 0$, we precisely recover
Galilei gravity as a non-relativistic limit of General Relativity. Let us now
consider the next order in expansion

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
M_{1}^{2}\lambda\left(-2\epsilon_{abc}R^{ab}(\omega)\wedge
e^{c}\wedge\tau\right)$ (3.3)
$\displaystyle+M_{1}^{2}\lambda^{3}\Big{(}-2\epsilon_{abc}\left(R^{ab}(\omega)\wedge
e^{c}\wedge m+R^{ab}(s)\wedge e^{c}\wedge\tau+R^{ab}(\omega)\wedge
t^{c}\wedge\tau\right)$
$\displaystyle\qquad\qquad+2\epsilon_{abc}R^{a}(\omega)\wedge e^{b}\wedge
e^{c}\Big{)}+\mathcal{O}(\lambda^{5})\,,$

where the fields are expanded in accordance with (2.21). Clearly, to single
out $\lambda^{3}$ action, it is not sufficient to rescale the mass parameter
$M_{1}^{2}\to M_{1}^{2}/\lambda^{3}$ and take the $\lambda\to 0$ limit since
the coefficient of the Galilei gravity diverges in that limit. Thus, we must
first annihilate the Galilei gravity action, then perform the proper scaling
of the mass parameter and the limit. This can be achieved by considering two
copies of the Einstein-Hilbert action

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
M_{1}^{2}\left[2\epsilon_{abc}\left(R(\Omega^{a})\wedge E^{b}\wedge
E^{c}-R(\Omega^{ab})\wedge E^{c}\wedge T\right)\right]$ (3.4)
$\displaystyle+M_{2}^{2}\left[2\epsilon_{abc}\left(R(\bar{\Omega}^{a})\wedge\bar{E}^{b}\wedge\bar{E}^{c}-R(\bar{\Omega}^{ab})\wedge\bar{E}^{c}\wedge\bar{T}\right)\right]\,,$

where the second set of gauge fields are
$\\{\bar{T}\,,\bar{E}^{a}\,,\bar{\Omega}^{a}\,,\bar{\Omega}^{ab}\\}$. To
cancel out the Galilei gravity, we may keep the definitions of the expansion
of the first set of fields as in (2.21) but expand the second set of the
fields as (2.16) with the following definitions to cancel out lowest order
divergences as long as $M_{1}^{2}=M_{2}^{2}$

$\displaystyle\bar{\tau}_{(0)}$ $\displaystyle=-\tau\,,$
$\displaystyle\bar{\tau}_{(2)}$ $\displaystyle=m\,,$
$\displaystyle\bar{\omega}^{ab}_{(0)}$ $\displaystyle=\omega^{ab}\,,$
$\displaystyle\bar{\omega}^{ab}_{(2)}$ $\displaystyle=-s^{ab}\,,$
$\displaystyle\bar{e}^{a}_{(1)}$ $\displaystyle=e^{a}\,,$
$\displaystyle\bar{e}^{a}_{(3)}$ $\displaystyle=-t^{a}\,,$
$\displaystyle\bar{\omega}^{a}_{(1)}$ $\displaystyle=\omega^{a}$
$\displaystyle\bar{\omega}^{a}_{(3)}$ $\displaystyle=b^{a}\,.$ (3.5)

With this choice for fields, the Galilei gravity comes with an opposite sign
and cancels the contribution from the first copy of the Einstein-Hilbert
action. On the other hand, the $\mathcal{O}(\lambda^{3})$ terms come with the
same numerical factors and signatures. Consequently, if we incorporate the
following scaling limit for the mass parameters

$\displaystyle M_{1}^{2}=M_{2}^{2}=\frac{M^{2}}{2\lambda^{3}}\,,$ (3.6)

we precisely recover the first-order formulation of Newtonian gravity after
taking the limit $\lambda\to 0$. In this section, our purpose is to define the
systematic of non/ultra-relativistic scaling limit of multimetric gravity in
this line of consideration. Before proceeding to the actual computation, we
remind the reader that the invariant non-relativistic actions in dimensions
$D\geq 4$ require an equal number of even and odd fields in the expansion
while $D=3$ is an exception, requiring two more even fields than the odd
fields [4]. Thus, we shall investigate these two cases separately.

The spacetime decomposed Einstein-Hilbert action in $D$-dimensions take the
following form

$\displaystyle\mathcal{L}_{EH}$ $\displaystyle=$ $\displaystyle
2\epsilon_{a_{1}a_{2}\ldots a_{d-1}a_{d}}\,E^{a_{1}}\wedge\ldots\wedge
E^{a_{d-1}}\wedge R(\Omega^{a_{d}})$ (3.7)
$\displaystyle+(d-1)\epsilon_{a_{1}a_{2}\ldots a_{d-1}a_{d}}\,T\wedge
E^{a_{1}}\wedge\ldots\wedge E^{a_{d-2}}\wedge R(\Omega^{a_{d-1}a_{d}})\,,$

where $d=D-1$ is the number of spatial dimensions. For the non-relativistic
expansion, the form of the relativistic fields (2.16) suggests that we should
investigate $d=2$ and $d>2$ cases separately since for $d=2$ all $E^{a}$ terms
drop out in the second term of the Lagrangian. The reason for this is the
following. Consider the expansion of the fields to order $(N_{0},N_{1})$ where
we either have $N_{0}=N_{1}$ or $N_{0}=N_{1}+1$. It is sufficient to consider
the appearance of the highest order even and odd fields to see which choice
include all terms at a given order $\mathcal{O}(\lambda^{n})$. If the highest
odd field is of order $\lambda^{2N+1}$, then we have two choices for the
highest even field depending on the chosen consistent truncation condition:

  * •

The highest even field is $\mathcal{O}(\lambda^{2N})$: In this case, the
second term in the Lagrangian (3.7) indicates that both the highest even and
the odd fields appears at order $d-2+2N$.

  * •

The highest even field is $\mathcal{O}(\lambda^{2N+2})$: In this case, the
second term in the Lagrangian (3.7) indicates that the highest odd field
appears at order $d-2+2N$ while the highest order even field appear at order
$d+2N$. This case cannot define a complete invariant Lagrangian since we
terminate the odd fields at $\mathcal{O}(\lambda^{2N+1})$, leading us to miss
the contributions of $\mathcal{O}(\lambda^{2N+3})$ odd fields to the action of
order $d+2N$.

Note that the situation changes dramatically for $d=2$. In that case, the
second term in (3.7) involves no odd field, and, as explained in Section 2.2,
the consistent truncation occurs by setting $N_{0}=N_{1}+1$. Thus, in this
section, we shall separately investigate $d=2$ and $d>2$. For $d>2$, we choose
$D=4$ as a representative example, however our arguments can be
straightforwardly generalized to $D>4$.

###  3.1 $D=4$

####

In $D=4$, the spacetime decomposed Einstein-Hilbert takes the following form

$\displaystyle\mathcal{L}_{EH}$ $\displaystyle=$ $\displaystyle
2M^{2}\epsilon_{abc}\left(R(\Omega^{a})\wedge E^{b}\wedge
E^{c}-R(\Omega^{ab})\wedge E^{c}\wedge T\right)\,.$ (3.8)

As mentioned, the Lie algebra expansion of this Lagrangian is formally given
by

$\displaystyle\mathcal{L}_{\rm EH}$ $\displaystyle=$ $\displaystyle\lambda
M^{2}\mathcal{L}_{1}+\lambda^{3}M^{2}\mathcal{L}_{3}+\lambda^{5}M^{2}\mathcal{L}_{5}+\ldots\,.$
(3.9)

where $\mathcal{L}_{n}$ represents the Lagrangian at the $\lambda^{n}$-order.
This form of the Lagrangian implies that if we want to single out a
$\mathcal{O}(\lambda^{N})$ Lagrangian by eliminating all terms of order $n<N$,
rescaling the mass parameter by $\lambda^{-N}$ and performing the $\lambda\to
0$ scaling limit, we need $N$-copies of the Einstein-Hilbert action. An
example of non-interacting bi-metric gravity and its limit as Newtonian
gravity was introduced in Section 3 by keeping the same expansion for the
first set of gauge fields (2.16) and properly choosing the second set to
annihilate the $\mathcal{O}(\lambda)$ action. For a more general result,
consider a non-interacting model of multimetric gravity that involves
$N$-number of spacetime decomposed vielbein and spin-connection

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[2M_{(i)}^{2}\epsilon_{abc}\left(R(\Omega^{a}_{(i)})\wedge
E^{b}_{(i)}\wedge E^{c}_{(i)}-R(\Omega^{ab}_{(i)})\wedge E_{(i)}^{c}\wedge
T_{(i)}\right)\right]\,,$ (3.10)

where $i,j=1,\ldots N$ label the set of fields
$\\{T_{(i)},E_{(i)},\Omega^{ab}_{(i)},\Omega^{a}_{(i)}\\}$. Following our
example for the Newtonian gravity, we keep the expansion of the first set of
fields $\\{T_{(1)},E_{(1)},\Omega^{a}_{(1)},\Omega^{ab}_{(1)}\\}$ the same as
(2.16), that is,

$\displaystyle T_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\tau_{(2n)}\,,$ $\displaystyle
E^{a}_{(1)}$ $\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}e^{a}_{(2n+1)}\,,$
$\displaystyle\Omega^{ab}_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\omega^{ab}_{(2n)}\,,$
$\displaystyle\Omega^{a}_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\omega^{a}_{(2n+1)}\,.$ (3.11)

However, for $i>1$, we can incorporate dimensionless free parameters
$\alpha_{(i)}$ as follows222 The expansion that we discuss here include the
previous example (3.5) as a special case up to an overall scaling and the
change of signature of the fields.

