# Losing the trace to find dynamical Newton or Planck constants

Pavel Jiroušek<EMAIL_ADDRESS>CEICO–Central European Institute for Cosmology
and Fundamental Physics,
FZU–Institute of Physics of the Czech Academy of Sciences,
Na Slovance 1999/2, 18221 Prague 8, Czech Republic  Institute of Theoretical
Physics, Faculty of Mathematics and Physics, Charles University,
V Holešovičkách 2, 180 00 Prague 8, Czech Republic  Keigo Shimada
<EMAIL_ADDRESS>Department of Physics, Tokyo Institute of
Technology, Tokyo 152-8551, Japan  Alexander Vikman<EMAIL_ADDRESS>CEICO–Central European Institute for Cosmology and Fundamental Physics,
FZU–Institute of Physics of the Czech Academy of Sciences,
Na Slovance 1999/2, 18221 Prague 8, Czech Republic  Masahide Yamaguchi
<EMAIL_ADDRESS>Department of Physics, Tokyo Institute of Technology,
Tokyo 152-8551, Japan

(August 27, 2024)

###### Abstract

We show that promoting the trace part of the Einstein equations to a trivial
identity results in the Newton constant being an integration constant. Thus,
in this formulation the Newton constant is a global dynamical degree of
freedom which is also a subject to quantization and quantum fluctuations. This
is similar to what happens to the cosmological constant in the unimodular
gravity where the trace part of the Einstein equations is lost in a different
way. We introduce a constrained variational formulation of these modified
Einstein equations. Then, drawing on analogies with the Henneaux-Teitelboim
action for unimodular gravity, we construct different general-covariant
actions resulting in these dynamics. The inverse of dynamical Newton constant
is canonically conjugated to the Ricci scalar integrated over spacetime.
Surprisingly, instead of the dynamical Newton constant one can formulate an
equivalent theory with a dynamical Planck constant. Finally, we show that an
axion-like field can play a role of the gravitational Newton constant or even
of the quantum Planck constant.

##  I Introduction and Our Main Idea

The origin of the cosmological constant and its relation to the vacuum
fluctuations of quantum fields persists to be one of the main puzzles of
contemporary physics, see e.g. (Zeldovich:1968ehl, ; Weinberg:1988cp, ;
Martin:2012bt, ; Padilla:2015aaa, ). However, since Einstein’s seminal paper
(Einstein:1919gv, ) it is well known that trace-free equations

$G_{\mu\nu}-\frac{1}{4}g_{\mu\nu}G=8\pi
G_{N}\left(T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T\right)\,,$ (1)

result in a cosmological constant (CC), $\Lambda$, as a constant of
integration, or _a global_ degree of freedom, see e.g. (Anderson:1971pn, ;
Weinberg:1988cp, ; Unruh:1988in, ; Unruh:1989db, ; Finkelstein:2000pg, ;
Ellis:2010uc, ; Ellis:2013eqs, ). For all other purposes, the trace-free
equations above are entirely classically equivalent to the usual General
Relativity (GR) equations with $\Lambda$

$G_{\mu\nu}=8\pi G_{N}T_{\mu\nu}+\Lambda g_{\mu\nu}\,.$ (2)

Crucially, the trace-free Einstein equations (1) are invariant under vacuum
shifts of the total energy-momentum tensor (EMT)

$T_{\mu\nu}\rightarrow T_{\mu\nu}+c\,g_{\mu\nu}\,,$ (3)

with $c=c\left(x^{\mu}\right)$, and in particular with $c=const$, which is not
the case for the usual Einstein equations (2), with or without $\Lambda$.

Nothing illustrates better the currently unexplainable small value of
$\Lambda$, as a comparison with $G_{N}$ — the only other dimensionful111We use
$\left(+,-,-,-\right)$ signature convention and units where $c=1$, while in
most parts of the text $\hbar=1$ and we restore $\hbar$ only to stress the
quantum nature of a formula. constant in (2):

$G_{N}\Lambda\sim 10^{-122}\,.$ (4)

The usual question to ask regarding these 122 orders of magnitude is: why is
$\Lambda$ so vanishingly small, but still not entirely zero? Perhaps one
should invert the question and ask instead: why is $G_{N}$ that tiny? Both
these questions require an ability to select values on these constants from
some ensemble. Thus, it makes sense to look for simple theories where $G_{N}$
would appear as an integration constant or a global degree of freedom. This is
a “landscape” of poor people.

With this motivation in mind, in this paper we propose a novel way to lose the
trace part of the Einstein equations. Indeed, instead of (1) one can write
_normalized_ or _trace-trivial_ equations

$\frac{G_{\mu\nu}}{G}=\frac{T_{\mu\nu}}{T}\,.$ (5)

These equations are _scale-free_ and invariant with respect to rescaling222It
is worth noting that transformations (3) and (6) could be performed on the
Einstein tensor instead of the EMT, see e.g. (Alexander:2018tyf, ). However,
it is not clear how to realize these transformations as transformations of the
metric. Moreover, the equations are completely blind to independent
transformations of both tensors.

$T_{\mu\nu}\rightarrow c\,T_{\mu\nu}\,,$ (6)

by a constant, or even an arbitrary function $c\left(x^{\mu}\right)$. As we
show in this paper, equations (5) are again equivalent to the usual GR,
however now $G_{N}$ becomes a constant of integration.

Following Einstein (Einstein:1919gv, ) on the cosmological constant, one can
either conclude that “… the new formulation has this great advantage, that the
quantity appears in the fundamental equations as a constant of integration,
and no longer as a universal constant peculiar to the fundamental law” or
rather that “Since the world exists as a single specimen, it is essentially
the same whether a constant is given the form of one belonging to the natural
laws or the form of an ‘integration constant’”, as in (Einstein:1918c, )333For
history of the cosmological constant see e.g. (ORaifeartaigh:2017yby, ).
Crucially, quantum cosmology and chaotic self-reproducing inflation
(Vilenkin:1983xq, ; Linde:1986fd, ) allow for a multiverse so that the world
would be not a “single specimen”, as thought by Einstein. Thus, one can
speculate that the value of the Newton constant may be a remnant from the
early universe quantum gravity era, similarly to what could happen to the
cosmological constant, as discussed in e.g. (Linde:1984ir, ; Linde:2015edk, ).

##  II Recap of unimodular gravity

Our proposal shares some structural similarities with the unimodular gravity.
Hence, it is worthwhile to recall its basic properties here. As described in
e.g. (Anderson:1971pn, ; Weinberg:1988cp, ; Unruh:1988in, ; Unruh:1989db, ;
Finkelstein:2000pg, ; Ellis:2010uc, ; Ellis:2013eqs, ), one can apply
covariant derivative $\nabla_{\mu}$ to both sides of (1) and by virtue of the
Bianchi identity, $\nabla_{\mu}G^{\mu\nu}=0$, along with the _assumed_
conservation of EMT, $\nabla_{\mu}T^{\mu\nu}=0$, one obtains444In some
formulations of gravity, the Bianchi identity may not hold which results in
non-vanishing right hand side of this differential consequence. The
implications of this have been recently explored in (Alexander:2019ctv, ;
Alexander:2019wne, ; Magueijo:2019vmk, ).

$\partial_{\mu}\left(G-\varkappa T\right)=0\,,$ (7)

where we introduced notation

$\varkappa=8\pi G_{N}\,,$ (8)

to make later expressions shorter. In this way

$G-\varkappa T=4\Lambda\,,$ (9)

where $\Lambda$ is a constant of integration. Substitution of this
differential consequence back into the trace-free Einstein equations (1)
yields normal Einstein’s General Relativity (GR) with the cosmological
constant $\Lambda$. Thus, loosing the trace-part of the standard Einstein
equations in this way results in promoting (or demoting) the cosmological
constant (CC) from the fundamental constants of nature, parameters or coupling
constants to a global dynamical degree of freedom. This modification of GR is
known under the name “unimodular gravity”.

One can obtain the traceless Einstein equations (1), provided one varies the
Einstein-Hilbert action under _constrained_ or _restricted_ variation of the
contravariant metric, $\bar{\delta}g^{\mu\nu}$, satisfying

$g_{\mu\nu}\bar{\delta}g^{\mu\nu}=0\,.$ (10)

This variational constraint implies that only _transverse_ variations
$\bar{\delta}g^{\mu\nu}$ are allowed. This general-covariant restriction (10)
of the variations $\delta g^{\mu\nu}$ can be obtained from an apparently non-
covariant “unimodular” constraint

$\sqrt{-g}\equiv\sqrt{-\text{det}g_{\mu\nu}}=\sigma\left(x\right)\,,$ (11)

where $\sigma\left(x^{\mu}\right)$ is an unspecified _function_ which is often
taken to be just unity:

$\sqrt{-g}=1\,.$ (12)

However, the later choice manifestly breaks general covariance and would
select preferred coordinate systems. Moreover, this selection clearly excludes
writing most important solutions of GR like the Schwarzschild and Friedmann
metrics in natural coordinates which have simple physical interpretation and
which can definitely be formed by test bodies and clocks. Clearly, for every
metric, $g_{\mu\nu}$, there is enough gauge freedom to find a coordinate
system to satisfy the “unimodular” constraint (12), but it is especially easy
if one does not specify the function $\sigma\left(x\right)$ at all. The
irrelevant choice of this function corresponds to a gauge freedom responsible
for the diffeomorphism invariance. In the manifestly general-covariant
formulation of the unimodular gravity by Henneaux-Teitelboim (Henneaux:1989zc,
)555For other diff invariant formulations of the unimodular gravity, see e.g.
(Kuchar:1991xd, ; Jirousek:2018ago, ; Hammer:2020dqp, ) the meaning of the
“unimodular” constraint is reversed: it is used not to find $\sqrt{-g}$ for a
given $\sigma\left(x\right)$, but to find instead a dynamical variable
composing $\sigma$ for a given $\sqrt{-g}$. Most importantly, restriction on
variations (10) is covariant and so are the resulting traceless equations of
motion (1). Thus, it is rather counterproductive to take $\sqrt{-g}=1$ and
break their general covariance by this unnecessary assumption, even though
this choice makes the action polynomial (vanderBij:1981ym, ) and may simplify
some calculations.

