aainstitutetext: Centro de Ciencias Exactas, Universidad del Bío-Bío, Avda.
Andrés Bello 720, 3800708, Chillán, Chilebbinstitutetext: Departamento de
Física, Universidade Federal do Maranhão, Campus Universitário do Bacanga, São
Luís (MA), 65085-580, Brazil

# Modified-gravity theories with nondynamical background fields

Carlos M. Reyes b and Marco Schreck<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the dynamics of a modified-gravity theory, which is supplemented by
an extended Gibbons-Hawking-York boundary term and incorporates diffeomorphism
violation through nondynamical background fields denoted as $u$ and
$s^{\mu\nu}$ in the literature. An ADM decomposition allows us to project the
modified Einstein equations into purely spacelike hypersurfaces, which implies
the field equations for the induced dynamical three-metric. We also obtain the
Hamilton-Jacobi equations of motion for the canonical variables of the theory
based on its Hamiltonian, which was derived in a previous work. The
computations show that the dynamical field equations obtained from the
Lagrangian and Hamiltonian approaches are consistent with each other.
Connections to Brans-Dicke theory and ghost-free massive gravity are
established.

###### Keywords:

Diffeomorphism violation, Modified theories of gravity, Classical differential
geometry

##  1 Introduction

The phenomena of perihelion advance of Mercury Einstein:1915bz , light
deflection Dyson:1920cwa , and gravitational redshift Pound:1960zz  compose
the three classical tests of General Relativity (GR) and successfully confirm
the latter as the correct gravity theory. Further significant validations of
GR are the detection of the geodetic and frame-dragging effects by Gravity
Probe B Everitt:2011hp , the first direct observation of gravitational waves
by LIGO LIGOScientific:2016aoc , and the image of the accretion disk of the
supermassive central black hole in the elliptic galaxy M87 recently taken by
the Event Horizon Telescope EventHorizonTelescope:2019dse . Apart from
linearized GR exhibiting a propagating massless spin-2 excitation,
diffeomorphism invariance is at the heart of Einstein’s theory. This symmetry
encodes the profound property of GR being invariant under nonlinear,
differentiable maps from a spacetime manifold onto itself. The field equations
of GR are also covariant under general coordinate transformations, but the
latter only operate on the atlas of a manifold as opposed to points, sets, and
curves on the manifold proper. The geometry of a curved spacetime manifold is
locally equivalent to the pseudo-Euclidean geometry of Minkowski spacetime,
whose symmetry group is the Poincaré group. Our focus is on diffeomorphisms
instead of the local symmetry structure of GR.

Any departure from diffeomorphism invariance in a modified-gravity theory is
expected to have significant theoretical and observational implications. In
particular, the constraint algebra Gambini:1996  and Dirac observables
Giesel:2008zz  of GR are likely to be modified. The exact physical
consequences of such deviations are challenging to identify and extremely
difficult to measure directly, as manipulations of the spacetime manifold are
beyond the capabilities of presently conceivable experiments. However,
parameterizing explicit diffeomorphism violation via nondynamical background
fields, which have nonstandard transformation properties, can provide a window
into probing associated phenomena.

In principle, diffeomorphism violation is as old as GR itself. While his
theory was still in the making, Einstein required that the determinant of the
spacetime metric correspond to unity, which led to vast simplifications of his
field equations Einstein:1916 . This condition is also incorporated in
unimodular gravity Buchmuller:1988wx ; Unruh:1988in ; Bufalo:2015wda , which
is not invariant under the full diffeomorphism group of GR, but only under
volume-preserving diffeomorphisms. Unimodular gravity corresponds to GR with
an arbitrary cosmological constant, although the latter arises as an
integration constant instead of a parameter introduced at the level of the
action Ng:1990xz ; Finkelstein:2000pg .

Despite the beauty of Einstein’s gravity theory and its astounding
achievements, alterations are expected when the latter is to be reconciled
with quantum theory. Modifications of the gravitational laws of physics have
been found to occur naturally at the Planck scale, where quantum effects are
expected to become dominant. Such settings are provided by string theory,
branes, and supergravity. For example, diffeomorphism violation in gravity can
arise from a spontaneous breaking of Lorentz invariance in particular string
field theories Kostelecky-Samuel . The potential mechanism gives rise to
problems related to compactified dimensions in the context of inflation,
suggesting the generation of mass terms for some metric components Phen1 ;
Phen2 . Recently, other diffeomorphism-violating gravity models were shown to
involve such mass terms Review1 ; Review2 .

Generically, both spontaneous and explicit diffeomorphism symmetry breaking
are parameterized within the comprehensive effective-field theory framework
known as the gravitational Standard-Model Extension (SME) Kostelecky:2003fs ;
Kostelecky:2020hbb . Searches for diffeomorphism violation — let it be
spontaneous or explicit — can constrain SME coefficients. The direct detection
of gravitational waves by LIGO LIGOScientific:2016aoc  ushered in the new era
of gravitational-wave astronomy. The latter constitutes one method to probe
deviations from GR, which has gained vast prominence and has been providing an
ever-increasing number of SME constraints Kostelecky:2016kfm ; Mewes:2019dhj ;
Shao:2020shv ; Wang:2021ctl ; ONeal-Ault:2021uwu ; Niu:2022yhr . The absence
of energy losses of particles by gravitational Cherenkov radiation
Moore:2001bv ; Kostelecky:2015dpa  provides an alternative possibility of
bounding SME coefficients. Some experimental constraints on background field
coefficients such as those obtained in, e.g., refs. Kostelecky:2021tdf ;
Ivanov:2021bvk  are interpreted as bounds on explicit diffeomorphism
violation. The data tables Kostelecky:2008bfz  can be consulted as an
extensive compilation of such constraints.

Diffeomorphism invariance is also violated explicitly in particular models of
massive gravity. An extension of GR implementing the propagation of a massive
spin-2 excitation is known as massive gravity and its construction has
received significant attention during the past years. The action initially
introduced by Fierz and Pauli FP  is ghost-free at the linearized level, but
does not agree with GR in the massless limit. The Vainshtein mechanism
Vainshtein:1972sx  solves this problem by taking into account nonlinear
effects. Unfortunately, with that settled, another severe issue was revealed
by Boulware and Deser: the occurrence of a ghost mode at the nonlinear level
at an unacceptably small scale BD .

Around ten years ago, the construction of de Rham, Gabadadze, and Tolley
(dRGT) deRham:2010kj  led to a breakthrough, since it avoids the Boulware-
Deser (BD) ghost in the decoupling limit to all orders. This realm of research
has uncovered many interesting aspects such as its interplay with the physics
of diffeomorphism violation Hassan ; Mattingly ; Kostelecky:2021xhb , extra
dimensions DGP1 ; DGP2 , the Stückelberg trick deRham:2011rn ; StuckHassan ;
Siegel ; ADMStuck , linearized FP ; Bergshoeff:2012ud  vs. nonlinear massive
gravity BD ; Vainshtein:1972sx ; deRham:2010kj ; Nonlinear1 ; Nonlinear2 ;
deRham:2011ca ; Hassan:2011vm , the helicity decomposition of the metric
perturbation deRham:2011qq , and the Ostrogradsky formalism as well as ghost
states Stelle ; BD ; Hassan:2011ea . It is worthwhile to consult the valuable
review deRham:2014zqa  for this research area.

The motivations for constructing gravity theories beyond GR are multifold.
Low-energy extensions of Einstein’s gravity are basically motivated by
cosmological observations. Understanding the late-time acceleration of the
Universe and inflation as well as the search for dark matter has inspired
various modifications of GR Buchdahl:1970ynr ; Starobinsky:1980te ; f(R) ;
DeFelice:2010aj ; Horn ; Galileons . Some alternatives such as Brans-Dicke
theory Brans:1961sx  intend to incorporate Mach’s principle into the
description of gravity. The latter was one of Einstein’s motivations for
developing his GR, but it seems that he abandoned this goal later on. Hořava-
Lifshitz gravity Horava:2008ih ; Horava:2009uw  is formulated to improve the
UV behavior of the graviton propagator, while the theory approaches Einstein’s
gravity in the IR regime. Another interesting model is cardinal gravity
Kostelecky:2009zr ; Seifert:2019kuz  that interprets long-range gravitational
forces as propagating Nambu-Goldstone modes arising from spontaneous symmetry
breaking. In general, some extensions of GR introduce operators with higher-
order derivatives LW-cosmology ; DHOST1 ; DHOST2 , others involve functions of
the Ricci curvature scalar Buchdahl:1970ynr ; Starobinsky:1980te ; f(R) ;
DeFelice:2010aj  or additional vector and tensor fields coupling to spacetime
curvature Gripaios:2004ms ; Bluhm:2004ep ; Bluhm:2007bd ; Seifert:2009vr ;
Colladay:2019lig ; Seifert:2019xan ; Bonder:2021gjo ; Delhom:2022 . Stating an
exhaustive list of references on extensions or modifications of Einstein’s
relativity lies beyond our scope. References Will:2014kxa ; Heisenberg:2018vsk
; Tasson:2016xib  provide elaborate reviews on modified-gravity theories and
experimental tests where ref. Tasson:2016xib  focuses on the SME, in
particular.

In light of the previously mentioned lines of research, many modified-gravity
theories exhibit diffeomorphism violation. While spontaneous symmetry
violation such as in Refs. Gripaios:2004ms ; Bluhm:2004ep ; Bluhm:2007bd ;
Seifert:2009vr ; Colladay:2019lig ; Seifert:2019xan ; Bonder:2021gjo ;
Delhom:2022  is known to be benign Bluhm:2014oua , the question may arise
whether or not explicit diffeomorphism violation implies internal
inconsistencies and under which circumstances. This paper intends to answer
that question for a particular class of modified-gravity theories. Its action
involves coordinate scalars beyond GR being noninvariant under
diffeomorphisms. Symmetry violation is parameterized by a nondynamical scalar
background field denoted as $u$ and a tensor-valued one called $s^{\mu\nu}$,
respectively. The latter settings and extensions thereof are part of the
gravitational SME Kostelecky:2003fs ; Kostelecky:2020hbb  mentioned before and
have been subject to phenomenological Bailey:2006fd ; Bailey:2009me ;
Tso:2011up ; Bailey:2013oda ; Bonder:2020fpn  as well as theoretical studies
Bonder:2015maa . As these alterations of GR are quite generic, relationships
to specific modified-gravity theories in the literature are expected. In
particular, we will refer to Brans-Dicke theory and dRGT massive gravity.

