# Locally Scale-Invariant Gravity

Meir Shimon  School of Physics and Astronomy, Tel Aviv University, Tel Aviv
69978, Israel<EMAIL_ADDRESS>
###### Abstract

We put forward the idea that in addition to diffeomorphism invariance of
general relativity (GR) the gravitational interaction is invariant under
arbitrary scale-deformations of the metric field. In addition, we assume that
the scaling field has an internal symmetry. The global charges that are
associated with this symmetry could potentially source the gravitational
field. In the case that isotropic deformations are considered, the theory
reduces to a Weyl-invariant (WI) version of GR. In the case that Minkowski
spacetime is deformed the vierbein formalism is recovered, rendering GR a
field theory on Minkowski spacetime. A few implications of a classical Weyl-
invariant scalar-tensor (WIST) generalization of general relativity (GR) are
considered. As an example, we recast the homogeneous and isotropic Friedmann-
Robertson-Walker (FRW) spacetime in the WIST form with static space and
monotonically evolving masses.

##  1 Introduction

It is hard to conceive any physically realistic description of the Universe in
the absence of either inertia or gravitation. Inertia is thought to be a
universal phenomenon; the Higgs field that is responsible for the generation
of particle masses in the electroweak sector is ubiquitous. Likewise,
gravitation is everywhere. It is normally identified with the omnipresence of
spacetime, the substratum of physical reality, at least down to spacetime
singularities, where the very concepts of space and time are believed to break
down.

Our current best theory of the (classical) gravitational interaction is
general relativity (GR). The latter has successfully passed a wide range of
tests in our solar system. GR is one of the simplest of all conceivable metric
theories; it obeys the metricity condition, it is torsionless and it is
described by a lagrangian linear in the Ricci scalar. GR provides a
diffeomorphism-invariant field theoretical framework that incorporates the
principle of relativity applied to non-inertial observers and is realized on
curved spacetimes. The curvature in the presence of a given energy-momentum
distribution is governed by $G$, the universal coupling constant of
gravitation that converts energy-momentum density units to length units. This
specific coupling between matter and geometry was adopted in GR so as to
guarantee consistency with Newton law of gravitation in the weak field limit,
assuming that the locally inferred $G$ is universal.

Naturally, GR is expected to depart from the Newtonian limit in the strong
gravity regime, i.e. in physical systems which are dominated by the
gravitational interaction while other interactions play a secondary role, e.g.
in black holes or on the largest cosmological scales. However, in contrast to
expectations Newtonian gravity does not adequately describe gravitational
phenomena on galactic scales and larger, e.g. at the outskirts of galaxies and
galaxy clusters, even though gravity is weak on these scales, i.e. the
dimensionless gravitational potential is as low as $\Phi\sim 10^{-4}$ or
$10^{-3}$. A straightforward remedy for this problem (and others in cosmology)
involves introducing some form of non-relativistic (NR) invisible matter, i.e.
‘dark matter’ (DM). By far the leading candidate is some yet-to-be-determined
beyond-the standard-model (SM) particle. According to another, less popular,
possibility primordial black holes may account for the missing non-luminous
matter.

As of yet, no DM particle candidate has been found in spite of decades of
intensive search. In addition, the status of beyond-the-SM theories of
particle physics – our best hope for new DM particles – is unclear with
persistent null results coming from the large hadron collider (LHC), although
certain anomalies have been reported recently with mild indications for new
physics. A recent high precision measurement of the Z boson mass at the
Fermilab Tevatron collider, that significantly departs from SM expectations,
calls the integrity of the SM into question. In any case, these are either
inconclusive as of yet or need independent corroborations by other
experiments, and even if eventually ratified considerable amount of effort is
still needed before the existence of a DM particle that adequately accounts
for DM phenomena on Hubble down to galactic scales is compellingly
established.

Another possibility is that particulate DM simply does not exist, and that on
galactic scales or larger gravitation is not strictly described by GR, but
rather by a modified version thereof. Indeed, the validity of GR coupled to
purely ordinary matter (described by the SM of particle physics) has been only
compellingly established within our own solar system. One possible extension
of GR, invokes a higher symmetry – Weyl invariance (WI), where $G$ is
determined by a scalar field (while the other three fundamental interactions
are assumed to have their standard forms, i.e. they are non-Weyl invariant).
In GR this scalar field is replaced by default with a fixed $G$ throughout
space and time. However, its local determination by Cavendish-type experiments
does not necessarily imply its universality. In a WI version of GR, $G$ can
vary in space and time, freely departing from its locally measured value, and
simultaneously inducing scalar metric perturbations. From this perspective,
there is a priori no preferred conformal frame, and whereas the choice
$G=const.$ seems to work impressively well in our solar system it clearly
fails on galactic scales and larger. It has been shown recently, that
fractional variations of $G$ and active gravitational masses at the
$O(10^{-3})$ or $O(10^{-4})$ level on galactic and galaxy cluster scales are
sufficient for explaining away DM phenomena without particulate DM at the
reasonable ‘cost’ of endowing GR with WI [1]. Another application of a WI
version of GR is a possible resolution of the ‘Hubble tension’: the apparent
statistically significant $\gtrsim 4\sigma$ tension between local and
cosmological inference of the Hubble constant, $H_{0}$ [2].

On the largest cosmological scales observations require some form of a non-
clustering vacuum-like DE, which is puzzling in a few respects. First, it is
$\sim 120$ orders of magnitude suppressed relative to naive expectations based
on our current understanding of particle physics. This might be hinting
towards a gravitational/geometric origin for DE rather than material one.
Second, it is found empirically to be comparable in energy density terms to
that of the NR matter at the present time although the two species greatly
differ in their evolution history. In this sense, there seems to be a fine-
tuning of the initial conditions that are associated with DE [3].

The present work focuses on the foundations of a certain generalization of GR
that allows for a fully anisotropic locally scale-invariant generalization of
GR by introducing a scale deformation (tensor) field $\Phi_{\mu\nu}$ living on
an arbitrary spacetime. In the special case that the latter is Minkowski
$\Phi_{\mu\nu}$ reduces to the well-known vierbein field. In the case of an
isotropic scale-invariance on an arbitrary background the theory becomes a WI
scalar-tensor (WIST) version of GR, hereafter referred to in abbreviation as
‘WIST’. In addition to local scale-invariance we endow $\Phi_{\mu\nu}$ with an
internal symmetry with which a global conserved charge is associated. The
latter provides an additional source for spacetime curvature.

Although various variants of WIST have been contemplated over the past fifty
years, e.g. [4-15], objections to this framework range between two extremes.
On the one hand, it is often claimed that the WIST is ‘GR in a guise’, that
extending the symmetry of GR to allow WI does not make it genuinely so. In
particular it is argued that there is classically no preferred conformal
frame, so WIST is equivalent to GR. On the other hand, it is sometimes implied
that dressing GR with WI brings about a ghost scalar field that inflicts
‘disastrous instabilities’ on the theory, so much so as to disqualify the
theory (while the same is not similarly claimed about the ‘equivalent’ GR). We
address these claims in the present work, and as a side we explicitly show
that, not surprisingly, the action of the Friedmann-Robertson-Walker (FRW)
spacetime, a legitimate solution of the Einstein equations, can be presented
in a WIST form. By doing so, we illustrate with a specific example, with well-
known general-relativistic description of spacetime, that there are no ‘fatal
instabilities’ associated with WIST that do not already exist in GR. Again, we
stress that it is our assumption that WIST is entirely classical to the extent
that GR is, and indeed the validity of GR has only been experimentally
established on macroscopic down to $\mu m$, i.e. mesoscopic, scales, and not
smaller.

The paper is organized as follows. The general anisotropic scale-invariant
version of GR is considered in section 2, followed by its special isotropic
case, the WIST theory, in section 3. The FRW example is discussed in section
4, followed by endowing the theory with global symmetries in section 5 and a
summary in section 6. In Appendix A the transformation from a purely
gravitational to non-gravitational description of the weak field approximation
is described, and in Appendix B the case of U(2) global symmetry is briefly
discussed. Throughout, we adopt a mostly-positive signature for the spacetime
metric $(-1,1,1,1)$, with the speed of light $c$ and reduced Planck constant
$\hbar$ set to unity.

##  2 Locally anisotropic scale-invariant theory of gravitation

GR is described by the diffeomorphism-invariant Einstein-Hilbert (EH) action

$\displaystyle\mathcal{I}_{EH}=\int\left(\frac{1}{16\pi
G}(R-2\Lambda)+\mathcal{L}_{M}\right)\sqrt{-g}d^{4}x,$ (1)

where $R$ is the Ricci curvature scalar (obtained from the metric field
$g_{\mu\nu}$ and its first and second derivatives in the standard way) and
$\Lambda$ is the cosmological constant, respectively, $\mathcal{L}_{M}$ is the
lagrangian density of matter, and $g$ is the metric determinant. The
cosmological constant, $\Lambda$, can be absorbed in $\mathcal{L}_{M}$ as a
species with vacuum-like properties. Diffeomorphism-invariance implies that
equation (1) is invariant under the transformation $g_{\mu\nu}\rightarrow
g_{\alpha\beta}\frac{\partial x^{\alpha}}{\partial x^{\mu}}\frac{\partial
x^{\beta}}{\partial x^{\nu}}$ (where Greek indices stand for spacetime
coordinates and repeated indices are summed over). This symmetry of equation
(1) under change of coordinate basis relies on the commutativity of
derivatives.