$\displaystyle T_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\alpha_{(i)}^{2n}\tau_{(2n)}\,,$
$\displaystyle E^{a}_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\alpha_{(i)}^{2n+1}e^{a}_{(2n+1)}\,,$
$\displaystyle\Omega^{ab}_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\alpha_{(i)}^{2n}\omega^{ab}_{(2n)}\,,$
$\displaystyle\Omega^{a}_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\alpha_{(i)}^{2n+1}\omega^{a}_{(2n+1)}\,.$
(3.12)

where $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$ and $\alpha_{i}\neq 1$ which
are the necessary conditions for having linearly independent set of gauge
fields. This expansion would yield the following expansion of
$\mathcal{L}_{(i)}$ that is the Einstein-Hilbert action for $i$-th set of
fields $\\{T_{(i)},E_{(i)},\Omega^{a}_{(i)},\Omega^{ab}_{(i)}\\}$

$\displaystyle\mathcal{L}_{EH,(i)}$ $\displaystyle=$
$\displaystyle\lambda\alpha_{i}M_{i}^{2}\mathcal{L}_{1}+\lambda^{3}\alpha_{i}^{3}M_{i}^{2}\mathcal{L}_{3}+\lambda^{5}\alpha_{i}^{5}M_{i}^{2}\mathcal{L}_{5}+\ldots\,.$
(3.13)

Note that the structure of the Lagrangian at each order $\lambda$ does not
change by the inclusion of the free parameters but we simply pick up a free
coefficient $\alpha_{i}^{2n+1}$ in front of the Lagrangians that are necessary
for cancellations. Let us now work out certain examples and then provide a
general formalism.

  1. 1.

Galilei Gravity: The Galilei gravity appears as $\mathcal{L}_{1}$ in the
expansion of Einstein-Hilbert action. As mentioned, it is sufficient to
consider a single copy of Einstein-Hilbert action and express the relativistic
fields and the mass parameter as

$\displaystyle T$ $\displaystyle=\tau\,$ $\displaystyle E^{a}$
$\displaystyle=\lambda e^{a}\,,$ $\displaystyle\Omega^{ab}$
$\displaystyle=\omega^{ab}\,,$ $\displaystyle\Omega^{a}$
$\displaystyle=\lambda\omega^{a}\,,$ $\displaystyle M^{2}$
$\displaystyle=\frac{m^{2}}{\lambda}\,.$ (3.14)

Upon taking the $\lambda\to 0$ limit, this choice would lead to the Galilei
gravity.

  2. 2.

Newtonian Gravity: The Newtonian gravity appears at the $\lambda^{3}$-order in
the Lie algebra expansion. Thus, we shall consider two copies of Einstein-
Hilbert action. For the first copy, we keep the same form as the Lie algebra
expansion, i.e.

$\displaystyle T_{1}$ $\displaystyle=\tau+\lambda^{2}m\,,$ $\displaystyle
E^{a}_{1}$ $\displaystyle=\lambda e^{a}+\lambda^{3}t^{a}\,,$
$\displaystyle\Omega^{ab}_{1}$
$\displaystyle=\omega^{ab}+\lambda^{2}s^{ab}\,,$ $\displaystyle\Omega^{a}_{1}$
$\displaystyle=\lambda\omega^{a}+\lambda^{3}b^{a}\,.$ (3.15)

For the second copy, we incorporate the free parameter $\alpha_{2}$

$\displaystyle T_{2}$ $\displaystyle=\tau+\lambda^{2}\alpha_{2}^{2}m\,,$
$\displaystyle E^{a}_{2}$
$\displaystyle=\lambda\alpha_{2}e^{a}+\lambda^{3}\alpha_{2}^{3}t^{a}\,,$
$\displaystyle\Omega^{ab}_{2}$
$\displaystyle=\omega^{ab}+\lambda^{2}\alpha_{2}^{2}s^{ab}\,,$
$\displaystyle\Omega^{a}_{2}$
$\displaystyle=\lambda\alpha_{2}\omega^{a}+\lambda^{3}\alpha_{2}^{3}b^{a}\,.$
(3.16)

The combination of these two Lagrangian with mass parameters $M_{1,2}$ gives
rise to

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\lambda
M^{2}\left(1+\frac{1}{2}\alpha_{2}\right)\mathcal{L}_{1}+\lambda^{3}M^{2}\left(1+\frac{1}{2}\alpha_{2}^{3}\right)\mathcal{L}_{3}+\mathcal{O}(\lambda^{5})\,.$
(3.17)

where we set $M_{1}^{2}=2M_{2}^{2}=M^{2}$ for simplicity. To cancel out the
first term, we fix $\alpha_{2}$ to be $\alpha_{2}=-2$. The coefficient of
$\mathcal{L}_{3}$ then becomes $-3$. Then, the following scaling of the mass
parameter

$\displaystyle M^{2}=\frac{1}{3\lambda^{3}}m^{2}\,.$ (3.18)

along with the scaling limit $\lambda\to 0$ recover the action principle for
the Newtonian gravity up to an overall minus sign.

  3. 3.

Beyond the Newtonian Gravity: The next-to-Newtonian gravity action arise at
order $\mathcal{O}(\lambda^{5})$. To perform a proper scaling limit, we need
three copies of the Einstein-Hilbert action. Using the expansion of the
relativistic fields (3.12), we obtain the following Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\lambda\left(M_{1}^{2}+M_{2}^{2}\alpha_{2}+M_{3}^{2}\alpha_{3}\right)\mathcal{L}_{1}+\lambda^{3}\left(M_{1}^{2}+M_{2}^{2}\alpha_{2}^{3}+M_{3}^{2}\alpha_{3}^{3}\right)\mathcal{L}_{3}$
(3.19)
$\displaystyle+\lambda^{5}\left(M_{1}^{2}+M_{2}^{2}\alpha_{2}^{5}+M_{3}^{2}\alpha_{3}^{5}\right)\mathcal{L}_{5}+\mathcal{O}(\lambda^{7})\,.$

Here, $\mathcal{L}_{5}$ represents the next-to-Newtonian gravity Lagrangian.
For convenience, let us choose $M_{1}^{2}=2M_{2}^{2}=2/3M_{3}^{2}=M^{2}$. In
that case, to cancel out the coefficients of the Galilei and Newtonian
gravity, we need to solve two equations

$\displaystyle 0$
$\displaystyle=1+\frac{1}{2}\alpha_{2}+\frac{3}{2}\alpha_{3}\,,$
$\displaystyle 0$
$\displaystyle=1+\frac{1}{2}\alpha_{2}^{3}+\frac{3}{2}\alpha_{3}^{3}\,.$
(3.20)

These two equations can be solved to eliminate $\alpha_{2}$ and $\alpha_{3}$
as $\alpha_{2}=-5/4$ and $\alpha_{3}=-1/4$. Then, the following scaling of the
mass parameter

$\displaystyle M^{2}=\frac{256}{135\lambda^{5}}m^{2}\,.$ (3.21)

along with the scaling limit $\lambda\to 0$ recover the Lagrangian for next-
to-Newtonian gravity up to an overall minus sign.

With these three examples, it is now evident that for a Lagrangian of
order-$2N+1$, we need to introduce $N$-copies of Einstein-Hilbert action.
Using the expansion of the fields, we obtain

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{N}\lambda^{2n+1}\left(M_{1}^{2}+\sum_{i=2}^{N+1}\alpha_{i}^{2n+1}M_{i}^{2}\right)\mathcal{L}_{2n+1}+\mathcal{O}(\lambda^{2N+3})\,.$
(3.22)

To single out the order-$2N+1$ Lagrangian, one needs to solve $N$ number of
algebraic equations that determines the values of $\alpha_{i}$ for
$i=2,3,\ldots N$

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{1}^{2}+\sum_{i=2}^{N+1}\alpha_{i}^{2n+1}M_{i}^{2}\,,\qquad\text{for}\qquad
n=0,1,\ldots,N-1\,.$ (3.23)

These values can finally be used in the coefficient of the order-$2N+1$
Lagrangian, along with the scaling of the mass parameters $M_{i}\to
M_{i}/\lambda^{2N+1}$. Finally, performing the $\lambda\to 0$ limit gets rid
of all higher order terms and yields the desired non-relativistic model.

