###### Abstract

We examine the basic conservation laws for diffeomorphism symmetry in the
context of spontaneous diffeomorphism and local Lorentz-symmetry breaking. The
conservation laws are used as constraints on a generic series of terms in an
expansion around a flat background. We find all such terms for a two-tensor
coupling to cubic order in the metric and tensor field fluctuations. The
results are presented in a form that can be used for phenomenological
calculations. One key result is that if we preserve the underlying
diffeomorphism symmetry in a spontaneous-symmetry breaking scenario, one
cannot decouple the two-tensor fluctuations from the metric fluctuations at
the level of the action, except in special cases of the quadratic actions.

###### keywords:

Lorentz violation, gravity, spontaneous-symmetry breaking

x 10.3390/—— xx Academic Editor: Quentin G. Bailey Received: date; Accepted:
date; Published: date Construction of higher-order metric fluctuation terms in
spacetime symmetry-breaking effective field theoryQuentin G. Bailey 1 Quentin
G. Bailey Correspondence<EMAIL_ADDRESS>Tel.: 928-777-3932

##  1 Introduction

The diffeomorphism, the operation of mapping points on a spacetime manifold,
is a fundamental symmetry transformation in gravitational theories of physics.
Together with local Lorentz transformations in the tangent space at each
point, these comprise the gauge symmetry groups that allow General Relativity
to be formulated as a local gauge theory of gravitation Hehl:1976 . Since
local gauge symmetries can be broken in quantum field theories, for instance
via spontaneous symmetry breaking, it is natural ask if the same could occur
for local Lorentz and diffeomorphism symmetries Kostelecky:1988 . Much recent
work has been devoted to the possibility of spacetime-symmetry breaking, and
its possible detection in sensitive tests Hees:2016 ; Kostelecky:2008 ;
Tasson:2016 ; Will:2014 ; Yunes:2016 . Among the theoretical mechanisms for
this symmetry breaking is the elegant spontaneous symmetry breaking one, in
which the underlying action maintains the local symmetries, but due to the
presence of a potential energy function with a nonzero minimum for a tensor
field, the solutions of the model break this symmetry.

A systematic approach to studying the theory and phenomenology of hypothetical
diffeomorphism and local Lorentz violation is the use of effective field
theory Kostelecky:1995 ; Colladay:1997 ; Colladay:1998 . In this approach
generic symmetry breaking terms are added to the action of General Relativity
and the Standard Model Kostelecky:2004 . These terms can generally be written
using coordinate invariant contractions of background tensor fields and
gravitational or matter field operators. The imposition of perfect symmetry
occurs when these background tensor fields vanish, and any statistically
significant nonzero value for their components would indicate a breaking of
diffeomorphism and local Lorentz symmetry. The use of a standard Sun-centered
coordinate system to express the experimental measurements of these background
coefficients allows cross comparisons between a wide variety of tests
Kostelecky:2008 . For example, the results of time-delay VLBI measurements in
the solar system already limited the coefficient $\overline{s}_{00}$ that
describes the variation of the speed of gravity compared to light at parts in
$10^{5}$ LePoncinLafitte:2016 , well beyond initial constraints obtainable
from the first gravitational wave observation. Later, of course, these
measurements were improvement from the simultaneous observations of
electromagnetic observations and gravitational wave observations Abbott:2017 .
In both tests, the same set of coefficients $\overline{s}_{\mu\nu}$ controls
the deviation from GR, at least in the lowest order description of the
effective field theory framework Bailey:2006 ; Bailey:2009 ; Kostelecky:2015 ;
Kostelecky:2016 .

So far, much of the development of effective field theory approaches to
diffeomorphism and Lorentz breaking have focused on the weak-field regime, in
which the spacetime metric can be expanded around a flat background
Bailey:2006 ; Kostelecky:2009a ; Kostelecky:2011 . For many measurements it is
appropriate to use this limit Bourgoin:2016 ; Mueller:2008 ; Flowers:2017 ;
Shao:2018 ; Hees:2015 ; LShao:2014 . However, it is of importance to extend
results beyond this level, and some work already exists in this regard
Bailey:2016 ; Bonder:2015 ; Bonder:2017 ; Bailey:2019 ; Bonder:2020 ;
O'Neal:2021 ; Xu:2020 ; Xu:2021 ; Bonder:2021 . A generic action quadratic in
the metric fluctuations $h_{\mu\nu}$ has already been constructed to describe
any Lorentz violation that can be written in the form ${\cal L}\sim
h{\hat{K}}h$, where ${\hat{K}}$ contains arbitrary coefficients and any number
of partial derivatives Bailey:2006 ; Kostelecky:2016 ; Kostelecky:2018 . This
form assumes that the coefficients are constant and the fluctuation around
their background values can be eliminated from the action by inserting their
solutions from the field equations. The class of models in which this
“decoupling" is possible has been studied in various toy models with vectors
and tensors Seifert:2009 ; Altschul:2010 ; Seifert:2018 . However, a full
study remains incomplete at higher order in small fluctuations $h_{\mu\nu}$
around a flat background.

We start the investigation in this work of higher-order terms in the weak-
field expansion around a flat background. A general cubic action is
constructed for the case of two-tensor couplings $s_{\mu\nu}$ under the
assumption of spontaneous-symmetry breaking. Using an expansion around the
background or vacuum values, we show that when proceeding to cubic order in
the action, terms that include the fluctuations $\tilde{s}_{\mu\nu}$ are
needed for proper adherance to the underlying diffeomorphism symmetry. Thus
one cannot generally decouple the fluctuations from the metric fluctuations at
the level of the action to cubic and higher order, in contrast to past
assumptions about the quadratic order terms Bailey:2006 . We show also that
these terms can be matched to a general covariant version of the effective
field theory at second and higher order in $s_{\mu\nu}$ O'Neal:2021 ;
Kostelecky:2021 .

The paper is organized as follows. First we consider the terms originally
proposed in Ref. Kostelecky:2004 , and argue for the inclusion of dynamical
terms for the fluctuations in section 2. In section 3, we perform a general
construction of all possible terms in the quadratic and cubic order Lagrange
densities. Then we discuss the match to “covariant" terms in section 4. In
section 5, we display the field equations resulting from the general action.
Finally we summarize in section 6, and comment on generalizations of this
work. We adopt natural units in this work, and conventions in past articles
Kostelecky:2004 ; Bailey:2006 .

##  2 Background on the action

The observer covariant form of the action for the effective field theory
framework in the gravity sector is

$S={\textstyle{{1}\over{16\pi G_{N}}}}\int
d^{4}x\sqrt{-g}\left((1-u)R+s_{\alpha\beta}R^{\alpha\beta}+t_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\right)+S^{\prime},$
(1)

where $t_{\alpha\beta\gamma\delta}$, $s_{\alpha\beta}$, and $u$ are the
coefficient fields for Lorentz violation, controlling the size of the mass
dimension $4$ ($M^{4}$) Lorentz-violating terms Kostelecky:2004  (the
gravitational constant $G_{N}$ has mass dimension $M^{-2}$). The term
$S^{\prime}$ contains additional dynamical contributions to the coefficients,
should the origin of the symmetry breaking be spontaneous.

In the spontaneous symmetry breaking scenario the following expansions are
used:

$\displaystyle g_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta}+h_{\alpha\beta},$ $\displaystyle u$
$\displaystyle=$ $\displaystyle\overline{u}+\tilde{u},$ $\displaystyle
s_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\overline{s}_{\alpha\beta}+\tilde{s}_{\alpha\beta},$
$\displaystyle t_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\overline{t}_{\alpha\beta\gamma\delta}+\tilde{t}_{\alpha\beta\gamma\delta}.$
(2)

The vacuum expectation value for the coefficients are denoted with bars, for
example, for $s_{\alpha\beta}$ it is $\langle
s_{\alpha\beta}\rangle=\overline{s}_{\alpha\beta}$, and we assume constancy of
these coefficients (e.g., $\partial_{\mu}\overline{s}_{\alpha\beta}=0$). A
flat background spacetime metric is assumed so that $\eta_{\alpha\beta}$ is
constant in Cartesian coordinates and we assume the partial derivatives of
$\overline{s}_{\alpha\beta}$ vanish to avoid global energy-momentum
conservation violation Bailey:2006 . From here on, we use the flat metric
$\eta_{\alpha\beta}$ and its inverse $\eta^{\alpha\beta}$ to raise and lower
indices on quantities unless specified. Note that converting expressions from
covariant form to linear, quadratic, cubic or beyond, must include appropriate
use of (2) Bailey:2018 . For example, for the metric inverse

$\displaystyle g^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta^{\alpha\beta}-\eta^{\alpha\gamma}\eta^{\beta\delta}h_{\gamma\delta}+\eta^{\alpha\gamma}\eta^{\epsilon\delta}\eta^{\beta\zeta}h_{\gamma\delta}h_{\epsilon\zeta}+O(h^{3}),$
(3) $\displaystyle=$
$\displaystyle\eta^{\alpha\beta}-h^{\alpha\beta}+h^{\alpha\gamma}h_{\gamma}^{\phantom{\gamma}\beta}+O(h^{3}),$

where in the second line it is understood that the indices are raised and
lowered with the Minkowski metric.