In the recent work Reyes:2021cpx , the formalism developed by Arnowitt, Deser,
and Misner (ADM) Arnowitt:1962hi ; Misner:1973 ; Hanson:1976 ; Poisson:2002 ;
Poisson:2004 ; Gourgoulhon:2007ue ; Arnowitt:2008 ; Gourgoulhon:2012  was
applied to the class of modified-gravity theories characterized by $u$ and
$s^{\mu\nu}$. The calculations showed that the background fields imply an
altered constraint structure in the Hamiltonian formulation. Incorporating an
extension of the Gibbons-Hawking-York (GHY) boundary term York:1972sj ;
Gibbons:1976ue  to eliminate additional second-order time derivatives of the
metric, which occur in the Lagrangian and couple to the background fields, was
an essential ingredient for constructing the Hamiltonian. Our objective in the
present article is to understand the dynamics of this theory better. We will
be working both within the covariant and the Hamiltonian formulation. We also
note in passing that the authors of ref. ONeal-Ault:2020ebv  investigate a
similar, though not equivalent, theory. Their interest and approach differ
from ours in certain aspects.

Our paper is organized as follows. In section 2 we recapitulate the modified-
gravity theory that forms the foundation of ref. Reyes:2021cpx  and define the
basic variables employed in the ADM decomposition. Furthermore, we emphasize
correspondences between the latter setting and Brans-Dicke theory and dRGT
massive gravity. In section 3, the canonical-momentum variables and the
Hamiltonians are compiled from ref. Reyes:2021cpx . Our new investigations
start in section 4 where the modified Einstein equations are projected into
purely spacelike hypersurfaces of the spacetime foliation. Section 5 is
dedicated to the Hamilton-Jacobi equations of motion. Our essential finding
here is that the latter correspond to the projected field equations in the
covariant formulation. Finally, via section 6 we would like to motivate
further SME-related studies along the same line by demonstrating which SME
background fields play a role in linearized dRGT theory and Hořava-Lifshitz
gravity. Our findings are concluded on in section 7. We employ natural units
with $c=1$ unless otherwise stated. Our metric signature is $(-,+,+,+)$. Greek
indices are spacetime indices, whereas Latin indices describe quantities
living in purely spacelike hypersurfaces of the ADM decomposition. The
Mathematica package xTensor xTensor:2020  provides computational support.

##  2 Modified gravity with background fields

We focus on the following modified Einstein-Hilbert (EH) action
Kostelecky:2003fs ; Kostelecky:2020hbb  without a cosmological constant:

$S=\int_{\mathcal{M}}\mathrm{d}^{4}x\frac{\sqrt{-g}}{2\kappa}\left[(1-u){}^{(4)}R+s^{\mu\nu}{}^{(4)}R_{\mu\nu}\right]\,,$
(1)

with $\kappa=8\pi G_{N}$, the Ricci tensor ${}^{(4)}R_{\mu\nu}$ and the
associated Ricci scalar ${}^{(4)}R:={}^{(4)}R^{\mu}_{\phantom{\mu}\mu}$ of the
four-dimensional spacetime manifold $\mathcal{M}$ with metric tensor
$g_{\mu\nu}$ and $g:=\det(g_{\mu\nu})$. Furthermore, $u=u(x)$ and
$s^{\mu\nu}=s^{\mu\nu}(x)$ are nondynamical background fields having a generic
spacetime dependence. The latter are considered to have zero fluctuations:
$\delta u=0$ and $\delta s^{\mu\nu}=0$. All fields are defined in the tangent
bundle of $\mathcal{M}$. The modified Einstein equations for eq. (1) were
first obtained in ref. Bailey:2006fd  and can be read from eqs. (6), (7) of
the latter reference:

$\displaystyle 0$
$\displaystyle=(1-u){}^{(4)}G^{\mu\nu}+\frac{1}{2}(\nabla^{\mu}\nabla^{\nu}u+\nabla^{\nu}\nabla^{\mu}u)-g^{\mu\nu}\square
u$
$\displaystyle\phantom{{}={}}-\frac{1}{2}\left(s^{\alpha\beta}{}^{(4)}R_{\alpha\beta}g^{\mu\nu}+\nabla_{\alpha}\nabla^{\mu}s^{\alpha\nu}+\nabla_{\alpha}\nabla^{\nu}s^{\alpha\mu}-\square
s^{\mu\nu}-g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}s^{\alpha\beta}\right)\,,$
(2)

where ${}^{(4)}G^{\mu\nu}$ is the Einstein tensor. Furthermore, $\nabla_{\mu}$
is the covariant derivative on the spacetime manifold $\mathcal{M}$ and
$\square:=\nabla_{\mu}\nabla^{\mu}$ the d’Alembertian. The Bianchi identities
of Riemannian geometry imply nontrivial restrictions on the form of the
background fields $u$ and $s^{\mu\nu}$, which is a well-known feature of this
theory that has been indicated in a series of papers such as refs.
Kostelecky:2003fs ; Kostelecky:2020hbb ; Bluhm:2014oua , amongst others. We
intend to study the consequences of these restrictions in a future paper, but
our current focus is on the purely dynamical aspects of eq. (1).

We use the ADM formalism expressed in terms of the configuration space
variables $X:=\\{N,N^{i},q_{ij}\\}$. Here, $N$ is the lapse function, $N^{i}$
are the components of the shift vector, and $q_{ij}$ are the components of the
induced metric on a spacelike hypersurface $\Sigma_{t}$ Arnowitt:1962hi ;
Misner:1973 ; Hanson:1976 ; Poisson:2002 ; Poisson:2004 ; Gourgoulhon:2007ue ;
Arnowitt:2008 ; Gourgoulhon:2012 . It then holds that $\sqrt{-g}=N\sqrt{q}$
with $q:=\det(q_{ij})$ and

$g_{00}=-N^{2}+q^{ij}N_{i}N_{j}\,,\quad g_{0i}=N_{i}\,,\quad g_{ij}=q_{ij}\,.$
(3)

Furthermore, the components of the contravariant metric amount to

$g^{00}=-\frac{1}{N^{2}}\,,\quad g^{0i}=\frac{N^{i}}{N^{2}}\,,\quad
g^{ij}=q^{ij}-\frac{N^{i}N^{j}}{N^{2}}\,.$ (4)

Since we will have to study projections of the modified Einstein equations
into $\Sigma_{t}$, recall also the following decomposition formula for the
Ricci tensor and scalar curvature that arise from the Gauss-Codazzi equation:

$\displaystyle
q^{i}_{\phantom{i}\alpha}q^{j}_{\phantom{j}\beta}{}^{(4)}R^{\alpha\beta}$
$\displaystyle=R^{ij}+\frac{1}{N}q^{ia}q^{jb}\mathcal{L}_{m}K_{ab}-\frac{1}{N}D^{i}D^{j}N+KK^{ij}-2K^{ik}K_{k}^{\phantom{k}j}$
$\displaystyle=\frac{1}{N}q^{ia}q^{jb}\mathcal{L}_{m}K_{ab}-(D^{i}a^{j}+a^{i}a^{j})+R^{ij}+KK^{ij}-2K^{ik}K_{k}^{\phantom{k}j}\,,$
(5a) $\displaystyle{}^{(4)}R$
$\displaystyle=R+K^{2}+K_{ij}K^{ij}-\frac{2}{N}D_{i}D^{i}N+\frac{2}{N}\mathcal{L}_{m}K\,,$
(5b)

in terms of the extrinsic curvature $K_{ab}:=D_{a}n_{b}$ with
$n_{\mu}=(-N,0,0,0)$, the Ricci tensor $R_{ij}$, and the Ricci scalar $R$ in a
spacelike hypersurface $\Sigma_{t}$. Furthermore, $\mathcal{L}_{m}$ denotes
the Lie derivative Carroll:1997ar  along the four-vector $m^{\mu}:=Nn^{\mu}$
with $n^{\mu}=(1/N,-N^{i}/N)$ and $a_{i}:=D_{i}\ln(N)$ is known as the
acceleration. Here, $D_{i}$ is the covariant derivative in $\Sigma_{t}$, which
is compatible with the induced metric $q_{ij}$.

The background field $s^{\mu\nu}$ is tensor-valued and decomposes into three
distinct sectors according to the foliation of the spacetime manifold in terms
of the hypersurfaces $\Sigma_{t}$. The following identity is valuable for
performing this decomposition:

$s^{\alpha\beta}=q^{\alpha}_{\phantom{\alpha}\mu}q^{\beta}_{\phantom{\beta}\nu}s^{\mu\nu}-(q^{\alpha}_{\phantom{\alpha}\nu}n^{\beta}+q^{\beta}_{\phantom{\beta}\nu}n^{\alpha})s^{\nu\mathbf{n}}+n^{\alpha}n^{\beta}s^{\mathbf{nn}}\,,$
(6)

where $q^{\mu}_{\phantom{\mu}\nu}$ projects a tensor or a part of it into
$\Sigma_{t}$. Here we define
$s^{ij}:=q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}s^{\mu\nu}$ as the purely
spacelike sector of $s^{\mu\nu}$ that lives in $\Sigma_{t}$ entirely.
Furthermore, let $s^{i\mathbf{n}}:=q^{i}_{\phantom{i}\mu}n_{\nu}s^{\mu\nu}$ be
the vector-valued piece and
$s^{\mathbf{n}\mathbf{n}}:=n_{\mu}n_{\nu}s^{\mu\nu}$ the scalar part.