Although we have so far discussed the non-relativistic limit of non-
interacting multimetric models, we can turn on potential terms that gives rise
to the interaction among $E_{(i)}^{a}$. Upon spacetime decomposition, they
typically take the following form in four dimensions

$\displaystyle\mathcal{L}_{CC}=\epsilon_{abc}J_{ijkl}T_{i}\wedge
E^{a}_{j}\wedge E^{b}_{k}\wedge E^{c}_{l}\,,$ (3.24)

where $J_{ijjk}$ is a matrix of constant coefficients of dimension mass-
squared. The off-diagonal elements of this matrix determine the interaction
between the temporal and spatial vielbein of different gravity sectors. Upon
implementing the $\alpha$-expansion, (3.11) and (3.12), we notice that the
lowest order contribution arise at $\mathcal{O}(\lambda^{3})$, i.e.

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\lambda^{3}\epsilon_{abc}\tau\wedge e^{a}\wedge e^{b}\wedge
e^{c}+\mathcal{O}(\lambda^{5})\,,$ (3.25)

where we assumed that the coefficients of $J_{ijkl}$ and $\alpha_{(i)}$ are
not chosen in a particular fashion to annihilate each other. This is the same
expansion order as the Newtonian gravity. As we will discuss the details in
Appendix A, this expansion implies that the non-relativistic limit of bi-
gravity with a non-vanishing potential is the Newtonian gravity with a
constant background mass density. For the three-metric model, we need to go to
the $\mathcal{O}(\lambda^{5})$

$\displaystyle\mathcal{L}\sim\lambda^{5}\epsilon_{abc}\left(m\wedge
e^{a}\wedge e^{b}\wedge e^{c}+3\tau\wedge t^{a}\wedge e^{b}\wedge
e^{c}\right)+\mathcal{O}(\lambda^{7})\,.$ (3.26)

Note that we have more than a sufficient number of free parameters to
annihilate the order-$\lambda^{3}$ Lagrangian, hence the details are not
presented.

###  3.2 $D=3$

####

In three-dimensions, the spacetime decomposed Einstein-Hilbert Lagrangian
takes the following form (2.23)

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
2M\left(R(\Omega)\wedge T+\epsilon_{ab}R(\Omega^{a})\wedge E^{b}\right)\,,$
(3.27)

which implies that upon Lie algebra expansion (2.16) we have the following
structure for the Lagrangian

$\displaystyle\mathcal{L}_{EH}$ $\displaystyle=$
$\displaystyle\lambda^{0}M\mathcal{L}_{0}+\lambda^{2}M\mathcal{L}_{2}+\lambda^{4}M\mathcal{L}_{4}+\ldots\,.$
(3.28)

Let us start our investigation with the zeroth-order Lagrangian which
describes the three-dimensional Galilei gravity

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$ $\displaystyle 2\tau\wedge
d\omega\,.$ (3.29)

This model only involves the gauge fields of time translation and rotations,
which, in the case of three-dimensions, is a $U(1)\times U(1)$ gauge theory.
Therefore, this model can be considered as an off-diagonal $U(1)\times U(1)$
Chern-Simons gauge theory

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle 2Z_{1}\wedge
dZ_{2}\,,$ (3.30)

which the identification $Z_{1}=\tau$ and $Z_{2}=\omega$, corresponding to the
off-diagonal invariant metric with $g(Z_{1},Z_{2})=1$. At this stage, it is
obvious that the elimination of lower-order Lagrangians to single out higher-
order is more subtle in $D=3$. It is not sufficient to consider multiple
copies of Einstein-Hilbert action, but the theory must be extended with a
$U(1)\times U(1)$ Chern-Simons gauge theory to cancel out the lowest order
Lagrangian, see [9, 12, 17] as particular examples.

Let’s now move on to the next order $\mathcal{L}_{2}$. In this case, we have

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
M\left(2R(\omega)\wedge\tau\right)+\lambda^{2}M\left(2\left(R(\omega)\wedge
m+R(s)\wedge\tau+\epsilon_{ab}R^{a}(\omega)\wedge
e^{b}\right)\right)+\mathcal{O}(\lambda^{4})\,.$ (3.31)

where the curvatures are as defined in (2.27). To obtain this model with as a
scaling limit, we begin with the three-dimensional Einstein-Hilbert action
with an additional $U(1)\times U(1)$ Chern-Simons gauge theory

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle 2MR(\Omega)\wedge
T+2M\epsilon_{ab}R(\Omega^{a})\wedge E^{b}+2MZ_{1}\wedge dZ_{2}\,.$ (3.32)

The zeroth-order Galilei gravity can then be annihilated by a proper choice of
$Z_{1}$ and $Z_{2}$, i.e.

$\displaystyle T$ $\displaystyle=\tau+\lambda^{2}m\,,$ $\displaystyle E^{a}$
$\displaystyle=\lambda e^{a}\,,$ $\displaystyle\Omega$
$\displaystyle=\omega+\lambda^{2}s\,,$ $\displaystyle\Omega^{a}$
$\displaystyle=\lambda\omega^{a}\,,$ $\displaystyle Z_{1}$
$\displaystyle=-\tau\,,$ $\displaystyle Z_{2}$ $\displaystyle=\omega\,,$
(3.33)

which precisely recovers the $\mathcal{O}(\lambda^{2})$ Lagrangian, known as
the extended Bargmann gravity [8] upon rescaling the mass parameter $M\to
M/\lambda^{2}$ and taking the limit $\lambda\to 0$. Note that this contraction
is much simpler than the one presented in [9], thank to the Lie algebra
expansion. Similarly, if we are to move on with the next order Lagrangian, we
have

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
M\left(2R(\omega)\wedge\tau\right)+\lambda^{2}M\left(2R(\omega)\wedge
m+2R(s)\wedge\tau-2R^{a}(\omega)\wedge e_{a}\right)$ (3.34)
$\displaystyle+\lambda^{4}M\left(R(s)\wedge m+R(z)\wedge\tau-R(\omega)\wedge
y-R^{a}(\omega)\wedge t_{a}-R^{a}(b)\wedge
e_{a}\right)+\mathcal{O}(\lambda^{6})$

where the $\mathcal{O}(\lambda^{4})$ Lagrangian is known as the extended
Newtonian gravity [12]. Note that the Lagrangian involves the redefinition of
the fields

$\displaystyle e^{a}$ $\displaystyle\to\epsilon^{ab}e_{b}\,,$ $\displaystyle
t^{a}$ $\displaystyle\to\epsilon^{ab}t_{b}\,,$ $\displaystyle y\to-y\,,$
(3.35)

in the Lie algebra expansion of the fields to match with the existing
literature [4]. As in the case of four-dimensions, we can reproduce this
Lagrangian by considering a three-dimensional bigravity with an additional
$U(1)\times U(1)$ Chern-Simons gauge theory

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
2M_{1}R(\Omega_{1})\wedge T_{1}+2M_{1}\epsilon_{ab}R(\Omega^{a}_{1})\wedge
E^{b}_{1}+2M_{2}R(\Omega_{2})\wedge T_{2}$ (3.36)
$\displaystyle+2M_{2}\epsilon_{ab}R(\Omega^{a}_{2})\wedge
E^{b}_{2}+2M_{3}Z_{1}\wedge dZ_{2}\,.$

Using the standard expansion for the first set of fields and implementing the
$\alpha$-expansion in the second set of fields

$\displaystyle T_{1}$ $\displaystyle=\tau+\lambda^{2}m-\lambda^{4}y\,,$
$\displaystyle\Omega_{1}$ $\displaystyle=\omega+\lambda^{2}s+\lambda^{4}z\,,$
$\displaystyle\Omega^{a}_{1}$
$\displaystyle=\lambda\omega^{a}+\lambda^{3}b^{a}\,,$ $\displaystyle
E^{a}_{1}$
$\displaystyle=\lambda\epsilon^{ab}e_{b}+\lambda^{3}\epsilon^{ab}t_{b}\,,$
$\displaystyle Z_{1}$ $\displaystyle=\beta_{1}\tau\,,$ $\displaystyle Z_{2}$
$\displaystyle=\omega\,,$ $\displaystyle T_{2}$
$\displaystyle=\tau+\alpha_{2}^{2}\lambda^{2}m-\alpha_{2}^{4}\lambda^{4}y\,,$
$\displaystyle\Omega_{2}$
$\displaystyle=\omega+\alpha_{2}^{2}\lambda^{2}s+\alpha_{2}^{4}\lambda^{4}z\,,$
$\displaystyle\Omega^{a}_{2}$
$\displaystyle=\alpha_{2}\lambda\omega^{a}+\alpha_{2}^{3}\lambda^{3}b^{a}\,,$
$\displaystyle E^{a}_{2}$
$\displaystyle=\alpha_{2}\lambda\epsilon^{ab}e_{b}+\alpha_{2}^{3}\lambda^{3}\epsilon^{ab}t_{b}\,,$
(3.37)

we obtain

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\left(M_{1}+\beta_{1}M_{3}+M_{2}\right)\left(2R(\omega)\wedge\tau\right)$
(3.38)
$\displaystyle+\lambda^{2}\left(M_{1}+\alpha_{2}^{2}M_{2}\right)\left(2R(\omega)\wedge
m+2R(s)\wedge\tau-2R^{a}(\omega)\wedge e_{a}\right)$
$\displaystyle+\lambda^{4}\left(M_{1}+\alpha_{2}^{4}M_{2}\right)\left(R(s)\wedge
m+R(z)\wedge\tau-R(\omega)\wedge y-R^{a}(\omega)\wedge t_{a}-R^{a}(b)\wedge
e_{a}\right)$ $\displaystyle+\mathcal{O}(\lambda^{6})\,.$