A natural generalization of diffeomorphism (that to the best of our knowledge
has not been considered so far in the literature) could be the following
replacement everywhere in equation (1)

$\displaystyle g_{\mu\nu}$
$\displaystyle\rightarrow\bar{g}_{\mu\nu}\equiv\Phi_{\mu}^{\
\alpha}g_{\alpha\beta}\Phi_{\ \nu}^{\beta},$ $\displaystyle\mathcal{L}_{M}$
$\displaystyle\rightarrow\bar{\mathcal{L}}_{M}/|\Phi|,$ (2)

where $\Phi_{\mu}^{\ \alpha}(x)$ is an arbitrary spacetime-dependent real
scale-deformation tensor of rank 2, and $|\Phi|\equiv\det(\Phi_{\mu}^{\
\nu})$. This transformation represents a local anisotropic rescaling of the
metric field and the lagrangian density (in which $\Lambda$ is absorbed),
thereby providing us with a generalization of the ‘isotropic’ local rescaling,
known also as WI, that will be discussed below and in the next section. In
this framework gravitation is described by both a tensor field $\Phi_{\mu\nu}$
and the spacetime metric $g_{\mu\nu}$ on which $\Phi_{\mu\nu}$ lives. We
assume that $\Phi_{\mu\nu}$ is all-pervading much like the Higgs field is
universal.

Naturally, the local (in general anisotropic) rescaling of the various fields
is by physical dimension, e.g. the dimensions of $g_{\mu\nu}$, $g^{\mu\nu}$
and $\mathcal{L}_{M}$ are $length^{2}$, $length^{-2}$ and $length^{-4}$,
respectively. The contravariant coordinates, $x^{\mu}$, are dimensionless, and
the covariant coordinates $x_{\mu}$ are of dimension $length^{-2}$.
Specifically, the anisotropic local scale transformations for the metric and
tensor fields, as well as the lagrangian density, are

$\displaystyle g_{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle\Omega_{\mu}^{\ \alpha}g_{\alpha\beta}\Omega^{\beta}_{\ \nu}$
$\displaystyle\Phi_{\mu}^{\ \alpha}$ $\displaystyle\rightarrow$
$\displaystyle\Phi_{\mu}^{\ \rho}(\Omega^{-1})_{\rho}^{\ \alpha}$
$\displaystyle\Phi_{\ \mu}^{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle(\Omega^{-1})_{\ \rho}^{\alpha}\Phi_{\ \mu}^{\rho}$
$\displaystyle\mathcal{L}_{M}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{L}_{M}/|\Omega(x)|,$ (3)

where $|\Omega(x)|$ is the determinant of an arbitrary spacetime-dependent
matrix $\Omega_{\mu}^{\ \nu}$, leaves $\bar{g}_{\mu\nu}$ and
$\bar{\mathcal{L}}_{M}/|\Phi|$ that are defined in equation (2), invariant.
Therefore, if we replace $g_{\mu\nu}$ and $\mathcal{L}_{M}$ with
$\bar{g}_{\mu\nu}$ and $\bar{\mathcal{L}}_{M}$, respectively, everywhere in
equation (1) then the latter is invariant under equation (3) in addition to
being diffeomorphism-invariant.

In the special case that the scaling matrix is isotropic, $\Phi_{\mu}^{\
\nu}=\phi(x)\delta_{\mu}^{\ \nu}$, i.e. the metric scale deformation is
isotropic, the proposed symmetry reduces to a WI version of GR, i.e. the
aforementioned WIST theory, in which case the replacement
$g_{\mu\nu}\rightarrow\tilde{g}_{\mu\nu}=g_{\mu\nu}\phi^{2}(x)$ and
$\mathcal{L}_{M}\rightarrow\mathcal{L}_{M}/\phi^{4}$ is made in equation (1),
where $\phi(x)$ is an arbitrary scalar function. This implies that the theory
is invariant under the Weyl transformation

$\displaystyle g_{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{g}_{\mu\nu}=g_{\mu\nu}\Omega^{2}(x)$ $\displaystyle\phi$
$\displaystyle\rightarrow$ $\displaystyle\tilde{\phi}=\phi\Omega^{-1}(x)$
$\displaystyle\mathcal{L}_{M}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\mathcal{L}}_{M}=\mathcal{L}_{M}\Omega^{-4}(x).$ (4)

This isotropic case will be further explored in the next section.

In the special case $\eta_{\mu\nu}\rightarrow e_{\mu}^{\
\alpha}\eta_{\alpha\beta}e_{\ \nu}^{\beta}=g_{\mu\nu}(x)$, where
$\eta_{\mu\nu}$ is the Minkowski metric, the tensor $\Phi_{\mu}^{\
\nu}=e_{\mu}^{\ \nu}$ is the vierbein field. Tensor-mode perturbations, i.e.
gravitational waves, could therefore describe massless tensor perturbations in
the $\Phi_{\mu}^{\ \nu}$ field, i.e. in ‘G’ and in active gravitational
masses, rather than in spacetime itself. When spacetime is fixed to be
Minkowski all the gravitational degrees of freedom that are normally
represented by the various components of the spacetime metric are now
accounted for by the tensor field, which in this case is merely the vierbein
field, $e_{\mu}^{\nu}$. In this choice the metric field itself, that describes
Minkowski spacetime, is non-dynamical, space and time are infinite and and all
the singularities are in $\Phi_{\mu}^{\ \nu}$, its derivatives, and
combinations thereof that replace, e.g. the standard spacetime curvature. This
differs from the standard interpretation of GR where distortions and
singularities are understood to be in spacetime itself, where e.g. the Big
Bang singularity is normally understood as the beginning of time, and perhaps
even of spacetime. To illustrate the description of gravitational phenomena
within the WIST theory on Minkowski spacetime we examine the transformation of
the geodesic equation for a point test particle in the ‘weak field’
approximation in Appendix A.

Setting $G=3/(8\pi)$ and replacing $\mathcal{L}_{M}$ with $V$ (the latter
contains no derivatives of the various fields), and $g_{\mu\nu}$ with
$\bar{g}_{\mu\nu}$ (see equation 2) everywhere in equation (1), we obtain

$\displaystyle\bar{\mathcal{I}}=\int\left(\frac{1}{6}(\bar{R}-2\Lambda)+V\right)\sqrt{-\bar{g}}d^{4}x.$
(5)

As will be detailed below we assume that $V$, the source of spacetime
curvature, is a potential in
$\Phi\equiv\sqrt{\Phi_{\alpha}^{\beta}\Phi_{\beta}^{\alpha}}$. In particular,
we make the assumption that it is an analytic polynomial in $\Phi$. Consider
the transformations described by equation (3). To first order, an
infinitesimal transformation results in

$\displaystyle\bar{g}_{\mu\nu}=\Omega_{\mu}^{\ \alpha}g_{\alpha\beta}\Omega_{\
\nu}^{\beta}\approx g_{\mu\nu}+\delta\Omega_{\mu}^{\
\alpha}g_{\alpha\nu}+g_{\mu\alpha}\delta\Omega_{\ \nu}^{\alpha},$ (6)

where we used $\Omega_{\mu}^{\ \nu}\approx\delta_{\mu}^{\
\nu}+\delta\Omega_{\mu}^{\ \nu}$. Similarly,

$\displaystyle\bar{\Phi}_{\mu}^{\ \nu}=(\Omega^{-1})_{\mu}^{\
\alpha}\Phi_{\alpha}^{\
\nu}\approx(\delta_{\mu}^{\alpha}-\delta\Omega_{\mu}^{\
\alpha})\Phi_{\alpha}^{\ \nu},$ (7)

from which it follows that

$\displaystyle\delta\Phi_{\mu}^{\ \nu}\approx-\delta\Omega_{\mu}^{\
\alpha}\Phi_{\alpha}^{\ \nu}.$ (8)

Employing the relation between variations of inverse matrices, $\delta
g^{\mu\nu}=-g^{\mu\alpha}\delta g_{\alpha\beta}g^{\beta\nu}$ and using
equation (6) we obtain

$\displaystyle\delta g^{\mu\nu}\approx-g^{\mu\alpha}\delta\Omega_{\alpha}^{\
\nu}-\delta\Omega_{\ \alpha}^{\mu}g^{\alpha\nu}.$ (9)