To quickly see that (10) results in (1) one writes first variation form of the
Einstein equations for _general_ variations of the metric which is also
satisfied by the restricted variations $\bar{\delta}g^{\mu\nu}$

$\left(G_{\mu\nu}-\varkappa T_{\mu\nu}\right)\bar{\delta}g^{\mu\nu}=0\,.$ (13)

However, due to the transversality constraint (10) only tensor components not
“proportional” to $g_{\mu\nu}$ had to be equal to zero now. This results in
appearance of the Lagrange multiplier $\lambda\left(x^{\mu}\right)$
responsible for the “reaction force” of the constraint

$G_{\mu\nu}-\varkappa T_{\mu\nu}=\lambda\left(x\right)g_{\mu\nu}\,.$ (14)

This is an analog of the Lagrange’s equations of the first kind in analytical
mechanics. One can exclude unspecified Lagrange multiplier by taking the trace
of (14), so that $\lambda\left(x\right)=\left(G-\varkappa T\right)/4,$ and
then substituting this expression back to (14) one obtains trace-free Einstein
equations (1). Of course, we could apply covariant derivative and use Bianchi
identity along with the covariant conservation of EMT directly in (14) to find
that $\lambda\left(x\right)=const$.

It is well known how to write an action for the trace-free Einstein equations
(1). The arguably simplest covariant action was introduced by Henneaux and
Teitelboim in (Henneaux:1989zc, )

$S\left[g,W,\Lambda\right]=\frac{1}{\varkappa}\int
d^{4}x\sqrt{-g}\left[-\frac{R}{2}+\Lambda\left(\nabla_{\mu}W^{\mu}-1\right)\right]\,.$
(15)

The second term in the brackets has a form of the Hamiltonian (or first order)
action. Furthermore, variation with respect to the spatial components $W^{i}$
yields $\partial_{i}\Lambda=0$, along every Cauchy hypersurface $\Sigma$.
Hence, for every imposed foliation, $\Lambda$ is space independent - is a
global quantity. Using this constraint and following Faddeev-Jackiw
(Faddeev:1988qp, ; Jackiw:1993in, ), one concludes “without tears” that
$\Lambda/\varkappa$ is a canonical momentum conjugated to the cosmic time

$\mathscr{T}\left(t\right)=\int_{\Sigma}d^{3}\mathbf{x}\sqrt{-g}\,W^{t}\left(t,\mathbf{x}\right)\,.$
(16)

This variable measures four-volume of space-time between Cauchy hypersurfaces
$\Sigma_{2}$ and $\Sigma_{1}$ as

$\mathscr{T}\left(t_{2}\right)-\mathscr{T}\left(t_{1}\right)=\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,.$
(17)

One can think about $\mathscr{T}$ as a charge which is not conserved, but
rather continuously produced by a unit source due to the constraint

$\nabla_{\mu}W^{\mu}=1\,.$ (18)

As $\Lambda/\varkappa$ and $\mathscr{T}$ are canonically conjugated, one can
apply Heisenberg uncertainty relation

$\delta\Lambda\times\delta\int_{\mathscr{V}}d^{4}x\sqrt{-g}\geq
4\pi\,\ell_{Pl}^{2}\,,$ (19)

where $\ell_{Pl}=\sqrt{\hbar G_{N}}$ is the Planck length. The presence of
these quantum fluctuations is the main difference of “unimodular” gravity from
the usual GR. These fluctuations may be relevant close to singularities.
Except of this phenomenon, perturbative “unimodular” gravity is equivalent to
GR also in quantum realm, for recent discussions see e.g. (Alvarez:2005iy, ;
Fiol:2008vk, ; Smolin:2009ti, ; Eichhorn:2013xr, ; Saltas:2014cta, ;
Padilla:2014yea, ; Bufalo:2015wda, ; Alvarez:2015sba, ; Alvarez:2015pla, ;
Percacci:2017fsy, ; Ardon:2017atk, ; Herrero-Valea:2018ilg, ; Herrero-
Valea:2020xaq, ; deBrito:2020rwu, ).

The constraint part of the action (15) can be written as

$\int
d^{4}x\,\Lambda\left[\partial_{\mu}\left(\sqrt{-g}W^{\mu}\right)-\sqrt{-g}\,\right]\,,$
(20)

so that $\partial_{\mu}\left(\sqrt{-g}W^{\mu}\right)$ plays the role of the
unspecified function $\sigma\left(x\right)$ in (11) or the
measure666Interestingly, this can be considered as a particular example of
theories with two different measures, see e.g. (Guendelman:1996qy, ;
Guendelman:1999qt, ; Guendelman:1999tb, ).  from (Anderson:1971pn, ). Though,
now the “unimodular” constraint (11) is used to find $W^{\mu}$ without
imposing any restrictions on the metric.

On the other hand, action (15) possesses a gauge redundancy

$W^{\mu}\rightarrow
W^{\mu}+\epsilon^{\mu}\,,\quad\text{where}\quad\nabla_{\mu}\epsilon^{\mu}=0\,.$
(21)

For every $W^{\mu}$ and every choice of coordinates
$\left(t^{\prime},x^{i}\right)$ one can find such $\epsilon^{\mu}$ that the
spatial components vanish $W^{\mu}=(W^{t^{\prime}},0)$. Further, one can
introduce new coordinates $\left(t,x^{i}\right)$ with
$t\left(t^{\prime},x^{i}\right)=\sqrt{-g^{\prime}}W^{t^{\prime}}$, so that

$W^{t}=\frac{\partial t}{\partial t^{\prime}}W^{t^{\prime}}=\frac{\partial
t}{\partial t^{\prime}}\frac{t}{\sqrt{-g^{\prime}}}\,,$ (22)

which through the transformation of measure $\sqrt{-g}\,\partial t/\partial
t^{\prime}=\sqrt{-g^{\prime}}$ implies that

$W^{\mu}=\delta_{t}^{\mu}\frac{t}{\sqrt{-g}}\,.$ (23)

If we fix this gauge and these coordinates _before_ performing the variation
(what is usually not a correct way to proceed) we obtain the often used
“unimodular” constraint777In terms of the ADM variables (Arnowitt:1962hi, ;
Poisson, ) the “unimodular” constraint relates the lapse $N$ to the
determinant of the spatial metric $\gamma$ as $N=\gamma^{-1/2}$. This relation
can be further extended (Barvinsky:2017pmm, ) to a general function
$N=f\left(\gamma\right)$, with rather peculiar consequences, e.g.
(Barvinsky:2019agh, ; Barvinsky:2020sxl, ; Barvinsky:2019qzx, ).

$\int d^{4}x\,\Lambda\left[1-\sqrt{-g}\,\right]\,,$ (24)

which is a usual starting point for a non-covariant formulation of the
unimodular gravity (Buchmuller:1988wx, ; Buchmuller:1988yn, ;
vanderBij:1981ym, ; Alvarez:2006uu, ). Sometimes it is claimed that this
constraint (24) can be satisfied just by a proper choice of the coordinates.
However, it is worth emphasizing that on the way to (24) we have already used
the diffeomorphism invariance once to enforce the unity of the density
$\partial_{\mu}\left(\sqrt{-g}W^{\mu}\right)$. If we could use the
diffeomorphism again to enforce the unity of $\sqrt{-g}$, this would imply
that just by a diffeomorphism one could enforce the unity of the scalar
$\nabla_{\mu}W^{\mu}$ which is clearly not possible. Thus, constraint (24) is
not just a mere gauge choice, but does change the dynamics by introducing a
global degree of freedom.

As showed in (Hammer:2020dqp, ), the vector field $W^{\mu}$ can be exchanged
with a more convenient Chern-Simons current of a (abelian or non-abelian)
gauge field $A_{\mu}$ so that

$S\left[g,A,\Lambda\right]=\frac{1}{\varkappa}\int
d^{4}x\sqrt{-g}\left[-\frac{R}{2}+\Lambda\left(F_{\alpha\beta}F^{\star\alpha\beta}-1\right)\right]\,,$
(25)

where $F_{\mu\nu}=D_{\mu}A_{\nu}-D_{\nu}A_{\mu}$ is the gauge field strength
or curvature of the covariant derivative $D_{\mu}=\nabla_{\mu}+iqA_{\mu}$,
while the Hodge dual is defined as

$F^{\star\alpha\beta}=\frac{1}{2}\cdot\frac{\epsilon^{\alpha\beta\mu\nu}}{\sqrt{-g}}\cdot
F_{\mu\nu}\equiv\frac{1}{2}\cdot E^{\alpha\beta\mu\nu}\cdot F_{\mu\nu}\,.$
(26)

Moreover, one could avoid the Lagrange multiplier and formulate unimodular
gravity as a higher-derivative and Weyl-invariant theory of a vector field
(Jirousek:2018ago, ) or of a gauge field (Hammer:2020dqp, ), see also
(Kimpton:2012rv, ). One can also describe this dynamics by a covariant action
containing a three-form as in e.g. (Aurilia:1980xj, ; Henneaux:1984ji, ), but
we will not expand on that.