Much work has been done by assuming a quadratic order and “decoupled" version
of 1. This implies that only the vacuum values of the coefficients
$\overline{s}_{\alpha\beta}$, $\overline{t}_{\alpha\beta\gamma\delta}$, and
$\overline{u}$ contribute to the gravitational effects of Lorentz violation in
the weak-field limit. The Lagrange density for the gravity sector in this
limit, up to overall constant scalings, can be written as

${\cal
L}={\textstyle{{1}\over{4\kappa}}}(-h_{\alpha\beta}G^{\alpha\beta}+\overline{s}_{\alpha\beta}h_{\gamma\delta}{\cal
G}^{\alpha\gamma\beta\delta}),$ (4)

with $\kappa=8\pi G_{N}$ and where the Einstein curvature tensor $G^{\mu\nu}$
and the curvature double dual ${\cal G}^{\alpha\gamma\beta\delta}$ are linear
order in $h_{\mu\nu}$. Despite the lack of fluctuations
$\tilde{u},\tilde{s},\tilde{t}$ in the second term, this remains invariant
under linearized gauge transformations of $h_{\mu\nu}$. Thus the fluctuations
of the coefficients $\tilde{u}$, $\tilde{s}_{\alpha\beta}$, and
$\tilde{t}_{\alpha\beta\gamma\delta}$ are assumed to have been decoupled via
their field equations. The latter process involves assumptions on the dynamics
in $S^{\prime}$ Bailey:2006 , which limits the class of possible models
included Seifert:2018 . On the other hand, (4) represents the simplest limit
in which the focus is on the effects acting on only the known gravitational
degrees of freedom in $h_{\mu\nu}$, thereby invoking Occam’s razor for the
already ambitious suggestion of Lorentz violation.

Note also that the coefficients $\overline{t}_{\alpha\beta\gamma\delta}$ do
not appear in the quadratic action (4). That this occurs can be viewed as a
consequence of the assumptions on the dynamics of the coefficients that leads
to the decoupled form of (4). Alternatively, (4) can be derived by considering
all possible operators in a general quadratic action of the form
$hK\partial\partial...h$ Kostelecky:2016 , and then truncating the expansion
to only mass dimension $4$ coefficients for Lorentz violation. Either way, a
deeper understanding of why the $\overline{t}_{\alpha\beta\gamma\delta}$
coefficients vanish in this limit represents a statement of the “t-puzzle"
Bailey:2015 . While alternative covariant actions have been countenanced
leading to physical effects of $t_{\alpha\beta\gamma\delta}$ Bonder:2017 ;
Bonder:2020 ; Bonder:2021 , it remains unexplained why it does not occur in
(4).

In this work, we include the fluctuation terms so that we relax the
assumptions made in (4), and thus consider a more general version of
$S^{\prime}$. To include the generality of possible $\tilde{s}_{\mu\nu}$
dynamics, we posit all possible kinetic, potential, and coupling terms with
the constraint that we consider only terms in the Lagrange density with no
higher than second order in derivatives, and first-order terms in the vacuum
coefficients $\overline{s}_{\mu\nu}$. We will attempt here the $s_{\mu\nu}$
case but other coefficients can similarly be accommodated.

##  3 General construction of Lagrange density

First we start with the quadratic action terms. These Lagrange density terms,
discarding those equivalent by partial integration, are constructed from
scalar density contractions of the following:

$\displaystyle{\cal L}^{(2)}$ $\displaystyle\supset$ $\displaystyle{\rm
All\,contractions\,of\,}\\{h_{\alpha\beta}\partial_{\gamma}\partial_{\delta}h_{\epsilon\zeta},\overline{s}_{\alpha\beta}h_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}h_{\eta\theta},\tilde{s}_{\alpha\beta}\partial_{\gamma}\partial_{\delta}\tilde{s}_{\epsilon\zeta},\tilde{s}_{\alpha\beta}\partial_{\gamma}\partial_{\delta}h_{\epsilon\zeta},$
(5)
$\displaystyle\phantom{allcontractionsof}\overline{s}_{\alpha\beta}\tilde{s}_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}h_{\eta\theta},\overline{s}_{\alpha\beta}\tilde{s}_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}\tilde{s}_{\eta\theta}\\}.$

In addition there are potential-type terms dealt with separately below. Note
that the form of the terms considered in (5) includes, in the second set of
terms $\sim\overline{s}h\partial\partial h$, the general structure of the
Lorentz-violating terms considered in the quadratic lagrangian construction of
Ref. Kostelecky:2016 , and it will include the General Relativity terms from
$\sim h\partial\partial h$. We truncate the series in this work by considering
terms of second order in derivatives, though the pattern could be generalized
to accommodate higher derivative terms.

For the cubic terms in the action, the construction follows similarly by
including all possible terms cubic in $h_{\alpha\beta}$, or quadratic in
$h_{\alpha\beta}$ and linear in $\tilde{s}_{\alpha\beta}$, etc. We truncate
the expansion to leading order in the vacuum values $\overline{s}_{ab}$, since
these terms will be of primary interest for phenomenology. The structure of
these terms, again discarding those equivalent by integration by parts, is

$\displaystyle{\cal L}^{(3)}$ $\displaystyle\supset$ $\displaystyle{\rm
All\,Contractions\,of\,}\\{h_{\alpha\beta}h_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}h_{\eta\theta},h_{\alpha\beta}\partial_{\gamma}h_{\delta\epsilon}\partial_{\zeta}h_{\eta\theta},\overline{s}_{\alpha\beta}h_{\gamma\delta}h_{\epsilon\zeta}\partial_{\eta}\partial_{\theta}h_{\kappa\lambda},\overline{s}_{\alpha\beta}h_{\gamma\delta}\partial_{\epsilon}h_{\zeta\eta}\partial_{\theta}h_{\kappa\lambda},$
(6) $\displaystyle
h_{\alpha\beta}\tilde{s}_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}\tilde{s}_{\theta\kappa},h_{\alpha\beta}\partial_{\gamma}\tilde{s}_{\delta\epsilon}\partial_{\zeta}\tilde{s}_{\theta\kappa},\overline{s}_{\alpha\beta}\tilde{s}_{\gamma\delta}h_{\epsilon\zeta}\partial_{\eta}\partial_{\theta}h_{\kappa\lambda},\overline{s}_{\alpha\beta}\tilde{s}_{\gamma\delta}\partial_{\epsilon}h_{\zeta\eta}\partial_{\theta}h_{\kappa\lambda},\tilde{s}_{\alpha\beta}h_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}h_{\eta\theta},$
$\displaystyle\tilde{s}_{\alpha\beta}\partial_{\gamma}h_{\delta\epsilon}\partial_{\zeta}h_{\eta\theta},\tilde{s}_{\alpha\beta}\tilde{s}_{\gamma\delta}\partial_{\epsilon}\partial_{\zeta}\tilde{s}_{\eta\theta},\tilde{s}_{\alpha\beta}\partial_{\gamma}\tilde{s}_{\delta\epsilon}\partial_{\zeta}\tilde{s}_{\eta\theta},\overline{s}_{\alpha\beta}h_{\gamma\delta}\tilde{s}_{\epsilon\zeta}\partial_{\eta}\partial_{\theta}\tilde{s}_{\kappa\lambda},\overline{s}_{\alpha\beta}h_{\gamma\delta}\partial_{\epsilon}\tilde{s}_{\zeta\eta}\partial_{\theta}\tilde{s}_{\kappa\lambda}\\}.$

Among all the terms considered in (5) and (6) are the would-be decoupled
terms, i.e., those terms that do not have $\tilde{s}$ in them. If such terms
are possible through cubic order, they will appear in this construction.

We next impose the requirement of diffeomorphism invariance of the action
without assuming the field equations are satisfied (this is an “off-shell"
requirement). The quantity constructed by the series

${\cal L}={\cal L}^{(2)}+{\cal L}^{(3)}+...,$ (7)

is a scalar density of weight $-1$. In the spontaneous-symmetry breaking
scenario, this function depends on the matter fields, the vacuum values of the
coefficients for Lorentz violation ($\overline{s}_{\mu\nu}$), the metric
fluctuations $h_{\mu\nu}$ and coefficient fluctuations $\tilde{s}_{\mu\nu}$.
To impose the symmetry requirement, we will need the infinitesimal
diffeomorphism symmetry transformations, which are given by the Lie
derivatives of the underlying fields $g_{\mu\nu}$ and $s_{\mu\nu}$ along an
arbitrary vector field $\xi^{\mu}$. These transformations are

$\displaystyle{\cal L}_{\xi}g_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\lambda\mu}\partial_{\nu}\xi^{\lambda}+g_{\lambda\nu}\partial_{\mu}\xi^{\lambda}+\xi^{\lambda}\partial_{\lambda}g_{\mu\nu},$
$\displaystyle{\cal L}_{\xi}s_{\mu\nu}$ $\displaystyle=$ $\displaystyle
s_{\lambda\mu}\partial_{\nu}\xi^{\lambda}+s_{\lambda\nu}\partial_{\mu}\xi^{\lambda}+\xi^{\lambda}\partial_{\lambda}s_{\mu\nu}.$
(8)

From this we can derive the effective Lie derivatives, or diffeomorphism
transformations for the field fluctuations:

$\displaystyle{\cal L}_{\xi}h_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}+h_{\mu\lambda}\partial_{\nu}\xi^{\lambda}+h_{\nu\lambda}\partial_{\mu}\xi^{\lambda}+\xi^{\lambda}\partial_{\lambda}h_{\mu\nu},$
$\displaystyle{\cal L}_{\xi}\tilde{s}_{\mu\nu}$ $\displaystyle=$
$\displaystyle(\overline{s}_{\lambda\mu}+\tilde{s}_{\lambda\mu})\partial_{\nu}\xi^{\lambda}+(\overline{s}_{\lambda\nu}+\tilde{s}_{\lambda\nu})\partial_{\mu}\xi^{\lambda}+\xi^{\lambda}\partial_{\lambda}\tilde{s}_{\mu\nu}.$
(9)

where we impose $\partial_{\lambda}\overline{s}_{\mu\nu}=0$ in the chosen
asymptotically flat cartesian coordinates. In this treatment, it is necessary
to keep all terms to linear order in $\xi_{\alpha}$, in order to result in the
correct conservation laws. This includes terms typically discarded like
$\sim\xi h$ in the linearized limit.