In our convention that $x^{\mu}$ are dimensionless (see discussion below
equation 2), $\int V\sqrt{-g}$ is dimensionless and so
$\delta(V\sqrt{-g})=\frac{(\delta V\sqrt{-g})}{\delta g^{\mu\nu}}\delta
g^{\mu\nu}+\frac{(\delta V\sqrt{-g})}{\delta\Phi_{\mu}^{\
\nu}}\delta\Phi_{\mu}^{\ \nu}+\frac{(\delta V\sqrt{-g})}{\delta\Phi_{\
\mu}^{\nu}}\delta\Phi_{\ \mu}^{\nu}=0$ under the transformations Eqs.(8) and
(9). Requiring invariance under arbitrary $\delta\Omega_{\mu}^{\ \nu}$ we
obtain

$\displaystyle S_{\mu}^{\ \nu}=\Phi_{\mu}^{\ \alpha}\frac{\delta
V}{\delta\Phi^{\alpha}_{\ \nu}}+\frac{\delta V}{\delta\Phi^{\
\alpha}_{\nu}}\Phi_{\ \mu}^{\alpha}$ (10)

where $S_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{(\delta V\sqrt{-g})}{\delta
g^{\mu\nu}}$. This provides a strong constraint on the potential if
anisotropic local scale invariance, Eqs. (1)-(3), is to be a symmetry of the
gravitational interaction. We will rederive the analog of this result in the
isotropic scaling case and delve into the implications in sections 3 and 4.

We emphasize that only gravitation is endowed with WI in the proposed
framework. All other fundamental interactions, namely the strong and
electroweak interactions are described by the SM of particle physics, with no
modifications. Since any realistic physical system is governed by both
gravitational and non-gravitational interactions, and since the fundamental
interactions are minimally-coupled to gravitation, it then follows that this
particular framework is clearly not merely a field re-definition of GR and the
SM of particle physics. In any case, introducing the global charge that is
generally associated with $\Phi_{\mu\nu}$ due to internal symmetry as
discussed in section 5 and in Appendix B renders the theory described here
inequivalent to the SM even in purely gravitational systems such as late time
cosmological scales.

Much like the nonvanishing and universality of VEV of the Higgs field
guarantees that inertia phenomena are ubiquitous so it is the case with
spacetime that never ceases to exist as is reflected by the fact that the
metric field can always be so chosen to be Minkowski in the most general
anisotropic locally scale-invariant gravitation. Since the SM of particle
physics is minimally coupled to gravitation then the tensor field
$\Phi_{\mu\nu}$ locally couples to the SM fields instead of them being living
on a curved spacetime. For example, $\mathcal{I}_{kin}=\int
g^{\mu\nu}(D_{\mu}H)^{\dagger}D_{\nu}H\sqrt{-g}d^{4}x$, the kinetic term of
the Higgs field $H$ (where $D_{\mu}$ is the gauge-covariant derivative),
transforms to
$\mathcal{I}_{kin}=\int(D_{\mu}H)^{\dagger}\bar{g}^{\mu\nu}D_{\nu}H\sqrt{-\bar{g}}d^{4}x$,
which in the Minkowski background simplifies to
$\mathcal{I}_{kin}=\int(D_{\mu}H)^{\dagger}\Phi^{\mu}_{\
\sigma}\Phi^{\sigma\nu}D_{\mu}H|\Phi|d^{4}x$. The tensor field $\Phi^{\mu\nu}$
then directly couples to $H$, (where $\Phi_{\mu}^{\ \nu}$ is actually the
vierbein field in the case of Minkowski background) thereby constituting a
field theory on Minkowski spacetime (spacetime indices are raised and lowered
by means of the Minkowski metric), as opposed to the standard interpretation
according to which $H$ interacts with the geometry of spacetime.

##  3 WIST theory

In this section we focus on the isotropic case, i.e.
$\Phi_{\mu}^{\nu}=\delta_{\mu}^{\nu}\phi(x)$, where $\phi(x)$ is a scale
deformation (scalar) field. Although it was assumed in equation (2) that
$\Phi_{\mu}^{\nu}$ is real (in order for the new metric field,
$\bar{g}_{\mu\nu}$, to be real) it is clear that at least in the special case
considered in this section, i.e. $\Phi_{\mu}^{\nu}=\delta_{\mu}^{\nu}\phi(x)$,
promoting $\phi(x)$ to a complex scalar field renders $\bar{g}_{\mu\nu}$ real.
Since the scalar field is assumed to be complex, it has a U(1) internal
symmetry. The more general case of an arbitrary internal symmetry is briefly
discussed in the section 5. Gravitation is then fundamentally endowed with WI,
i.e. an isotropic locally scale-invariance, within a WIST theory, and standard
GR is recovered in a particular conformal frame, where $\phi=constant$.

Rather than using the more general results from the previous section we derive
them directly in the isotropic case. The WIST theory readily follows from
locally rescaling the spacetime metric
$g_{\mu\nu}\rightarrow\phi\phi^{*}g_{\mu\nu}$ everywhere in equation (1). The
resulting theory is then described by the following action

$\displaystyle\mathcal{I}_{WIST}$ $\displaystyle=$
$\displaystyle\int\left(\frac{1}{6}|\phi|^{2}R-\phi^{*}\Box\phi+V(|\phi|,\\{\Psi\\})\right)\sqrt{-g}d^{4}x$
(11) $\displaystyle=$
$\displaystyle\int\left(\frac{1}{6}|\phi|^{2}R+\phi_{\mu}\phi^{*\mu}+V(|\phi|,\\{\Psi\\})\right)\sqrt{-g}d^{4}x,$

where $\phi_{\mu}\equiv\frac{\partial\phi}{\partial x^{\mu}}$, $V$ that
replaces $\mathcal{L}_{M}$ is now allowed to explicitly depend on $|\phi|$ but
not on its derivatives, and the second equality follows from integration by
parts of the kinetic term associated with the scalar field. All other fields
(including $g_{\mu\nu}$) are collectively denoted by $\\{\Psi\\}$. We
emphasize that in the new formulation, equation (11), gravitation is sourced
by $V$ rather than by ‘matter’, $\mathcal{L}_{M}$. In general, GR is
consistent with observations within the solar system and so on these scales
$V$ can be replaced by $\mathcal{L}_{M}$, i.e. ‘ordinary’ matter. However, on
larger scales, where both DM and DE are required by observations we favor the
form equation (11) which does not imply the existence of DM particles. In
addition, in equation (11) geometry and matter are not presented as two
different entities. Rather, the existence of $\phi$ in both the source, $V$,
and the kinetic term, reflects the viewpoint that they are actually
intertwined. They are ‘unified’ in the sense that WI of equation (11), a
symmetry that is not recognized by GR, depends on both the scalar and metric
fields.

With $V$ being explicitly $\phi$-dependent, equation (11) is now a scalar-
tensor theory of the Bergmann-Wagoner type [16], [17]. It was first obtained
by Deser [4] and later in [10] with $\mathcal{L}_{M}$ [referred to as V in
equation (11)] that does not necessarily depend on $\phi$. In [4] $\phi$ was
assumed to be real. A similar procedure to the replacement
$g_{\mu\nu}\rightarrow\phi\phi^{*}g_{\mu\nu}$ that transforms equation (1) to
equation (10) was employed by Chamseddine and Mukhanov in their Mimetic
Gravity [18] but in the latter case the scalar field was an irotational
velocity potential $\varphi$, i.e.
$g_{\mu\nu}\rightarrow\varphi_{,\mu}\varphi^{,\mu}g_{\mu\nu}$.

Formally, the kinetic term associated with the scalar field that appears in
equation (11), $\mathcal{L}_{\phi}\equiv\phi_{,\mu}\phi^{*,\mu}$ or
$\mathcal{L}_{\phi}\equiv-\phi^{*}\Box\phi$, can be viewed as a new source of
the gravitational interaction. This term is completely ignored in GR where
$\phi$ is set to a constant value, $\phi_{0}\equiv\sqrt{\frac{3}{8\pi G}}$.
Invariance of equation (11) implies that $V\rightarrow|\phi|^{-4}V$ under
$g_{\mu\nu}\rightarrow|\phi|^{2}g_{\mu\nu}$. This simple derivation also
underscores the origin of the ‘ghost’ scalar field whose kinetic term,
$\mathcal{L}_{\phi}\equiv\phi_{,\mu}\phi^{*,\mu}$, appears with the ‘wrong’
sign in equation (11); it simply stems from locally stretching/squeezing the
spacetime metric, or equivalently stretching/squeezing the yardstick with
which distances are measured, in our case it is the Planck length
$l_{p}\propto\sqrt{G}$ in vacuum, or $\propto(G\rho)^{-1/2}$ where $\rho$ is
the energy density in non-vacuum configurations. This observation is crucial
for evaluation of the significance of the ghost field. We see that $\phi$ is
tightly related to $g_{\mu\nu}$ (merely a local stretch of the spacetime
metric), and as $g_{\mu\nu}$ is treated classically so should be $\phi$ and
$V$; the ghostly nature of $\phi$ derives solely from the specific form in
which $g_{\mu\nu}$ and its derivatives appear in the EH action. Below, we
explicitly show (although it should be clearly evident already from the very
local scaling procedure $g_{\mu\nu}\rightarrow\phi\phi^{*}g_{\mu\nu}$ that the
spacetime dependence of $\phi$ is arbitrary) that the classical field $\phi$
is not determined by any dynamical equation (indeed not unexpected in a WI
theory). In particular, it is not driven dynamically to unstable field
configurations (which is the generic problem with ghost fields in non-WI
theories). The latter are considered disastrous in the quantum regime because
the Hamiltonian is unbounded from below.