Finally, we would like to mention that it is the absence of the usual kinetic
term $F_{\alpha\beta}F^{\alpha\beta}$ which forces $\Lambda$ to be constant.
Hence, it is easy to further extend this action to resemble the one of the
usual axion, c.f. (Wilczek:1983as, ) and see (Hammer:2020dqp, ):

$\displaystyle S\left[g,A,\theta\right]=\int
d^{4}x\sqrt{-g}\left[-\frac{R}{2\varkappa}+\right.$ (27)
$\displaystyle\left.+\frac{1}{2}\left(\partial\theta\right)^{2}+\frac{\theta}{f_{\Lambda}}F_{\alpha\beta}F^{\star\alpha\beta}-V_{\lambda}\left(\theta\right)\right]\,,$

where now $\theta$ is a canonically normalized pseudoscalar and $f_{\Lambda}$
is some mass scale emulating axion decay constant and
$V_{\lambda}\left(\theta\right)$ is a “potential”. The presence of the
standard kinetic term for $\theta$ does not change anything, as on-shell
$\theta$ is always a constant $\theta_{\star}$ corresponding to the vacuum
energy density $V_{\lambda}$$\left(\theta_{\star}\right)$. In particular, this
action is parity symmetric. This gives a hope to find unimodular gravity as a
particular dynamical regime (maybe strongly coupled) of some more usual high
energy system.

##  III Scale-Free Equations

In this paper we propose a novel way to lose the trace part of the Einstein
equations. Indeed, instead of (1) one can write _normalized_ or _scale-free_
and still _trace-trivial_ equations

$\frac{G_{\mu\nu}}{G}=\frac{T_{\mu\nu}}{T}\,,$ (28)

so that, the trace part of these normalized Einstein equations becomes a
useless identity $1=1$ instead of $0=0$ in the previous “unimodular” case.
This _unitrace_ formulation does not require any dimensional parameter, as
both sides are dimensionless. If again $T_{\mu\nu}$ is covariantly conserved,
one can follow the same procedure as for the trace-free Einstein equations
(1). We apply covariant derivative to both sides of (28) and use Bianchi
identity to obtain

$\frac{G_{\mu\nu}}{G^{2}}\nabla^{\mu}G=\frac{T_{\mu\nu}}{T^{2}}\nabla^{\mu}T\,.$
(29)

Substituting here the normalized Einstein equations (28) one obtains instead
of (7)

$\partial_{\mu}\log\frac{G}{T}=0\,,$ (30)

so that

$G=\beta T\,,$ (31)

where $\beta$ is constant of integration. Now we can substitute this
differential consequence (31) back into the original normalized Einstein
equations (28) to obtain the standard Einstein equations

$G_{\mu\nu}=\beta T_{\mu\nu\,.}$ (32)

Thus, this novel trace-trivial formulation (28) allows to have the Newton
constant

$G_{N}=\frac{\beta}{8\pi}\,,$ (33)

as a constant of integration or global dynamical degree of freedom. Similarly
to $\Lambda$ in the “unimodular” case, this constant of integration $\beta$ is
allowed to be both positive and negative.

Here a cautionary remark is necessary: not only we assumed that the traces $T$
and $G$ are non-vanishing, but also we required a stronger condition that
$G_{\mu\nu}$ (or $T_{\mu\nu}$) has an inverse for interesting solutions. Of
course gravity waves in empty space would violate this assumption. However,
conformal anomaly can motivate the non-vanishing trace of the EMT888In
particular, the trace of total EMT and the corresponding Ricci scalar are not
entirely vanishing during the early universe radiation domination époque, see
e.g. (Caldwell:2013mox, ). , while an existence of an arbitrary small
cosmological constant (included into this EMT) would imply desired
invertibility for all solutions, except of measure zero when this CC is
exactly compensated by some matter.

Interestingly, one can obtain _normalized_ , _scale-free_ and _trace-trivial_
equations (28) from a restricted variation of the Einstein-Hilbert action.
However, now instead of the transversality condition (10) one should take
either “Einstein transversality” condition:

$G_{\mu\nu}\bar{\delta}g^{\mu\nu}=0\,,$ (34)

or the “energy transversality” condition:

$T_{\mu\nu}\bar{\delta}g^{\mu\nu}=0\,.$ (35)

These conditions are mutually complementary and interchangeable. Indeed,
assuming one of them results through the general unrestricted variational
relation (13) in the other one. Either of them would be deadly for
unrestricted variations, but here we should merely require equality of only
those parts of $T_{\mu\nu}$ which are not proportional to $G_{\mu\nu}$.
Namely, imposing the “Einstein transversality” condition (13) yields the
Lagrange equations of the first kind

$\varkappa T_{\mu\nu}=\lambda\left(x\right)G_{\mu\nu}\,,$ (36)

where $\lambda\left(x\right)$ is a Lagrange multiplier. Now we can again
multiply both sides of this equation with $g^{\mu\nu}$ and exclude the
Lagrange multiplier to obtain (28). Clearly, the “energy transversality” case
is treated completely analogously. Similarly to the unimodular gravity we
could apply covariant derivative and use Bianchi identity along with the
_assumed_ covariant conservation of EMT directly in (36) to find that
$\lambda\left(x\right)=const$.

##  IV Many faces of Lagrange

The proportionality between $G_{\mu\nu}$ and $T_{\mu\nu}$ in our scale-free
formulation, or proportionality between the metric $g_{\mu\nu}$ and
$\left(G_{\mu\nu}-\varkappa T_{\mu\nu}\right)$ in the case of the unimodular
gravity, can be expressed without any use of the Lagrange multiplier. One can
follow an analogy with usual vectors. There, any two vectors are proportional,
provided their vector (wedge) product vanishes and vice versa. Thus, for our
novel scale-free gravity one can write analogous expression

$G_{\mu\nu}T_{\alpha\beta}=G_{\alpha\beta}T_{\mu\nu}\,,$ (37)

instead of the Lagrange’s equations of the first kind for the scale-free
formulation (36), while for the unimodular case one would write

$\left(G_{\mu\nu}-\varkappa
T_{\mu\nu}\right)g_{\alpha\beta}=\left(G_{\alpha\beta}-\varkappa
T_{\alpha\beta}\right)g_{\mu\nu}\,,$ (38)

instead of (14).

Further, it is noteworthy that one can exclude the Lagrange multipliers from
the Lagrange equations of the first kind (14), (36) in many different ways.
The multiplication of (14), (36) with respect to the contravariant metric has
a clear advantage that this inverse metric always exists and that it’s trace
is always equal to the number of spacetime dimensions. However, one could use
other equivalent ways to eliminate the Lagrange multiplier and write equations
in terms of known fields in different forms.

####  IV.0.1 Unimodular gravity

For example, in unimodular gravity one could multiply (14) with $G^{\mu\nu}$
and, under the assumption that $G\neq 0$, obtain

$\lambda=\frac{G^{\mu\nu}}{G}\left(G_{\mu\nu}-\varkappa T_{\mu\nu}\right)\,,$
(39)

so that

$G_{\mu\nu}-\varkappa
T_{\mu\nu}=\frac{G^{\alpha\beta}}{G}\left(G_{\alpha\beta}-\varkappa
T_{\alpha\beta}\right)g_{\mu\nu}\,.$ (40)

Or repeating the same procedure with $T_{\mu\nu}$

$G_{\mu\nu}-\varkappa
T_{\mu\nu}=\frac{T^{\alpha\beta}}{T}\left(G_{\alpha\beta}-\varkappa
T_{\alpha\beta}\right)g_{\mu\nu}\,.$ (41)

Clearly, one could take any second rank tensor $M^{\alpha\beta}$ constructed
from the matter fields and the metric and then use it instead of
$T^{\alpha\beta}$ or $G^{\alpha\beta}$, provided the trace $M\neq 0$.

On the other hand, instead of taking the trace one can take the determinant of
(14) to obtain

$\lambda^{4}=\frac{\text{det}\left(G_{\mu\nu}-\varkappa
T_{\mu\nu}\right)}{\text{det}g_{\alpha\beta}}\,.$ (42)

Consequently for positive $\lambda$ instead of trace-free Einstein equations
(1) of the unimodular gravity one acquires equations on unimodular parts of
the corresponding tensors

$\frac{G_{\mu\nu}-\varkappa
T_{\mu\nu}}{\left(-\text{det}\left(G_{\alpha\beta}-\varkappa
T_{\alpha\beta}\right)\right)^{1/4}}=\frac{g_{\mu\nu}}{\left(-\text{det}g_{\alpha\beta}\right)^{1/4}}\,.$
(43)

Of course, taking the trace of these equations one excludes determinants and
arrives back to trace-free Einstein equations (1).