Now, the total ADM-decomposed action based on eq. (1) reads Reyes:2021cpx

$S=\int_{\mathcal{M}}\mathrm{d}t\mathrm{d}^{3}x\,\mathcal{L}_{\mathrm{ADM}}\,,\quad\mathcal{L}_{\mathrm{ADM}}=\mathcal{L}^{(0)}+\mathcal{L}^{(u)}+\sum_{i=1,2,3}\mathcal{L}^{(s)}_{i}\,,$
(7a) where $\mathcal{L}^{(0)}$ is the ADM-decomposed EH Lagrange density,
$\mathcal{L}^{(0)}=\frac{N\sqrt{q}}{2\kappa}\left(\frac{2}{N}\mathcal{L}_{m}K-\frac{2}{N}D_{i}D^{i}N+R+K^{2}+K_{ij}K^{ij}\right)\,,$
(7b) and the modifications are $\displaystyle\mathcal{L}^{(u)}$
$\displaystyle=\frac{N\sqrt{q}}{2\kappa}\,\left[-u(R-K^{2}+K_{ij}K^{ij})+\frac{2}{N}(K\mathcal{L}_{m}u+uD_{i}D^{i}N)\right]\,,$
(7c) $\displaystyle\mathcal{L}^{(s)}_{1}$
$\displaystyle=\frac{N\sqrt{q}}{2\kappa}\left[-\frac{1}{N}(K_{ij}\mathcal{L}_{m}s^{ij}+s^{ij}D_{i}D_{j}N)+s^{ij}(R_{ij}-2K_{i}^{\phantom{i}l}K_{lj})\right]\,,$
(7d) $\displaystyle\mathcal{L}^{(s)}_{2}$
$\displaystyle=\frac{N\sqrt{q}}{2\kappa}\left[s^{\mathbf{nn}}\left(\frac{1}{N}D_{i}D^{i}N-K^{ij}K_{ij}+K^{2}\right)+\frac{1}{N}K\mathcal{L}_{m}s^{\mathbf{nn}}\right]\,,$
(7e) $\displaystyle\mathcal{L}^{(s)}_{3}$
$\displaystyle=\frac{N\sqrt{q}}{2\kappa}\left[2s^{i\mathbf{n}}(D_{i}K-D_{l}K^{l}_{\phantom{l}i})\right]\,.$
(7f)

By defining new shift vector components as
$\tilde{N}^{i}:=N^{i}-Ns^{i\mathbf{n}}$, the Lagrange density
$\mathcal{L}_{3}^{(s)}$, which involves the vector-valued coefficients
$s^{i\mathbf{n}}$ only, can be reproduced at first order in $s^{i\mathbf{n}}$
from the ADM decomposition of the EH Lagrange density. This reason is a strong
argument for $s^{i\mathbf{n}}$ corresponding to mere gauge degrees of freedom
Reyes:2021cpx . Thus, they will be completely discarded in the remainder of
the paper. Then, we also take into account an extended GHY boundary term
Reyes:2021cpx  of the form

$S_{\begin{subarray}{c}\text{ext}\\\
\text{GHY}\end{subarray}}=\frac{\varepsilon}{2\kappa}\oint_{\partial\mathcal{M}}\mathrm{d}^{3}y\,\sqrt{q}\,\left[2(1-u)K-s^{\mathbf{nn}}K+K_{ij}s^{ij}\right]\,,$
(8)

where the parameter $\varepsilon=\mp 1$ for a spacelike (timelike) boundary
$\partial\mathcal{M}$ of the spacetime manifold $\mathcal{M}$ and the integral
runs over the coordinates $y^{a}$ defined on this boundary. Last but not
least, we introduce a second boundary term for $u$ and $s^{\mathbf{nn}}$ that
is of plainly different nature compared to that of eq. (8):

$S_{\partial\Sigma}=-\frac{1}{2\kappa}\oint_{\partial\Sigma_{t}}\mathrm{d}^{2}z\,\sqrt{q}r_{l}\left[ND^{l}(2u+s^{\mathbf{nn}})\right]\,,$
(9)

with the coordinates $z^{a}$ given on the boundary of a spacelike hypersurface
$\Sigma_{t}$ and a suitably normalized vector $r_{l}$ orthogonal to the
boundary. In earnest, such boundary terms can emerge when integrations by
parts are performed over $\Sigma_{t}$ while computing the Hamiltonian from the
corresponding Lagrangian via a Legendre transformation Reyes:2021cpx . The
purpose of these boundary contributions will become clear later on.

Any modified-gravity theory that is formulated at the level of an action is
expected to have a corresponding background field in the comprehensive
parameterization of the gravitational SME recently developed in ref.
Kostelecky:2020hbb . However, it may be challenging to establish such a
correspondence for a specific model under consideration. In the following, a
number of popular examples are provided, which shall serve as a motivation for
developing a better understanding of nondynamical background fields in the
gravitational SME.

###  2.1 Brans-Dicke and $f(R)$ theories

An early motivation for modifying Einstein’s GR was to incorporate Mach’s
principle. One among many possibilities of formulating the latter is via a
gravitational “constant” that is promoted to a dynamical field with an
explicit dependence on the spacetime coordinates Bondi:1996md . In the
formulation by Brans and Dicke Brans:1961sx  the inverse of the gravitational
constant is replaced by a scalar field $\phi$, whereupon the modified field
equations are based on the variation of a modified action as follows:

$0=\delta\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{-g}\left(\phi R+16\pi
L_{m}-\omega\frac{(\nabla\phi)^{2}}{\phi}\right)\,.$ (10)

Here, $\omega$ is a dimensionless free parameter and $L_{m}$ contains the
matter fields. The action can then be cast into the form

$S_{\mathrm{BD}}=\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{-g}\left(\frac{G_{N}}{2\kappa}\phi
R+L_{m}-\frac{\omega G_{N}}{2\kappa}\frac{(\nabla\phi)^{2}}{\phi}\right)\,,$
(11)

where $[G_{N}]=[\kappa]=-2$ and $[\phi]=[R]=2$. We identify

$G_{N}\phi=1-u\,,$ (12)

with the scalar background field $u$ in $\mathcal{L}^{(u)}$ of eq. (7c). A
reformulation of the dynamical term in eq. (11) amounts to

$S_{\mathrm{BD}}\supset\int_{\mathcal{M}}\mathrm{d}^{4}x\,\sqrt{-g}\left(-\frac{\omega
G_{N}}{2\kappa}\frac{(\nabla\phi)^{2}}{\phi}\right)=-\int_{\mathcal{M}}\mathrm{d}^{4}x\,\frac{\sqrt{-g}}{2\kappa}\frac{(\nabla
u)^{2}}{1-u}\,.$ (13)

Now, the modified field equations Brans:1961sx  following from eq. (11) read

$\displaystyle G_{N}\phi G_{\mu\nu}$ $\displaystyle=8\pi
G_{N}T_{\mu\nu}+\frac{\omega}{G_{N}\phi}\left[\nabla_{\mu}(G_{N}\phi)\nabla_{\nu}(G_{N}\phi)-\frac{1}{2}g_{\mu\nu}(\nabla
G_{N}\phi)^{2}\right]$
$\displaystyle\phantom{{}={}}+\nabla_{\mu}\nabla_{\nu}G_{N}\phi-
g_{\mu\nu}\square G_{N}\phi\,,$ (14)

and expressing $\phi$ in terms of $u$ according to eq. (12) implies

$\displaystyle(1-u)G_{\mu\nu}$ $\displaystyle=8\pi
G_{N}T_{\mu\nu}+\frac{\omega}{1-u}\left[\nabla_{\mu}u\nabla_{\nu}u-\frac{1}{2}g_{\mu\nu}(\nabla
u)^{2}\right]-\nabla_{\mu}\nabla_{\nu}u+g_{\mu\nu}\square u\,.$ (15)

Dropping the energy-momentum tensor as well as the dynamical contributions
proportional to $\omega$ leads to the modified Einstein equations for the $u$
term. They are deduced from eq. (2) by setting $s^{\mu\nu}=0$ and will be
recalled in eq. (24) to come. Thus, Brans-Dicke theory without the dynamical
term can be identified with the nondynamical $u$ sector of the gravitational
SME. To the best of our knowledge, this correspondence has not been stated in
the literature, so far.

Another observation is that the Lagrange density $\mathcal{L}^{(u)}$ of eq.
(7c) corresponds, in principle, to a particular $f(R)$ theory involving a
nondynamical background field such that $f(R)=(1-u)R$. It is not difficult to
check that the field equations quoted in eq. (24) are equivalent to eqs.
(1.1), (A4) of ref. Buchdahl:1970ynr  with the energy-momentum tensor
$T_{\mu\nu}=0$.

###  2.2 dRGT massive gravity

As GR, massive-gravity theories constructed before the seminal work
deRham:2010kj  did not involve canonical momenta associated with the lapse
function and the shift vector, which gives rise to 4 primary first-class
constraints. Thus, counting the number of physical degrees of freedom
Henneaux:1992  amounts to 6, which is one more than expected for a massive
graviton. The additional degree of freedom corresponds to the BD ghost mode
propagating at energy scales proportional to the minuscule graviton mass,
which is unacceptable. dRGT theory emerged in an attempt to conceive a
massive-gravity theory that is devoid of the BD ghost. To do so, the Einstein-
Hilbert term is supplemented by a particular potential of the graviton that
has a highly peculiar form. The theory is constructed in a manner to provide
another primary and an additional secondary constraint Hassan:2011ea  on the
three-metric components. Thereupon, a counting of the physical degrees of
freedom indicates that the ghost mode is eliminated.

The potential of dRGT theory involves traces of coordinate two-tensors
$\mathds{X}^{\mu}_{\phantom{\mu}\nu}:=\sqrt{g^{\mu\alpha}f_{\alpha\nu}}$
expressed in terms of the dynamical spacetime metric $g_{\mu\nu}$ and a
nondynamical reference metric, which is frequently denoted as $f_{\mu\nu}$
deRham:2014zqa . At the level of the pure-gravity action of dRGT theory it is
challenging to identify the nondynamical background $f_{\mu\nu}$ or
contractions thereof with SME coefficients. After all, the potential is not a
simple sum of contributions involving $f_{\mu\nu}$ contracted with dynamical
fields or curvature components in a linear fashion. The two-tensor
$\mathds{X}^{\mu}_{\phantom{\mu}\nu}$ cannot be evaluated generically, but
must be computed based on particular choices of the dynamical metric and the
background field.

Finding an appropriate map in the matter sector is plainly simpler. Coupling a
particular set of matter fields to the gravity sector via both metrics
$g_{\mu\nu}$ and $f_{\mu\nu}$ was demonstrated to reintroduce the BD ghost
deRham:2014naa . A reemergence of the ghost can only be avoided by coupling in
a more sophisticated way via an effective metric given by

$g_{\mu\nu}^{(\mathrm{eff})}=\alpha^{2}g_{\mu\nu}+2\alpha\beta
g_{\mu\alpha}\mathds{X}^{\alpha}_{\phantom{\alpha}\nu}+\beta^{2}f_{\mu\nu}\,,$
(16)

with arbitrary dimensionless parameters $\alpha,\beta$ that may be tuned to
either couple with the dynamical metric or the reference metric or both. Now,
a background vierbein $\overline{v}_{\mu}^{\phantom{\mu}a}$ is introduced such
that
$f_{\mu\nu}=\overline{v}_{\mu}^{\phantom{\mu}a}\overline{v}_{\nu}^{\phantom{\nu}b}\eta_{ab}$
with the Minkowski metric $\eta_{ab}$ in a local inertial frame. If the
dynamical vierbein $e_{\mu}^{\phantom{\mu}a}$ and the background vierbein
satisfy $e^{\mu}_{\phantom{\mu}a}\overline{v}_{\mu
b}=e^{\mu}_{\phantom{\mu}b}\overline{v}_{\mu a}$, it holds that
$\mathds{X}^{\mu}_{\phantom{\mu}\nu}=e^{\mu}_{\phantom{\mu}a}\overline{v}_{\nu}^{\phantom{\nu}a}=:\gamma^{\mu}_{\phantom{\mu}\nu}$
Hinterbichler:2012cn . In the context of the gravitational SME, a redefinition
of the (inverse) metric according to
$g^{\mu\nu}\simeq(1+u)\tilde{g}^{\mu\nu}+s^{\mu\nu}$ with a new metric
$\tilde{g}_{\mu\nu}$ can be carried out to remove $u$ and $s^{\mu\nu}$ from
the pure-gravity sector. Expressing the former metric $g_{\mu\nu}$ in terms of
the new metric $\tilde{g}_{\mu\nu}$ in the matter sector, introduces
appropriate SME coefficients in the matter sector. By taking $s^{\mu\nu}$ as
traceless, for the effective metric of eq. (16) it is possible to identify
Bluhm:2019ato

$u=-\frac{1}{2}\beta\gamma^{\sigma}_{\phantom{\sigma}\sigma}\,,\quad
s^{\mu\nu}=-2\beta\left(\gamma^{\mu\nu}-\frac{1}{4}\gamma^{\sigma}_{\phantom{\sigma}\sigma}g^{\mu\nu}\right)\,.$
(17)

Thus, the theory of eq. (7) has a direct link to matter-gravity couplings in
dRGT massive gravity.