Note that the three-dimensional case is not complicated than the scaling limit
in four-dimensions. The $U(1)\times U(1)$ gauge theory only annihilates the
zeroth-order Lagrangian but does not interfere with higher-order ones.
Consequently, $\beta_{1}$ can be solved by using the coefficient of the
Galilei gravity and the remainder of the problem is the same as the four-
dimensional case. To present a solution, let us set $M_{1}=M_{3}=-4M_{2}=M$.
In that case, we can set the coefficients of the Galilei and the extended
Bargmann gravity by the following choices for $\alpha_{2}$ and $\beta_{1}$

$\displaystyle\alpha_{2}$ $\displaystyle=2\,,$ $\displaystyle\beta_{1}$
$\displaystyle=-\frac{3}{4}\,.$ (3.39)

Finally, rescaling the mass parameter $M$ as

$\displaystyle M=-\frac{m}{3\lambda^{4}}\,,$ (3.40)

and performing the scaling limit $\lambda\to 0$ precisely recover the extended
Newtonian gravity [12]. Once again, the scaling limit presented in (3.37) is
much simpler than the one found in [12].

With these three examples, the systematic of three-dimensional non-
relativistic scaling limit is obvious. To start with , one needs an $N$-copy
of Einstein-Hilbert action with an additional $U(1)\times U(1)$ Chern-Simons
gauge theory

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[2M_{(i)}R(\Omega_{(i)})\wedge
T_{(i)}+2M_{(i)}\epsilon_{ab}R(\Omega^{a}_{(i)})\wedge
E^{b}_{(i)}\right]+2M_{N+1}Z_{1}\wedge dZ_{2}\,.$ (3.41)

The fields can then be expanded using the standard expansion for the first set
of fields (2.16) and the $\alpha$-expansion for the other set of fields
(3.12). In addition, the gauge fields $Z_{1}$ and $Z_{2}$ must be chosen as

$\displaystyle Z_{1}=$ $\displaystyle\beta_{1}\tau\,,$ $\displaystyle Z_{2}$
$\displaystyle=\omega\,.$ (3.42)

These choices would lead us to the following Lagrangian for the $N$-metric
theory

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\left(\beta_{1}M_{N+1}+\sum_{i=1}^{N}M_{i}\right)\mathcal{L}_{0}+\sum_{n=1}^{N}\lambda^{2n}\left(M_{1}+\sum_{i=2}^{N}\alpha_{i}^{2n+1}M_{i}\right)\mathcal{L}_{2n}+\mathcal{O}(\lambda^{2N+2})\,.$
(3.43)

The solution for $\beta_{1}$ is simply

$\displaystyle\beta_{1}=-\frac{1}{M_{N+1}}\sum_{i=1}^{N}M_{(i)}$ (3.44)

For the remaining coefficients, we can follow our footsteps in $D=4$ and solve
the coefficients of the $\mathcal{O}(\lambda^{2N-2})$ models to single out the
Lagrangian at order $\lambda^{2N}$. In the last step, we rescale the mass
parameters $M_{(i)}\to M_{(i)}/\lambda^{2N}$ and perform the scaling limit
$\lambda\to 0$ and obtain the desired non-relativistic Lagrangian.

###  3.3 Non-Relativistic Algebras as a Contraction of Multiple Poincaré
Algebras

####

The Lie algebra expansion is not just the expansion of the gauge fields but
also the expansion of the generators (2.5). Thus, it is expected that the
scaling limit that we establish here for the Lagrangians can be extended to a
relation between $N$-copies of the Poincaré algebra and the non-relativistic
algebra at the relevant order. The Galilei and the Bargmann algebra have been
known to arise from the contraction of the Poincaré and Poincaré $\oplus$ U(1)
algebras, respectively [57]. In this section, we show that the higher-order
non-relativistic algebras that admit an invariant Lagrangian arise from the
contraction of multiple copies of Poincaré algebra. There is of course an
exception in three-dimension with requires an addition of $U(1)\times U(1)$
sector to cancel out divergences.

Let us start our discussion with two representative examples: The Galilei
algebra (2.13) as a contraction of the Poincaré algebra and the non-
relativistic algebra of Newtonian gravity (2.14) as a contraction of two
copies of the Poincaré algebra. Consider the spacetime decomposed Poincaré
algebra (2.10). If we rescale the odd generators as

$\displaystyle P_{a}$ $\displaystyle\to\lambda P_{a}\,,$ $\displaystyle
G_{a}\to\lambda G_{a}\,,$ (3.45)

and take the $\lambda\to\infty$ limit, the $[P_{a},G_{b}]$ and $[G_{a},G_{b}]$
commutators vanish, giving rise to the Galilei algebra (2.13). Next, consider
two copies of the spacetime decomposed Poincaré algebra, the first set being
labeled as $\\{H^{1},P_{a}^{1},G_{a}^{1},J_{ab}^{1}\\}$ and the second set
with $\\{H^{2},P_{a}^{2},G_{a}^{2},J_{ab}^{2}\\}$. Expressing the non-
relativistic generators as

$\displaystyle J_{ab}$ $\displaystyle=J_{ab}^{1}+J_{ab}^{2}\,,$ $\displaystyle
S_{ab}$
$\displaystyle=\lambda^{2}\left(\alpha_{1}^{2}J_{ab}^{1}+\alpha_{2}^{2}J_{ab}^{2}\right)\,,$
$\displaystyle H$ $\displaystyle=H^{1}+H^{2}\,,$ $\displaystyle M$
$\displaystyle=\lambda^{2}\left(\alpha_{1}^{2}H^{1}+\alpha_{2}^{2}H^{2}\right)\,,$
$\displaystyle P_{a}$
$\displaystyle=\lambda\left(\alpha_{1}P_{a}^{1}+\alpha_{2}P_{a}^{2}\right)\,,$
$\displaystyle T_{a}$
$\displaystyle=\lambda^{3}\left(\alpha_{1}^{3}P_{a}^{1}+\alpha_{2}^{3}P_{a}^{2}\right)\,,$
$\displaystyle G_{a}$
$\displaystyle=\lambda\left(\alpha_{1}G_{a}^{1}+\alpha_{2}G_{a}^{2}\right)\,,$
$\displaystyle B_{a}$
$\displaystyle=\lambda^{3}\left(\alpha_{1}^{3}G_{a}^{1}+\alpha_{2}^{3}G_{a}^{2}\right)\,,$
(3.46)

we precisely recover the commutation relations for the algebra (2.14) in the
limit $\lambda\to 0$ as long as $\alpha_{1},\alpha_{2}\neq 1$ and
$\alpha_{1}\neq\alpha_{2}$. Note that we perform the $\lambda\to 0$ limit
rather than $\lambda\to\infty$ since the relations are inverted. We can use
these inverted relations to establish the generators of the Poincaré algebra
in terms of the generators of the algebra (2.14), in which case
$\lambda\to\infty$ limit must be taken. However this relation is much easier
to see that the necessary commutation relations are satisfied. As a matter of
fact, the $N$-th order non-relativistic algebra (2.12) can be established by
using the $N$ copies of the Poincaré algebra

$\displaystyle J_{ab}^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}J_{ab}^{i}\,,$
$\displaystyle H^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}H^{i}\,,$
$\displaystyle P_{a}^{2n+1}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}P_{a}^{i}\,,$
$\displaystyle G_{a}^{2n+1}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}G_{a}^{i}\,,$
(3.47)

once the $\lambda\to 0$ limit is taken, given that $N_{0}=N_{1}$. Note that
this expansion makes sense as long as we impose $\alpha_{i}\neq 1$ and
$\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. In this generator expansion these
alpha parameters are free and not necessary to be fixed. Once again, these
relations can be inverted to construct the elements of the Poincaré algebras
in terms of the generators of the non-relativistic algebra. However, with this
form of the relation, it is trivial that the commutation relations (2.12) are
satisfied in the $\lambda\to 0$ limit.