The total action will take the form

$S=S_{M}+{\textstyle{{1}\over{2\kappa}}}\int d^{4}x({\cal L}^{(2)}+{\cal
L}^{(3)}+...),$ (10)

where the first term is the matter sector, which we assume to be minimally
coupled to gravity. The remaining terms are those that we seek to construct
and are the focus in what follows. Under a diffeomorphism, one can show the
total change in the action takes the form

$\delta S={\textstyle{{1}\over{2\kappa}}}\int d^{4}x\left({{\delta{\cal
L}}\over{\delta h_{\alpha\beta}}}{\cal L}_{\xi}h_{\alpha\beta}+{{\delta{\cal
L}}\over{\delta\tilde{s}_{\alpha\beta}}}{\cal
L}_{\xi}\tilde{s}_{\alpha\beta}\right).$ (11)

At this stage we make no assumption about whether the field equations are
satisfied. Insertion of (9) and using integration by parts gives

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle-{\textstyle{{1}\over{2\kappa}}}\int
d^{4}x\bigg{\\{}\partial_{\beta}\left({{\delta{\cal L}}\over{\delta
h_{\gamma\beta}}}\right)+\Gamma^{\gamma}_{\phantom{\beta}\alpha\beta}{{\delta{\cal
L}}\over{\delta
h_{\alpha\beta}}}+g^{\delta\gamma}s_{\delta\alpha}\partial_{\beta}\left({{\delta{\cal
L}}\over{\delta\tilde{s}_{\alpha\beta}}}\right)$ (12)
$\displaystyle\phantom{{\textstyle{{1}\over{4\kappa}}}\int
d^{4}x}+g^{\delta\gamma}{\textstyle{{1}\over{2}}}(\partial_{\alpha}\tilde{s}_{\beta\delta}+\partial_{\beta}\tilde{s}_{\alpha\delta}-\partial_{\delta}\tilde{s}_{\alpha\beta}){{\delta{\cal
L}}\over{\delta h_{\alpha\gamma}}}\bigg{\\}}g_{\gamma\epsilon}\xi^{\epsilon}.$

The quantity in braces will vanish for “off-shell" diffeomorphism invariance
of the action in the spontaneous-symmetry breaking scenario. For more details,
the reader is referred to expositions on Noether identities in the context of
Lorentz and diffeomorphism breaking presented in Refs. Kostelecky:2004 ;
Bluhm:2005 ; Bluhm:2008 ; Bluhm:2015 ; Bluhm:2016 ; Bluhm:2019 ; Bluhm:2021 .

We take the general expansion in equations (5) and (6) and insert an arbitrary
parameter in front of each independent contraction. The starting lagrange
density for quadratic order takes the form

${\cal
L}^{(2)}=C_{1}h^{\alpha\beta}\overline{s}^{\gamma\delta}\partial_{\alpha}\partial_{\beta}h_{\gamma\delta}+C_{2}h^{\alpha\beta}\overline{s}^{\gamma}_{\phantom{\gamma}\gamma}\partial_{\alpha}\partial_{\beta}h^{\delta}_{\phantom{\delta}\delta}+C_{3}h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\partial_{\gamma}\partial_{\beta}h^{\delta}_{\phantom{\delta}\delta}+...,$
(13)

with the remaining terms not displayed here for brevity. For the cubic terms,
these take the form

${\cal
L}^{(3)}=A_{1}h_{\alpha}^{\phantom{\alpha}\gamma}h^{\alpha\beta}\overline{s}^{\delta\epsilon}\partial_{\gamma}\partial_{\beta}h_{\delta\epsilon}+A_{2}hh^{\beta\gamma}\overline{s}^{\delta\epsilon}\partial_{\gamma}\partial_{\beta}h_{\delta\epsilon}+A_{3}h_{\alpha}^{\phantom{\alpha}\gamma}h^{\alpha\beta}\overline{s}^{\delta}_{\phantom{\delta}\delta}\partial_{\gamma}\partial_{\beta}h+...,$
(14)

displaying just the first few terms. Note that because the Lagrange density
has mass dimension $4$, and $s_{\mu\nu}$ is taken as dimensionless, the
parameters $C_{n}$ and $A_{n}$ have mass dimension $2$.

The idea is to insert the total lagrange density into the vector combination
of terms in brackets (12) above, and demand the result be zero. All
tensorially independent terms must vanish and this imposes many relations
among the arbitrary parameters $\\{C_{n}\\}$ and $\\{A_{n}\\}$. When one
imposes these relations on the parametrized lagrange densities (13) and (14),
a vastly smaller subset of these terms remains, each obeying the off-shell
conservation law expected from the underlying diffeomorphism symmetry. This
entire calculation is carried out within the Mathematica set of packages in
xAct, including in particular, the xTras package xTras:2014 .

We now display the Lagrange density result in several pieces. Firstly, the
entire process produces the quadratic and cubic lagrange densities of General
Relativity, up to integration by parts. Secondly, scalings of GR by the trace
$\overline{s}^{\alpha}_{\phantom{\alpha}\alpha}$ occur, which we leave out of
our tables below because these are unobservable scalings, as expected.
Neglected also are terms that satisfy the constraints (12), but do not
contribute to the gravitational field equation for $h_{\mu\nu}$ like, for
example, terms in equation (5) of the form
$\overline{s}\partial\tilde{s}\partial\tilde{s}$. We find the following
quadratic terms organized in the generic form

${\cal L}^{(2)}={\cal L}^{(2)}_{GR}+\sum^{10}_{n=1}\alpha_{n}{\cal
M}^{(2)}_{n},$ (15)

where the ${\cal M}^{(2)}_{n}$ terms are listed in the Table 1. Where
necessary for clarity, terms with the parentheses label of $(1)$ are first
order in the metric $h_{\alpha\beta}$ and $\tilde{s}_{\alpha\beta}$
fluctuations while those with the label $(2)$ are second order. All
Christoffel symbols are taken to first order in $h_{\alpha\beta}$ with indices
raised and lowered with the flat metric. It is also useful to use a linearized
Christoffel symbol ${\tilde{\Gamma}}$ with the substitution
$h_{\alpha\beta}\rightarrow\tilde{s}_{\alpha\beta}$. Thus

$\displaystyle\Gamma_{\alpha\beta\gamma}$ $\displaystyle=$
$\displaystyle{\textstyle{{1}\over{2}}}(\partial_{\beta}h_{\gamma\alpha}+\partial_{\gamma}h_{\beta\alpha}-\partial_{\alpha}h_{\beta\gamma}),$
$\displaystyle{\tilde{\Gamma}}_{\alpha\beta\gamma}$ $\displaystyle=$
$\displaystyle{\textstyle{{1}\over{2}}}(\partial_{\beta}\tilde{s}_{\gamma\alpha}+\partial_{\gamma}\tilde{s}_{\beta\alpha}-\partial_{\alpha}\tilde{s}_{\beta\gamma}).$
(16)

Two different contractions of the Christoffel symbols with one free index are
defined by

$\displaystyle\Gamma_{A\mu}$ $\displaystyle=$
$\displaystyle\Gamma^{\alpha}_{\phantom{\alpha}\alpha\mu},$
$\displaystyle\Gamma_{B}^{\mu}$ $\displaystyle=$
$\displaystyle\Gamma^{\mu\alpha}_{\phantom{\mu\alpha}\alpha},$ (17)

with the same definitions holding for ${\tilde{\Gamma}}$. We employ some
shorthands like $\Gamma_{A}^{2}=(\Gamma_{A})^{\alpha}(\Gamma_{A})_{\alpha}$,
$h=h_{\alpha\beta}\eta^{\alpha\beta}$, and
$\tilde{s}=\tilde{s}_{\alpha\beta}\eta^{\alpha\beta}$. Finally, we use
shorthands for curvature-like quantities built from
${\tilde{\Gamma}}_{\alpha\beta\gamma}$, like, for example,
$\tilde{G}^{(1)}_{\mu\nu}=\partial_{\alpha}{\tilde{\Gamma}}^{\alpha}_{\phantom{\alpha}\mu\nu}-\partial_{\mu}{\tilde{\Gamma}}_{A\nu}-{{1}\over{2}}\eta_{\mu\nu}(\partial^{\alpha}{\tilde{\Gamma}}_{B\alpha}-\partial^{\alpha}{\tilde{\Gamma}}_{A\alpha})$.