##  3 Canonical momenta and Hamiltonians

We are now interested in understanding the dynamics of the modified-gravity
theory based on eq. (1). Both the dynamical equations of motion and the
constraints are encoded in the modified Einstein equations. In principle, each
constraint arises from eq. (2) by applying suitable normal and tangential
projections with respect to the hypersurfaces $\Sigma_{t}$ defined in the ADM
formalism.

We now consider the set of momenta $\Pi:=\\{\pi_{N},\pi_{i},\pi^{ij}\\}$
associated with the canonical variables. They are defined as

$\pi_{N}:=\frac{\partial\mathcal{L}_{\mathrm{ADM}}}{\partial\dot{N}}\,,\quad\pi_{i}:=\frac{\partial\mathcal{L}_{\mathrm{ADM}}}{\partial\dot{N}^{i}}\,,\quad\pi^{ij}:=\frac{\partial\mathcal{L}_{\mathrm{ADM}}}{\partial\dot{q}_{ij}}\,,$
(18)

where a dot stands for the time derivative. Since the Lagrangian of eq. (7)
does not involve velocities of the gauge variables, $\dot{N}$ and
$\dot{N}^{i}$, the canonical momentum conjugates to the lapse function and the
shift vector, respectively, are found to be weakly equal to zero:

$\pi_{N}\approx 0\,,\quad\pi_{i}\approx 0\,.$ (19)

Furthermore, we impose the following fundamental equal-time Poisson brackets
between the canonically conjugate variables (see ref. Bertschinger:2002 ):

$\displaystyle\\{N(\mathbf{x}),\pi_{N}(\mathbf{x}^{\prime})\\}$
$\displaystyle=\delta^{(3)}(\mathbf{x}-\mathbf{x}^{\prime})\,,$ (20a)
$\displaystyle\\{N^{j}(\mathbf{x}),\pi_{i}(\mathbf{x}^{\prime})\\}$
$\displaystyle=\delta_{i}^{\phantom{i}j}\delta^{(3)}(\mathbf{x}-\mathbf{x}^{\prime})\,,$
(20b) $\displaystyle\\{q_{kl}(\mathbf{x}),\pi^{ij}(\mathbf{x}^{\prime})\\}$
$\displaystyle=(\delta^{i}_{\phantom{i}k}\delta^{j}_{\phantom{j}l}-\delta^{i}_{\phantom{i}l}\delta^{j}_{\phantom{j}k})\delta^{(3)}(\mathbf{x}-\mathbf{x}^{\prime})\,,$
(20c)

where the Poisson bracket is defined as

$\\{Q(\mathbf{x}),P(\mathbf{x}^{\prime})\\}:=\int_{\Sigma_{t}}\mathrm{d}^{3}\tilde{x}\,\left[\frac{\delta
Q(\mathbf{x})}{\delta X_{i}(\tilde{\mathbf{x}},t)}\frac{\delta
P(\mathbf{x}^{\prime})}{\delta\Pi^{i}(\tilde{\mathbf{x}},t)}-\frac{\delta
Q(\mathbf{x})}{\delta\Pi^{i}(\tilde{\mathbf{x}},t)}\frac{\delta
P(\mathbf{x}^{\prime})}{\delta X_{i}(\tilde{\mathbf{x}},t)}\right]\,,$ (21)

with the functional derivatives $\delta/\delta X_{i}$, $\delta/\delta\Pi^{i}$
for the canonical variables and momenta, respectively.

The canonical Hamiltonian that results from the Legendre transformation of the
Lagrange density given in eq. (7) was obtained in ref. Reyes:2021cpx . A wise
approach is to explore the scalar $u$ as well as the scalar and purely
spacelike sectors of $s^{\mu\nu}$ separately. The canonical momenta for each
of these sectors then follow from eqs. (7b) – (7e):

$\displaystyle\pi^{ij}$
$\displaystyle:=\frac{\partial(\mathcal{L}^{(0)}+\mathcal{L}^{(u)})}{\partial\dot{q}_{ij}}=\frac{\sqrt{q}}{2\kappa}\left[(1-u)(K^{ij}-q^{ij}K)+\frac{1}{N}q^{ij}\mathcal{L}_{m}u\right]\,,$
(22a) $\displaystyle p^{ij}$
$\displaystyle:=\frac{\partial(\mathcal{L}^{(0)}+\mathcal{L}_{1}^{(s)})}{\partial\dot{q}_{ij}}=\frac{\sqrt{q}}{2\kappa}\left[(1-s^{\mathbf{nn}})(K^{ij}-q^{ij}K)+\frac{1}{2N}q^{ij}\mathcal{L}_{m}s^{\mathbf{nn}}\right]\,,$
(22b) $\displaystyle P^{ij}$
$\displaystyle:=\frac{\partial(\mathcal{L}^{(0)}+\mathcal{L}^{(s)}_{2})}{\partial\dot{q}_{ij}}=\frac{\sqrt{q}}{2\kappa}\left[K^{ij}-q^{ij}K-(s^{il}K_{l}^{\phantom{l}j}+s^{jl}K_{l}^{\phantom{l}i})-\frac{1}{2N}\mathcal{L}_{m}s^{ij}\right]\,.$
(22c)

For the $u$, $s^{\mathbf{nn}}$, and $s^{ij}$ sector each, the canonical
Hamiltonians have the form

$\displaystyle{H_{u}}$
$\displaystyle=\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left[-\frac{\sqrt{q}}{2\kappa}N\left((1-u)R+2D^{i}D_{i}u\right)+\frac{\mathcal{L}_{m}u}{1-u}\left(\pi-\frac{3}{4}\frac{\sqrt{q}}{\kappa
N}\mathcal{L}_{m}u\right)\right.$ $\displaystyle\phantom{{}={}}\hskip
39.83368pt\left.{}+\frac{2\kappa
N}{\sqrt{q}(1-u)}\left(\pi^{ij}\pi_{ij}-\frac{\pi^{2}}{2}\right)-2(D_{i}\pi^{ij})N_{j}\right]\,,$
(23a) $\displaystyle H_{1}$
$\displaystyle=\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left[-\frac{\sqrt{q}}{2\kappa}N(R+D^{i}D_{i}s^{\mathbf{nn}})+\frac{\mathcal{L}_{m}s^{\mathbf{nn}}}{2(1-s^{\mathbf{nn}})}\left(p-\frac{3}{8}\frac{\sqrt{q}}{\kappa
N}\mathcal{L}_{m}s^{\mathbf{nn}}\right)\right.$
$\displaystyle\phantom{{}={}}\hskip 39.83368pt\left.{}+\frac{2\kappa
N}{\sqrt{q}(1-s^{\mathbf{nn}})}\left(p^{ij}p_{ij}-\frac{p^{2}}{2}\right)-2(D_{i}p^{ij})N_{j}\right]\,,$
(23b) $\displaystyle H_{2}$
$\displaystyle=\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\bigg{[}-\frac{\sqrt{q}}{2\kappa}N\left(R+s^{ij}R_{ij}-D_{j}D_{i}s^{ij}\right)+\left(P_{ij}-\frac{P}{2}q_{ij}\right)\mathcal{L}_{m}s^{ij}$
$\displaystyle\phantom{{}={}}\hskip 42.67912pt+\frac{2\kappa
N}{\sqrt{q}}\left(P^{ij}P_{ij}-(1-s^{i}_{\phantom{i}i})\frac{P^{2}}{2}-2s^{ij}(P_{ij}P-P_{i}^{\phantom{i}k}P_{kj})\right)$
$\displaystyle\phantom{{}={}}\hskip
42.67912pt-2(D_{i}P^{ij})N_{j}\bigg{]}+\mathcal{O}[(s^{ij})^{2}]\,.$ (23c)

Note that $H_{2}$ is valid at first order in the coefficients $s^{ij}$. Since
the latter parameterize a modification of GR, the coefficients are expected to
be $\ll 1$ — at least in regions with weak gravitational fields. Studying
contributions to $H_{2}$ that are of higher order in $s^{ij}$ may be
worthwhile, but it is very reasonable to understand the properties of this
modification at leading order first.