####  IV.0.2 Scale-free gravity

Similarly one can write the _scale-free_ Einstein equations __(28) in many
different ways. For example, analogously to the unimodular case one can use
determinant to solve for the Lagrange multiplier in the unitrace case and
write

$\frac{G_{\mu\nu}}{\left(\text{det}G_{\alpha\beta}\right)^{1/4}}=\frac{T_{\mu\nu}}{\left(\text{det}T_{\alpha\beta}\right)^{1/4}}\,,$
(44)

so that again there is no scale - no dimensionful parameter. It looks like it
is a very different equation. However, taking the trace one acquires that

$\frac{\left(\text{det}T_{\mu\nu}\right)^{1/4}}{\left(\text{det}G_{\mu\nu}\right)^{1/4}}=\frac{T}{G}\,,$
(45)

so that one can get rid of the ratio of the determinants and obtains in this
way the same (28). This different parametrization can be as general as

$\frac{G_{\mu\nu}}{f\left(G_{\alpha\beta}\right)}=\frac{T_{\mu\nu}}{f\left(T_{\alpha\beta}\right)}\,,$
(46)

where $f$ is general homogeneous function of the first degree of the tensor
argument. In that case, there is no scale in this equation. This function can
also depend on the metric. Instead of the built-in identity $1=1$ with respect
to taking the trace999Note that the operation of taking trace can be
considered as a particular example of such function on tensors,
$f\left(\mathcal{O}_{\alpha\beta}\right)=g^{\alpha\beta}\mathcal{O}_{\alpha\beta}$,
which is homogeneous of the first degree. of (28), here the corresponding
identity is $f\left(G_{\mu\nu}/f\left(G_{\alpha\beta}\right)\right)=1\,.$
However, taking the trace one gets back (28) again. For example,

$\frac{G_{\mu\nu}}{\left(-g\right)^{1/4}G+\left(\text{det}G_{\alpha\beta}\right)^{1/4}}=\frac{T_{\mu\nu}}{\left(-g\right)^{1/4}T+\left(\text{det}T_{\alpha\beta}\right)^{1/4}}\,,$
(47)

still reproduces exactly the same dynamics as (28). Similarly to the
unimodular case, one can also use $G^{\mu\nu}$ to express the Lagrange
multiplier in (36) as

$\lambda\left(x\right)=\varkappa\frac{T_{\mu\nu}G^{\mu\nu}}{G^{\alpha\beta}G_{\alpha\beta}}\,.$
(48)

In this way, instead of _unitrace_ equations __(28) one arrives to scale-free
equations

$T_{\mu\nu}=\frac{T_{\sigma\lambda}G^{\sigma\lambda}}{G^{\alpha\beta}G_{\alpha\beta}}\,G_{\mu\nu}\,,$
(49)

with built-in identity with respect to multiplication with $G^{\mu\nu}$. On
the other hand, multiplying (36) with $T^{\mu\nu}$ one obtains

$\frac{T_{\sigma\lambda}G^{\sigma\lambda}}{T^{\alpha\beta}T_{\alpha\beta}}\,T_{\mu\nu}=G_{\mu\nu}\,.$
(50)

Again, instead of $T_{\mu\nu}$ or $G_{\mu\nu}$ one could employ any second
rank tensor $M^{\alpha\beta}$ constructed from the matter fields and metric,
provided the trace $M^{\alpha\beta}G_{\alpha\beta}\neq 0$, so that

$\lambda\left(x\right)=\varkappa\frac{T_{\mu\nu}M^{\mu\nu}}{M^{\alpha\beta}G_{\alpha\beta}}\,,$
(51)

which yields

$T_{\mu\nu}=\frac{T_{\sigma\lambda}M^{\sigma\lambda}}{M^{\alpha\beta}G_{\alpha\beta}}\,G_{\mu\nu}\,.$
(52)

In particular, one could even use inverse tensors
$\left(T^{-1}\right)^{\mu\nu}$ or $\left(G^{-1}\right)^{\mu\nu}$ or
$\left(R^{-1}\right)^{\mu\nu}$ : defined as
$\left(T^{-1}\right)^{\mu\lambda}T_{\lambda\nu}=\delta_{\nu}^{\mu}$ etc.,
provided they exist101010For a very recent discussion of other use of inverse
Ricci tensor or anti-curvature in cosmology, see (Amendola:2020qho, ).. In
this case, one would write

$G_{\mu\nu}=\frac{G_{\sigma\lambda}\left(T^{-1}\right)^{\lambda\sigma}}{4}\,T_{\mu\nu}\,,$
(53)

instead of (28).

##  V Einstein-Transversality in Action for Scale-Free Gravity

By analogy with the Henneaux-Teitelboim formulation of the unimodular gravity
(15), it is easy to write similar actions for a global degree of freedom
representing the Newton constant:

$S\left[g,C,\alpha\right]=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left(\nabla_{\mu}C^{\mu}-R\right)\alpha\,.$ (54)

An action similar to (54) was used in (Kaloper:2015jra, ; Kaloper:2018kma, )
for local version of the vacuum energy sequester (Kaloper:2013zca, ). On the
other hand, following (25) one can exchange the vector $C^{\mu}$ with the
Chern-Simons current of a gauge field $\mathcal{A}_{\mu}$ and write

$S\left[g,\mathcal{A},\alpha\right]=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left(\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}-R\right)\alpha\,.$
(55)

Here we assume standard general-covariant and minimally coupled action for all
usual matter fields.

Checking the equations of motion for (54) we have

$\frac{2}{\sqrt{-g}}\cdot\frac{\delta S}{\delta
C^{\mu}}=-\partial_{\mu}\alpha=0\,,$ (56)

and for the metric variation

$\displaystyle\delta_{g}S=\frac{1}{2}\int
d^{4}x\left[-\alpha\delta_{g}\left(\sqrt{-g}R\right)+\alpha\delta_{g}\partial_{\mu}\left(\sqrt{-g}C^{\mu}\right)\right]=$
$\displaystyle=\frac{1}{2}\int
d^{4}x\left[-\alpha\delta_{g}\left(\sqrt{-g}R\right)-\left(\delta\sqrt{-g}\right)C^{\mu}\partial_{\mu}\alpha\right]\,,$
(57)

so that on equation of motion for $C^{\mu}$ the divergence term does not
contribute to the metric equation of motion. In this way, for the standard
action of the minimally coupled matter fields one obtains

$\alpha\,G_{\mu\nu}=T_{\mu\nu}\,,$ (58)

and consequently

$\alpha=\frac{1}{8\pi G_{N}}\,.$ (59)

On the other hand, $\sqrt{-g}\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}$
does not contribute to the tensor equations of motion, see (26). Whereas for
abelian $\mathcal{A}_{\beta}$

$\frac{2}{\sqrt{-g}}\cdot\frac{\delta
S}{\delta\mathcal{A}_{\gamma}}=4\mathcal{F}^{\star\gamma\mu}\partial_{\mu}\alpha=0\,.$
(60)

However, to conclude here that $\partial_{\mu}\alpha=0$ one has to assume the
existence of $\left(\mathcal{F}^{\star\gamma\mu}\right)^{-1}$. The Hodge dual
of the field tensor has an inverse provided
$\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}\neq 0$, for a recent discussion
see (DeFelice:2019hxb, ), hence one should assume that $R$ is not identically
vanishing111111It is worth noting here again that in the early universe $R$ is
not vanishing (Caldwell:2013mox, )..

The canonical structure is more transparent for the action (54). There one can
either follow a direct analogy with the Henneaux and Teitelboim formulation of
the unimodular gravity (Henneaux:1989zc, ), and Faddeev-Jackiw
(Faddeev:1988qp, ; Jackiw:1993in, ) procedure or look at the Hamiltonian
analysis performed for the vacuum energy sequester, see (Bufalo:2016omb, ;
Kluson:2014tma, ; Svesko:2018cbo, ). Similarly to (16), in this action
$\alpha$ is a canonical momentum conjugated to the global quantity

$\mathscr{R}\left(t\right)=\frac{1}{2}\int_{\Sigma}d^{3}\mathbf{x}\sqrt{-g}\,C^{t}\left(t,\mathbf{x}\right)\,,$
(61)

which measures integrated Ricci scalar between the Cauchy hypersurfaces
$\Sigma_{2}$ and $\Sigma_{1}$

$\mathscr{R}\left(t_{2}\right)-\mathscr{R}\left(t_{1}\right)=\frac{1}{2}\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,R\,.$
(62)

Again one can think about $\mathscr{R}$ as a charge which is not conserved,
but rather continuously produced by Ricci curvature which plays the role of a
source

$\nabla_{\mu}C^{\mu}=R\,.$ (63)

Thus, the inverse Newton constant $\alpha$ is canonically conjugated121212For
thermodynamical arguments in favor of this relation, see (Volovik:2020qtp, ;
Klinkhamer:2008ff, ).  to the spacetime averaged Ricci curvature
$\mathscr{R}\left(t\right)$. On the other hand, $\mathscr{R}$ is clearly
proportional to the Einstein-Hilbert (volume) part of the gravitational
action. Thus, this resembles action-angle variables. As $\alpha$ is
canonically conjugated to $\mathscr{R}$ one can directly apply the Heisenberg
uncertainty relation to obtain

$\frac{\delta
G_{N}}{G_{N}}\times\frac{\delta\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,R}{\ell_{Pl}^{2}}\geq
8\pi\,.$ (64)

It is instructive to express this inequality in terms of fluctuations of the
corresponding Planck length $\ell_{Pl}=\sqrt{\hbar G_{N}}$

$\frac{\delta\ell_{Pl}}{\ell_{Pl}}\times\frac{\delta\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,R}{\ell_{Pl}^{2}}\geq
4\pi\,.$ (65)

Similarly to (19), these inequalities may have nontrivial consequences close
to singularities. Interestingly, $\hbar$ is not explicit in these uncertainty
relations above.