In three-dimensions, the spacetime decomposed Poincaré algebra takes a simpler
form

$\displaystyle\left[G_{a},H\right]$ $\displaystyle=P_{a}\,,$
$\displaystyle\left[J,P_{a}\right]$ $\displaystyle=-\epsilon_{ab}P^{b}\,,$
$\displaystyle\left[J,G_{a}\right]$ $\displaystyle=-\epsilon_{ab}G^{b}$
$\displaystyle\left[G_{a},P_{b}\right]$ $\displaystyle=\delta_{ab}H\,,$
$\displaystyle\left[G_{a},G_{b}\right]$ $\displaystyle=-\epsilon_{ab}J\,,$
(3.48)

To obtain the extended Bargmann algebra (2.28), we need to introduce two
central generators $M$ and $S$. Then, if we make the following scaling and
redefinition

$\displaystyle H$ $\displaystyle\to H+\lambda^{2}M\,,$ $\displaystyle J$
$\displaystyle\to J+\lambda^{2}S\,,$ $\displaystyle P_{a}$
$\displaystyle\to\lambda P_{a}\,,$ $\displaystyle G_{a}$
$\displaystyle\to\lambda G_{a}\,,$ (3.49)

we precisely recover the extended Bargmann algebra. For the next order
extended Newtonian algebra [12], we need to consider two copies of the
Poincaré algebra along with two $U(1)$ generators $Y$ and $Z$. In this case,
the following definitions precisely recover the extended Newtonian algebra

$\displaystyle J$ $\displaystyle=J^{1}+J^{2}\,,$ $\displaystyle S$
$\displaystyle=\lambda^{2}\left(\alpha_{1}^{2}J^{1}+\alpha_{2}^{2}J^{2}\right)\,,$
$\displaystyle H$ $\displaystyle=H^{1}+H^{2}\,,$ $\displaystyle M$
$\displaystyle=\lambda^{2}\left(\alpha_{1}^{2}H^{1}+\alpha_{2}^{2}H^{2}\right)\,,$
$\displaystyle P_{a}$
$\displaystyle=\lambda\left(\alpha_{1}P_{a}^{1}+\alpha_{2}P_{a}^{2}\right)\,,$
$\displaystyle T_{a}$
$\displaystyle=\lambda^{3}\left(\alpha_{1}^{3}P_{a}^{1}+\alpha_{2}^{3}P_{a}^{2}\right)\,,$
$\displaystyle G_{a}$
$\displaystyle=\lambda\left(\alpha_{1}G_{a}^{1}+\alpha_{2}G_{a}^{2}\right)\,,$
$\displaystyle B_{a}$
$\displaystyle=\lambda^{3}\left(\alpha_{1}^{3}G_{a}^{1}+\alpha_{2}^{3}G_{a}^{2}\right)\,,$
$\displaystyle Y$
$\displaystyle=\lambda^{4}\left(\alpha_{1}^{4}H^{1}+\alpha_{2}^{4}H^{2}\right)$
$\displaystyle Z$
$\displaystyle=\lambda^{4}\left(\alpha_{1}^{4}J^{1}+\alpha_{2}^{4}J^{2}\right)\,.$
(3.50)

Once again, these relations can be generalized as

$\displaystyle J^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}J^{i}\,,$
$\displaystyle H^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}H^{i}\,,$
$\displaystyle P_{a}^{2n+1}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}P_{a}^{i}\,,$
$\displaystyle G_{a}^{2n+1}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}G_{a}^{i}\,,$
(3.51)

where truncation condition is now $N_{0}=N_{1}+1$, and we impose that
$\alpha_{i}\neq 1$ and $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. Note that
$J^{2N_{1}+2}$ and $H^{2N_{1}+2}$ are the central charges of the extended non-
relativistic algebra that we inherit from the $U(1)\times U(1)$ extension of
$N$-copies of the Poincaré algebra.

##  4 Ultra-Relativistic Scaling Limit of Multimetric Gravity

####

The main difference between the non-relativistic and ultra-relativistic
expansion is the change of the character of the generator of time translations
$H$, and spatial translations $P_{a}$. In the ultra-relativistic expansion of
the Poincaré algebra, $H$ is expanded in odd-powers of $\lambda$ while $P_{a}$
is expanded in even powers. This is also reflected in the expansion of the
corresponding gauge fields, see (2.32). As a result, the $D$-dimensional
Einstein-Hilbert action captures all $\mathcal{O}(\lambda^{2N+1})$ terms,
giving rise to an invariant Lagrangian as long as $N_{0}=N_{1}$. This
statement is true for $D=3$ as well, hence we do not need to present a
separate treatment for $D=3$ but can provide a representative example for
$D=4$ which can be generalized to arbitrary dimensions.

In four dimensions, the structure of the spacetime decomposed Einstein-Hilbert
action (3.8) implies that the ultra-relativistic expansion with (2.32) yields
the following structure

$\displaystyle\mathcal{L}_{\rm EH}$ $\displaystyle=$ $\displaystyle\lambda
M^{2}\mathcal{L}_{1}+\lambda^{3}M^{2}\mathcal{L}_{3}+\lambda^{5}M^{2}\mathcal{L}_{5}+\ldots\,.$
(4.1)

The lowest order Lagrangian can be isolated by the lowest order expressions
for the relativistic fields (2.35) which can be followed by the rescaling the
mass parameter $M^{2}\to M^{2}/\lambda$ and the scaling limit $\lambda\to 0$.
To isolate the higher-order Lagrangians, we need to find an $\alpha$-expansion
which we discussed previously for the non-relativistic models. Following our
footsteps, we start with a non-interacting $N$-metric theory

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[2M_{(i)}^{2}\epsilon_{abc}\left(R(\Omega^{a}_{(i)})\wedge
E^{b}_{(i)}\wedge E^{c}_{(i)}-R(\Omega^{ab}_{(i)})\wedge E_{(i)}^{c}\wedge
T_{(i)}\right)\right]\,,$ (4.2)

where $i,j=1,\ldots N$ label the set of fields
$\\{T_{(i)},E_{(i)},\Omega^{ab}_{(i)},\Omega^{a}_{(i)}\\}$. Then, we keep the
expansion of the first set of fields
$\\{T_{(1)},E_{(1)},\Omega^{a}_{(1)},\Omega^{ab}_{(1)}\\}$ the same as (2.32)

$\displaystyle T_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\tau_{(2n+1)}\,,$
$\displaystyle E^{a}_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}e^{a}_{(2n)}\,,$
$\displaystyle\Omega^{ab}_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\omega^{ab}_{(2n)}\,,$
$\displaystyle\Omega^{a}_{(1)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\omega^{a}_{(2n+1)}\,.$ (4.3)

while for $i>1$, we incorporate the dimensionless parameters $\alpha_{(i)}$

$\displaystyle T_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\alpha_{(i)}^{2n+1}\tau_{(2n+1)}\,,$
$\displaystyle E^{a}_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\alpha_{(i)}^{2n}e^{a}_{(2n)}\,,$
$\displaystyle\Omega^{ab}_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{0}}\lambda^{2n}\alpha_{(i)}^{2n}\omega^{ab}_{(2n)}\,,$
$\displaystyle\Omega^{a}_{(i)}$
$\displaystyle=\sum_{n=0}^{N_{1}}\lambda^{2n+1}\alpha_{(i)}^{2n+1}\omega^{a}_{(2n+1)}\,.$
(4.4)

where $\alpha_{i}\neq 1$ and $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. This
yield the following expansion of $\mathcal{L}_{(i)}$ that is the Einstein-
Hilbert action for $i$-th set of fields
$\\{T_{(i)},E_{(i)},\Omega^{a}_{(i)},\Omega^{ab}_{(i)}\\}$

$\displaystyle\mathcal{L}_{EH(i)}$ $\displaystyle=$
$\displaystyle\lambda\alpha_{i}M_{i}^{2}\mathcal{L}_{1}+\lambda^{3}\alpha_{i}^{3}M_{i}^{2}\mathcal{L}_{3}+\lambda^{5}\alpha_{i}^{5}M_{i}^{2}\mathcal{L}_{5}+\ldots\,.$
(4.5)

Keeping the first set of fields with the standard expansion while using the
$\alpha$-expansion for the reminder of the fields, the Einstein-Hilbert action
becomes

$\displaystyle\mathcal{L}_{EH}$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{N}\lambda^{2n+1}\left(M_{1}^{2}+\sum_{i=2}^{N+1}\alpha_{i}^{2n+1}M_{i}^{2}\right)\mathcal{L}_{2n+1}+\mathcal{O}(\lambda^{2N+3})\,.$
(4.6)

Once again, we isolate the order-$2N+1$ Lagrangian by solving the $N$ number
of algebraic equations that determines the values of $\alpha_{i}$ for
$i=2,3,\ldots N$, i.e.

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
M_{1}^{2}+\sum_{i=2}^{N+1}\alpha_{i}^{2n+1}M_{i}^{2}\,,\qquad\text{for}\qquad
n=0,1,\ldots,N-1\,.$ (4.7)

Finally, by rescaling of the mass parameters $M_{i}^{2}\to
M_{i}^{2}/\lambda^{2N+1}$ and performing the $\lambda\to 0$ limit we eliminate
any lower-order divergences and higher order Lagrangians to obtain the desired
ultra-relativistic model. Let us now provide examples.