##  4 Field equations in the hypersurface

Coordinate invariance allows us to work in coordinates with a nontrivial lapse
function, $N=N(t,\mathbf{x})$, and a zero shift vector, $N^{i}=0$, without a
loss of generality. In the following, we will focus on the $u$ sector first
and set $s^{\mu\nu}=0$. Later on, we will be studying the complementary
$s^{\mu\nu}$ sector with $u=0$. Now, the modified Einstein equations for a
nonzero $u$ read Bailey:2006fd

$Q^{\mu\nu}:=(1-u){}^{(4)}G^{\mu\nu}+\nabla^{\mu}\nabla^{\nu}u-g^{\mu\nu}\square
u=0\,.$ (24)

Their projection into the hypersurface $\Sigma_{t}$ is obtained from

$(1-u)q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}{}^{(4)}G^{\mu\nu}+q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}\nabla^{\mu}\nabla^{\nu}u-q^{ij}\Box
u=0\,.$ (25)

Carrying out the projection of the part proportional to the Einstein tensor is
straightforward and the result is readily found to be

$\displaystyle q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}{}^{(4)}G^{\mu\nu}$
$\displaystyle=R^{ij}-\frac{R}{2}q^{ij}+q^{ia}q^{jb}\frac{1}{N}\mathcal{L}_{m}K_{ab}-\frac{1}{N}D^{i}D^{j}N+KK^{ij}-2K^{ik}K_{k}^{\phantom{k}j}$
$\displaystyle\phantom{{}={}}-\frac{1}{2}q^{ij}\left(K^{2}+K_{kl}K^{kl}+\frac{2}{N}\mathcal{L}_{m}K-\frac{2}{N}D_{l}D^{l}N\right)\,.$
(26)

The covariant-derivative pieces are projected in the following way. Consider

$q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}\nabla^{\mu}\nabla^{\nu}u=D^{i}D^{j}u-\frac{1}{N}K^{ij}\mathcal{L}_{m}u\,,$
(27)

as well as

$\square
u=D^{2}u-\frac{1}{N}\left[K\mathcal{L}_{m}u+\mathcal{L}_{m}\left(\frac{1}{N}\mathcal{L}_{m}u\right)\right]+a^{\mu}\nabla_{\mu}u\,.$
(28)

We then intend to express the projected field equations in terms of the
canonical momentum density $\pi^{ij}$ that is given by eq. (22a). Inverting
the latter for the extrinsic-curvature tensor $K^{ij}$ gives

$K^{ij}=\frac{1}{1-u}\left[\frac{2\kappa}{\sqrt{q}}\left(\pi^{ij}-\frac{\pi}{2}q^{ij}\right)+\frac{q^{ij}}{2N}\mathcal{L}_{m}u\right]\,.$
(29)

Taking into consideration the previous ingredients, the left-hand side of the
modified Einstein equations completely projected into $\Sigma_{t}$ reads

$\displaystyle(\vec{\bm{q}}^{*}\mathbf{Q})^{ij}$
$\displaystyle=\frac{2\kappa}{N\sqrt{q}}\dot{\pi}^{ij}+(1-u)\left(R^{ij}-\frac{R}{2}q^{ij}\right)$
$\displaystyle\phantom{{}={}}+\frac{1}{N}\left(q^{ij}D_{k}D^{k}[(1-u)N]-D^{i}D^{j}[(1-u)N]\right)$
$\displaystyle\phantom{{}={}}+q^{ij}a^{k}D_{k}u-(a^{i}D^{j}u+a^{j}D^{i}u)$
$\displaystyle\phantom{{}={}}+\frac{4\kappa^{2}}{q(1-u)}\left[2\pi^{ik}\pi_{k}^{\phantom{k}j}-\pi\pi^{ij}-\frac{1}{2}\left(\pi_{kl}\pi^{kl}-\frac{\pi^{2}}{2}\right)q^{ij}\right]$
$\displaystyle\phantom{{}={}}+\frac{\mathcal{L}_{m}u}{N(1-u)}\left(\frac{2\kappa}{\sqrt{q}}\pi^{ij}-\frac{3}{4N}q^{ij}\mathcal{L}_{m}u\right)\,,$
(30a) with the time derivative of the canonical momentum stated in eq. (22a):
$\displaystyle\dot{\pi}^{ij}$
$\displaystyle=\frac{\sqrt{q}}{2\kappa}\bigg{[}N(1-u)K(3K^{ij}-q^{ij}K)+(1-u)(\mathcal{L}_{m}K^{ij}-q^{ij}\mathcal{L}_{m}K)$
$\displaystyle\phantom{{}={}}\hskip
22.76228pt+(2q^{ij}K-3K^{ij})\mathcal{L}_{m}u+q^{ij}\mathcal{L}_{m}\left(\frac{1}{N}\mathcal{L}_{m}u\right)\bigg{]}\,.$
(30b)

Here, we took over the convenient notation from refs. Gourgoulhon:2007ue ;
Gourgoulhon:2012  describing the projection of a generic spacetime tensor into
$\Sigma_{t}$:

$(\vec{\bm{q}}^{*}\mathbf{T})^{\alpha_{1}\dots\alpha_{s}}_{\phantom{\alpha_{1}\dots\alpha_{s}}\beta_{1}\dots\beta_{t}}:=q^{\alpha_{1}}_{\phantom{\alpha_{1}}\gamma_{1}}\dots
q^{\alpha_{s}}_{\phantom{\alpha_{s}}\gamma_{s}}q^{\delta_{1}}_{\phantom{\delta_{1}}\beta_{1}}\dots
q^{\delta_{t}}_{\phantom{\delta_{t}}\beta_{t}}T^{\gamma_{1}\dots\gamma_{s}}_{\phantom{\gamma_{1}\dots\gamma_{s}}\delta_{1}\dots\delta_{t}}\,.$
(31)

The next step is to consider the modified Einstein equations for the
$s^{\mu\nu}$ coefficients Bailey:2006fd :

$J^{\mu\nu}:={}^{(4)}G^{\mu\nu}-\frac{1}{2}s^{\alpha\beta}R_{\alpha\beta}g^{\mu\nu}-\frac{1}{2}\nabla_{\alpha}\nabla^{\mu}s^{\alpha\nu}-\frac{1}{2}\nabla_{\alpha}\nabla^{\nu}s^{\alpha\mu}+\frac{1}{2}\square
s^{\mu\nu}+\frac{1}{2}g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}s^{\alpha\beta}=0\,.$
(32)

The projection of the Einstein tensor is quickly obtained from eq. (4). As we
remarked before, we distinguish between the different sectors of $s^{\mu\nu}$
mentioned below eq. (6). Recall that the coefficients $s^{i\mathbf{n}}$ are
pure gauge degrees of freedom, which is why they were discarded right from the
start. To compute the projection of eq. (32) for the purely timelike sector
composed of the single scalar coefficient $s^{\mathbf{nn}}$, the following
results are indispensable:

$\displaystyle q^{ij}s^{\mathbf{nn}}R_{\mathbf{nn}}$
$\displaystyle=q^{ij}s^{\mathbf{nn}}\left(-\frac{1}{N}\mathcal{L}_{m}K+D_{l}a^{l}+a_{l}a^{l}-K^{kl}K_{kl}\right)\,,$
(33a) $\displaystyle
q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}\nabla_{\alpha}\nabla^{\mu}(s^{\mathbf{nn}}n^{\alpha}n^{\nu})$
$\displaystyle=(D^{i}s^{\mathbf{nn}})a^{j}-s^{\mathbf{nn}}a^{i}a^{j}+s^{\mathbf{nn}}(KK^{ij}-K^{ki}K^{j}_{\phantom{j}k})$
$\displaystyle\phantom{{}={}}+\frac{1}{N}(\mathcal{L}_{m}s^{\mathbf{nn}})K^{ij}+\frac{s^{\mathbf{nn}}}{N}q^{ik}q^{jl}\mathcal{L}_{m}K_{kl}\,,$
(33b) $\displaystyle
q^{i}_{\phantom{i}\alpha}q^{j}_{\phantom{j}\beta}\square(s^{\mathbf{nn}}n^{\mu}n^{\nu})$
$\displaystyle=2s^{\mathbf{nn}}(K^{ik}K^{j}_{\phantom{j}k}-a^{i}a^{j})\,,$
(33c) $\displaystyle
q^{i}_{\phantom{i}\mu}q^{j}_{\phantom{j}\nu}(g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}s^{\alpha\beta})$
$\displaystyle=q^{ij}\left[\frac{1}{N}\mathcal{L}_{m}\left(\frac{1}{N}\mathcal{L}_{m}s^{\mathbf{nn}}\right)+a^{k}D_{k}s^{\mathbf{nn}}+\frac{2}{N}(\mathcal{L}_{m}s^{\mathbf{nn}})K\right.$
$\displaystyle\phantom{{}={}}\hskip
14.22636pt\left.{}+s^{\mathbf{nn}}\left(\frac{1}{N}\mathcal{L}_{m}K+K^{2}\right)+s^{\mathbf{nn}}(D_{l}a^{l}+a_{l}a^{l})\right]\,.$
(33d)

Here, the extrinsic curvature expressed in terms of the canonical momentum
reads

$K^{ij}=\frac{1}{1-s^{\mathbf{nn}}}\left[\frac{2\kappa}{\sqrt{q}}\left(p^{ij}-\frac{p}{2}q^{ij}\right)+\frac{q^{ij}}{4N}\mathcal{L}_{m}s^{\mathbf{nn}}\right]\,.$
(34)

The projected field equations are simpler in comparison to those of the $u$
sector, as there is no term of the form $s^{\mathbf{nn}}R$ (cf. eqs. (7c),
(7c)). Now we are in a position to evaluate the left-hand side of eq. (32) for
$s^{i\mathbf{n}}=s^{ij}=0$ completely projected into $\Sigma_{t}$. It is
useful to define $(J_{1})^{\mu\nu}:=J^{\mu\nu}|_{s^{i\mathbf{n}}=s^{ij}=0}$
with $J^{\mu\nu}$ given in eq. (32). Then,

$\displaystyle(\vec{\bm{q}}^{*}\mathbf{J}_{1})^{ij}$
$\displaystyle=\frac{2\kappa}{N\sqrt{q}}\dot{p}^{ij}+R^{ij}-\frac{R}{2}q^{ij}+\frac{1}{N}(q^{ij}D_{k}D^{k}N-D^{i}D^{j}N)$
$\displaystyle\phantom{{}={}}+\frac{1}{2}\left[q^{ij}a^{k}D_{k}s^{\mathbf{nn}}-(a^{i}D^{j}s^{\mathbf{nn}}+a^{j}D^{i}s^{\mathbf{nn}})\right]$
$\displaystyle\phantom{{}={}}+\frac{4\kappa^{2}}{q(1-s^{\mathbf{nn}})}\left[2p^{ik}p_{k}^{\phantom{k}j}-pp^{ij}-\frac{1}{2}\left(p_{kl}p^{kl}-\frac{p^{2}}{2}\right)q^{ij}\right]$
$\displaystyle\phantom{{}={}}+\frac{\mathcal{L}_{m}s^{\mathbf{nn}}}{N(1-s^{\mathbf{nn}})}\left(\frac{\kappa}{\sqrt{q}}\pi^{ij}-\frac{3}{16N}q^{ij}\mathcal{L}_{m}s^{\mathbf{nn}}\right)\,,$
(35a) with the following time derivative of the canonical momentum of eq.
(22b): $\displaystyle\dot{p}^{ij}$
$\displaystyle=\frac{\sqrt{q}}{2\kappa}\bigg{[}N(1-s^{\mathbf{nn}})K(3K^{ij}-q^{ij}K)+(1-s^{\mathbf{nn}})(\mathcal{L}_{m}K^{ij}-q^{ij}\mathcal{L}_{m}K)$
$\displaystyle\phantom{{}={}}\hskip
19.91684pt+\left(\frac{3}{2}q^{ij}K-2K^{ij}\right)\mathcal{L}_{m}s^{\mathbf{nn}}+\frac{q^{ij}}{2}\mathcal{L}_{m}\left(\frac{1}{N}\mathcal{L}_{m}s^{\mathbf{nn}}\right)\bigg{]}\,.$
(35b)