Following discussion around equations (22), (23) and (24) and applying them to
$C^{\mu}$ instead of $W^{\mu}$ one can try to fix

$C^{\mu}=\delta_{t}^{\mu}\frac{t}{\sqrt{-g}}\,M_{\alpha}^{2}\,,$ (66)

in the action, _before_ variation. Here $M_{\alpha}$ is some mass scale
introduced for dimensional reasons. Then, in units where $M_{\alpha}=1$, the
action takes a non-covariant form

$S\left[g,\alpha\right]=\frac{1}{2}\int
d^{4}x\left(1-\sqrt{-g}R\right)\alpha\,,$ (67)

thus instead of the “unimodular” constraint (24) we have

$\sqrt{-g}R=1\,,$ (68)

or the “unicurvature” condition. Similarly to what is often done in the
unimodular gravity one can forget about gauge and coordinate fixing steps
resulting in (66) and take this non-covariant action (67), as a starting
point.

Following a nice discussion in (vanderBij:1981ym, ), one can always construct
such coordinates where (68) is fulfilled. Suppose in some coordinates
$\left(t^{\prime},x^{\prime i}\right)$ we have
$\sqrt{-g^{\prime}}R^{\prime}\neq 1$. Then using that the Ricci density
transforms as

$\sqrt{-g^{\prime}}R^{\prime}=\left|\frac{\partial x}{\partial
x^{\prime}}\right|\sqrt{-g}R\,,$ (69)

we can find such coordinates $\left(t,x^{i}\right)$ that $x^{\prime i}=x^{i}$,
but

$t=\int^{t^{\prime}}dt^{\prime\prime}\sqrt{-g^{\prime}\left(t^{\prime\prime},x^{\prime
i}\right)}\,R^{\prime}\left(t^{\prime\prime},x^{\prime i}\right)\,,$ (70)

so that one obtains

$\left|\frac{\partial x}{\partial x^{\prime}}\right|=\frac{\partial
t}{\partial t^{\prime}}=\sqrt{-g^{\prime}}R^{\prime}\,,$ (71)

which enforces the “unicurvature” condition (68) in coordinates
$\left(t,x^{i}\right)$. However, similarly to the non-covariant formulation of
unimodular gravity one cannot just find coordinates where both conditions (66)
and (68) hold simultaneously. Therefore the latter condition is an equation of
motion implying nontrivial dynamics.

Let us check that, despite the illegitimate step of fixing the gauge directly
in the action before variation, still the action (67) reproduces the same
dynamics as the original general-covariant action (54). Variation with respect
to $\delta g^{\mu\nu}$ of (67) accompanied with the action for minimally
coupled matter yields

$\alpha
G_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\alpha+g_{\mu\nu}\Box\alpha=T_{\mu\nu}\,,$
(72)

where $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$. Here we are obviously missing
(56) which enforces that $\alpha$ is a constant. However, applying
$\nabla^{\mu}$ to both sides of the equation (72) one obtains

$G_{\mu\nu}\nabla^{\mu}\alpha+\nabla_{\nu}\Box\alpha-\Box\nabla_{\nu}\alpha=0\,,$
(73)

where we used the Bianchi identity and EMT conservation. Further, invoking the
identity

$\nabla_{\nu}\Box\alpha-\Box\nabla_{\nu}\alpha=-R_{\mu\nu}\nabla^{\mu}\alpha\,,$
(74)

yields $R\partial_{\nu}\alpha=0,$ so that we are back to (58).

Interestingly, varying the “unicurvature” constraint (68) one obtains

$G_{\mu\nu}\bar{\delta}g^{\mu\nu}=\nabla_{\mu}\nabla_{\nu}\bar{\delta}g^{\mu\nu}-g_{\mu\nu}\Box\bar{\delta}g^{\mu\nu}\,,$
(75)

reproducing the “Einstein transversality” condition (34) only up to a
divergence.

It is worth mentioning, that similarly to what is sometimes done in the
unimodular gravity one can also substitute $\sqrt{-g}=R^{-1}$ into matter part
of the action. This would produce an apparently new and crazily looking theory
without changing any physics.

Finally, we would like to mention that it is the absence of the usual kinetic
term $\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}$ which forces $\alpha$ to be
constant. Hence, similarly to (27), one can write

$\displaystyle S\left[g,\mathcal{A},\nu\right]=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}\nu^{2}R+\right.$ (76)
$\displaystyle\left.+\frac{1}{2}\left(\partial\nu\right)^{2}+\frac{\nu}{f_{\alpha}}\mathcal{F}_{\gamma\sigma}\mathcal{F}^{\star\gamma\sigma}-V_{\alpha}\left(\nu\right)\right]\,,$

where now $\nu$ is canonically normalized pseudoscalar and $f_{\alpha}$ is
some mass scale emulating the axion would-be decay constant, while
$V_{\alpha}\left(\nu\right)$ is a potential. Again, the presence of the
standard kinetic term for $\nu$ does not change anything, as on-shell $\nu$
always remains an arbitrary constant $\nu_{\star}$. In particular, the kinetic
term may be non-canonical. Another advantage is that this action has usual
properties regarding parity transformations. The action is written in the
Jordan frame. The potential induces cosmological constant
$V_{\alpha}\left(\nu_{\star}\right)$ whose presence is also useful for
invertibility of $\mathcal{F}^{\star\alpha\beta}$. This construction gives
hope to find the _scale-free_ gravity (28) as a particular dynamical regime
(maybe strongly coupled) of a more usual QFT system.

##  VI Energy-Transversality in Action for Scale-Free Gravity

On the other hand, one can attain the effective Newton constant as a global
degree of freedom by changing the usual action for matter fields $\Phi_{m}$,

$S_{0}\left[g,\Phi_{m}\right]=\int d^{4}x\sqrt{-g}\,\mathscr{L}_{m}\,,$ (77)

to the one (c.f. (Carroll:2017gqo, )) similar to the Henneaux and Teitelboim
construction (15)

$S\left[g,\beta,L,\Phi_{m}\right]=\int
d^{4}x\sqrt{-g}\,\beta\left(\mathscr{L}_{m}-\nabla_{\lambda}L^{\lambda}\right)\,.$
(78)

Or instead of the vector field $L^{\lambda}$ one can again use a gauge field
$\mathscr{A}_{\mu}$

$S\left[g,\beta,\mathscr{A},\Phi_{m}\right]=\int
d^{4}x\sqrt{-g}\,\beta\left(\mathscr{L}_{m}-\mathscr{F}_{\mu\nu}\mathscr{F}^{\star\mu\nu}\right)\,,$
(79)

so that the “axionic” field $\beta$ _universally_ couples to _all_ matter
fields. Here we assume that the gravitational sector is described by the
standard Einstein-Hilbert action. We could slightly modify the action in order
to have better transformation properties with respect to parity symmetry and
write

$S\left[g,\eta,\mathscr{A},\Phi_{m}\right]=\int
d^{4}x\sqrt{-g}\,\eta\left(\eta\mathscr{L}_{m}-\mathscr{F}_{\mu\nu}\mathscr{F}^{\star\mu\nu}\right)\,.$
(80)

Under parity $\eta$ (and $\beta$) is a pseudoscalar which transforms as
$\eta\rightarrow-\eta$. Lest the normal matter become ghosty under parity
transformation, one should use this improved version of the action (80).

Of course, completely analogously to previous cases, the variation with
respect to the vector field $L^{\mu}$ (or $\mathscr{A}_{\mu}$) results in
$\partial_{\mu}\beta=0$ (or $\partial_{\mu}\eta=0$). On shell
$\eta=\eta_{\star}$, which opens a path to further modification of this action
to make it look more familiar

$\displaystyle S\left[g,\eta,\mathscr{A},\Phi_{m}\right]=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}\left(\partial\eta\right)^{2}-V_{\beta}\left(\eta\right)+\right.$
$\displaystyle\qquad\qquad\qquad\left.+\frac{\eta^{2}}{M_{m}^{2}}\mathscr{L}_{m}-\frac{\eta}{f_{\beta}}\mathscr{F}_{\mu\nu}\mathscr{F}^{\star\mu\nu}\right]\,,$
(81)

where $V_{\beta}\left(\eta\right)$ is a potential and the usual dimensions of
$\eta$ are restored by introduction of two mass scales $M_{m}$ and
$f_{\beta}$. And again the presence of the potential introduces a cosmological
constant $V_{\beta}\left(\eta_{\star}\right)$. This guarantees the existence
of $\left(\mathscr{F}^{\star\mu\nu}\right)^{-1}$ required to infer the
constancy of $\eta$. It is crucial that the axion _$\eta$ _is _universally_
coupled to all other matter fields. While gravity is assumed to be described
by the standard Einstein-Hilbert action so that the description is in the
Einstein frame. Clearly, in our setup the mass scales and even the form of the
potential $V_{\beta}\left(\eta\right)$ and the form of the kinetic term (which
can be again non-canonical) are rather irrelevant. However, in any attempt to
embed this theory into more usual QFT setup these will be crucial bits of
information. This action appears from standard Yang-Mills construction
provided one takes the limit when one can neglect the usual kinetic term
$-\mathscr{F}_{\mu\nu}\mathscr{F}^{\mu\nu}/4\text{g}^{2}$ for the gauge field.
Naively, this limit corresponds to a confinement or an infinitely strong
coupling $\text{g}\rightarrow\infty$.

It is worth noting that combining action (81) or (76) with the (27) provides
an axionic formulation of the vacuum energy sequestering (Kaloper:2013zca, ;
Kaloper:2015jra, ; Padilla:2015aaa, ). This novel axionic formulation of
sequester is much closer to the usual particle physics models.