  1. 1.

Carroll Gravity: The Carroll gravity appears as $\mathcal{L}_{1}$ in the
ultra-relativistic expansion of Einstein-Hilbert action (4.1). It is, thus,
sufficient to consider the Poincaré algebra, a single set its gauge fields and
the Einstein-Hilbert action. Using the lowest order definitions for the
relativistic fields

$\displaystyle T$ $\displaystyle=\lambda\tau\,$ $\displaystyle E^{a}$
$\displaystyle=e^{a}\,,$ $\displaystyle\Omega^{ab}$
$\displaystyle=\omega^{ab}\,,$ $\displaystyle\Omega^{a}$
$\displaystyle=\lambda\omega^{a}\,,$ $\displaystyle M^{2}$
$\displaystyle=\frac{m^{2}}{\lambda}\,.$ (4.8)

we obtain the Carroll gravity

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=$
$\displaystyle-2\epsilon_{abc}R^{ab}(\omega)\wedge
e^{a}\wedge\tau+2\epsilon_{abc}R^{a}(\omega)\wedge e^{b}\wedge e^{c}\,,$ (4.9)

upon rescaling the mass parameter with $M^{2}\to M^{2}/\lambda$ and taking the
scaling limit $\lambda\to 0$.

  2. 2.

Beyond the Carroll Gravity: The next order ultra-relativistic model appears at
the $\lambda^{3}$-order in the Lie algebra expansion. Thus, we shall consider
two copies of the Einstein-Hilbert action. As for the non-relativistic scaling
limit, we keep the same form as the Lie algebra expansion for the first set of
fields,

$\displaystyle T_{1}$ $\displaystyle=\lambda\tau+\lambda^{3}m\,,$
$\displaystyle E^{a}_{1}$ $\displaystyle=e^{a}+\lambda^{2}t^{a}\,,$
$\displaystyle\Omega^{ab}_{1}$
$\displaystyle=\omega^{ab}+\lambda^{2}s^{ab}\,,$ $\displaystyle\Omega^{a}_{1}$
$\displaystyle=\lambda\omega^{a}+\lambda^{3}b^{a}\,.$ (4.10)

For the second copy, we incorporate the free parameter $\alpha_{2}$

$\displaystyle T_{2}$
$\displaystyle=\lambda\tau+\lambda^{3}\alpha_{2}^{3}m\,,$ $\displaystyle
E^{a}_{2}$ $\displaystyle=\alpha_{2}e^{a}+\lambda^{2}\alpha_{2}^{2}t^{a}\,,$
$\displaystyle\Omega^{ab}_{2}$
$\displaystyle=\omega^{ab}+\lambda^{2}\alpha_{2}^{2}s^{ab}\,,$
$\displaystyle\Omega^{a}_{2}$
$\displaystyle=\lambda\alpha_{2}\omega^{a}+\lambda^{3}\alpha_{2}^{3}b^{a}\,.$
(4.11)

The combination of these two Lagrangian with mass parameters $M_{1,2}$ gives
rise to

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\lambda
M^{2}\left(1+\frac{1}{2}\alpha_{2}\right)\mathcal{L}_{1}+\lambda^{3}M^{2}\left(1+\frac{1}{2}\alpha_{2}^{3}\right)\mathcal{L}_{3}+\mathcal{O}(\lambda^{5})\,.$
(4.12)

where we set $M_{1}^{2}=2M_{2}^{2}=M^{2}$ for simplicity. Here.
$\mathcal{L}_{1}$ refer to the Carroll gravity (4.9) and $\mathcal{L}_{3}$ is
what we refer to as the beyond the Carroll gravity

$\displaystyle\mathcal{L}_{3}$ $\displaystyle=$
$\displaystyle-2\epsilon_{abc}R^{ab}(s)\wedge
e^{c}\wedge\tau-2\epsilon_{abc}R^{ab}(\omega)\wedge
t^{c}\wedge\tau-2\epsilon_{abc}R^{ab}(\omega)\wedge e^{c}\wedge m$ (4.13)
$\displaystyle+2\epsilon_{abc}R^{a}(b)\wedge e^{b}\wedge
e^{c}+4\epsilon_{abc}R^{a}(\omega)\wedge e^{b}\wedge t^{c}\,.$

Note that this is the same form of the non-relativistic limit, which is no
surprising since in four dimensions, both the non-relativistic and the ultra-
relativistic expansion of the Einstein-Hilbert action take the same form in
series expansion in $\lambda$, see (3.9) and (4.1). We remind the reader that
this is not longer true for $D\neq 4$. Nevertheless, we can fix $\alpha_{2}$
to be $\alpha_{2}=-2$ which cancels out the coefficient of the Carroll gravity
and fixed the coefficient of $\mathcal{L}_{3}$ to be $-3$. Then, the following
scaling of the mass parameter

$\displaystyle M^{2}=\frac{1}{3\lambda^{3}}m^{2}\,.$ (4.14)

along with the scaling limit $\lambda\to 0$ recover the beyond the Carroll
gravity model up to an overall minus sign.

The multi-gravity models that we discussed so far needs to include a potential
term for the vielbeine to be physically viable. Unlike the non-relativistic
case, however, these terms appear at the $\mathcal{O}(\lambda)$ Lagrangian due
to the changing expansion character of the spatial and the temporal vielbein

$\displaystyle\mathcal{L}_{\rm{pot}}$ $\displaystyle=$
$\displaystyle\lambda\left(\epsilon_{abc}\tau\wedge e^{a}\wedge e^{b}\wedge
e^{c}\right)$ (4.15) $\displaystyle+\lambda^{3}\left(e_{abc}m\wedge
e^{a}\wedge e^{b}\wedge e^{c}+3e_{abc}\tau\wedge e^{a}\wedge e^{b}\wedge
t^{c}\right)+\mathcal{O}(\lambda^{5})\,.$

This is precisely the same expansion of the non-relativistic models except
that the Lagrangians are now shifted by $\lambda^{-2}$. Nevertheless, our
arguments for the elimination of the lower-order cosmological terms still hold
since the structure of the potential terms, (3.24), contains more than
necessary the number of free coefficients.

We end this section with a brief discussion on how to obtain the ultra-
relativistic higher-order algebras by contracting the multiple copies of the
Poincaré algebra. As with the non-relativistic algebras for $D\geq 4$, the
ultra-relativistic algebras that admit an invariant action formulation arise
from the Lie algebra expansion for $N_{0}=N_{1}$. This implies that for an
algebra of order $(N_{1},N_{1})$, one has the same number of generators as
$(N_{1}+1)$-copies of Poincaré algebra. Furthermore, the expansion does not
change the structure constant of the smaller core algebra, indicating that we
can combine the generators of multiple copies of the Poincaré algebra in a
linearly independent way to exhibit the generators of the larger non/ultra-
relativistic algebras. Based on the $\alpha$-expansion of the relativistic
fields, we introduce the following expressions for the ultra-relativistic
generators in terms of the generators of the Poincaré algebra

$\displaystyle J_{ab}^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}J_{ab}^{i}\,,$
$\displaystyle H^{(2n+1)}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}H^{i}\,,$
$\displaystyle P_{a}^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}P_{a}^{i}\,,$
$\displaystyle G_{a}^{(2n+1)}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}G_{a}^{i}\,,$
(4.16)

where $\alpha_{i}\neq 1$ and $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. Note
that in this case these alpha parameters are free and not necessary to be
fixed. This direct sum structure trivially satisfy the algebra (2.30). Once
again, these relations can be inverted to express the relativistic generators
in terms of the ultra-relativistic ones. Furthermore, we remind the reader
that the truncation condition is $N_{0}=N_{1}$.

##  5 Discussion

####

In this work, we have presented non-relativistic and ultra-relativistic
scaling limits of multimetric gravity theories. By the field content of the
multimetric gravity, it is expected that these models contain more number of
degrees of freedom compared to that of Galilei/Carroll gravity, which are the
limits of General Relativity. We have shown that the limits of multimetric
gravity correspond to non/ultra-relativistic gravity with extended symmetries.
In particular, we have shown that the non-relativistic limit of bi-metric
gravity is the recent formulation of an action principle for the Newtonian
gravity when no potential terms are present. On the other hand, turning on the
potential terms yield a constant background mass density in the non-
relativistic sector. We expect that the scaling limit that we provide in this
paper will be helpful in phenomenological studies for both the multigravity
and the non/ultra-relativistic gravity. Especially, given the fact that the
Newtonian gravity has a spherically symmetric vacuum solution [42, 44, 43, 45]
and passes the three classical tests of General Relativity, it would be
interesting to see the bimetric origin of this solution and relevant
phenomenological consequences.