Finally, we turn to the $s^{ij}$ sector. We need

$\displaystyle
q^{i}_{\phantom{i}\alpha}q^{j}_{\phantom{j}\beta}g^{\alpha\beta}s^{\mu\nu}R_{\mu\nu}$
$\displaystyle=q^{ij}s^{kl}\bigg{[}\frac{1}{N}\mathcal{L}_{m}K_{kl}-(D_{k}a_{l}+a_{k}a_{l})+R_{kl}$
$\displaystyle\phantom{{}={}}\hskip
31.2982pt+KK_{kl}-2K_{kn}K^{n}_{\phantom{n}l}\bigg{]}\,,$ (36a) $\displaystyle
q^{i}_{\phantom{i}\alpha}q^{j}_{\phantom{j}\beta}\nabla_{\nu}\nabla^{\alpha}(q^{\nu}_{\phantom{\nu}\gamma}q^{\beta}_{\phantom{\beta}\delta}s^{\gamma\delta})$
$\displaystyle=KK^{i}_{\phantom{i}k}s^{kj}+2K^{i}_{\phantom{i}k}K^{k}_{\phantom{k}l}s^{lj}+D_{k}D^{i}s^{kj}-a^{i}a_{k}s^{kj}+a_{k}D^{i}s^{kj}$
$\displaystyle\phantom{{}={}}+\frac{1}{N}(\mathcal{L}_{m}K^{il})q_{lk}s^{kj}+K^{i}_{\phantom{i}k}s^{kl}K_{l}^{\phantom{l}j}\,,$
(36b) $\displaystyle
q^{i}_{\phantom{i}\alpha}q^{j}_{\phantom{j}\beta}\square(q^{\alpha}_{\phantom{\alpha}\gamma}q^{\beta}_{\phantom{\beta}\delta}s^{\gamma\delta})$
$\displaystyle=D_{k}D^{k}s^{ij}-K\left(\frac{1}{N}\mathcal{L}_{m}s^{ij}+s^{ik}K_{k}^{\phantom{k}j}+s^{jk}K_{k}^{\phantom{k}i}\right)$
$\displaystyle\phantom{{}={}}+K^{i}_{\phantom{i}k}K^{k}_{\phantom{k}l}s^{lj}+K^{j}_{\phantom{j}k}K^{k}_{\phantom{k}l}s^{il}$
$\displaystyle\phantom{{}={}}-\bigg{[}\frac{1}{N}\mathcal{L}_{m}\left(\frac{1}{N}\mathcal{L}_{m}s^{ij}\right)+\frac{1}{N}\left(K_{k}^{\phantom{k}j}\mathcal{L}_{m}s^{ik}+K^{i}_{\phantom{i}k}\mathcal{L}_{m}s^{kj}\right)$
$\displaystyle\phantom{{}={}}\hskip
14.22636pt+\left(\frac{1}{N}\mathcal{L}_{m}s^{ik}+s^{il}K_{l}^{\phantom{l}k}+s^{kl}K_{l}^{\phantom{l}i}\right)K_{k}^{\phantom{k}j}$
$\displaystyle\phantom{{}={}}\hskip
14.22636pt+s^{ik}q_{kl}\left(\frac{1}{N}\mathcal{L}_{m}K^{lj}+2K^{lm}K_{m}^{\phantom{m}j}\right)$
$\displaystyle\phantom{{}={}}\hskip
14.22636pt+\left(\frac{1}{N}\mathcal{L}_{m}s^{jk}+s^{jl}K_{l}^{\phantom{l}k}+s^{kl}K_{l}^{\phantom{l}j}\right)K_{k}^{\phantom{k}i}$
$\displaystyle\phantom{{}={}}\hskip
14.22636pt+s^{jk}q_{kl}\left(\frac{1}{N}\mathcal{L}_{m}K^{li}+2K^{lm}K_{m}^{\phantom{m}i}\right)$
$\displaystyle\phantom{{}={}}\hskip 14.22636pt-
a_{k}D^{k}s^{ij}+a^{i}s^{jk}a_{k}+a^{j}s^{ik}a_{k}\bigg{]}\,,$ (36c)
$\displaystyle
q^{i}_{\phantom{i}\alpha}q^{j}_{\phantom{j}\beta}(g^{\alpha\beta}\nabla_{\mu}\nabla_{\nu}s^{\mu\nu})$
$\displaystyle=q^{ij}\bigg{[}\frac{1}{N}\mathcal{L}_{m}(K_{kl}s^{kl})+s^{kl}D_{k}a_{l}+2a_{k}D_{l}s^{kl}$
$\displaystyle\phantom{{}={}}\hskip
17.07182pt+a_{i}a_{j}s^{ij}+D_{k}D_{l}s^{kl}+KK_{kl}s^{kl}\bigg{]}\,.$ (36d)

Furthermore, the extrinsic-curvature tensor is given by

$\displaystyle K^{ij}$
$\displaystyle=\frac{2\kappa}{\sqrt{q}}\left(P^{ij}-\frac{1}{2}\left[(1-s^{k}_{\phantom{k}k})P+2s^{kl}P_{kl}\right]q^{ij}+s^{ik}P_{k}^{\phantom{k}j}+s^{jk}P_{k}^{\phantom{k}i}-s^{ij}P\right)$
$\displaystyle\phantom{{}={}}+\frac{1}{2N}\left(\mathcal{L}_{m}s^{ij}-\frac{q^{ij}}{2}q_{kl}\mathcal{L}_{m}s^{kl}\right)+\mathcal{O}[(s^{ij})^{2}]\,.$
(37)

Let $(J_{2})^{\mu\nu}:=J^{\mu\nu}|_{s^{\mathbf{nn}}=s^{i\mathbf{n}}=0}$ be the
left-hand side of the modified Einstein equations of eq. (32) for the purely
spacelike sector composed of the coefficients $s^{ij}$. Considering their
projection into $\Sigma_{t}$ and incorporating eq. (36) as well as expressing
the extrinsic curvature in terms of the canonical momentum according to eq.
(22c) results in

$\displaystyle(\vec{\bm{q}}^{*}\mathbf{J}_{2})^{ij}$
$\displaystyle=\frac{2\kappa}{N\sqrt{q}}\dot{P}^{ij}+R^{ij}-\frac{R}{2}q^{ij}+\frac{1}{N}(q^{ij}D_{k}D^{k}N-D^{i}D^{j}N)$
$\displaystyle\phantom{{}={}}+\frac{1}{2}\left[-q^{ij}(s^{kl}R_{kl}-D_{k}D_{l}s^{kl})-D_{k}D^{i}s^{kj}-D_{k}D^{j}s^{ki}+D_{k}D^{k}s^{ij}\right]$
$\displaystyle\phantom{{}={}}+\frac{1}{2}\left(2q^{ij}\left[a_{k}D_{l}s^{kl}+s^{kl}(D_{k}a_{l}+a_{k}a_{l})\right]-a_{k}(D^{i}s^{kj}+D^{j}s^{ki})+a_{k}D^{k}s^{ij}\right)$
$\displaystyle\phantom{{}={}}+\frac{\kappa^{2}}{q}\Big{(}\left[(1-s^{k}_{\phantom{k}k})P^{2}-2P^{kl}P_{kl}+4s^{kl}(PP_{kl}-P_{k}^{\phantom{k}m}P_{lm})\right]q^{ij}$
$\displaystyle\phantom{{}={}}\hskip
34.14322pt+8\bigg{[}P^{ik}P_{k}^{\phantom{k}j}+s^{ik}P_{kl}P^{lj}+s^{jk}P_{kl}P^{li}-s^{ik}PP_{k}^{\phantom{k}j}-s^{jk}PP_{k}^{\phantom{k}i}$
$\displaystyle\phantom{{}={}}\hskip
34.14322pt+s^{kl}(P^{i}_{\phantom{i}k}P_{l}^{\phantom{l}j}-P_{kl}P^{ij})\Big{]}+2P^{2}s^{ij}-4(1-s^{k}_{\phantom{k}k})PP^{ij}\Big{)}$
$\displaystyle\phantom{{}={}}+\frac{\kappa}{N\sqrt{q}}\left[2\left(P^{i}_{\phantom{i}k}\mathcal{L}_{m}s^{jk}+P^{j}_{\phantom{j}k}\mathcal{L}_{m}s^{ik}\right)-\left(P\mathcal{L}_{m}s^{ij}+P^{ij}q_{kl}\mathcal{L}_{m}s^{kl}\right)\right]\,,$
(38a) where the time derivative of the canonical momentum of eq. (22c) reads
$\displaystyle\dot{P}^{ij}$
$\displaystyle=\frac{\sqrt{q}}{2\kappa}\left[NK(3K^{ij}-q^{ij}K)-N(s^{il}K_{l}^{\phantom{l}j}+s^{jl}K_{l}^{\phantom{l}i})K+\mathcal{L}_{m}K^{ij}-(s^{il}\mathcal{L}_{m}K_{l}^{\phantom{l}j}+s^{jl}\mathcal{L}_{m}K_{l}^{\phantom{l}i})\right.$
$\displaystyle\phantom{{}={}}\hskip
17.07182pt\left.{}-q^{ij}\mathcal{L}_{m}K-\frac{K}{2}\mathcal{L}_{m}s^{ij}-(\mathcal{L}_{m}s^{il})K_{l}^{\phantom{l}j}-(\mathcal{L}_{m}s^{jl})K_{l}^{\phantom{l}i}-\frac{1}{2}\mathcal{L}_{m}\left(\frac{1}{N}\mathcal{L}_{m}s^{ij}\right)\right]\,.$
(38b)

With the latter result at hand, the modified Einstein equations for the scalar
sectors $u$ and $s^{\mathbf{nn}}$ as well as the purely spacelike sector
$s^{ij}$ have been successfully projected into spacelike hypersurfaces
$\Sigma_{t}$.