The canonical structure of the system is more transparent for the action (78).
In this formulation of the theory, on the right hand side of the Einstein
equations one obtains a rescaled stress tensor for matter fields

$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta
g^{\mu\nu}}=\beta\,T_{\mu\nu}^{\left(m\right)},$ (82)

where

$T_{\mu\nu}^{\left(m\right)}=\frac{2}{\sqrt{-g}}\frac{\delta S_{0}}{\delta
g^{\mu\nu}}=\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathscr{L}_{m}\right)}{\delta
g^{\mu\nu}}\,,$ (83)

is defined exclusively through the usual Lagrangian density or from the
original action (77). In the last equality we assumed that matter sector is
standard and in particular that the Lagrangian does not depend on the
derivatives of the metric. In fact, this last equality would not be applicable
for _Kinetic Gravity Braiding_ (Deffayet:2010qz, ) and _G-Inflation_
(Kobayashi:2010cm, ).

Similarly to (16) and (61) the dynamical degree of freedom canonically
conjugated to $\beta$ is

$I\left(t\right)=-\int_{\Sigma}d^{3}\mathbf{x}\sqrt{-g}\,L^{t}\left(t,\mathbf{x}\right)\,,$
(84)

which measures the usual matter action between the Cauchy hypersurfaces
$\Sigma_{2}$ and $\Sigma_{1}$

$I\left(t_{2}\right)-I\left(t_{1}\right)=-\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,\mathscr{L}_{m}\,.$
(85)

Also here one can think about $I\left(t\right)$ as a charge which is not
conserved, but rather continuously produced by the total matter Lagrangian
which plays the role of a source

$\nabla_{\mu}L^{\mu}=\mathscr{L}_{m}\,.$ (86)

On the other hand, the introduction of $\beta$ rescales the commutation
relations for usual matter. In the same way, in the approach with the
Einstein-transverse condition described in (V) a similar rescaling (89) with
$1/G_{N}$ instead of $\beta$ occurs for the commutator of the _dimensionless_
spatial metric and it’s conjugated momentum given by the trace-free extrinsic
curvature density, see (Poisson, ).

For example, for the usual scalar field $\phi$ the canonical momentum gets
rescaled in the analogous way to the EMT:

$\pi=\beta\pi^{\left(m\right)}=\beta\sqrt{-g}\frac{\partial\mathscr{L}_{m}}{\partial\dot{\phi}}\,.$
(87)

Hence, if the canonical commutator is

$\left[\phi\left(\mathbf{x}\right),\pi\left(\mathbf{y}\right)\right]=i\hbar\delta\left(\mathbf{x}-\mathbf{y}\right)\,,$
(88)

the usual commutator gets rescaled as

$\left[\phi\left(\mathbf{x}\right),\pi^{\left(m\right)}\left(\mathbf{y}\right)\right]=\frac{i\hbar}{\beta}\delta\left(\mathbf{x}-\mathbf{y}\right)\,,$
(89)

so that the effective Planck constant is

$\bar{\hbar}=\frac{\hbar}{\beta}\,.$ (90)

This is ideologically similar to what is discussed in (Carroll:2017gqo, ),
though our frameworks are different.

For the usual Einstein-Hilbert action with fixed $G_{N}$ one obtains rescaled
Einstein equations

$\frac{G_{\mu\nu}}{8\pi G_{N}}=\beta T_{\mu\nu}^{\left(m\right)}\,,$ (91)

so that the effective Newton constant, $\bar{G}_{N}$, is rescaled as

$\bar{G}_{N}=G_{N}\beta\,.$ (92)

Interestingly, one obtains

$\bar{\hbar}\bar{G}_{N}=\hbar G_{N}\,,$ (93)

so that the Planck length (and time) $\ell_{Pl}=\sqrt{\hbar G_{N}}$ remain
invariant. This discussion is applicable if the universe is in an eigenstate
of $\beta$.

On the other hand, in generic state, $\beta$ can have its own quantum
fluctuations. In that case, we again write the Heisenberg uncertainty relation
for the fluctuation of the effective Newton constant (92)

$\delta\bar{G}_{N}\times\delta\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,\mathscr{L}_{m}\geq\frac{1}{2}\ell_{Pl}^{2}\,,$
(94)

or equivalently for the effective Planck constant (90)

$\delta\bar{\hbar}\times\delta\int_{\mathscr{V}}d^{4}x\sqrt{-g}\,\mathscr{L}_{m}\geq\frac{1}{2}\bar{\hbar}^{2}\,.$
(95)

We should remark that in case of the action (80) instead of $\beta$ one should
use $\eta^{2}$ in all rescaling above, while for (81) one uses
$\eta^{2}/M_{m}^{2}$.

Interestingly, the invariance of $\ell_{Pl}$ given by (93) does not allow for
fluctuations of $\ell_{Pl}$, contrary to those (65) present in the approach
with the Einstein-transverse condition.

One can also combine both actions (78) and (54) and obtain the Einstein
equations written as

$\alpha\,G_{\mu\nu}=\beta\,T_{\mu\nu}^{\left(m\right)}\,,$ (96)

so that the effective Newton constant is

$\bar{G}_{N}=\frac{1}{8\pi}\,\frac{\beta}{\alpha}\,.$ (97)

###  VI.1 “Noncovariant” or “unimatter” formulation.

Finally, similarly to fixing $W^{\mu}$ as in Eq. (24) for the non-covariant
formulation of the unimodular gravity and to fixing $C^{\mu}$ as in Eq. (66)
for the non-covariant “unicurvature” gravity, one can try to illegitimately
fix

$L^{\mu}=\delta_{t}^{\mu}\frac{t}{\sqrt{-g}}\,M_{\beta}^{4}\,,$ (98)

_before_ varying the action (78). Here $M_{\beta}$ is some mass scale. Then,
again in units where this mass scale is unity, $M_{\beta}=1$, the action takes
an unusual non-covariant form

$S\left[g,\beta,\Phi_{m}\right]=\int
d^{4}x\,\beta\left(\sqrt{-g}\mathscr{L}_{m}-1\right)\,,$ (99)

implying unity of the matter Lagrangian density, which is clearly different
from Eq. (4) from (Carroll:2017gqo, ). On top, we again assume the usual
Einstein-Hilbert action with fixed $G_{N}$. One could call this “unimatter
gravity”, as

$\sqrt{-g}\mathscr{L}_{m}=1\,,$ (100)

which corresponds to a coordinate choice or a partial gauge fixing, modulo the
point which we have already discussed after Eq. (24). We can again repeat the
same procedure to find proper coordinates where “unimatter” condition holds by
applying formulas (69), (70) and (71) with exchanging $R$ with
$\mathscr{L}_{m}$.

From the usual definition of the EMT (83) we have

$\delta\left(\sqrt{-g}\mathscr{L}_{m}\right)=\frac{\sqrt{-g}}{2}\,T_{\mu\nu}^{\left(m\right)}\bar{\delta}g^{\mu\nu}=0\,,$
(101)

thus the “unimatter” condition (100) reproduces the “energy transversality”
condition (35) restricting the variations $\bar{\delta}g^{\mu\nu}$.

If, as it is the case for the Standard Model fields, $\mathscr{L}_{m}$ does
not contain derivatives of the metric, then the variation with respect to the
metric yields (91). However, contrary to the general-covariant formulation,
now $\beta\left(x\right)$ is a free function similarly to (36), as we do not
have an equation of motion for $L^{\mu}$ to impose $\partial_{\mu}\beta=0$. In
our restricted variational formulation with energy-transversality condition
(35) it was the joint work of the Bianchi identity and the _assumed_
conservation of the EMT which resulted in $\partial_{\mu}\beta=0$. However,
similarly to what happens (Buchmuller:1988wx, ) (c.f. (Percacci:2017fsy, )) in
the non-covariant formulation (24) of the unimodular gravity, due to the non-
covariant formulation one cannot just assume the conservation of
$T_{\mu\nu}^{\left(m\right)}$.

To illustrate this point, lets first consider just one scalar field $\phi$, as
general as k-essence (ArmendarizPicon:2000ah, ; Chiba:1999ka, ). This can be
also considered as a proxy for irrotational hydrodynamics. The Euler-Lagrange
equation for $\phi$ from the action (99) is

$\sqrt{-g}\beta\frac{\partial\mathscr{L}_{m}}{\partial\phi}=\partial_{\mu}\left(\sqrt{-g}\beta\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\mu}\phi}\right)\,,$
(102)

and can be rewritten as

$\nabla_{\mu}\left(\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\mu}\phi}\right)-\frac{\partial\mathscr{L}_{m}}{\partial\phi}=-\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\mu}\phi}\frac{\partial_{\mu}\beta}{\beta}\,.$
(103)

Further, for the scalar field the Noether form of the EMT is already symmetric
and can be written as

$T_{\quad\beta}^{\alpha\left(m\right)}=\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\alpha}\phi}\partial_{\beta}\phi-\mathscr{L}_{m}\delta_{\beta}^{\alpha}\,,$
(104)

while its divergence is

$\nabla_{\alpha}T_{\quad\beta}^{\alpha\left(m\right)}=\left(\nabla_{\alpha}\left(\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\alpha}\phi}\right)-\frac{\partial\mathscr{L}_{m}}{\partial\phi}\right)\partial_{\beta}\phi\,.$
(105)

In the usual case, the expression in front of $\partial_{\beta}\phi$ would be
the equation of motion so that the EMT would be conserved. However, here the
equation of motion (103) is modified by the presence of $\beta$ so that there
is a non-conservation:

$\nabla_{\alpha}T_{\quad\mu}^{\alpha\left(m\right)}=-\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\alpha}\phi}\,\frac{\partial_{\alpha}\beta}{\beta}\,\partial_{\mu}\phi\,.$
(106)

Now we can apply covariant derivative to the Einstein equations (91) and
obtain

$\displaystyle\frac{\nabla_{\alpha}G_{\mu}^{\alpha}}{8\pi
G_{\text{N}}}=\nabla_{\alpha}\left(\beta
T_{\quad\mu}^{\alpha\left(m\right)}\right)=$
$\displaystyle=T_{\quad\mu}^{\alpha\left(m\right)}\partial_{\alpha}\beta+\beta\nabla_{\alpha}T_{\quad\mu}^{\alpha\left(m\right)}=-\mathscr{L}_{m}\partial_{\mu}\beta\,,$
(107)

where we used (106) and (104). Hence, the Bianchi identity, indeed, again
implies that $\beta$ is a constant. This, of course, also results in the
conservation of $T_{\mu\nu}^{\left(m\right)}$.