Term | Expression
---|---
${\cal M}^{(2)}_{\rm GR}$ | $-\tfrac{1}{4\kappa}h^{\alpha\beta}G^{(1)}_{\alpha\beta}$
${\cal M}^{(2)}_{1}$ | $\overline{s}_{\alpha\beta}\tilde{s}^{\alpha\beta}R^{(1)}$
${\cal M}^{(2)}_{2}$ | $\tilde{s}^{\alpha\beta}\tilde{G}^{(1)}_{\alpha\beta}-{\textstyle{{1}\over{2}}}\tilde{s}\partial^{\beta}{\tilde{\Gamma}}_{A\beta}+\tilde{s}\,\overline{s}^{\alpha\beta}\partial_{\alpha}\Gamma_{B\beta}$
${\cal M}^{(2)}_{3}$ | $\overline{s}_{\alpha}^{\phantom{\alpha}\beta}\tilde{s}^{\alpha\gamma}(R^{(1)}_{\beta\gamma}-\partial^{\delta}\Gamma_{\beta\gamma\delta})+{\textstyle{{1}\over{2}}}\tilde{s}^{\alpha\beta}\partial^{\gamma}{\tilde{\Gamma}}_{\alpha\beta\gamma}$
${\cal M}^{(2)}_{4}$ | $-\tilde{s}^{\alpha\beta}\tilde{G}^{(1)}_{\alpha\beta}$
${\cal M}^{(2)}_{5}$ | $\overline{s}^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}(R^{(1)}_{\beta\gamma}+\partial^{\delta}\Gamma_{\beta\gamma\delta})-{\textstyle{{1}\over{2}}}\tilde{s}^{\alpha\beta}\partial^{\gamma}{\tilde{\Gamma}}_{\alpha\beta\gamma}$
${\cal M}^{(2)}_{6}$ | $2\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}(R^{(1)}_{\gamma\alpha\beta\delta}-\partial_{\delta}\Gamma_{\alpha\beta\gamma})+\tilde{s}^{\alpha\beta}\partial_{\gamma}{\tilde{\Gamma}}^{\gamma}_{\phantom{\gamma}\alpha\beta}$
${\cal M}^{(2)}_{7}$ | $-\tilde{s}^{\alpha\beta}\partial_{\gamma}{\tilde{\Gamma}}^{\gamma}_{\phantom{\gamma}\alpha\beta}+2\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}\partial_{\delta}\Gamma_{\alpha\beta\gamma}$
${\cal M}^{(2)}_{8}$ | $-\tilde{s}\partial^{\alpha}{\tilde{\Gamma}}_{A\alpha}+2\overline{s}^{\alpha\beta}\tilde{s}\partial^{\delta}\Gamma_{\alpha\beta\delta}$
${\cal M}^{(2)}_{9}$ | $h^{\alpha\beta}\overline{s}^{\gamma\delta}{\cal G}^{(1)}_{\alpha\gamma\beta\delta}-\overline{s}^{\alpha\beta}h_{\alpha\beta}R^{(1)}+2\tilde{s}^{\alpha\beta}R^{(1)}_{\alpha\beta}$
${\cal M}^{(2)}_{10}$ | $(\tilde{s}-\overline{s}^{\alpha\beta}h_{\alpha\beta})R^{(1)}$
Table 1: Quadratic order terms in (15).

Similarly, the cubic terms can be organized as

${\cal L}^{(3)}=\sum^{10}_{n=1}\alpha_{n}{\cal M}^{(3)}_{n},$ (18)

and they are listed in Table 2.