##  5 Hamilton-Jacobi equations of motion

With the projected altered Einstein equations of the modified-gravity theory
based on eq. (7) at our disposal, in this section we will be working within
the Hamiltonian formalism. The latter is based on the following set of field
equations that are of first order in time:

$\displaystyle\dot{q}_{ij}$ $\displaystyle=\\{q_{ij},H\\}\,,$ (39a)
$\displaystyle\dot{\pi}^{ij}$ $\displaystyle=\\{\pi^{ij},H\\}\,.$ (39b)

The first set of equations corresponds to a geometrical identity Gambini:1996
, whereas the second set describes the dynamics of the theory under
consideration Bertschinger:2002 . In the latter, we employ the canonical
Hamiltonians of the $u$, $s^{\mathbf{nn}}$, and $s^{ij}$ sectors given in eq.
(23). Our goal is to compare the projected Einstein equations stated in eqs.
(30), (35), and (38) to eq. (39b) evaluated for each of the canonical
Hamiltonians of eq. (23).

The canonical Hamiltonian associated with the $u$ sector is given by eq. (23).
First of all, we will investigate the first set of Hamilton’s equations in eq.
(39a), i.e.,

$\dot{q}_{ij}=\\{q_{ij},{H_{u}}\\}\,.$ (40)

Evaluating the Poisson bracket leads to

$\left\\{q_{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{1}{1-u}\left[\pi\mathcal{L}_{m}u+\frac{2\kappa
N}{\sqrt{q}}\left(\pi^{kl}\pi_{kl}-\frac{\pi^{2}}{2}\right)\right]\right\\}=2NK_{ij}\,,$
(41)

where we used the extrinsic-curvature tensor of eq. (29) given as a function
of the canonical momentum. After confirming the outcome expected for the
Hamiltonian describing the gravitational system, we devote ourselves to the
second set of Hamilton’s equations in eq. (39b):

$\displaystyle-\left\\{\pi^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\sqrt{q}}{2\kappa}N(1-u)R\right\\}$
$\displaystyle=-\frac{\sqrt{q}}{2\kappa}\Big{(}q^{ij}D_{l}D^{l}[(1-u)N]-D^{i}D^{j}[(1-u)N]$
$\displaystyle\phantom{{}={}}\hskip 31.2982pt+N(1-u)G^{ij}\Big{)}\,,$ (42a)
$\displaystyle\left\\{\pi^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{2\kappa
N}{\sqrt{q}(1-u)}\left(\pi^{ij}\pi_{ij}-\frac{\pi^{2}}{2}\right)\right\\}$
$\displaystyle=-\frac{2\kappa
N}{\sqrt{q}(1-u)}\bigg{[}2\pi^{ia}\pi^{j}_{\phantom{j}a}-\pi\pi^{ij}$
$\displaystyle\phantom{{}={}}\hskip
62.59596pt-\frac{1}{2}\left(\pi_{ab}\pi^{ab}-\frac{\pi^{2}}{2}\right)q^{ij}\bigg{]}\,,$
(42b)
$\displaystyle-\left\\{\pi^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{N\sqrt{q}}{\kappa}D^{i}D_{i}u\right\\}$
$\displaystyle=-\frac{N\sqrt{q}}{2\kappa}q^{ij}a_{l}D^{l}u\,,$ (42c)
$\displaystyle\left\\{\pi^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\mathcal{L}_{m}u}{1-u}\left(\pi-\frac{3}{4}\frac{\sqrt{q}}{\kappa
N}\mathcal{L}_{m}u\right)\right\\}$
$\displaystyle=-\frac{\mathcal{L}_{m}u}{1-u}\left(\pi^{ij}-\frac{3}{8}\frac{\sqrt{q}}{\kappa
N}q^{ij}\mathcal{L}_{m}u\right)\,,$ (42d)

with the Einstein tensor $G^{ij}$ in $\Sigma_{t}$. Organizing the individual
parts, a multiplication by $2\kappa/(N\sqrt{q})$ reproduces eq. (30) apart
from the last two terms in the third line, which depend on the acceleration.
The boundary term of eq. (9) solves this issue. The part proportional to $u$
is interpreted as an additional contribution to ${H_{u}}$ of eq. (23) that
contains a total derivative:

$\Delta_{u}:=-\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\sqrt{q}}{\kappa}D_{l}(ND^{l}u)=-\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{N\sqrt{q}}{\kappa}(D_{l}D^{l}u+a_{l}D^{l}u)\,.$
(43)

Its Poisson bracket with the canonical momentum density reads

$\\{\pi^{ij},\Delta_{u}\\}=\frac{N\sqrt{q}}{2\kappa}(a^{i}D^{j}u+a^{j}D^{i}u)\,,$
(44)

and provides the missing pieces to eq. (30).

Next, our focus is on the $s^{\mathbf{nn}}$ and $s^{ij}$ sectors. A
computation reveals that the first set of Hamilton’s equations in eq. (39a) is
satisfied for these sectors, too. Thus, for the Hamiltonians $H_{1}$ and
$H_{2}$ of eq. (23) and eq. (23), respectively, it holds that

$\displaystyle\dot{q}_{ij}$
$\displaystyle=\\{q_{ij},H_{1}\\}=\left\\{q_{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{1}{1-s^{\mathbf{nn}}}\left[\frac{p}{2}\mathcal{L}_{m}s^{\mathbf{nn}}+\frac{2\kappa
N}{\sqrt{q}}\left(p^{kl}p_{kl}-\frac{p^{2}}{2}\right)\right]\right\\}=2NK_{ij}\,,$
(45a) $\displaystyle\dot{q}_{ij}$
$\displaystyle=\\{q_{ij},H_{2}\\}=\left\\{q_{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left[\frac{2\kappa
N}{\sqrt{q}}\left(P^{kl}P_{kl}-(1-s^{k}_{\phantom{k}k})\frac{P^{2}}{2}-2s^{kl}(P_{kl}P-P_{k}^{\phantom{k}n}P_{nl})\right)\right.\right.$
$\displaystyle\phantom{{}={}}\left.\left.\hskip
119.50148pt{}+\left(P_{kl}-\frac{P}{2}q_{kl}\right)\mathcal{L}_{m}s^{kl}\right]\right\\}=2NK_{ij}\,,$
(45b)

where the right-hand sides are given by the extrinsic-curvature tensors of eq.
(34) and eq. (4), respectively. With that settled, we take a look at the
second set of Hamilton’s equations of eq. (39b) for the $s^{\mathbf{nn}}$
sector. The individual contributions to the Poisson bracket are

$\displaystyle-\left\\{p^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\sqrt{q}}{2\kappa}NR\right\\}$
$\displaystyle=-\frac{\sqrt{q}}{2\kappa}\left(q^{ij}D_{l}D^{l}N-D^{i}D^{j}N+NG^{ij}\right)\,,$
(46a)
$\displaystyle\left\\{p^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{2\kappa
N}{\sqrt{q}(1-s^{\mathbf{nn}})}\left(p^{ij}p_{ij}-\frac{p^{2}}{2}\right)\right\\}$
$\displaystyle=-\frac{2\kappa
N}{\sqrt{q}(1-s^{\mathbf{nn}})}\bigg{[}2p^{ia}p^{j}_{\phantom{j}a}-pp^{ij}$
$\displaystyle\phantom{{}={}}\hskip
73.97733pt-\frac{1}{2}\left(p_{ab}p^{ab}-\frac{p^{2}}{2}\right)q^{ij}\bigg{]}\,,$
(46b)
$\displaystyle-\left\\{p^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{N\sqrt{q}}{2\kappa}D^{i}D_{i}s^{\mathbf{nn}}\right\\}$
$\displaystyle=-\frac{N\sqrt{q}}{4\kappa}q^{ij}a_{l}D^{l}s^{\mathbf{nn}}\,,$
(46c)
$\displaystyle\left\\{p^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\mathcal{L}_{m}s^{\mathbf{nn}}}{2(1-s^{\mathbf{nn}})}\left(p-\frac{3}{8}\frac{\sqrt{q}}{\kappa
N}\mathcal{L}_{m}s^{\mathbf{nn}}\right)\right\\}$
$\displaystyle=-\frac{\mathcal{L}_{m}s^{\mathbf{nn}}}{2(1-s^{\mathbf{nn}})}\left(p^{ij}-\frac{3}{16}\frac{\sqrt{q}}{\kappa
N}q^{ij}\mathcal{L}_{m}s^{\mathbf{nn}}\right)\,.$ (46d)

By using these findings and multiplying by $2\kappa/(N\sqrt{q})$, we arrive at
eq. (35) with the exception of the last two terms in the second line depending
on the acceleration. Recall that an analogous issue occurred previously in the
$u$ sector. The boundary term of eq. (9) again solves with problem. The part
proportional to $s^{\mathbf{nn}}$ contributes to $H_{1}$ of eq. (23) with a
total derivative:

$\Delta_{1}:=-\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\sqrt{q}}{2\kappa}D_{l}(ND^{l}s^{\mathbf{nn}})=-\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{N\sqrt{q}}{2\kappa}(D_{l}D^{l}s^{\mathbf{nn}}+a_{l}D^{l}s^{\mathbf{nn}})\,.$
(47)

Computing the Poisson bracket of the latter contribution with the canonical
momentum density results in

$\\{\pi^{ij},\Delta_{1}\\}=\frac{N\sqrt{q}}{4\kappa}(a^{i}D^{j}s^{\mathbf{nn}}+a^{j}D^{i}s^{\mathbf{nn}})\,,$
(48)

which is related to the missing parts in eq. (35).