Now let us consider usual free electrodynamics131313supplemented by a
cosmological constant $\Lambda$ to avoid that $\mathscr{L}_{m}=0$ on
electromagnetic waves which are solutions of the free electrodynamics.

$\mathscr{L}_{m}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\Lambda\,,$ (108)

so that

$\frac{\partial\mathscr{L}_{m}}{\partial\partial_{\mu}A_{\nu}}=-F^{\mu\nu}\,,$
(109)

and we have the same equations of motion (103), where just instead of $\phi$
we have $A_{\mu}$, so that

$\nabla_{\lambda}F^{\mu\lambda}=-F^{\mu\lambda}\frac{\partial_{\lambda}\beta}{\beta}\,.$
(110)

The symmetric Belinfante-Rosenfeld / Hilbert EMT relevant for gravity

$T_{\quad\nu}^{\mu\left(m\right)}=F_{\>\,\lambda}^{\mu}F_{\>\,\nu}^{\lambda}-\mathscr{L}_{m}\delta_{\nu}^{\mu}\,,$
(111)

deviates from the canonical Noether one (104) which is not even covariantly
conserved in the presence of gravity. In the absence of the electromagnetic
current one obtains

$\nabla^{\mu}T_{\mu\nu}^{\left(m\right)}=-F_{\lambda\nu}\nabla_{\mu}F^{\lambda\mu}=F_{\lambda\nu}F^{\lambda\alpha}\frac{\partial_{\alpha}\beta}{\beta}\,.$
(112)

Substituting this expression for non-conservation of EMT (112) along with
(111) into the left hand side of (107) we again obtain the last equality there
which, in turn, implies $\beta=const$.

In both examples above we essentially used explicit forms (104) and (111) of
the EMT. The case with the scalar field is easily generalizable to many scalar
fields, but this is not a generic matter source. Unfortunately, for generic
matter with internal spin, the canonical Noether EMT is not symmetric and is
not a source of gravity and does not coincide with the proper Hilbert EMT
obtained through the variation of the action with respect to the metric. Below
we provide the proof of the last equality in (107) for generic usual matter,
whose Lagrangian does not involve derivatives of the metric. However, this
proof does not cover fermions, as we do not want to overcomplicate the
discussion by involving tetrads.

The “unimatter” action (99) is written in a particular class of coordinates
where (98) holds. Let us denote one particular system of such coordinates
$X^{\mu}$. Now we can write this action (99) in arbitrary coordinates
$y^{\mu}$ by noting that $\mathscr{L}_{m}$, $d^{4}x\sqrt{-g}$ and $\beta$ are
scalar objects:

$S_{X}\left[g,\beta,\Phi_{m}\right]=\int
d^{4}y\,\beta\left(\sqrt{-g}\mathscr{L}_{m}-\left|\frac{\partial X}{\partial
y}\right|\right)\,,$ (113)

where $\left|\partial X/\partial y\right|$ is the Jacobian. We should stress
that in this action the original coordinates $X^{\mu}\left(y\right)$ are _not_
dynamical variables and remain “frozen” like external sources or _background_
fields. One could promote them to four Stückelberg scalar fields similarly to
how it is done for the unimodular gravity in (Kuchar:1991xd, ), but we refrain
from doing this. Further we can recall that the measure transforms as

$d^{4}X\sqrt{-g\left(X\right)}=d^{4}y\left|\frac{\partial X}{\partial
y}\right|\sqrt{-g\left(X\right)}=d^{4}y\sqrt{-g\left(y\right)}\,,$ (114)

so that

$\left|\frac{\partial X}{\partial
y}\right|=\frac{\sqrt{-g\left(y\right)}}{\sqrt{-g\left(X\right)}}\,.$ (115)

In this case the action (99) assumes the _generally covariant_ form

$S_{X}\left[g,\beta,\Phi_{m}\right]=\int
d^{4}y\sqrt{-g}\,\beta\left(\mathscr{L}_{m}-\frac{1}{\sqrt{-g\left(X\right)}}\right)\,,$
(116)

because with respect to further coordinate transformations $y\rightarrow
y\left(y^{\prime}\right)$ the _background_ field $\sqrt{-g\left(X\right)}$
transforms as a _scalar_. Now we can consider an infinitesimal change of
coordinates

$y^{\prime\mu}=y^{\mu}-\xi^{\mu}\left(y\right)\,,$ (117)

which leads to the variations given by the Lie derivatives

$\pounds_{\xi}\left(-g\left(X\right)\right)^{-1/2}=\xi^{\mu}\partial_{\mu}\left(-g\left(X\right)\right)^{-1/2}\,,$
(118) $\pounds_{\xi}g^{\mu\nu}=-\xi^{\mu;\nu}-\xi^{\nu;\mu}\,,$ (119)
$\pounds_{\xi}\left(\sqrt{-g}\mathscr{L}_{m}\right)=-\frac{\sqrt{-g}}{2}T_{\mu\nu}^{\left(m\right)}\left(\xi^{\mu;\nu}+\xi^{\nu;\mu}\right)\,,$
(120)
$\pounds_{\xi}\sqrt{-g}=\frac{1}{2}\sqrt{-g}g_{\mu\nu}\left(\xi^{\mu;\nu}+\xi^{\nu;\mu}\right)\,,$
(121)

where $\left(\>\right)_{;\mu}=\nabla_{\mu}\left(\>\right)$. We do not need
variations of $\beta$ and of $\Phi_{m}$, as we assume that their equations of
motion hold and result in vanishing variations of the action. Moreover, at
this step we can use equation of motion for $\beta$:
$\left(-g\left(X\right)\right)^{-1/2}=\mathscr{L}_{m}$. With respect to the
diffeomorphisms (117) the action remains invariant so that

$\int
d^{4}y\sqrt{-g}\beta\left[-T_{\mu\nu}^{\left(m\right)}\xi^{\mu;\nu}-\mathscr{L}_{m}g_{\mu\nu}\xi^{\mu;\nu}-\xi^{\mu}\partial_{\mu}\mathscr{L}_{m}\right]=0\,.$

Now we can integrate by parts the first two terms, what, under the assumption
that $\xi^{\mu}$ vanishes on the boundary, provides

$\int d^{4}y\sqrt{-g}\left[\nabla^{\nu}\left(\beta
T_{\mu\nu}^{\left(m\right)}\right)+\partial_{\mu}\left(\beta\mathscr{L}_{m}\right)-\beta\partial_{\mu}\mathscr{L}_{m}\right]\xi^{\mu}=0\,.$

Taking into account arbitrariness of $\xi^{\mu}$ this expression yields

$\nabla^{\nu}\left(\beta
T_{\mu\nu}^{\left(m\right)}\right)=-\mathscr{L}_{m}\partial_{\mu}\beta\,,$
(122)

which should hold on all equations of motion. Hence, we again obtained last
equality in (107) which now holds for generic matter with metric EMT.

Here we should mention that one could substitute the “unimatter” constraint
(100) directly into the Einstein-Hilbert action, similarly to what is
sometimes done with the unimodular constraint. However, contrary to the latter
case this does not enforces a polynomial form of this action in the metric.

##  VII Einstein Frame

As we have seen in the variational formulation “Einstein transversality”
condition was equivalent to the “Energy transversality” condition. In the
integral formulation of the “Einstein transversality” we assumed that the
matter is minimally coupled. In these actions (54) and (55) one can make a
Weyl transformation

$g_{\mu\nu}=e^{2\omega}\gamma_{\mu\nu}\,,$ (123)

in order to pass to an Einstein frame. The Weyl transformation above will not
change the form of the axionic term
$\alpha\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}\sqrt{-g}$, while, after a
field redefinition (Weyl transformation) of $C^{\mu}$, the term
$\alpha\nabla_{\mu}C^{\mu}\sqrt{-g}$ will preserve its form as well. However,
as it is well known

$R\left(g\right)=e^{-2\omega}\left(R\left(\gamma\right)-6\Box_{\gamma}\omega-6\gamma^{\mu\nu}\partial_{\mu}\omega\partial_{\nu}\omega\right)\,,$
(124)

where
$\Box_{\gamma}=\gamma^{\mu\nu}\widetilde{\nabla}_{\mu}\widetilde{\nabla}_{\nu}$
and $\widetilde{\nabla}_{\mu}$ is the Levi-Civita connection compatible with
the transformed metric

$\widetilde{\nabla}_{\mu}\gamma_{\alpha\beta}=0\text{~.}$ (125)

Further, by taking

$\omega=-\frac{1}{2}\log\alpha\,,$ (126)

and introducing

$\chi=\sqrt{6}\omega=-\sqrt{\frac{3}{2}}\log\alpha\,,$ (127)

one obtains the action (55) in the Einstein frame

$\displaystyle S\left[\gamma,\mathcal{A},\chi\right]=\int
d^{4}x\sqrt{-\gamma}\left[-\frac{1}{2}R\left(\gamma\right)+\frac{1}{2}\gamma^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi+\right.$
$\displaystyle\qquad\qquad\left.+e^{-\sqrt{2/3}\chi}\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}\right]\,.$
(128)

The usual matter is now coupled to the metric

$g_{\mu\nu}=e^{2\omega}\gamma_{\mu\nu}=e^{\sqrt{2/3}\chi}\gamma_{\mu\nu}\,.$
(129)

Here we omitted boundary terms. Interestingly, the field $\chi$ which is just
a mere field redefinition of the Lagrange multiplier $\alpha$, looks now as a
healthy scalar field with a usual non-ghosty canonical kinetic term, though
this term does not play any role as it is zero due to the vector equations of
motion, similarly to what happens in the action (81).