Term | Expression
---|---
$2\kappa{\cal M}^{(3)}_{\rm GR}$ | $h^{\alpha\beta}(\Gamma_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}\Gamma_{B\gamma}+\Gamma^{\gamma}_{\phantom{\gamma}\alpha\beta}\Gamma_{A\gamma}-\Gamma^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}\Gamma_{\delta\gamma\beta}-3\Gamma_{\alpha\gamma\delta}\Gamma^{\gamma\delta}_{\phantom{\gamma\delta}\beta}+\Gamma_{A\alpha}\Gamma_{B\beta}+{{1}\over{2}}\Gamma_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}\Gamma_{A\gamma}-{{1}\over{2}}\Gamma^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}\Gamma_{\gamma\delta\beta}-{{1}\over{2}}\Gamma_{\alpha\gamma\delta}\Gamma_{\beta}^{\phantom{\beta}\gamma\delta})$
|
$+h({{3}\over{4}}\Gamma^{\alpha\beta\gamma}\Gamma_{\gamma\alpha\beta}-{{1}\over{2}}\Gamma_{A\alpha}\Gamma_{B}^{\alpha}+{{1}\over{4}}\Gamma^{\alpha\beta\gamma}\Gamma_{\alpha\beta\gamma})+{{1}\over{2}}({{1}\over{2}}hh^{\alpha\beta}-h^{\alpha\gamma}h^{\beta}_{\phantom{\alpha}\gamma})\partial^{\delta}\Gamma_{\alpha\beta\delta}$
${\cal M}^{(3)}_{1}$ | ${{1}\over{2}}\tilde{s}_{\alpha\beta}\tilde{s}^{\alpha\beta}R^{(1)}-2h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\tilde{s}_{\beta\gamma}R^{(1)}+\overline{s}_{\alpha\beta}\tilde{s}^{\alpha\beta}(\Gamma_{\alpha\beta\gamma}\Gamma^{\alpha\beta\gamma}-\Gamma_{B}^{2}+{{1}\over{2}}hR^{(1)}-2h^{\alpha\beta}R^{(1)}_{\alpha\beta})$
${\cal M}^{(3)}_{2}$ | $2\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}(\Gamma_{\alpha\gamma\delta}\Gamma_{B\beta}-\Gamma_{\alpha\gamma}^{\phantom{\alpha\gamma}\eta}\Gamma_{\beta\delta\eta})-\overline{s}^{\alpha\beta}\tilde{s}(2\Gamma_{\alpha}^{\phantom{\alpha}\delta\eta}\Gamma_{\delta\eta\beta}+\Gamma_{B\delta}\Gamma_{\alpha\beta}^{\phantom{\alpha\beta}\delta}+\Gamma_{B\delta}\Gamma^{\delta}_{\phantom{\delta}\alpha\beta}+\Gamma_{B\alpha}\Gamma_{B\beta})$
|
$+h^{\alpha\beta}(2{\tilde{\Gamma}}_{\alpha}^{\phantom{\alpha}\gamma\delta}{\tilde{\Gamma}}_{\gamma\delta\beta}-{\tilde{\Gamma}}_{\alpha}^{\phantom{\alpha}\gamma\delta}{\tilde{\Gamma}}_{\beta\gamma\delta}+{\tilde{\Gamma}}^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}{\tilde{\Gamma}}_{\delta\gamma\beta}-2{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{B\gamma}+{\tilde{\Gamma}}_{B\alpha}{\tilde{\Gamma}}_{B\beta})-h({\tilde{\Gamma}}^{\alpha\beta\gamma}{\tilde{\Gamma}}_{\beta\alpha\gamma}+{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{A}^{2})$
|
$+({\textstyle{{1}\over{2}}}h\tilde{s}\overline{s}^{\alpha\beta}-2\tilde{s}h^{\gamma\alpha}\overline{s}_{\gamma}^{\phantom{\gamma}\beta}-h^{\gamma\delta}\tilde{s}_{\gamma\delta}\overline{s}^{\alpha\beta})(R^{(1)}_{\alpha\beta}+\partial^{\eta}\Gamma_{(\alpha\beta)\eta})+2h^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}\partial_{\gamma}{\tilde{\Gamma}}_{A\beta}-h\tilde{s}^{\alpha\beta}\partial_{\alpha}{\tilde{\Gamma}}_{A\beta}-h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}\partial_{\delta}\Gamma_{\gamma\alpha\beta}$
${\cal M}^{(3)}_{3}$ | $\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}\Gamma_{\alpha\gamma\epsilon}\Gamma_{\beta\delta}^{\phantom{\beta\delta}\epsilon}+\overline{s}^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}(2\Gamma_{\beta\delta\epsilon}\Gamma^{\delta\epsilon}_{\phantom{\delta\epsilon}\gamma}+2\Gamma_{\delta\epsilon\gamma}\Gamma^{\delta\epsilon}_{\phantom{\delta\epsilon}\beta}+\Gamma_{\beta\gamma\delta}\Gamma_{B}^{\delta}-\Gamma_{B\delta}\Gamma^{\delta}_{\phantom{\delta}\beta\gamma})$
|
$+h^{\alpha\beta}({\textstyle{{1}\over{2}}}{\tilde{\Gamma}}^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}{\tilde{\Gamma}}_{\delta\gamma\beta}-{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{\alpha\gamma\delta}{\tilde{\Gamma}}_{\beta}^{\phantom{\beta}\gamma\delta}-{\tilde{\Gamma}}_{\alpha\gamma\delta}{\tilde{\Gamma}}^{\gamma\delta}_{\phantom{\gamma\delta}\beta}+{\tilde{\Gamma}}_{\alpha\beta\gamma}{\tilde{\Gamma}}^{\gamma}_{B}+{\tilde{\Gamma}}_{B\gamma}{\tilde{\Gamma}}^{\gamma}_{\phantom{\gamma}\alpha\beta}+{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{A\gamma}+{\tilde{\Gamma}}^{\gamma}_{\phantom{\gamma}\alpha\beta}{\tilde{\Gamma}}_{A\gamma})$
|
$-h({\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{\beta\gamma\delta}{\tilde{\Gamma}}^{\beta\gamma\delta}+{\tilde{\Gamma}}_{\beta\gamma\delta}{\tilde{\Gamma}}^{\gamma\delta\beta}+{\textstyle{{1}\over{4}}}{\tilde{\Gamma}}_{B}^{2}+{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{A\beta}{\tilde{\Gamma}}_{B}^{\beta}+{\textstyle{{1}\over{4}}}{\tilde{\Gamma}}_{A}^{2})+h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}_{\gamma}^{\phantom{\gamma}\epsilon}(\partial_{\alpha}\Gamma_{\delta\beta\epsilon}+R^{(1)}_{\delta\alpha\beta\epsilon})$
|
$+h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}_{\alpha\gamma}(\partial^{\epsilon}\Gamma_{\delta\beta\epsilon}-R^{(1)}_{\beta\delta})+{\textstyle{{1}\over{2}}}h\overline{s}^{\beta\gamma}\tilde{s}_{\beta}^{\phantom{\beta}\delta}(R^{(1)}_{\gamma\delta}-\partial^{\epsilon}\Gamma_{\gamma\delta\epsilon})-{\textstyle{{1}\over{2}}}h\tilde{s}^{\beta\gamma}(\partial^{\delta}{\tilde{\Gamma}}_{\beta\gamma\delta}+\partial_{\delta}{\tilde{\Gamma}}^{\delta}_{\phantom{\delta}\beta\gamma})$
|
$+h^{\alpha\beta}\overline{s}_{\alpha\gamma}\tilde{s}^{\gamma\delta}(\partial^{\epsilon}\Gamma_{\beta\delta\epsilon}-R^{(1)}_{\beta\delta})+h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\tilde{s}_{\beta}^{\phantom{\beta}\delta}(\partial^{\epsilon}\Gamma_{\gamma\delta\epsilon}-R^{(1)}_{\gamma\delta})+2h^{\alpha\beta}(\tilde{s}^{\gamma\delta}\partial_{\delta}{\tilde{\Gamma}}_{(\alpha\gamma)\beta}-\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}\partial^{\delta}{\tilde{\Gamma}}_{(\beta\gamma)\delta})$
${\cal M}^{(3)}_{4}$ | $-h^{\alpha\beta}\overline{s}_{\alpha\beta}\tilde{s}^{\gamma\delta}R^{(1)}_{\gamma\delta}+2\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}(\Gamma_{\alpha\gamma\epsilon}\Gamma_{\beta\delta}^{\phantom{\beta\gamma}\epsilon}-\Gamma_{B\alpha}\Gamma_{\beta\gamma\delta})+\overline{s}^{\alpha\beta}\tilde{s}(\Gamma_{B\alpha}\Gamma_{B\beta}-\Gamma_{\alpha\delta\epsilon}\Gamma_{\beta}^{\phantom{\beta}\delta\epsilon})+h({\tilde{\Gamma}}_{\beta\gamma\delta}{\tilde{\Gamma}}^{\gamma\delta\beta}-{\tilde{\Gamma}}_{A\beta}{\tilde{\Gamma}}_{B}^{\beta})$
|
$+\tilde{s}\Gamma_{A\beta}({\tilde{\Gamma}}_{A}^{\beta}+{\tilde{\Gamma}}_{B}^{\beta})+h^{\alpha\beta}({\tilde{\Gamma}}_{A\alpha}{\tilde{\Gamma}}_{A\beta}+2{\tilde{\Gamma}}_{B\alpha}{\tilde{\Gamma}}_{A\beta}+2{\tilde{\Gamma}}^{\gamma}_{\phantom{\gamma}{\alpha\beta}}{\tilde{\Gamma}}_{A\gamma}-2{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{A\gamma}-{\tilde{\Gamma}}_{B\alpha}{\tilde{\Gamma}}_{B\beta}$
|
$+2{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{B\gamma}-2{\tilde{\Gamma}}_{\gamma\delta\alpha}{\tilde{\Gamma}}_{\beta}^{\phantom{\beta}\gamma\delta}-{\tilde{\Gamma}}_{\gamma\delta\alpha}{\tilde{\Gamma}}^{\delta\gamma}_{\phantom{\delta\gamma}\beta}+{\tilde{\Gamma}}_{\alpha\gamma\delta}{\tilde{\Gamma}}_{\beta}^{\phantom{\beta}\gamma\delta}+\tilde{s}\partial^{\delta}{\tilde{\Gamma}}_{\delta\alpha\beta})$
${\cal M}^{(3)}_{5}$ | $-\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}\Gamma_{\alpha\gamma}^{\phantom{\alpha\gamma}\epsilon}\Gamma_{\beta\delta\epsilon}-\overline{s}^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}(2\Gamma_{\beta}^{\phantom{\beta}\delta\epsilon}\Gamma_{\delta\epsilon\gamma}+\Gamma_{\beta\gamma\delta}\Gamma_{B}^{\delta}+\Gamma_{B}^{\delta}\Gamma_{\delta\beta\gamma})+h^{\alpha\beta}({\tilde{\Gamma}}_{\alpha}^{\phantom{\alpha}\gamma\delta}{\tilde{\Gamma}}_{\gamma\delta\beta}-{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{\alpha\gamma\delta}{\tilde{\Gamma}}_{\beta}^{\phantom{\beta}\gamma\delta}+{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}{\tilde{\Gamma}}_{\delta\gamma\beta})$
|
$-{\textstyle{{1}\over{2}}}h({\tilde{\Gamma}}^{\beta\gamma\delta}{\tilde{\Gamma}}_{\gamma\delta\beta}+{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{B}^{2}+{\tilde{\Gamma}}_{A\gamma}{\tilde{\Gamma}}^{\gamma}_{B}+{\textstyle{{1}\over{2}}}{\tilde{\Gamma}}_{A}^{2})-h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}_{\alpha\gamma}(R^{(1)}_{\beta\delta}+\partial^{\epsilon}\Gamma_{\delta\beta\epsilon})-h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\tilde{s}_{\gamma}^{\phantom{\gamma}\delta}(\partial^{\epsilon}\Gamma_{\beta\delta\epsilon}+R^{(1)}_{\beta\delta})$
|
$+{\textstyle{{1}\over{2}}}h\overline{s}^{\beta\gamma}\tilde{s}_{\beta}^{\phantom{\beta}\delta}(\partial^{\epsilon}\Gamma_{\gamma\delta\epsilon}-\partial_{\delta}\Gamma_{A\gamma})-{\textstyle{{1}\over{2}}}h\tilde{s}^{\beta\gamma}(\partial^{\delta}{\tilde{\Gamma}}_{\beta\gamma\delta}+\partial_{\delta}{\tilde{\Gamma}}^{\delta}_{\phantom{\delta}\beta\gamma})+h^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}(\partial^{\delta}{\tilde{\Gamma}}_{\gamma\beta\delta}+\partial_{\delta}{\tilde{\Gamma}}^{\delta}_{\phantom{\delta}\beta\gamma})$