Finally, the most intricate sector of the $s^{ij}$ coefficients is on the
menu. Here, we benefit from the results

$\displaystyle-\left\\{P^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{\sqrt{q}}{2\kappa}NR\right\\}$
$\displaystyle=-\frac{\sqrt{q}}{2\kappa}\left(NG^{ij}+q^{ij}D_{l}D^{l}N-D^{i}D^{j}N\right)\,,$
(49a)
$\displaystyle-\left\\{P^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{N\sqrt{q}}{2\kappa}(s^{kl}R_{kl}-D_{l}D_{k}s^{kl})\right\\}$
$\displaystyle=\frac{N\sqrt{q}}{4\kappa}\Big{(}-2q^{ij}\left[a_{k}D_{l}s^{kl}+s^{kl}(D_{k}a_{l}+a_{k}a_{l})\right]$
$\displaystyle\phantom{{}={}}\hskip
34.14322pt+a_{k}(D^{i}s^{kj}+D^{j}s^{ki})-a_{k}D^{k}s^{ij}$
$\displaystyle\phantom{{}={}}\hskip
34.14322pt+q^{ij}(s^{kl}R_{kl}-D_{k}D_{l}s^{kl})+D_{k}D^{i}s^{kj}$
$\displaystyle\phantom{{}={}}\hskip
34.14322pt+D_{k}D^{j}s^{ki}-D_{k}D^{k}s^{ij}\Big{)}\,,$ (49b)
$\displaystyle\left\\{P^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\left(P_{kl}-\frac{P}{2}q_{kl}\right)\mathcal{L}_{m}s^{kl}\right\\}$
$\displaystyle=\frac{1}{2}(P^{ij}q_{kl}\mathcal{L}_{m}s^{kl}+P\mathcal{L}_{m}s^{ij})$
$\displaystyle\phantom{{}={}}-(P^{i}_{\phantom{i}k}\mathcal{L}_{m}s^{kj}+P^{j}_{\phantom{j}k}\mathcal{L}_{m}s^{ki})\,,$
(49c) as well as
$\displaystyle\left\\{P^{ij},\int_{\Sigma_{t}}\mathrm{d}^{3}x\,\frac{2\kappa
N}{\sqrt{q}}\left[P^{ij}P_{ij}-\frac{1}{2}(1-s^{k}_{\phantom{k}k})P^{2}-2s^{kl}(PP_{kl}-P_{k}^{\phantom{k}n}P_{ln})\right]\right\\}$
$\displaystyle=-\frac{\kappa
N}{2\sqrt{q}}\Big{(}\left[(1-s^{k}_{\phantom{k}k})P^{2}-2P^{kl}P_{kl}+4s^{kl}(PP_{kl}-P_{k}^{\phantom{k}m}P_{lm})\right]q^{ij}$
$\displaystyle\phantom{{}={}}\hskip
36.98866pt+8\Big{[}P^{ik}P_{k}^{\phantom{k}j}+s^{ik}P_{kl}P^{lj}+s^{jk}P_{kl}P^{li}-s^{ik}PP_{k}^{\phantom{k}j}-s^{jk}PP_{k}^{\phantom{k}i}$
$\displaystyle\phantom{{}={}}\hskip
36.98866pt+s^{kl}(P^{i}_{\phantom{i}k}P_{l}^{\phantom{l}j}-P_{kl}P^{ij})\Big{]}\Big{)}+2P^{2}s^{ij}-4(1-s^{k}_{\phantom{k}k})PP^{ij}\,.$
(49d)

We employ the latter findings and multiply by $2\kappa/(N\sqrt{q})$, which
yields eq. (38). Hence, our conclusion is that the Lagrangian and Hamiltonian
formulation of eq. (7) lead to the same dynamics. The lengthy computations
necessary to demonstrate this outcome corroborate the nontrivial nature of
this property.

##  6 Outlook: Other modified-gravity theories and the SME

After demonstrating the dynamical consistency of eq. (1), two additional
correspondences between modified-gravity theories and particular sectors of
the gravitational SME shall be presented. By doing so, we intend to foster
further interest in SME-related physics.

###  6.1 Linearized dRGT massive gravity

The recent paper Kostelecky:2021xhb  is dedicated to gaining a better
theoretical understanding of dRGT theory with an emphasis on the stability of
static solutions as well as certain properties of the linearized theory such
as gravitational-wave propagation and the gravitational energy between two
pointlike masses. To be able to study the propagation of gravitational-wave
modes, the linearized field equations are necessary. In momentum space, they
are cast into the form

$\tilde{O}_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}\tilde{h}_{\alpha\beta}=0\,,$
(50)

where $\tilde{h}_{\alpha\beta}$ corresponds to the Fourier transform of the
dynamical-metric perturbation $h_{\mu\nu}:=g_{\mu\nu}-\eta_{\mu\nu}$ and
$\tilde{O}_{\mu\nu}^{\phantom{\mu\nu}\alpha\beta}$ is a tensor-valued operator
in momentum space given by their eq. (82). Furthermore, the authors describe
the reference metric via a small deviation from the frequently taken Minkowski
metric $\eta_{\mu\nu}$, i.e., they define $\delta
f^{\mu\nu}:=f^{\mu\nu}-\eta^{\mu\nu}\ll 1$. All results are stated at first
order in the perturbation $\delta f^{\mu\nu}$.

The modifications of GR are proportional to five dimensionless parameters
$c_{i}$ with $i\in\\{1\dots 5\\}$. Now, all possible background fields with an
arbitrary mass dimension preserving or violating diffeomorphism invariance in
linearized gravity are classified in ref. Kostelecky:2017zob . Appropriate
background fields violating diffeomorphism invariance can be associated with
each of these contributions via their symmetry properties and momentum
dependences. The only suitable coefficients are those of mass dimension 2 that
are totally symmetric and not contracted with additional momenta:
$k^{(2,1)\mu\nu\varrho\sigma}$. For these SME coefficients we refer to line 11
in Table 1 of ref. Kostelecky:2017zob . Then,

$\displaystyle k^{(2,1)\mu\nu\varrho\sigma}$
$\displaystyle=-\frac{m^{2}}{2}\bigg{(}c_{1}(\eta^{\varrho(\mu}\eta^{\nu)\sigma}-\eta^{\mu\nu}\eta^{\varrho\sigma})+c_{2}\eta^{(\varrho\mu}\eta^{\sigma\nu)}+c_{3}(\eta^{-1}\delta
f)\eta^{\varrho(\mu}\eta^{\nu)\sigma}$ $\displaystyle\phantom{{}={}}\hskip
31.2982pt+c_{4}\left[\delta f^{\mu\nu}-\eta^{\mu\nu}(\eta^{-1}\delta
f)\right]\eta^{\varrho\sigma}+c_{5}\eta^{\mu\nu}\delta
f^{\varrho\sigma}\bigg{)}\,,$ (51)

where $m$ is the graviton mass and indices enclosed by a pair of parentheses
indicate that a symmetrization is performed in these indices. The latter
coefficients characterize the linear regime of dRGT massive gravity
completely.

###  6.2 Hořava-Lifshitz gravity

Hořava-Lifshitz gravity Horava:2008ih ; Horava:2009uw  is a possible UV
completion of GR. Its formulation relies on a foliation of spacetime into
purely spacelike hypersurfaces via the ADM decomposition, while the leading
contributions read Blas:2009qj

$S_{\mathrm{HL}}=\frac{1}{16\pi
G_{H}}\int_{\mathcal{M}}\mathrm{d}t\mathrm{d}^{3}x\,N\sqrt{q}(\xi
R+K_{ij}K^{ij}-\lambda K^{2}+\alpha a_{i}a^{i})\,,$ (52)

with the extrinsic curvature $K_{ij}$ and the acceleration $a_{i}$ defined
directly under eq. (5). Furthermore, the action involves dimensionless
dynamical parameters $\xi,\lambda,\alpha$ as well as the dimensionful
gravitational constant $G_{H}$ of this theory. A significant property of
Hořava-Lifshitz gravity are different weight factors associated with the
contributions $K_{ij}K^{ij}$ and $K^{2}$. As it turns out, it is this property
that makes a match to the $u$ and $s^{\mu\nu}$ sectors of eq. (7) presumably
impossible. The authors of ref. ONeal-Ault:2020ebv  also observed this issue,
which is why they added further would-be dynamical terms to their action to
enable a match. An alternative is to match eq. (52) with a setting known as
Einstein-aether theory Jacobson:2004ts ; Elliott:2005va ; Jacobson:2010mx . By
doing so, the corresponding SME-based Lagrange density is readily obtained as

$\mathcal{L}^{(2)}_{\mathrm{gravity}}=\frac{1}{2\kappa}\breve{k}^{(2)}\,,\quad\breve{k}^{(2)}=k_{\kappa\lambda\mu\nu}g^{\kappa\mu}g^{\lambda\nu}+k_{\mu\nu}g^{\mu\nu}\,,$
(53a) with the background fields $\displaystyle k_{\kappa\lambda\mu\nu}$
$\displaystyle=-\left(1+\frac{\lambda}{\xi}\right)\nabla_{\kappa}u_{\mu}\nabla_{\lambda}u_{\nu}+\left(1+\frac{1}{\xi}\right)\nabla_{\kappa}u_{\lambda}\nabla_{\nu}u_{\mu}\,,$
(53b) $\displaystyle k_{\mu\nu}$
$\displaystyle=-\frac{\alpha}{\xi}u^{\kappa}u^{\lambda}\nabla_{\kappa}u_{\mu}\nabla_{\lambda}u_{\nu}\,,$
(53c)

where $\xi=G_{H}/G_{N}$ and $u^{\mu}$ is a unit vector orthogonal to the
spacelike hypersurfaces of the spacetime foliation. Here we refer to the
second line of Table XVI and eq. (41) of ref. Kostelecky:2021xhb . Note that
the author of ref. Jacobson:2010mx  uses the metric signature $(+,-,-,-)$,
which is different from that of ref. Kostelecky:2021xhb .

##  7 Conclusions

This work was dedicated to a modified-gravity theory incorporating
nondynamical background fields that break diffeomorphism invariance
explicitly. We focused on two distinct types of background fields: a scalar
field denoted as $u$ and a tensor-valued one known as $s^{\mu\nu}$ in the SME
literature Kostelecky:2003fs ; Kostelecky:2020hbb . The latter decomposes into
a purely timelike and a purely spacelike sector. The canonical Hamiltonian had
already been obtained in the previous article Reyes:2021cpx  and formed the
base of the current paper.

On the one hand, the ADM decomposition allowed us to project the modified
Einstein equations into purely spacelike hypersurfaces. On the other hand, the
Hamilton-Jacobi equations gave rise to another set of field equations for the
theory. We have found that both approaches are consistent with each other.
This outcome is highly nontrivial, as nondynamical background fields transform
unconventionally under diffeomorphisms. Hence, the structure of such theories
is rendered substantially obscure as opposed to GR or extensions thereof with
either diffeomorphism symmetry intact or broken spontaneously. Such an
analysis in the context of the gravitational SME is the first of its kind in
the literature. In principle, the Dirac procedure could imply additional sets
of constraints beyond those of GR. However, our findings indicate that the
canonical Hamiltonian completely encodes the dynamical properties of the
gravitational theory under study. Furthermore, they corroborate the results
derived previously in ref. Reyes:2021cpx .

The setting investigated is quite generic and links to several popular
modified-gravity models proposed in the literature such as Brans-Dicke theory
and dRGT massive gravity. Hence, this investigation could also arouse interest
of scientists not having had contact to the gravitational SME, so far.
Additional explorations of the constraint structure are beyond the scope of
this paper and will be pursued in a future work. Gaining a better
understanding of the SME sectors identified in sections 6.1, 6.2 by utilizing
the methodology employed here may be further worthwhile projects.

##  8 Acknowledgments

It is a pleasure to thank P. Sundell for valuable discussions on several
aspects of this project. C.M.R acknowledges partial support by the research
project Fondecyt Regular 1191553. M.S. is indebted to FAPEMA Universal
00830/19, CNPq Produtividade 312201/2018-4, and CAPES/Finance Code 001.

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