Since equation (11) was obtained from equation (1) by affecting the
transformation $g_{\mu\nu}\rightarrow\phi\phi^{*}g_{\mu\nu}$, invariance of
the former is guaranteed insofar $|\phi|^{2}g_{\mu\nu}$ is invariant, i.e.

$\displaystyle\phi$ $\displaystyle\rightarrow$ $\displaystyle\phi/\Omega$
$\displaystyle g_{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle\Omega^{2}g_{\mu\nu},$ (12)

for an arbitrary $\Omega(x)>0$, i.e. equation (4). Equation (12) are the Weyl
transformations of metric and scalar fields, respectively. It is thanks to
this freedom that $\phi=\phi_{0}/\Omega(x)$ is arbitrary, and so whereas it is
spacetime-dependent it is not dynamical in the sense that it is not determined
by a dynamical differential equation which has an attractor solution, that,
e.g. drives the kinetic term,
$\mathcal{L}_{\phi}\equiv\phi_{,\mu}\phi^{*,\mu}$, to arbitrary negative
values, as is the case with generic ghost fields. Actually, we will see in
section 4 that exactly the opposite takes place in our expanding Universe when
the action of the FRW spacetime is recast in the form of equation (11) instead
of equation (1); the negative kinetic term evolves from negative infinity at
the Big Bang towards vanishingly negative values at the remote future assuming
a monotonic expansion history, compatible with the concordance cosmological
model.

In fact, the presence of the kinetic term in
$\mathcal{L}_{\phi}\equiv\phi_{,\mu}\phi^{*,\mu}$ is expected once $G$ is
promoted to a field. More specifically, since the curvature scalar depends on
derivatives of the metric field it transforms inhomogeneously under equation
(12)

$\displaystyle
R\rightarrow\Omega^{-2}\left(R-6\frac{\Box\Omega}{\Omega}\right),$ (13)

where $\Box$ is the d’Alambertian. Invariance of equation (11) under equation
(12) then requires the presence of the kinetic term
$\mathcal{L}_{\phi}\equiv\phi_{,\mu}\phi^{*,\mu}$ that transforms
inhomogeneously and guarantees the mutual cancellation of derivatives of
$\Omega$ (provided that the appropriate integration by parts has been carried
out). The prefactor $\frac{1}{6}$ in front of the curvature coupling term in
equation (11) guarantees that the inhomogeneous term on the right hand side of
equation (13) is compensated by a similar inhomogeneous term from the
transformation of the kinetic term, $\phi^{*}_{\mu}\phi^{\mu}$. In other
words, under the transformation equation (12) the combination of the first two
terms in equation (11) transforms as,
$\frac{1}{6}|\phi|^{2}R-\phi\Box\phi^{*}\rightarrow\Omega^{-4}(\frac{1}{6}|\phi|^{2}R-\phi\Box\phi^{*})$,
i.e. it has the appropriate well-defined conformal weight of $length^{-4}$,
such that its product with the volume element in equation (11) is WI.

In the special case that $\phi$ is a real field and V is independent of $\phi$
equation (11) can be cast in the form of a Brans-Dicke (BD) theory [19]
$\mathcal{I}_{BD}=\int\Big{(}\Phi
R-\omega_{BD}\frac{\Phi_{\mu}\Phi^{\mu}}{\Phi}+\mathcal{L}_{M}\Big{)}\sqrt{-g}d^{4}x$
with $\Phi$ (not to be confused with the norm of $\Phi_{\mu\nu}$ that is
defined just below equation 5) and $\omega_{BD}$ being the BD scalar field and
dimensionless BD parameter, respectively, where $\Phi\equiv\phi^{2}/6$ and
$\omega_{BD}\equiv-3/2$. Whereas a tight lower limit, $\omega_{BD}>40000$, has
been reported in [20], it should be stressed that it was obtained from
satellite observations directly exploring the impact of varying $G$ within our
own solar system, and not beyond. In addition, in the (generally non-WI) BD
framework the source term $\mathcal{L}_{M}$ does not in general depend on
$\phi$ while in WIST the potential $V$ does.

It is constructive at this point to further elucidate the issue of stability
of Eq. (11) unlike what is sometimes claimed or implied. While it is true that
the kinetic term appears in equation (11) with the ‘wrong’ sign relative to
$V(|\phi|)$ and $\phi$ is formally a ‘ghost’ field it is merely a gauge
artifact that can be seen when the latter is set to a constant in the unitary
gauge [21]. The ‘fatal instability’ usually ascribed to quantum ghost fields
stems from the fact that in their presence there is either no stable vacuum
state, or non-conservation of probability takes place, e.g. [22]. However, GR
is classical, and so is WIST. As was already mentioned above, in the special
case of WIST, it is exactly WI that prevents classical instability from taking
place because there is essentially no equation of motion for $\phi$ to
dynamically drive it towards instability. This is not surprising since at
least classically, two conformally-related theories are equivalent, e.g. [23],
[24]. Actually, it has been argued that WI is a sham/fake symmetry of GR [25],
[26], i.e. that equation (11) is equivalent to GR. On face value, the
conclusion that the scalar field is a ‘spurion’ [25] implies that its formal
ghostly nature is of no practical significance. As mentioned above, adding a
global symmetry to $\Phi_{\mu\nu}$ renders both the anisotropic and isotropic
(equation 11) locally scale invariant theories inequivalent to GR. We also re-
iterate in this context that in the framework proposed here gravitation is WI,
whereas the SM of particle physics is not, and since in general the other
three interactions are non-negligible in typical physical systems, then the SM
of particle physics coupled to WIST is not equivalent to the SM of particle
physics coupled to GR. This was a key observation that was exploited in [1] in
explaining away particulate DM in galaxies and clusters of galaxies. Had the
SM of particle physics been WI as well then the proposed remedy for the DM
problem would gauge away.

Ultimately, the ‘appropriate’ conformal frame to be used in the description of
a physical system should be determined by observations, much like the
‘appropriate’ coordinate system employed in the description of, e.g., a black
hole in GR is determined by observations, e.g. [24]. For example, a
spherically symmetric static black hole as seen from the perspective of a
static observer would be most naturally described by a Schwarzschild metric,
while a freely falling observer towards the same black hole will likely employ
a different, time-dependent, metric in describing the same black hole. The
diffeomorphism symmetry of GR allows observers to describe the system in the
most symmetric fashion, which clearly depends on the (a priori arbitrary)
observers states. In the same fashion, at the classical level, theory cannot
pre-select the ‘appropriate’ units to be used in the description of a physical
system. This freedom is clearly not a property of GR, but is certainly a
desirable tenet of its WI generalization described by equation (11) or the
more general, locally anisotropic scale-invariant, theory described in the
previous section.

In the quantum regime the situation generally changes as quantization and Weyl
transformations do not in general commute, e.g. [23], [24], and in fact ghosts
are known to violate unitarity, e.g. [27]. However, quantization of GR has
never been demonstrated to be feasible (at least this is the consensus), nor
are there available measurements indicating that quantum gravitational effects
are in play so as to necessitate its quantization. Actually, the standard
theoretical arguments in favor of its quantization might be strong, yet
inconclusive (e.g. [28] and references within), and at the very best GR could
be considered an effective low energy limit of a more fundamental quantum
theory of gravity, if it should indeed be ultimately quantized. Therefore, in
the same fashion that the non-renormalizability of GR does not disqualify it
from being the backbone of the standard cosmological model what might be a
ghost ‘problem’ of equation (11) at the quantum level is not an issue for
physical systems that are adequately described by GR, e.g. [29]. Therefore,
the inequivalence between conformal frames at the quantum level need not
concern us in the present work.