The work we present can be regarded as a starting point for various further
studies. On the technical side, it would be interesting to extend the analysis
that we presented here for the string limit and corresponding algebras, i.e.,
string limit of Einstein-Hilbert action, see [4, 59]. Furthermore, here we
focus on three and higher dimensions. The reason for this is that the two-
dimensional gravitational theories in their $BF$-formalism cannot be
established just by the gauge fields of the Poincaré (or (A)dS) algebra, but
they require the presence of matter fields transforming in the coadjoint
representation. We believe that the same scaling limits can be defined in the
presence of the matter sector, however, a careful analysis is indeed
essential.

A rather interesting continuation of our work would be to include
supersymmetry. Although the non/ultra-relativistic superalgebras are now well-
understood, thanks to the Lie algebra expansion, we also need theories that
contain both gauge fields and matter fields. At present, the existing
techniques, which are based on the Lie algebra expansion, can produce
reducible representations for the matter multiplets of supersymmetric theories
[46]. Furthermore, these multiplets are rigid and their extensions to local
theories have been an open problem. We hope that the scaling limit that we
present in this paper can shed light on this open problem.

Another interesting point is that although we have achieved the scaling limit
with multiple copies of the Poincaré algebra, the same limit can also be
established with the Poincaré and the Euclidean algebras, see [12] for a
three-dimensional example. When the Euclidean algebra is considered, the
difference arises in the commutation relations that include $H\equiv P_{0}$
and $G_{a}\equiv J_{0a}$, i.e.

$\displaystyle\left[G_{a},P_{b}\right]$ $\displaystyle=\delta_{ab}H\,,$
$\displaystyle\left[G_{a},H\right]$ $\displaystyle=-P_{a}\,,$
$\displaystyle\left[J_{ab},P_{c}\right]$
$\displaystyle=\delta_{bc}P_{a}-\delta_{ac}P_{b}\,,$
$\displaystyle\left[J_{ab},G_{c}\right]$
$\displaystyle=\delta_{bc}G_{a}-\delta_{ac}G_{b}\,,$
$\displaystyle\left[J_{ab},J_{cd}\right]$
$\displaystyle=4\delta_{[a[c}J_{d]b]}\,,$
$\displaystyle\left[G_{a},G_{b}\right]$ $\displaystyle=-J_{ab}\,.$ (5.1)

In the contraction process to obtain the non/ultra-relativistic algebras, the
minus factor that appears in the $[G_{a},H]$ and $[G_{a},G_{b}]$ commutators
can be handled by introducing a minus sign in the definitions of the
non/ultra-relativistic generators of extended algebras. For instance, in the
case of non-relativistic extended algebras, we have the following definitions
for the generators

$\displaystyle J^{(2n)}_{ab}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}\sigma_{i}^{n}J^{i}_{ab}\,,$
$\displaystyle H^{(2n)}$
$\displaystyle=\lambda^{2n}\bigoplus\limits_{i=1}^{N_{0}}\alpha_{i}^{2n}\sigma_{i}^{n}H^{i}$
$\displaystyle P^{(2n+1)}_{a}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}\sigma_{i}^{n+1}P^{i}_{a}\,,$
$\displaystyle G^{(2n+1)}_{a}$
$\displaystyle=\lambda^{2n+1}\bigoplus\limits_{i=1}^{N_{1}}\alpha_{i}^{2n+1}\sigma_{i}^{n}G^{i}_{a}\,,$
(5.2)

where $\alpha_{i}\neq 1$ and $\alpha_{i}\neq\alpha_{j}$ for $i\neq j$. The
necessary sign is introduced by the $\sigma_{i}$ which is defined as
$\sigma_{i}=1$ for Lorentizan and $\sigma_{i}=-1$ for Euclidean algebras. With
this relation in hand, it is tempting to propose a bimetric theory of gravity
that is the sum of Einstein-Hilbert action in the Lorentizan and the Euclidean
signatures as a simple, worthwhile phenomenological model. To our knowledge,
such a model has not been investigated in the literature. We hope to study the
phenomenological aspects of this model in a near future. The multiple copy
structure of the Poincaré algebra also suggest a possible connection with our
construction and colored gravity [61, 62], which may be another interesting
avenue to investigate.

As an important remark, in Appendix A, we have explicitly shown the
equivalence of the first and the second order formulation of the Newtonian
gravity. It would be interesting to see if the same is true between the ultra-
relativistic Beyond Carroll gravity action (4.13) and the metric formulation
[58]. Finally, the coadjoint Poincaré (or AdS) algebra, which is a Lie algebra
expansion of the Poincaré (or AdS) algebra for $\\{P_{A},J_{AB}\\}\subset
V_{0}$, is also known to reproduce certain three and four-dimensional extended
non-relativistic algebras [60]. It would be interesting to see if there is a
relation between the coadjoint algebras and multiple copies of Poincaré and
Euclidean algebra.

Acknowledgements We thank Eric Bergshoeff and Johannes Lahnsteiner for
discussions. The work of O.K. and C.B.S. is supported by TUBITAK grant
121F064. M.O. is supported in part by TUBITAK grant 121F064 and Istanbul
Technical University Research Fund under grant number TGA-2020-42570. M.O.
acknowledges the support by the Distinguished Young Scientist Award BAGEP of
the Science Academy. M.O. also acknowledges the support by the Outstanding
Young Scientist Award of the Turkish Academy of Sciences (TUBA-GEBIP). U.Z. is
supported by TUBITAK - 2218 National Postdoctoral Research Fellowship Program
with grant number 118C512.

##  Appendix A Equivalence of the First Order and the Second Order Formulation
of the Newtonian Gravity

####

In this Appendix, we show that the first order formulation of Newtonian
gravity that arises from the Lie algebra expansion is equivalent to the second
order formulation that arises from the $1/c^{2}$ expansion of the General
Relativity. The first order formulation of Newtonian gravity appears at
$\mathcal{O}(\lambda^{3})$ in the non-relativistic expansion of the Einstein-
Hilbert action and is given as

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle\int d^{4}x\
\epsilon^{\mu\nu\rho\sigma}\epsilon_{abc}\Big{(}e_{\mu}{}^{a}e_{\nu}{}^{b}R_{\rho\sigma}{}^{c}(\omega)-e_{\mu}{}^{a}\tau_{\nu}R_{\rho\sigma}{}^{bc}(s)-e_{\mu}{}^{a}m_{\nu}R_{\rho\sigma}{}^{bc}(\omega)$
(A.1) $\displaystyle\qquad\qquad\qquad\quad-
t_{\mu}{}^{a}\tau_{\nu}R_{\rho\sigma}{}^{bc}(\omega)\Big{)}\,,$

where the group-theoretical curvatures are given in (2.22). Using the inverse
spatial and the temporal vielbein, this action can be put in a more useful
form

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle\int d^{4}x\ e\
\bigg{(}-4\tau^{\mu}e^{\nu}{}_{a}R_{\mu\nu}{}^{a}(\omega)+2e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(s)-2\tau^{\rho}m_{\rho}e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)\,$
(A.2)
$\displaystyle+4e^{\rho}{}_{a}m_{\rho}\tau^{\mu}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)+2e^{\rho}{}_{c}t_{\rho}{}^{c}e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)-4e^{\rho}{}_{a}t_{\rho}{}^{b}e^{\mu}{}_{b}e^{\nu}{}_{c}R_{\mu\nu}{}^{ac}(\omega)\bigg{)}\,,\quad$

where $e=\det(\tau_{\mu},e_{\mu}{}^{a})$. Note that we made a slight change of
notation $\omega^{a}\rightarrow-\omega^{a}\,,s^{ab}\rightarrow-s^{ab}$ to
match with the conventions of [7]. The inverse elements satisfy the following
properties

$\displaystyle\tau^{\mu}e_{\mu}{}^{a}$ $\displaystyle=0\,,$
$\displaystyle\tau_{\mu}e^{\mu}{}_{a}$ $\displaystyle=0\,,$
$\displaystyle\tau^{\mu}\tau_{\mu}$ $\displaystyle=-1\,,$ $\displaystyle
e_{\mu}{}^{a}e^{\mu}{}_{b}$ $\displaystyle=\delta^{a}{}_{b}\,,$ $\displaystyle
e_{\mu}{}^{a}e^{\nu}{}_{a}$
$\displaystyle=\delta_{\mu}{}^{\nu}+\tau_{\mu}\tau^{\nu}\,.$ (A.3)

Note here that the inverse temporal vielbein have non-trivial Galilean
transformations. However, a Galilean invariant timelike vector can be
constructed by using the gauge field $m_{\mu}$ and vielbein as
$\hat{\tau}^{\mu}=\tau^{\mu}-h^{\mu\nu}m_{\nu}$. Note that $\hat{\tau}^{\mu}$
also satisfies the invertibility conditions. To show the equivalence of the
first and the second order formulation, we first to use the following Bianchi
identity