|
$-h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\tilde{s}_{\beta}^{\phantom{\beta}\delta}(R^{(1)}_{\gamma\delta}+\partial^{\epsilon}\Gamma_{\gamma\delta\epsilon})-h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}_{\gamma}^{\phantom{\gamma}\epsilon}\partial_{\epsilon}\Gamma_{\delta\alpha\beta}+{\textstyle{{1}\over{2}}}h\overline{s}^{\beta\gamma}\tilde{s}_{\beta}^{\phantom{\beta}\delta}\partial_{\epsilon}\Gamma^{\epsilon}_{\phantom{\epsilon}\gamma\delta}$
${\cal M}^{(3)}_{6}$ | $2\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}(2\Gamma_{\alpha\gamma\epsilon}\Gamma^{\epsilon}_{\phantom{\epsilon}\beta\delta}+\Gamma_{\alpha\beta}^{\phantom{\alpha\beta}\epsilon}\Gamma_{\epsilon\gamma\delta}+\Gamma_{\epsilon\gamma\delta}\Gamma^{\epsilon}_{\phantom{\epsilon}\alpha\beta}+\Gamma_{\alpha\gamma\delta}\Gamma_{B\beta})+2h^{\alpha\beta}({\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{B\gamma}+{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{B\gamma}-{\tilde{\Gamma}}_{\alpha}^{\phantom{\alpha}\gamma\delta}{\tilde{\Gamma}}_{\beta\gamma\delta})$
|
$-h{\tilde{\Gamma}}^{\beta\gamma\delta}{\tilde{\Gamma}}_{\gamma\delta\beta}+2h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\tilde{s}^{\delta\epsilon}(2R^{(1)}_{\delta\beta\epsilon\gamma}+\partial_{\epsilon}\Gamma_{\beta\gamma\delta}+\partial_{\epsilon}\Gamma_{\gamma\delta\beta})+h\overline{s}^{\beta\gamma}\tilde{s}^{\delta\epsilon}(R^{(1)}_{\delta\beta\gamma\epsilon}-\partial_{\epsilon}\Gamma_{\beta\gamma\delta})$
|
$+2h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}_{\alpha}^{\phantom{\alpha}\epsilon}(R^{(1)}_{\beta\gamma\epsilon\delta}+\partial_{\delta}\Gamma_{\gamma\beta\epsilon}+\partial_{\epsilon}\Gamma_{\gamma\beta\delta})+2h^{\alpha\beta}\tilde{s}^{\gamma\delta}\partial_{\delta}{\tilde{\Gamma}}_{\alpha\beta\gamma}-2h^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}(\partial_{\delta}{\tilde{\Gamma}}_{\beta\gamma}^{\phantom{\beta\gamma}\delta}+\partial_{\delta}{\tilde{\Gamma}}^{\delta}_{\phantom{\delta}\beta\gamma})$
${\cal M}^{(3)}_{7}$ | $-2\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}(2\Gamma_{\alpha\gamma}^{\phantom{\alpha\gamma}\epsilon}\Gamma_{\epsilon\beta\delta}+\Gamma_{\alpha\beta}^{\phantom{\alpha\beta}\epsilon}\Gamma_{\epsilon\gamma\delta}+\Gamma_{\epsilon\beta\delta}\Gamma^{\epsilon}_{\phantom{\epsilon}\alpha\gamma}+\Gamma_{\alpha\gamma\delta}\Gamma_{B\beta})$
|
$+h^{\alpha\beta}({\tilde{\Gamma}}_{\alpha}^{\phantom{\alpha}\gamma\delta}{\tilde{\Gamma}}_{\beta\gamma\delta}-2{\tilde{\Gamma}}_{\alpha}^{\phantom{\alpha}\gamma\delta}{\tilde{\Gamma}}_{\gamma\delta\beta}-{\tilde{\Gamma}}^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}{\tilde{\Gamma}}_{\delta\gamma\beta}+2{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{B\gamma}+2{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{A\gamma}-{\tilde{\Gamma}}_{B\alpha}{\tilde{\Gamma}}_{B\beta}-2{\tilde{\Gamma}}_{B\alpha}{\tilde{\Gamma}}_{A\beta}-{\tilde{\Gamma}}_{A\alpha}{\tilde{\Gamma}}_{A\beta})$
|
$+h{\tilde{\Gamma}}_{\alpha\beta\gamma}{\tilde{\Gamma}}^{\beta\gamma\alpha}+h^{\alpha\beta}\tilde{s}_{\alpha\beta}(\partial_{\gamma}{\tilde{\Gamma}}^{\gamma}_{B}+\partial_{\gamma}{\tilde{\Gamma}}^{\gamma}_{A})-4h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\tilde{s}^{\delta\epsilon}\partial_{\epsilon}\Gamma_{(\beta\gamma)\delta}+h\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}\partial_{\delta}{\tilde{\Gamma}}_{\alpha\beta\gamma}-4h^{\alpha\beta}\overline{s}^{\gamma\delta}\tilde{s}_{\alpha}^{\phantom{\alpha}\epsilon}\partial_{\epsilon}\Gamma_{\gamma\beta\delta}$
${\cal M}^{(3)}_{8}$ | $4\overline{s}^{\alpha\beta}\tilde{s}^{\gamma\delta}\Gamma_{\alpha\beta}^{\phantom{\alpha\beta}\epsilon}\Gamma_{\gamma\delta\epsilon}+4h^{\alpha\beta}\tilde{s}^{\gamma\delta}(\Gamma_{\gamma\delta}^{\phantom{\gamma\delta}\epsilon}{\tilde{\Gamma}}_{\alpha\beta\epsilon}+\Gamma_{\gamma\alpha\delta}{\tilde{\Gamma}}_{A\beta})-2h^{\alpha\beta}(2{\tilde{\Gamma}}_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}{\tilde{\Gamma}}_{A\gamma}+{\tilde{\Gamma}}_{A\alpha}{\tilde{\Gamma}}_{A\beta})$
|
$-4\tilde{s}^{\alpha\beta}\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\gamma}{\tilde{\Gamma}}_{A\gamma}+h{\tilde{\Gamma}}_{A}^{2}-2h\overline{s}^{\alpha\beta}\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\gamma}{\tilde{\Gamma}}_{A\gamma}+8h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}\Gamma_{(\beta\gamma)}^{\phantom{(\beta\gamma)}\delta}{\tilde{\Gamma}}_{A\delta}$
${\cal M}^{(3)}_{9}$ | $h^{\alpha\beta}\overline{s}^{\gamma\delta}(4\Gamma_{\alpha\gamma}^{\phantom{\alpha\gamma}\epsilon}\Gamma_{[\beta\delta]\epsilon}+2\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\epsilon}\Gamma_{\gamma\delta\epsilon}-3\Gamma_{\gamma\alpha}^{\phantom{\gamma\alpha}\epsilon}\Gamma_{\delta\beta\epsilon}-2\Gamma_{\gamma\delta}^{\phantom{{\gamma\delta}}\epsilon}\Gamma_{\epsilon\alpha\beta}+2\Gamma_{\alpha\gamma}^{\phantom{\alpha\gamma}\epsilon}\Gamma_{\epsilon\beta\delta}+2\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\epsilon}\Gamma_{\epsilon\gamma\delta}$
|
$-2\Gamma_{\epsilon\gamma\delta}\Gamma^{\epsilon}_{\phantom{\epsilon}\alpha\beta}+\Gamma_{\epsilon\beta\delta}\Gamma^{\epsilon}_{\phantom{\epsilon}\alpha\gamma}-8\Gamma_{(\alpha\gamma)\delta}\Gamma_{A\beta}+4\Gamma_{\gamma\alpha\beta}\Gamma_{A\delta}-8\Gamma_{(\alpha\gamma)\delta}\Gamma_{B\beta})$
|
$+h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}(2\Gamma_{\delta\epsilon\gamma}\Gamma^{\delta\epsilon}_{\phantom{\delta\epsilon}\beta}+2\Gamma^{\delta\epsilon}_{\phantom{\delta\epsilon}\beta}\Gamma_{\epsilon\delta\gamma}-2\Gamma_{B\beta}\Gamma_{B\gamma}-6\Gamma_{(\beta\gamma)}^{\phantom{(\beta\gamma)}\delta}\Gamma_{B\delta}-4\Gamma_{B\delta}\Gamma^{\delta}_{\phantom{\delta}\beta\gamma}-2\Gamma_{B\gamma}\Gamma_{A\beta}$
|
$+4\Gamma_{B\beta}\Gamma_{A\gamma}-8\Gamma_{(\beta\gamma)}^{\phantom{(\beta\gamma))}\delta}\Gamma_{A\delta}-8\Gamma^{\delta}_{\phantom{\delta}\beta\gamma}\Gamma_{A\delta})$
|
$+h\overline{s}^{\alpha\beta}(-\Gamma_{(\delta\epsilon)\beta}\Gamma^{\delta\epsilon}_{\phantom{\delta\epsilon}\alpha}+{\textstyle{{1}\over{2}}}\Gamma_{B\alpha}\Gamma_{B\beta}+\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\gamma}\Gamma_{B\gamma}+\Gamma_{B\gamma}\Gamma^{\gamma}_{\phantom{\gamma}\alpha\beta}+\Gamma_{B\alpha}\Gamma_{A\beta}+{\textstyle{{3}\over{2}}}\Gamma_{A\alpha}\Gamma_{A\beta}-\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\gamma}\Gamma_{A\gamma}-\Gamma^{\delta}_{\phantom{\delta}\alpha\beta}\Gamma_{A\delta})$
|
$+2\tilde{s}^{\alpha\beta}(\Gamma_{\gamma\delta\beta}\Gamma^{\gamma\delta}_{\phantom{\gamma\delta}\alpha}-\Gamma_{B\gamma}\Gamma^{\gamma}_{\phantom{\gamma}\alpha\beta})+2h^{\alpha\beta}\tilde{s}^{\gamma\delta}R^{(1)}_{\gamma\alpha\beta\delta}-4h^{\alpha\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma}R^{(1)}_{\beta\gamma}+h\tilde{s}^{\alpha\beta}R^{(1)}_{\alpha\beta}-2h^{\alpha\beta}h^{\gamma\delta}\overline{s}_{\alpha}^{\phantom{\alpha}\epsilon}\partial_{\delta}\Gamma_{(\beta\epsilon)\gamma}$
|
$-3h^{\alpha\beta}h^{\gamma\delta}\overline{s}_{\alpha\gamma}\partial_{\delta}\Gamma_{A\beta}-4h_{\alpha}^{\phantom{\alpha}\gamma}h^{\alpha\beta}\overline{s}_{\beta}^{\phantom{\beta}\delta}\partial^{\epsilon}\Gamma_{(\gamma\delta)\epsilon}+hh^{\alpha\beta}\overline{s}_{\beta}^{\phantom{\beta}\delta}\partial^{\epsilon}\Gamma_{(\gamma\delta)\epsilon}$
${\cal M}^{(3)}_{10}$ | $\tilde{s}(\Gamma_{\alpha\beta\gamma}\Gamma^{\alpha\beta\gamma}-\Gamma_{B\alpha}\Gamma_{B}^{\alpha})+h^{\alpha\beta}\overline{s}^{\gamma\delta}(2\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\epsilon}\Gamma_{\gamma\delta\epsilon}-\Gamma_{\gamma\delta}^{\phantom{{\gamma\delta}}\epsilon}\Gamma_{\epsilon\alpha\beta}+\Gamma_{\gamma\delta\alpha}\Gamma_{A\beta}-3\Gamma_{B\alpha}\Gamma_{\gamma\delta\beta})$
|
$-{\textstyle{{1}\over{2}}}h^{\alpha\beta}\overline{s}_{\alpha\beta}(\Gamma_{B\gamma}\Gamma_{B}^{\gamma}+\Gamma_{A\gamma}\Gamma_{A}^{\gamma})+4h^{\alpha\beta}\overline{s}_{\alpha}^{\phantom{\alpha}\gamma}(\Gamma_{(\beta\gamma)}^{\phantom{(\beta\gamma)}\delta}\Gamma_{A\delta}-\Gamma_{(\beta\gamma)}^{\phantom{(\beta\gamma)}\delta}\Gamma_{B\delta})+h\overline{s}^{\alpha\beta}(\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\gamma}\Gamma_{B\gamma}-\Gamma_{\alpha\beta}^{\phantom{{\alpha\beta}}\gamma}\Gamma_{A\gamma})$
|
$-2h^{\alpha\beta}\tilde{s}R^{(1)}_{\alpha\beta}+({\textstyle{{1}\over{2}}}h\tilde{s}-h^{\alpha\beta}\tilde{s}_{\alpha\beta})R^{(1)}-{\textstyle{{1}\over{2}}}h^{\alpha\beta}h^{\gamma\delta}\overline{s}_{\alpha\beta}\partial^{\epsilon}\Gamma_{\gamma\delta\epsilon}$
Table 2: Cubic order terms in (18). Each term, ${\cal M}^{(2)}_{n}+{\cal
M}^{(3)}_{n}$, is invariant under the diffeomorphism transformations of the
metric fluctuations $h_{\alpha\beta}$ and the fluctuations
$\tilde{s}_{\alpha\beta}$ in equations (8) and (9).