It has been claimed in [27] that fixing the scalar field [and thereby
rendering equation (11) equivalent to GR] in order to go around the ‘ghost
problem’ in practice ‘robbs’ equation (11) from its claimed WI. Again, while
this argument might be valid at the quantum level it definitely does not apply
classically. Moreover, and as discussed below, the set of fields $\phi$ and
$g_{\mu\nu}$ that appears in equation (11) is under-determined and $\phi$
could be chosen to be any arbitrary function, not necessarily of a fixed
value, thereby still ‘manifesting’ WI, while at the same time avoiding driving
the field configuration away from its equilibrium state. In summary, the
‘wrong’ sign of the scalar field in a WI theory is not an issue at the
classical regime, which is the only domain at which equation (11), exactly as
GR, can be trusted.

It is constructive at this point to explore a slightly more general case of a
dynamical scalar field which is non-minimally coupled to spacetime curvature

$\displaystyle\mathcal{I}_{ST}=\int\Big{(}\xi|\phi|^{2}R+\phi^{*}_{\mu}\phi^{\mu}+V(|\phi|)\Big{)}\sqrt{-g}d^{4}x,$
(14)

where $\xi$ is an arbitrary dimensionless coupling parameter, thereby
generalizing equation (11), i.e the theory is in general not WI, unless
$\xi=\frac{1}{6}$.

The field equations derived from variation of equation (14) with respect to
the metric and scalar field are, respectively

$\displaystyle 2\xi|\phi|^{2}G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
S_{M,\mu\nu}+\Theta_{\mu\nu},$ (15)

and

$\displaystyle\xi\phi R-\Box\phi+\frac{\partial V}{\partial\phi}$
$\displaystyle=$ $\displaystyle 0,$ (16)

where

$\displaystyle 3\Theta_{\mu\nu}\equiv
6\xi\phi^{*}\phi_{;\mu;\nu}+(6\xi-3)\phi^{*}_{\mu}\phi_{\nu}$ (17)
$\displaystyle-$ $\displaystyle 6\xi
g_{\mu\nu}\left[\phi^{*}\Box\phi-\left(\frac{1}{4\xi}-1\right)\phi^{*}_{\alpha}\phi^{\alpha}\right]+c.c.$

Here and throughout
$S_{M,\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}V)}{\delta
g^{\mu\nu}}$ is the analog of the energy-momentum tensor. Equation (15) is a
generalization of Einstein equations with $2\xi|\phi|^{2}$ replacing $1/(16\pi
G)$, and $\Theta_{\mu\nu}$ is an effective contribution to the energy-momentum
tensor essentially due to gradients of G and active gravitational masses, the
latter appear in the source term, $\mathcal{L}_{M}$. Multiplying equation (16)
by $\phi^{*}$, adding the result to its complex conjugate and to the trace of
equation (15) results in

$\displaystyle\phi\frac{\partial V}{\partial\phi}+\phi^{*}\frac{\partial
V}{\partial\phi^{*}}=S_{M}+(1-6\xi)(\phi^{*}\Box\phi+\phi^{*}_{\alpha}\phi^{\alpha}+c.c).$
(18)

Together, equation (15) and equation (18) provide a system of dynamical
equations for the ten metric components and the complex scalar field. In the
BD form ($\phi$ is real and $\mathcal{L}_{M}$, that is here replaced by $V$,
is independent of $\phi$) equation (18) reduces to the well-known form
$(3+2\omega_{BD})\Box\Phi=T_{M}$. It follows from equation (18) that for any
$\xi\neq\frac{1}{6}$ (equivalently $\omega_{BD}\neq-3/2$) the scalar field is
dynamically determined by the matter distribution via a differential equation,
which thereby embodies the Machianity of BD theory of gravity.

In contrast, in the special case $\xi=\frac{1}{6}$ (equivalently
$\omega_{BD}=-3/2$ [4-15]) Equation (14) is identical to (11), and equation
(18) reduces to a constraint that only determines the dependence of $V$ on the
scalar field (as in the more general equation 10). In this special case,
$\phi$ is indeed non-dynamical; it can be set to any desired function,
irrespective of the matter distribution, and Machianity is lost. Although
$\phi$ still appears in equation (15) we now have only ten differential
equations for ten metric components and a scalar field – this under-determined
system simply reflects the freedom to locally rescale the fields in the
special case $\xi=\frac{1}{6}$, merely a manifestation of WI, equation (12),
[4-15]. The latter implies invariance of equation (11) under local-rescaling
of fields such that $V\rightarrow|\phi|^{-4}V$. This requires in particular
that, e.g.,

$\displaystyle\psi$ $\displaystyle\rightarrow$
$\displaystyle\Omega^{-\frac{3}{2}}\psi$ $\displaystyle A_{\mu}$
$\displaystyle\rightarrow$ $\displaystyle A_{\mu},$

where $\psi$ and $A_{\mu}$ are Dirac and vector fields, respectively, in
addition to the transformation of the scalar and metric fields described in
equation (12) – each field is rescaled by its mass/length dimension. In the
special case $\xi=\frac{1}{6}$, equation (18) could be derived solely from the
requirement that $V\sqrt{-g}$ is WI, as was done in the previous section in
the more general case, in equation (10).

Any mass terms that appear in $V$ are by definition active gravitational
masses, which need not be equivalent to inertial or passive gravitational
masses. While the three types of mass are not necessarily equivalent [30], the
notion of a passive gravitational mass in any theory of gravity that satisfies
the equivalence principle is vacuous. If the ratio of the latter two mass
types is a universal constant then the equivalence principle is satisfied – an
assumption that we indeed make in the present work. Again, we emphasize that
the other three fundamental interactions are described by the lagrangian of
the SM of particle physics, $\mathcal{L}_{SM}$, where all masses are inertial,
whether generated via the Higgs mechanism in the electroweak sector or via the
explicitly broken chiral symmetry in QCD.

Since $\phi(x)=\Omega^{-1}(x)\phi_{0}$, where $\phi_{0}$ is the fixed GR
value, then for any given such a choice $\Theta_{\mu\nu}$ can be calculated
according to equation (17) and used in equation (15) to solve for the
corresponding $g_{\mu\nu}$. In practice, however, it is easier to start from
any known solution of the Einstein equations, $\phi_{0}$ and $g_{\mu\nu}$, and
transform it to $\Omega^{-1}(x)\phi_{0}$ and $\Omega^{2}(x)g_{\mu\nu}$ with
arbitrary $\Omega(x)$ by virtue of WI. We emphasize, once again, that this
latter procedure is only possible assuming (as is virtually always assumed)
that there are no global charges associated with the field as is discussed in
section 5.

In the theory described by equation (11), the scalar field $\phi$ determines
not only $G$ but also active gravitational masses. Only pure radiation
$S_{rad}=0$ is consistent with the theory described by equation (14) with
$\xi=\frac{1}{6}$ unless $V$ explicitly depends on $|\phi|$. In the case of
perfect fluid $S=V$ on shell [31], and with EOS $w_{M}$ the trace is
$S=-\rho_{M}(1-3w_{M})$. It then follows from equation (18) that
$V\propto|\phi|^{1-3w_{M}}$ in case that $\xi=\frac{1}{6}$, i.e it is linear
and quartic in $|\phi|$ in cases of NR- and vacuum-like terms, respectively,
and is independent of $|\phi|$ in case of pure radiation. Linearity of $V$ in
$|\phi|$ in the case of vanishing EOS, $w_{M}=0$, suggests that active
gravitational masses are regulated by $|\phi|$. Not only that the same $\phi$
determines both Planck mass and active gravitational masses is a necessary
condition for the consistency of non-radiation sources with this WI theory, it
is also a conceptually natural “conclusion” as the concept of active
gravitational mass is meaningless unless it couples to G, and in this sense it
seems natural that both quantities are determined by the same scalar field.
This clearly does not have necessarily to be the case in general (as in e.g,
BD theory), but it is a nice merit of the WI model described by equation (11).

We emphasize that in the original BD proposal [19] the matter lagrangian,
$\mathcal{L}_{M}$, does not explicitly depend on the scalar field. Notably,
Brans & Dicke required $\omega_{BD}>0$ to guarantee the positivity of the
Hamiltonian in their original proposal [19], a fact that was emphasized and
reinterpreted in [32]. The instability of BD theories with $\omega_{BD}<0$ was
further emphasized in [33], but as we argued above, the non-positive kinetic
term of the scalar field is not an issue in the special case
$\omega_{BD}=-3/2$ and $V=V(|\phi|)$ due to WI. The observational lower limit
$\omega_{BD}\geq 40000$ reported in [20] implies that the BD field $\Phi$ is
very nearly a constant, essentially reducing the theory to GR. But, again,
this conclusion applies to the BD theory, where $\mathcal{L}_{M}$ is
independent of the scalar field, and in addition (perhaps not less important
and relevant to astrophysics and cosmology) this tight limit has been obtained
in our solar system and clearly does not readily apply on, e.g., galactic
scales.