$\displaystyle\tau\wedge R^{a}(\omega)-e_{b}\wedge R^{ab}(\omega)=0\,,$ (A.4)

which is a consequence of the curvature constraints that are necessary to
express $\omega_{\mu}{}^{ab},\omega_{\mu}{}^{a}$ in terms of the independent
fields of the Newtonian gravity $\\{e_{\mu}{}^{a}\,,\tau_{\mu}\,,m_{\mu}\\}$
[57]. Utilizing the Bianchi identity and using the Galilean invariant timelike
vector, the first-order action becomes

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle-\frac{1}{32\pi G}\int
d^{4}x\ e\
\bigg{(}-4\hat{\tau}^{\mu}e^{\nu}{}_{a}R_{\mu\nu}{}^{a}(\omega)+2e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(s)-2\tau^{\rho}m_{\rho}e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)\,$
(A.5)
$\displaystyle\qquad\qquad\qquad\qquad+2e^{\rho}{}_{c}t_{\rho}{}^{c}e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)-4e^{\rho}{}_{a}t_{\rho}{}^{b}e^{\mu}{}_{b}e^{\nu}{}_{c}R_{\mu\nu}{}^{ac}(\omega)\bigg{)}\,.$

where we introduce proper numerical prefactor and the Newton’s constant for
future purposes. The Riemann tensor is then given by [57]

$\displaystyle
R_{\mu\nu\sigma}{}^{\rho}=e^{\rho}{}_{a}\tau_{\sigma}R_{\mu\nu}{}^{a}(\omega)-e_{\sigma
a}e^{\rho}{}_{b}R_{\mu\nu}{}^{ab}(\omega)\,,$ (A.6)

from which we read-off the following contraction that is an important quantity
when relating the first and the second order formulations

$\displaystyle-\hat{\tau}^{\mu}e^{\nu}{}_{a}R_{\mu\nu}{}^{a}(\omega)$
$\displaystyle=$
$\displaystyle\hat{\tau}^{\mu}\hat{\tau}^{\nu}\bar{R}_{\mu\nu}\,.$ (A.7)

Here, we use the $\bar{R}_{\mu\nu}$ notation in order to indicate that the
connection is the Galilean invariant to stay compatible with the notation of
[7]. With these results in hand, let us focus back to the action (A.5), in
particular the term that involve the curvature of $s^{ab}$

$\displaystyle\epsilon^{\mu\nu\rho\sigma}\epsilon_{abc}e_{\mu}{}^{a}\tau_{\nu}R_{\rho\sigma}{}^{bc}(s)$
$\displaystyle=$
$\displaystyle\epsilon^{\mu\nu\rho\sigma}\epsilon_{abc}e_{\mu}{}^{a}\tau_{\nu}(2\partial_{\rho}s_{\sigma}{}^{bc}+2\omega_{\rho}{}^{bd}s_{\sigma
d}{}^{c}-2\omega_{\rho}{}^{cd}s_{\sigma d}{}^{b}$ (A.8)
$\displaystyle-\omega_{\rho}{}^{b}\omega_{\sigma}{}^{c}+\omega_{\sigma}{}^{b}\omega_{\rho}{}^{c})\,.$

The integration by parts of the derivative term yield the curvature
constraints for $\tau_{\mu}$ and $e_{\mu}{}^{a}$ which are already imposed.
Thus, what remains is the last two terms that involve $\omega^{a}$

$\displaystyle\epsilon^{\mu\nu\rho\sigma}\epsilon_{abc}e_{\mu}{}^{a}\tau_{\nu}\left(-\omega_{\rho}{}^{b}\omega_{\sigma}{}^{c}+\omega_{\sigma}{}^{b}\omega_{\rho}{}^{c}\right)=-2e\
\left(\nabla_{\mu}\tau^{\mu}\nabla_{\nu}\tau^{\nu}-\nabla_{\mu}\tau^{\nu}\nabla_{\nu}\tau^{\mu}\right)\,.$
(A.9)

which can be expressed as follows after partial integration

$\displaystyle-2e\
\left(\nabla_{\mu}\tau^{\mu}\nabla_{\nu}\tau^{\nu}-\nabla_{\mu}\tau^{\nu}\nabla_{\nu}\tau^{\mu}\right)$
$\displaystyle=$
$\displaystyle-2\hat{\tau}^{\mu}\hat{\tau}^{\nu}\bar{R}_{\mu\nu}-2h^{\mu\sigma}h^{\lambda\rho}m_{\sigma}m_{\lambda}R_{\mu\rho}\,,$
(A.10)

Here, we used the definition $\hat{\tau}^{\mu}$ as well as the identities [45]

$\displaystyle\left[\nabla_{\mu},\nabla_{\nu}\right]X_{\sigma}$
$\displaystyle=-R_{\mu\nu\sigma}{}^{\rho}X_{\rho}\,,$
$\displaystyle\bar{R}_{\mu\nu\lambda}{}^{\rho}h^{\lambda\sigma}$
$\displaystyle=\bar{R}_{\mu\nu\lambda}{}^{\sigma}h^{\lambda\rho}\,,$ (A.11)

where $h^{\mu\nu}=e^{\mu}{}_{a}e^{\nu\,a}$. Note that we assume the vanishing
of torsion since it is not required to establish the action principle that
gives rise to the Poisson equation for Newtonian gravity as an equation of
motion. Finally, focusing on the $R^{ab}(\omega)$ terms in the Lagrangian
(A.5), we have

$\displaystyle-2\Phi
h^{\mu\nu}\bar{R}_{\mu\nu}-\Phi_{\mu\nu}h^{\mu\nu}h^{\rho\sigma}\bar{R}_{\rho\sigma}+2\Phi_{\mu\lambda}h^{\mu\sigma}h^{\nu\lambda}\bar{R}_{\nu\sigma}$
$\displaystyle=2\tau^{\rho}m_{\rho}e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)$
$\displaystyle+2e^{\rho}{}_{c}t_{\rho}{}^{c}e^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}(\omega)$
$\displaystyle-4e^{\rho}{}_{a}t_{\rho}{}^{b}e^{\mu}{}_{b}e^{\nu}{}_{c}R_{\mu\nu}{}^{ac}(\omega)\,,$
(A.12)

where we used the following definitions

$\displaystyle\Phi=-\tau^{\mu}m_{\mu}\,,\qquad\Phi_{\mu\nu}=\delta_{ab}\left(e_{\mu}{}^{a}t_{\nu}{}^{b}+e_{\nu}{}^{a}t_{\mu}{}^{b}\right)\,.$
(A.13)

We now introduce the Galilean invariant definitions of $\Phi$ and
$\Phi_{\mu\nu}$ as adding and subtracting the appropriate terms as

$\displaystyle-2\Phi
h^{\mu\nu}\bar{R}_{\mu\nu}-\Phi_{\mu\nu}h^{\mu\nu}h^{\rho\sigma}\bar{R}_{\rho\sigma}+2\Phi_{\mu\lambda}h^{\mu\sigma}h^{\nu\lambda}\bar{R}_{\nu\sigma}$
$\displaystyle=2\bar{\Phi}_{\mu\lambda}h^{\mu\sigma}h^{\nu\lambda}\bar{R}_{\nu\sigma}-\bar{\Phi}_{\mu\nu}h^{\mu\nu}h^{\rho\sigma}\bar{R}_{\rho\sigma}$
$\displaystyle-2\bar{\Phi}h^{\mu\nu}\bar{R}_{\mu\nu}+2h^{\mu\sigma}h^{\nu\rho}m_{\sigma}m_{\rho}R_{\mu\nu}\,,$
(A.14)

where the Galilean invariant quantities are defined as [45],

$\displaystyle\bar{\Phi}=\Phi+\frac{1}{2}h^{\mu\nu}m_{\mu}m_{\nu}\,,\qquad\bar{\Phi}_{\mu\nu}=\Phi_{\mu\nu}-m_{\mu}m_{\nu}\,.$
(A.15)

Substituting these results back in the action (A.5), we obtain

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi G}\int
d^{4}x\ e\
\bigg{(}-\hat{\tau}^{\mu}\hat{\tau}^{\nu}\bar{R}_{\mu\nu}+\bar{\Phi}h^{\mu\nu}\bar{R}_{\mu\nu}-\bar{\Phi}_{\rho\sigma}h^{\mu\rho}h^{\nu\sigma}(\bar{R}_{\mu\nu}-\frac{1}{2}h_{\mu\nu}h^{\kappa\lambda}\bar{R}_{\kappa\lambda})\bigg{)}\,,\qquad\,,$
(A.16)

which is the action principle that gives rise to the Poisson equation for
Newtonian gravity as an equation of motion [7].

As mentioned in Section 3.1, we can add a cosmological constant to this action
$e\Lambda$, which arise from the contraction of the bimetric gravity with a
non-vanishing potential terms for vielbein. This would give rise to the
Poisson equation as an equation of motion [7]

$\displaystyle\nabla^{2}\Phi$ $\displaystyle=$ $\displaystyle 4\pi
G\Lambda\,,$ (A.17)

which represents a constant background mass density.

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