One consequence of this construction is that there are no terms ${\cal
M}^{(2)}_{n}+{\cal M}^{(3)}_{n}$ that are devoid of the fluctuations
$\tilde{s}_{\alpha\beta}$, save for the GR terms. In other words, in equations
(5) and (6), terms of the form $\overline{s}h\partial\partial h$ with cubic
counterparts $h\overline{s}\partial h\partial h$,
$h\overline{s}h\partial\partial h$ would be decoupled, but such terms do not
occur in the results in the Tables 1 and 2. Furthermore, no linear combination
of these terms can be taken to eliminate the fluctuations. This shows it is
not possible to construct terms in the action at cubic order, and second order
in derivatives, that arise from spontaneous symmetry breaking that also fully
decouple the metric fluctuations and the coefficient fluctuations.

For the potential terms for the two-tensor $s_{\alpha\beta}$, these are
functions of the 4 possible independent scalar traces of products of the
tensor $s_{\alpha\beta}g^{\beta\gamma}$ Kostelecky:2009b . Note that this
includes the possibility of having "steep" potentials that vanish in the
linearized or higher-order approximations. For example, one useful potential
is a function of $s^{\alpha}_{\phantom{\alpha}\alpha}$ via

$V={{1}\over{4}}\lambda\left(s^{\alpha}_{\phantom{\alpha}\alpha}-x\right)^{4},$
(19)

where $x$ is a constant. When expanding around a flat background, it can be
shown that the leading order effects in the field equations are of cubic order
in the fluctuations.

##  4 Covariant construction of Lagrange density

As an alternative method to section 3, we can construct an fully covariant
lagrangian for the gravitational sector including the $s_{\alpha\beta}$ terms.
We seek kinetic lagrangian terms that would produce the ones in the expansion
in the previous section. These terms were included as a second order in
$s_{\alpha\beta}$ expansion in Ref. O'Neal:2021 . These are labeled
$a_{1}-a_{11}$ and obtained by considering all possibilities of scalars
contractions of $s_{\alpha\beta}s_{\gamma\delta}R_{\mu\nu\kappa\lambda}$ and
$\nabla_{\gamma}s_{\alpha\beta}\nabla_{\delta}s_{\mu\nu}$, and discarding
surface terms:

$\displaystyle{\cal L}_{\rm cov}$ $\displaystyle=$
$\displaystyle{{\sqrt{-g}}\over{2\kappa}}\Big{[}a_{1}s^{\lambda}_{\phantom{\lambda}\lambda}R+a_{2}s_{\mu\nu}R^{\mu\nu}+a_{3}\tfrac{1}{2}(\nabla_{\mu}s_{\nu\lambda})(\nabla^{\mu}s^{\nu\lambda})+a_{4}\tfrac{1}{2}(\nabla_{\mu}s^{\mu\lambda})(\nabla_{\lambda}s^{\beta}_{\phantom{\beta}\beta})$
(20)
$\displaystyle+a_{5}\tfrac{1}{2}(\nabla_{\mu}s^{\mu\lambda})(\nabla_{\nu}s^{\nu}_{\phantom{\beta}\lambda})+a_{6}\tfrac{1}{2}(\nabla_{\mu}s^{\nu}_{\phantom{\beta}\nu})(\nabla^{\mu}s^{\lambda}_{\phantom{\beta}\lambda})+a_{7}s_{\mu\nu}s_{\kappa\lambda}R^{\mu\kappa\nu\lambda}+a_{8}s_{\mu\nu}s^{\mu}_{\phantom{\mu}\lambda}R^{\nu\lambda}$
$\displaystyle+a_{9}s^{\lambda}_{\phantom{\lambda}\lambda}s_{\mu\nu}R^{\mu\nu}+a_{10}s^{\mu\nu}s_{\mu\nu}R+a_{11}s^{\lambda}_{\phantom{\lambda}\lambda}s^{\mu}_{\phantom{\mu}\mu}R\Big{]}.$

The covariant terms can be matched to the cubic and quadratic terms by
expanding them to the relevant orders. The expansion, however, is cumbersome,
hence the advantage of the earlier construction. We find the following matches
at quadratic order in the expansion:

$\displaystyle{\cal L}_{a_{1}}$ $\displaystyle=$
$\displaystyle{\textstyle{{a_{1}}\over{2\kappa}}}{\cal
M}^{(2)}_{10}+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{2}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{2}}\over{2\kappa}}}{\cal
M}^{(2)}_{9}+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{3}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{3}}\over{2\kappa}}}\left({\cal
M}^{(2)}_{5}-{\cal M}^{(2)}_{3}\right)+O(3)+O(\overline{s}^{2})$
$\displaystyle{\cal L}_{a_{4}}$ $\displaystyle=$
$\displaystyle{\textstyle{{a_{4}}\over{4\kappa}}}\left({\cal
M}^{(2)}_{2}+{\cal M}^{(2)}_{7}+{\textstyle{{1}\over{2}}}{\cal
M}^{(2)}_{8}\right)+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{5}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{5}}\over{2\kappa}}}\left({\cal
M}^{(2)}_{5}+{\textstyle{{1}\over{2}}}{\cal
M}^{(2)}_{7}\right)+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{6}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{6}}\over{2\kappa}}}{\cal
M}^{(2)}_{8}+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{7}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{7}}\over{2\kappa}}}\left({\cal
M}^{(2)}_{6}+{\textstyle{{1}\over{2}}}{\cal
M}^{(2)}_{7}\right)+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{8}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{8}}\over{2\kappa}}}\left({\cal
M}^{(2)}_{3}+{\cal M}^{(2)}_{5}\right)+O(3)+O(\overline{s}^{2})$
$\displaystyle{\cal L}_{a_{9}}$ $\displaystyle=$
$\displaystyle{\textstyle{{a_{9}}\over{2\kappa}}}\left({\cal
M}^{(2)}_{2}+{\cal M}^{(2)}_{4}-2{\cal
M}^{(2)}_{8}+{\textstyle{{1}\over{2}}}\overline{s}^{\alpha}_{\phantom{\alpha}\alpha}{\cal
M}^{(2)}_{9}\right)+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{10}}$
$\displaystyle=$ $\displaystyle{\textstyle{{a_{10}}\over{\kappa}}}{\cal
M}^{(2)}_{1}+O(3)+O(\overline{s}^{2})$ $\displaystyle{\cal L}_{a_{11}}$
$\displaystyle=$
$\displaystyle{\textstyle{{a_{11}}\over{\kappa}}}\overline{s}^{\alpha}_{\phantom{\alpha}\alpha}{\cal
M}^{(2)}_{10}+O(3)+O(\overline{s}^{2})$ (21)

Here $O(3)$ denotes terms of cubic order in fluctuations $h^{3}$,
$\tilde{s}^{3}$, $h^{2}\tilde{s}$, etc. while the $O(\overline{s}^{2})$
includes terms of second order in the vacuum values $\overline{s}_{\mu\nu}$.
Note that one could have started with the covariant expansion instead of going
through the process of the earlier section in the work. However, the quadratic
and cubic expansions are more useful for approximate solutions for weak
fields, for which one has to expand, order by order, in any case. Furthermore,
the conservation law construction can be used for more general tensor
couplings, for which it may be challenging to construct the covariant terms.
The terms in (20) are a subset of the recent construction in Ref.
Kostelecky:2021 .