##  4 The FRW spacetime In WIST representation

In the present section we cast the FRW action in the form of equation (11) in
a particular conformal frame where spacetime is static. The conventional
initial curvature singularity at the Big Bang is replaced in this alternative
description by the vanishing of $\phi$, i.e. of Planck and active
gravitational masses, at $t=0$. The redshifting Universe is then a reflection
not of space expansion, but rather of temporal evolution of masses, i.e. the
redshifting Universe is a manifestation of shrinking Planck length and Compton
wavelengths. The latter is a result of the fact that the time coordinate is
now the conformal time, $\eta$, rather than the cosmic time, $t$, as will be
discussed below.

The FRW action

$\displaystyle\mathcal{I}_{FRW}$ $\displaystyle=$
$\displaystyle\int[-a^{\prime 2}+Ka^{2}-\Lambda
a^{4}/3+a^{4}\mathcal{L}_{M}(a)]\sqrt{-\gamma}d^{4}x,$ (20)

is obtained from equation (1) in units where $3/(8\pi G)\equiv 1$ and assuming
that the metric is $g_{\mu\nu}=a^{2}\gamma_{\mu\nu}$, where the two metrics
$g_{\mu\nu}$ and $\gamma_{\mu\nu}$ are conformally related, and
$\gamma_{\mu\nu}\equiv diag(-1,\frac{1}{1-Kr^{2}},r^{2},r^{2}\sin^{2}\theta)$
is the metric in the comoving frame, $\gamma\equiv det(\gamma_{\mu\nu})$, and
after the term proportional to the corresponding curvature scalar
$a^{4}R=6a^{4}(a^{\prime\prime}/a+K)/a^{2}$ is integrated by parts. Here, $K$
is the spatial curvature parameter, $a(\eta)$ is the purely time-dependent
scale factor, a prime denotes derivatives with respect to $\eta$, where
$d\eta\equiv dt/a$ and $d^{4}x=d\eta drd\theta d\phi$.

Since null geodesics are blind to conformal metric transformations we expect
light to still be redshifted in the WIST description with the spatially-static
metric $\gamma_{\mu\nu}$ now replacing the standard expanding space metric
$g_{\mu\nu}$. This is explained by the fact that
$-d\eta^{2}=-\frac{dt^{2}}{a^{2}}$, and so the the lapse function,
$\gamma_{00}$, is $a^{-2}$ if the metric $\gamma_{\mu\nu}$ is presented in
cosmic time, $t$, coordinates, rather than conformal $\eta$. Incoming photon
wavelengths are not stretched by space expansion in this description (space is
static) but rather by the temporaly-evolving gravitational potential,
$\gamma_{00}(t)$, which is $\propto a^{-2}(\eta)$, which as we see below is
$\propto\phi^{-2}(\eta)$. In other words, cosmological redshift is
(indirectly) due to the growing (decreasing) Planck mass (length). A more
direct way to explain redshift is by the fact that in the comoving frame,
described by $\gamma_{\mu\nu}$, inertial masses $M_{0}$ are transformed to
$M=a(\eta)M_{0}$, i.e. the Rydberg constant monotonically increases with time.

We thus see that the ratio between either the Planck or active gravitational
mass to the inertial mass is time-independent on cosmological scales where the
distribution of matter is assumed to be homogeneous and isotropic and Eq. (20)
applies. However, we stress that this is not the case in general spacetimes
and matter distributions since the inertial mass is independent of the scalar
field. It is only due to the specific symmetries of the cosmological model
making it amenable to a particular combination of Weyl (e.g.
$g_{\mu\nu}\rightarrow\gamma_{\mu\nu}=g_{\mu\nu}/a^{2}$) and coordinate (i.e.
$dt\rightarrow d\eta=dt/a$) transformations that, e.g. the ratio of inertial-
to-Planck mass is fixed at the background level. However, in the presence of
small perturbations of the scalar field the Planck and active gravitational
mass are perturbed while inertial masses are not. This difference is critical
for certain applications. For example, it guarantees that CDM perturbations
can be ascribed in the present framework to scalar field perturbations
(essentially to $\lesssim 10^{-3}-10^{-4}$ fractional variations of $G$) on
galactic and galaxy cluster scales, [1], [34]. Otherwise they could be gauged
away and DM phenomena cannot be explained this way.

It can be readily verified that the Euler-Lagrange equation for the scale
factor $a(\eta)$ that extremizes the action $\mathcal{I}_{FRW}$ is indeed the
Friedmann equation $\mathcal{H}^{2}+K=a^{2}\rho_{M}+\frac{\Lambda
a^{2}}{3}+const./a^{2}$, where $\mathcal{H}\equiv a^{\prime}/a$ is the
conformal Hubble function, and it is assumed that $\mathcal{L}_{M}(a)$ is a
power-law in $a$, much like $\rho_{M}$ is a power-law in standard cosmology.
Here, $\mathcal{L}_{M}(a)$ accounts for all sources (e.g. dust, radiation,
etc.) other than spatial curvature and cosmological constant (which are
characterized by effective equations of state -1/3 and -1, respectively).
Completing the derivation requires that the integration constant is related to
the energy density of radiation $\rho_{r}=const./a^{4}$. Since $\rho_{M}>0$ it
is clear that the kinetic and potential terms in equation (20) have the
‘wrong’ relative sign (unless $K>0$ or $\Lambda<0$), yet the GR-based FRW
model is not considered to be ‘plagued with ghosts’, or ‘disastrous
instabilities’ at the classical level. On the contrary, FRW is the backbone of
the remarkably successful standard cosmological model.

Re-defining the scale factor $\phi\equiv a$, the FRW action is reformulated as
a WIST theory with a non-positive kinetic term, defined on a static background

$\displaystyle\mathcal{I}_{FRW}$ $\displaystyle=$
$\displaystyle\int\left(\frac{1}{6}R\phi^{2}-\phi^{\prime
2}-\lambda\phi^{4}+\tilde{\mathcal{L}}_{M}(\phi)\right)\sqrt{-\gamma}d^{4}x.$
(21)

Here, $\tilde{\mathcal{L}}_{M}=a^{4}\mathcal{L}_{M}$, $R=6K$ and
$\lambda\equiv\Lambda/3$. The single degree of freedom of standard FRW
spacetime, $a(\eta)$, is here replaced by the scalar field $\phi(\eta)$.
Clearly, renaming the scale factor, $a\rightarrow\phi$, in going from the
general relativistic equation (20) to the WIST form, equation (21), does not
introduce any new dynamics, in particular no unacceptable new instabilities
that are not already present in the FRW spacetime arise due to this field
redefinition. The relative ‘wrong’ sign between the kinetic and potential term
is simply manifested in that the integration of the Friedmann equation results
in $a(\eta)$ [or equivalently $\phi(\eta)$] that evolves monotonically with
$\eta$ rather than having oscillatory behavior.

As equation (20) and equation (21) are equivalent any perturbation is either
$a(\eta)$ of equation (20) or $\phi(\eta)$ of equation (21) will have
precisely the same dynamics. Exactly as in the standard cosmological model
$\delta a$ is never considered as a dynamical degree of freedom, so should be
the case with $\delta\phi$. This is so since
$g_{\mu\nu}=a^{2}\gamma_{\mu\nu}$, and therefore any perturbation in $a$ can
be absorbed in $\delta\gamma_{\mu\nu}$, a perturbation of $\gamma_{\mu\nu}$.
Considering scalar metric perturbations only – these are the Newtonian
potentials. Therefore, invoking WI implies that any perturbation in $G$
induces scalar metric perturbations [2], and vice versa any excessive
gravitational potential could be explained by corresponding perturbations in
$G$ [1].

As we just saw, local isotropic scaling has sufficient freedom to allow for
reformulation of the (background) FRW model on a static background. However,
this is not generally the case with Bianchi Universes. The latter are
homogeneous but not isotropic. As such, they may have up to three different
expansion rates along the three principal axes for which a corresponding
number of functional degrees of freedom are required. For that purpose, the
more general anisotropic local rescaling freedom described in section 2 is
required. In addition, at the perturbation level, vector and tensor
perturbation modes cannot be accounted for by a single function $\phi$, and
for that purpose the more general theory described in section 2 is required as
well, as was already mentioned above.

##  5 Global Symmetry

As reviewed in section 3 one of the often-made claims about WIST is that the
procedure of ‘conformalizing’ GR does not really result in modified physics.
Specifically, it is argued, the conformal frame can always be so chosen so as
to recover GR from equation (11) by setting $\phi$ to a constant. Irrespective
of the validity of this claim in light of the discussion towards the end of
section 2, it is clear that if a global symmetry is associated with the tensor
field $\Phi_{\mu}^{\nu}$ then the presence of a global conserved charge
definitely modifies the theory. Specifically, the field $\Phi_{\mu}^{\ \nu}$
itself can be generalized to carry the indices of an internal (non-coordinate)
symmetry, i.e. $\Phi_{\mu}^{\ \nu}\rightarrow{\bf\Phi}_{\mu}^{\
\nu}=(\Phi_{I}^{J})_{\mu}^{\ \nu}$ where boldface letters indicate that the
tensor field carries internal indices (capital Latin letters) of a (typically
unitary) group. The metric on (internal) group space is denoted
$\mathcal{G}_{IJ}$, such that e.g., ${\bf\Phi}_{\mu}^{\
\alpha}{\bf\Phi}_{\alpha}^{\ \nu}$ is a shorthand notation for
$(\Phi_{I}^{K})_{\mu}^{\ \alpha}\mathcal{G}_{KL}(\Phi_{M}^{L})_{\alpha}^{\
\nu}$, and it is understood that contracted capital Latin letters are summed
over. Equation (2) is then generalized to

$\displaystyle g_{\mu\nu}$
$\displaystyle\rightarrow\bar{g}_{\mu\nu}\equiv{\bf\Phi}_{\mu}^{\
\alpha}g_{\alpha\beta}{\bf\Phi}_{\ \nu}^{\beta},$
$\displaystyle\mathcal{L}_{M}$
$\displaystyle\rightarrow\bar{\mathcal{L}}_{M}/|{\bf\Phi}|.$ (22)

When expanded, the first of equations (22) is
$g_{\mu\nu}\rightarrow\bar{g}_{\mu\nu}\equiv(\Phi_{I}^{K})_{\mu}^{\
\alpha}g_{\alpha\beta}\mathcal{G}_{KL}(\Phi^{LI})_{\ \nu}^{\beta}$. In the
special case that the internal symmetry is described by a unitary group (whose
elements are $U_{IJ}$) then no stretch and contraction of the fundamental
units takes place ($|U|=1$), i.e. no local scale invariance, but rather only a
global charge is added to the theory. In this case, derivatives of the new
metric field generically result in quadratic kinetic terms of phases that
function as ‘cyclic coordinates’ on the group manifold, thereby resulting in
conserved charges.

We stress that the internal symmetry is assumed to apply only to the field
${\bf\Phi}_{\mu}^{\ \nu}$, and not to any other fields. There is a conserved
global charge that is associated with this global symmetry that physically
alters the field equations of gravitation. The presence of this charge
modifies the kinetic term associated with the field, as is demonstrated in
e.g., the U(2) case in Appendix B. An example of this within a WIST-based
bouncing cosmological model is described in [34].

##  6 Summary

Symmetry plays a central role in our current understanding of the fundamental
interactions. The SM of particle physics is based on a local
SU(3)$\times$SU(2)$\times$U(1) gauge symmetry, and GR is diffeomorphism-
invariant. In the present work, the idea that GR accommodates a broader
symmetry, anisotropic local scale invariance (of which WI is just a special
case), is explored first. The underlying idea is that not only that the
gravitational interaction (and by induction also all the other interactions
which take place in the spacetime arena) does not depend on the reference
frame of the observer (as is manifested by diffeomorphism invariance of, e.g.,
the Einstein and geodesic equations) it also does not depend on (a priori
arbitrary) choices of units, i.e. yardsticks, e.g. the Planck length scale,
$l_{P}$. The latter is the natural yardstick in vacuum. In the non-vacuum case
the natural meter stick would be the dynamical gravitational time scale,
$(G\rho_{M})^{-1/2}\propto l_{P}^{-1}\rho_{M}^{-1/2}$ where $G$ is Newton
constant and $\rho_{M}$ is the energy density of matter. The standard units
choice adopted in GR is that both $G$ and particle masses (active
gravitational masses included) are fixed constants.

Newton constant, $G$, has been measured in our solar system at a reasonable
precision, and is purportedly also deduced (at the few percent precision)
outside of our solar system from Big Bang nucleosynthesis (BBN) via
observational inference of light element abundances , etc. However, all these
estimates of $G$ are based on the premise that all other universal constants
are truly constants, as is clearly manifested in the Brans-Dicke program. In
addition, Newtonian gravity fails to account for a few observed phenomena on
galactic scales and beyond in spite of the fact that the gravitational pull is
weak in these systems and Newtonian gravity is expected, at least naively, to
provide a reasonably accurate description. The standard solution to this
problem involves an assumption about the existence of a non-luminous DM
substance, presumably exotic electrically neutral particles. Another way,
albeit less popular, to tackle the problem has involved modifying the
Newtonian dynamics on sufficiently large scales. Both remedies assume that $G$
is a fixed universal constant across space and time; in the absence of
compelling direct evidence for the universality of $G$ this has to be assumed.

However, positing WI as an additional symmetry of (what is usually considered
GR) opens up new possibilities. Promoting $G$ to a tensor (section 2) or
scalar (section 3) scale-deformation field necessitates that active
gravitational masses are regulated by the same fields as well. Clearly, it is
assumed here that (anisotropic or isotropic) local scale invariance applies
only within the domain of validity of GR. This is of course not the case on
microscopic scales. While GR is assumed here to be locally scale-invariant,
the SM of particle physics is assumed to be as is, i.e. non-scale-invariant,
with no modifications. The effective energy and pressure contained in spatio-
temporal variations of the scalar field, i.e. $G$ and active gravitational
masses, then provide additional (non-particulate) source for spacetime
curvature, thereby accounting for, e.g. ‘DM phenomena’ with no recourse to DM.
In the present work we explored a few key aspects of the underlying locally
scale-invariant generalizations of GR. In addition to local scale invariance,
we equip the ‘scaling’ fields ($\phi$ or $\Phi_{\mu\nu}$) with internal
symmetry and generally non-vanishing global charge.

It should be emphasized that whereas the case isotropic local rescaling has
long been explored, its generalization, i.e., the anisotropic case that is
discussed in section 2, has not been considered in the literature in full
generality (except for the special case of Minkowski background where it is
known as the ‘vierbein formalism’) to the best of our knowledge.

While GR is modified in this framework, the SM is not. We stress that unlike
in other frameworks that ‘conformalize’ all the four fundamental interactions
and so any ‘new physics’ can be entirely gauged away by means of an
appropriate Weyl transformation (in the absence of new global charges), this
cannot be done in the framework adopted in the present work. It should be also
emphasized that even if all the four fundamental interactions are
‘conformalized’ the presence of internal symmetry and global charge do modify
the field equations, i.e. the dynamics, and consequently give rise to ‘new
physics’.

Whereas GR is recovered from the locally anisotropic scale-invariant or WIST
theory in a given conformal frame (in case that $G$ is set to a constant), the
additional functional degrees of freedom (obtained by promoting $m_{p}$ to a
scalar $\phi$ or tensor $\Phi_{\mu\nu}$ field) provides an ample freedom that
allows doing away with clustered DM as mentioned above. A notable property of
$\phi$ is that it is formally a ‘ghost’ field, i.e. its kinetic and potential
terms have the ‘wrong’ relative sign. However, this property by itself does
not disqualify the theory because the problem with ghost-afflicted theories is
that the field configuration dynamically runs away from its ground state, or
equilibrium, by ever lowering the kinetic energy while increasing the
potential term. However, as was explicitly shown in section 3, and as is
equally clear from the underlying WI, the spacetime-dependent $\phi$ in the
case of WIST, or $\Phi_{\mu}^{\nu}$ in the anisotropic case, is non-dynamical;
it is simply not required to obey any specific Euler-Lagrange equation.
Consequently, physical systems described by locally scale-invariant theories
of gravity are not doomed to runaway from stable field configurations. We also
illustrated this conclusion with the redshifting Universe in section 4. The
FRW spacetime can be recast in the canonical WIST form on a static background.
Cosmological redshift is then explained not by space expansion but rather by
contraction of our fundamental yardsticks, e.g. the Rydberg ‘constant’, which
are (directly and indirectly) regulated by $\phi$. Whereas $\phi$ can be
locally rescaled along with a corresponding rescaling of $g_{\mu\nu}$,
adopting a specific conformal frame in which the metric is static implies that
$\phi$ satisfies exactly the same Friedmann equation which is normally
satisfied by $a$, the scale factor that describes space expansion in the
standard cosmological model. The conformal Hubble function
$\mathcal{H}=a^{\prime}/a$ is now replaced by $\phi^{\prime}/\phi$. In the
WIST version of the FRW action the negative kinetic term has been actually
increasing ($\phi^{\prime}/\phi$ monotonically decreases) asymptotically
towards zero over the entire cosmic history between the ‘Big Bang’ and the
present time. This is a counter-example to often-raised concerns about
instabilities associated with WIST. It is true that on the face of it we could
have found ourselves in a contracting Universe with an ever decreasing
negative kinetic term of either $a$ or $\phi$ (in the GR and WIST formulations
respectively) culminating in a ‘big crunch’, but even in this case the WIST
dynamics is not ‘worse’ than that of GR. More general cosmological models,
e.g. Bianchi-types models, that are described by non-isotropic expansion rates
are in general described by three scale factors. This family of models can be
easily described by the anisotropic locally scale-invariant theory.