##  5 Field equations

We turn to the field equations from the action above. Considering the large
number of terms in tables above, abbreviations will be used for different
pieces of the field equations. Multiple partial derivatives will be
abbreviated as
$\partial_{\alpha\beta...}=\partial_{\alpha}\partial_{\beta}...$. Variation of
the entire Lagrange density (through cubic order) with respect to the metric
fluctuations yields the field equations in the form

$\displaystyle{{\delta{\cal L}}\over{\delta h_{\mu\nu}}}$ $\displaystyle=$
$\displaystyle-{\textstyle{{1}\over{2\kappa}}}(G^{(1)})^{\mu\nu}+{\textstyle{{1}\over{2}}}T_{M}^{\mu\nu}+{\textstyle{{1}\over{2}}}\tau^{\mu\nu}+{\textstyle{{\delta{\cal
L}_{V}}\over{\delta
h_{\mu\nu}}}}+\alpha_{1}\overline{s}^{\alpha\beta}(\partial^{\mu\nu}-\eta^{\mu\nu}\Box)\tilde{s}_{\alpha\beta}+\alpha_{2}(\overline{s}^{\alpha(\mu}\partial^{\nu)}\partial_{\alpha}-{\textstyle{{1}\over{2}}}\eta^{\mu\nu}\overline{s}^{\alpha\beta}\partial_{\alpha\beta})\tilde{s}$
(22)
$\displaystyle+\alpha_{3}(-\overline{s}^{\alpha(\mu}\Box\tilde{s}^{\nu)}_{\phantom{\nu}\alpha}+\overline{s}^{\alpha\beta}\partial_{\beta}\partial^{(\mu}\tilde{s}^{\nu)}_{\phantom{\nu}\alpha})-{\textstyle{{1}\over{2}}}\eta^{\mu\nu}\overline{s}^{\alpha\beta}\partial_{\gamma\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma})+\alpha_{5}(\overline{s}^{\alpha(\mu}\partial^{\nu)}\partial_{\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\beta}-{\textstyle{{1}\over{2}}}\eta^{\mu\nu}\overline{s}^{\alpha\beta}\partial_{\gamma\beta}\tilde{s}_{\alpha}^{\phantom{\alpha}\gamma})$
$\displaystyle+\alpha_{6}(\overline{s}^{\alpha\beta}\partial_{\alpha\beta}-2\overline{s}^{\alpha(\mu}\partial_{\beta\alpha}\tilde{s}^{\nu)\beta})+\alpha_{7}\overline{s}^{\mu\nu}\partial_{\alpha\beta}\tilde{s}^{\alpha\beta}+\alpha_{8}\overline{s}^{\mu\nu}\Box\tilde{s}$
$\displaystyle+\alpha_{9}(2(\tilde{G}^{(1)})^{\mu\nu}+2\overline{s}_{\alpha\beta}({\cal
G}^{(1)})^{\alpha\mu\beta\nu})+(\alpha_{9}+\alpha_{10})[(\partial^{\mu\nu}-\eta^{\mu\nu}\Box)(\tilde{s}-\overline{s}^{\alpha\beta}h_{\alpha\beta})-\overline{s}^{\mu\nu}R^{(1)})],$

where $\tau^{\alpha\beta}$ contains the full variation of the cubic terms
$\tau^{\mu\nu}=2\delta{\cal L}^{(3)}/\delta h_{\mu\nu}$, the factor of $2$
chosen so that it matches the GR pseudo tensor in the correct limit. Here
${\cal L}_{V}$ is the portion of the Lagrange density with the potential
energy function for $\tilde{s}_{\mu\nu}$, and $T_{M}^{\mu\nu}$ comes from the
variation of the matter sector with respect to the metric fluctuations.
Similarly, the variation of the Lagrange density with respect to the two-
tensor fluctuations $\tilde{s}_{\alpha\beta}$ is

$\displaystyle{{\delta{\cal L}}\over{\delta\tilde{s}_{\mu\nu}}}$
$\displaystyle=$ $\displaystyle(\tau_{s})^{\mu\nu}+{\textstyle{{\delta{\cal
L}_{V}}\over{\delta\tilde{s}_{\mu\nu}}}}+\alpha_{1}\overline{s}^{\mu\nu}R^{(1)}+\alpha_{2}[2(\tilde{G}^{(1)})^{\mu\nu}-{\textstyle{{1}\over{2}}}\eta^{\mu\nu}(\Box\tilde{s}+\overline{s}^{\alpha\beta}(\partial_{\alpha\beta}h-2\partial_{\gamma\beta}h_{\alpha}^{\phantom{\alpha}\gamma}))]$
(23)
$\displaystyle+\alpha_{3}[{\textstyle{{1}\over{2}}}\Box\tilde{s}^{\alpha\beta}-{\textstyle{{1}\over{2}}}\overline{s}^{\alpha(\mu}\partial^{\nu)}\partial_{\alpha}h+\overline{s}^{\alpha(\mu}\partial_{\alpha\beta}h^{\nu)\beta}-\overline{s}^{\alpha(\mu}\Box
h^{\nu)}_{\phantom{\nu}\alpha}]-2\alpha_{4}(\tilde{G}^{(1)})^{\mu\nu}$
$\displaystyle+\alpha_{5}[-{\textstyle{{1}\over{2}}}\Box\tilde{s}^{\alpha\beta}-{\textstyle{{1}\over{2}}}\overline{s}^{\alpha(\mu}\partial^{\nu)}\partial_{\alpha}h+\overline{s}^{\alpha(\mu}\partial^{\nu)}\partial_{\beta}h_{\alpha}^{\phantom{\alpha}\beta}]$
$\displaystyle+\alpha_{6}[-\Box\tilde{s}^{\mu\nu}+2\partial_{\alpha}\partial^{(\mu}\tilde{s}^{\nu)\alpha}+\overline{s}^{\alpha\beta}(\partial_{\alpha\beta}h^{\mu\nu}-\partial_{\beta}\partial^{(\mu}h^{\nu)}_{\phantom{\nu}\alpha})]$
$\displaystyle+\alpha_{7}(\Box\tilde{s}^{\mu\nu}-2\partial_{\alpha}\partial^{(\mu}\tilde{s}^{\nu)\alpha}+\overline{s}^{\alpha\beta}\partial^{\mu\nu}h_{\alpha\beta})-\alpha_{8}\eta^{\mu\nu}\Box(\tilde{s}-\overline{s}^{\alpha\beta}h_{\alpha\beta})$
$\displaystyle+2\alpha_{9}(R^{(1)})^{\mu\nu}+\alpha_{10}\eta^{\mu\nu}R^{(1)},$

where $(\tau_{s})^{\mu\nu}=\delta{\cal L}^{(3)}/\delta\tilde{s}_{\mu\nu}$. The
matter sector will have its own field equations which depend on the system to
be modeled. We note here that when the field equations hold, the stress-energy
tensor for matter will be conserved, a fact that can be exploited in solving
the post-Newtonian limit.

The generic case of equations (22) and (23) with arbitrary parameters contains
“ghost" kinetic terms for the fluctuations $\tilde{s}_{\alpha\beta}$, when
treated as a field theory on a flat background. For example, a kinetic term of
the form ${\cal L}\sim\tilde{s}^{\alpha\beta}\Box\tilde{s}_{\alpha\beta}$ will
contain non-positive definite contributions to the Hamiltonian density, and
could thereby result in unstable solutions. It is of general interest to study
this issue in detail, but we do not attempt it here.

We note that this effective field theory construction represents a test
framework for Lorentz and diffeomorphism breaking. Rather than just the $9$
coefficients in (4), we now have up to $10$ arbitrary parameters
$\alpha_{1}-\alpha_{10}$ and $9$ coefficients $\overline{s}_{\mu\nu}$, for a
total of $19$ quantities describing Lorentz and diffeomorphism violation for
gravity.

##  6 Summary and Outlook

In this work we described the method to construct quadratic and cubic
effective actions for weak-field gravity studies of spontaneous symmetry
breaking. The main results are the Lagrange density terms in equations (15)
and (18) and the Tables 1 and 2. Furthermore, we showed that complete
decoupling of the fluctuations from the metric fluctuations cannot occur at
the level of the action with only second derivatives when going to cubic order
in the fluctuations.

The field equations for the metric fluctuations and the two-tensor
fluctuations are given in equations (23) and (22). Future work will solve
these equations in the post-Newtonian limit, with near-field and radiation
zone solutions for applications in astrophysics. Some preliminary analysis of
the field equations in (23) and (22) shows that the near-field post-Newtonian
metric contains many of the same fluid potential function terms that were
found in Ref. Bailey:2006 . We conjecture that including all of the extra
terms in the Lagrange density could potentially provide an interesting re-
analysis of the match to the Parametrized Post-Newtonian metric Will:1971 ;
Will:1973 . The match of the action above to vector models of Lorentz
violation should also be revisited Kostelecky:1989 ; Jacobson:2001 ;
Seifert:2009 . Finally, the method adopted in this work for the two-tensor
$s_{\mu\nu}$ could be continued and applied to other gravity sector
coefficients, such at the $t_{\alpha\beta\gamma\delta}$ coefficients, or non-
minimal coefficients.

Analysis could also proceed with the general covariant action (20), such as
exploring black hole solutions or cosmology solutions. Progress has already
been made studying strong-field gravity solutions with vector and tensor
models of spontaneous Lorentz-symmetry breaking Casana:2018 ; Maluf:2021 .
Complimentary analysis with (20) would be of interest.

##  7 Acknowledgements

This work was supported in part by the National Science Foundation under Grant
No. 1806871. K. O’Neal-Ault and M. Mewes provided valuable comments on the
manuscript.

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