## Appendix A: The Weak gravitational field limit as an inertial force

The classical trajectories followed by particles in GR are described by
geodesics

$\displaystyle\ddot{x}^{\mu}+\Gamma^{\mu}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma}=0,$
(23)

where $\Gamma^{\mu}_{\rho\sigma}$ is the Christoffel symbol that is calculated
from the metric field and its derivatives in the usual way, and
$\dot{x}^{\mu}\equiv\frac{dx^{\mu}}{d\lambda}$, with $\lambda$ an affine
parameter. Equation (23) is subject to the constraint
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=c$ where $c$ equals -1 or 0 in case of
massive or massless particles, respectively. In the presence of additional
non-gravitational forces, $F^{\mu}$, equation (23) generalizes to

$\displaystyle\ddot{x}^{\mu}+\Gamma^{\mu}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma}=\frac{F^{\mu}}{m},$
(24)

in the case of a massive particle of mass $m$. Now, the Christoffel symbol
transforms inhomogeneously under Weyl transformations (equation 12)

$\displaystyle\Gamma^{\mu}_{\rho\sigma}\rightarrow\tilde{\Gamma}^{\mu}_{\rho\sigma}=\Gamma^{\mu}_{\rho\sigma}+\Omega^{-1}(\delta_{\rho}^{\mu}\Omega_{\sigma}+\delta_{\sigma}^{\mu}\Omega_{\rho}-g_{\rho\sigma}g^{\mu\delta}\Omega_{\delta}).$
(25)

Therefore, in the case $F^{\mu}=0$, equation (23) is rewritten in the new
conformal frame as

$\displaystyle\ddot{x}^{\mu}+\tilde{\Gamma}^{\mu}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma}=\frac{2\Omega_{\sigma}}{\Omega}\dot{x}^{\sigma}\dot{x}^{\mu}-cg^{\mu\nu}\Omega_{\nu}/\Omega,$
(26)

or equivalently

$\displaystyle\ddot{x}^{\mu}+\tilde{\Gamma}^{\mu}_{\rho\sigma}\dot{x}^{\rho}\dot{x}^{\sigma}=-\frac{2\phi_{\sigma}}{\phi}\dot{x}^{\sigma}\dot{x}^{\mu}+cg^{\mu\nu}\phi_{\nu}/\phi.$
(27)

We see that what is considered as a pure gravitational force in one conformal
frame is interpreted as not pure gravitational in other frames. A striking
example is the case where spacetime is described by Minkowski metric in the
new frame, and in cartesian coordinates then
$\tilde{\Gamma}^{\mu}_{\rho\sigma}$ vanishes. In this particular case equation
(27) describes a purely non-gravitational interaction. The particles (massive
or massless) follow exactly the same geodesics (that describe e.g.
gravitational clustering of massive particles or light bending of massless
photons) in the two conformal frames, but while in the first frame it is due
to spacetime curvature in the other frame it is due to an inertial ‘force’,
$F\equiv-2\dot{x}^{\mu}\dot{x}^{\nu}\phi_{\nu}/\phi+c\eta^{\mu\nu}\phi_{\nu}/\phi$,
where $\eta^{\mu\nu}$ is Minkowski metric.

For example, the line element that describes a weakly perturbed Minkowski
spacetime in Cartesian coordinates and in case that stress is negligible, and
neglecting vector and tensor modes, is
$ds^{2}=-(1+2\varphi)dt^{2}+(1-2\varphi)dx^{i}dx^{j}$. Here $\varphi(x)$ is
the gravitational potential. Applying a Weyl transformation $\Omega=1+\varphi$
the metric in the new conformal frame becomes purely Minkowski, and
$\tilde{\Gamma}^{\mu}_{\rho\sigma}$ vanishes in Cartesian coordinates.
However, the r.h.s of equation (27) does not vanish now. The entire kinematics
of test particles is now due to the fact that
$\phi_{0}\rightarrow\tilde{\phi}\approx\phi_{0}[1-\varphi(x)]$. In other
words, light bending and gravitational clustering are not due to spacetime
curvature (which is flat in this case) but rather due to the spacetime
dependence of $G$ and active gravitational masses. In the presence of vector
and/or tensor modes the single functional degree of freedom, $\phi$, added by
the WIST theory is insufficient for entirely flattening the spacetime geometry
and the more general locally scale invariant theory that is described in
section 2 is required. The tensor field $\Phi_{\alpha\beta}$ can be used
reformulate GR on a Minkowski spacetime, in which case it reduces to the
vierbein field. The generalization of equation (27) in that case is
straightforward.

## Appendix B: The Case of U(2) Global Symmetry

As an example for possibilities opened up by introducing internal symmetries
associated with the gravitational interaction, we consider the case that the
scalar field $\boldsymbol{\phi}$ is a U(2) multiplet. Elements of this group
are represented by

$\boldsymbol{\phi}=\rho\left({\begin{array}[]{cc}\cos(\theta)e^{i\varphi}&\sin(\theta)e^{i\chi}\\\
-\sin(\theta)e^{-i\chi}&\cos(\theta)e^{-i\varphi}\\\ \end{array}}\right),$
(28)

where $\rho$ is the modulus of the field and $\theta$, $\varphi$ and $\chi$
are three phases. The line element on this group manifold is
$ds_{\boldsymbol{\phi}}^{2}=\mathcal{G}_{IJ}d\phi^{I}d\phi^{J}$, where
contracted capital Latin indices are summed over. Since this is a unitary
group then,
$ds_{\boldsymbol{\phi}}^{2}=\frac{1}{2}Tr(d\boldsymbol{\phi}^{\dagger}d\boldsymbol{\phi})=-\frac{1}{2}Tr(\boldsymbol{\phi}^{-1}d\boldsymbol{\phi})^{2}=d\rho^{2}+\rho^{2}\left[\cos^{2}(\theta)d\varphi^{2}+\sin^{2}(\theta)d\chi^{2}+d\theta^{2}\right]$.
The lagrangian of the kinetic term,
$\mathcal{L}_{kin}=\mathcal{G}_{IJ}\phi_{\mu}^{I}\phi^{\mu,J}$ (which is a
straightforward generalization of the kinetic term in Equation 11) is
therefore
$\mathcal{L}_{kin}=-\dot{\rho}^{2}-\rho^{2}[\cos^{2}(\theta)\dot{\phi}^{2}+\sin^{2}(\theta)\dot{\chi}^{2}+\dot{\theta}^{2}]$.
The Euler-Lagrange equations associated with $\chi$ and $\varphi$ that are
obtained from variations of $\mathcal{L}_{kin}$ can be each integrated once to
give

$\displaystyle\dot{\chi}$ $\displaystyle=$
$\displaystyle\frac{c_{\chi}}{\rho^{2}\sin^{2}\theta}$
$\displaystyle\dot{\varphi}$ $\displaystyle=$
$\displaystyle\frac{c_{\varphi}}{\rho^{2}\cos^{2}\theta},$ (29)

where $c_{\chi}$ and $c_{\varphi}$ are integration constants. Employing these
in the field equation for $\theta$ and integrating once we obtain

$\displaystyle\rho^{4}\dot{\theta}^{2}=-\frac{c_{\chi}^{2}}{\sin^{2}\theta}-\frac{c_{\varphi}^{2}}{\cos^{2}\theta}+c_{\theta}^{2},$
(30)

where $c_{\theta}$ is another integration constant. Employing all these in the
kinetic term
$\phi_{\mu}\phi^{\mu}=-\dot{\rho}^{2}-\rho^{2}[\cos^{2}(\theta)\dot{\varphi}^{2}+\sin^{2}(\theta)\dot{\chi}^{2}+\dot{\theta}^{2}]$
we obtain $-\dot{\rho}^{2}-\frac{c_{\theta}^{2}}{\rho^{2}}$, and both
integration constants $c_{\chi}$ and $c_{\varphi}$ vanish from our final
expression. All this amounts to replacing $\phi$ with $\rho$ everywhere in
equation (11) in addition to introducing a negative contribution to the
potential, $-\frac{c_{\theta}^{2}}{\rho^{2}}$.

In the cosmological context this new term competes with the radiation term at
sufficiently small $\rho$. This term, obtained due to the U(2) symmetry of the
scalar field (a symmetry which is not recognized by GR) rather than having to
recourse to exotic forms of matter with negative energy density, is
responsible to a bounce, where the cosmological expansion rate momentarily
halts. As expected, this is exactly the result obtained in [34] in case of
U(1) symmetry. The constant $c_{\theta}$ is simply a global charge associated
with the U(1) subgroup, which is the relevant symmetry in a ‘central’
potential, i.e. a potential which is independent on the phase, and depends
only on $\rho$. In cosmology the term $\propto c_{\theta}$ serves as an
effective stiff matter with a negative energy density, thereby replacing the
initial Big Bang singularity with a Big Bounce [34].

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