Clearly, for the vector field action one would get

$\displaystyle S\left[\gamma,\widetilde{C},\chi\right]=\int
d^{4}x\sqrt{-\gamma}\left[-\frac{1}{2}R\left(\gamma\right)+\frac{1}{2}\gamma^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi+\right.$
$\displaystyle\qquad\qquad\left.+e^{-\sqrt{2/3}\chi}\widetilde{\nabla}_{\mu}\widetilde{C}^{\mu}\right]\,,$
(130)

where the vector $C^{\mu}$ got transformed according to it’s conformal weight
4:

$\widetilde{C}^{\mu}=e^{4\omega}C^{\mu}=\alpha^{-2}C^{\mu}\,.$ (131)

Of course we could omit the kinetic term for $\chi$ in these Einstein frame
actions. Then these actions deviate from the (78) and (79) only by matter
fields rescaling. The rescaling of the matter metric is in accord with what
was discussed in the vacuum sequester (Kaloper:2013zca, ; Kaloper:2014dqa, ;
Kaloper:2015jra, ; Kaloper:2016jsd, ; Kaloper:2018kma, ; Ben-Dayan:2015nva, ).

## One constant of integration to fix both $G_{N}$ and $\Lambda$

One can further extend (76) by writing

$\displaystyle S\left[g,\mathcal{A},\varphi\right]=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{2}\mu^{2}\left(\varphi\right)R+\alpha\left(\varphi\right)\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}+\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\left(\partial\varphi\right)^{2}-V\left(\varphi\right)\right]$
(132)

where $\mu\left(\varphi\right)$ , $\alpha\left(\varphi\right)$ and
$V\left(\varphi\right)$ are arbitrary functions of the scalar field $\varphi$
and the normal matter is minimally coupled. One should assume that $\varphi$
is a pseudoscalar so that
$\alpha\left(-\varphi\right)=-\alpha\left(\varphi\right)$. The gauge field
equation of motion (60) gets an irrelevant modification

$\frac{1}{\sqrt{-g}}\cdot\frac{\delta
S}{\delta\mathcal{A}_{\gamma}}=4\alpha^{\prime}\mathcal{F}^{\star\gamma\mu}\partial_{\mu}\varphi=0\,,$
(133)

where prime denotes derivative with respect to $\varphi$. This equation has an
obvious solution

$\varphi=\varphi_{\star}=\text{const}\,.$ (134)

While the variation with respect to $\varphi$ yields constraint

$\alpha^{\prime}\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}-V^{\prime}=\mu\mu^{\prime}R\,.$
(135)

where we used (134). This constraint allows one to express the Pontryagin
invariant, $\mathcal{F}_{\mu\nu}\mathcal{F}^{\star\mu\nu}$, through the Ricci
scalar. The later is to obtain from the tensor equations of motion. The
functional dependence $\mu\left(\varphi\right)$, $\alpha\left(\varphi\right)$
reduces to a mere coefficient of proportionality while the “potential”,
$V\left(\varphi\right)$ only provides a linear shift, as all these functions
are evaluated at $\varphi_{\star}$. This shift is important because it allows
to have a nonvanishing Pontryagin invariant even for a vanishing Ricci scalar.

In this Jourdan frame one obtains that the effective Planck mass is given by

$M_{Pl}=\frac{\mu\left(\varphi_{\star}\right)}{\sqrt{8\pi}}\,,$ (136)

while the potential provides the effective cosmological constant

$\Lambda=V\left(\varphi_{\star}\right)\,.$ (137)

Thus one integration constant provides the values for both coupling constants
of GR. Let us briefly discuss the functions which can appear.

Simplest option from dimensional reasons would be
$\mu\left(\varphi\right)=\varphi.$ If the gauge filed is abelian, it can have
arbitrary dimensionality as we neglect the standard kinetic term anyway. In
particular, $\left[\mathcal{A}\right]=\left[m^{1/2}\right]$ would allow for
the simplest coupling $\alpha=\varphi.$ On the other hand, the form of the
covariant derivative $D_{\mu}=\partial_{\mu}+i\text{g}\mathcal{A}_{\mu}$ with
the self-coupling constant g dictates that either the coupling constant g is
dimensionless and $\left[\mathcal{A}\right]=\left[m\right]$ so that one has to
introduce a scale $M$ into the axion coupling $\alpha=\varphi/M$, or that
$\left[\mathcal{A}\right]=\left[m^{1/2}\right]$ and
$\left[\text{g}\right]=\left[m^{1/2}\right]$. Thus the non-abelian realization
provides a scale which can be used in the potential. In the later case one
could write the potential as

$V\left(\varphi\right)=\text{g}^{8}f\left(\frac{\varphi}{\text{g}^{2}}\right)\,.$
(138)

Appealing forms of the potential would resemble instanton contribution

$V_{1}\left(\varphi\right)=\frac{1}{4}\varphi^{4}\exp\left(-\frac{\varphi}{\text{g}^{2}}\right)\,,\qquad$
(139)

and

$V_{2}\left(\varphi\right)=\frac{1}{4}\text{g}^{8}\exp\left(-\frac{\varphi}{\text{g}^{2}}\right)\,.$
(140)

It would be great if the appearance of the scale in the coupling constant g
could be related to the scale of strong coupling or dimensional transmutation.
The strong coupling may hint to an explanation of the absence of the usual
kinetic term for the gauge field in the action (132). To obtain 120 orders of
magnitude difference one would need to assume: for $V_{1}\left(\varphi\right)$
that $\text{g}^{2}\simeq\varphi_{\star}/277$, whereas for
$V_{2}\left(\varphi\right)$ that $\text{g}^{2}\simeq\varphi_{\star}/254$.

Finally we would like to note that the sign of the kinetic term for $\varphi$
can also be negative. Moreover, one can write an arbitrary function
$P\left(\left(\partial\varphi\right)^{2},\varphi\right)$. Similarly to
(Lim:2010yk, ) the functional dependence on kinetic term does not change
anything here.

## Conclusions

In this paper we proposed a new very simple scale-free Einstein equations (5)
which promote the Newton constant to a global degree of freedom. In this way
the trace part of the usual Einstein equations is exchanged to a useless
identity $1=1$. This is similar to trace–free Einstein equations of the
unimodular gravity where this identity is $0=0$. Then we discussed different
equivalent formulations of these equations and their variational formulations.
After that we introduced different generally covariant actions (54), (55),
(78), (79), (81), (132) describing Newton constant as a global degree of
freedom. From these actions, we derived and discussed novel non-covariant
“unicurvature” (67) and “unimatter” (99) actions following the same ideology
as many works on unimodular gravity. It is interesting whether such non-
covariant formulations can be further meaningfully generalized, as it was done
in the case of unimodular gravity in (Barvinsky:2017pmm, ; Barvinsky:2019agh,
; Barvinsky:2020sxl, ). Some of our frameworks for the dynamical Newton
constant (78), (79), (81) actually imply that the effective Planck constant
$\hbar$ becomes a global degree of freedom, similarly to claims of
(Carroll:2017gqo, ) where a different system was discussed. We have also
written the Heisenberg uncertainty relations (64), (94) for the effective
Newton constant $G_{N}$; (19) for cosmological constant $\Lambda$; (95) for
the Planck constant $\hbar$ and (65) for the Planck length $\ell_{Pl}$. Most
probably these relations can only be physically relevant close to
singularities either at the very beginning of our universe or at the final
stages of gravitational collapse inside black holes. We left any detailed
discussion of these uncertainty relations for future work.

Intriguingly, the unimodular gravity naturally appears in the thermodynamic or
emergent gravity setup (Jacobson:1995ab, ; Padmanabhan:2013nxa, ;
Padmanabhan:2019art, ; Alonso-Serrano:2020dcz, ). There are already some hints
(Volovik:2020qtp, ; Klinkhamer:2008ff, ) that the scale-free gravity (5) with
the Newton constant as a constant of integration, may also emerge in this way.
It is very interesting to further investigate this option.

On the other hand, combining unimodular gravity with theories where the Newton
constant is another global degree of freedom is the key element of the vacuum
sequester proposal, (Kaloper:2013zca, ; Kaloper:2015jra, ; Padilla:2015aaa, ).
A novelty with respect to vacuum sequester works (Kaloper:2013zca, ;
Kaloper:2014dqa, ; Kaloper:2015jra, ; Kaloper:2016jsd, ; Kaloper:2018kma, ;
Ben-Dayan:2015nva, ; Carroll:2017gqo, ) is that following (Hammer:2020dqp, )
we also use normal Yang-Mills gauge fields instead of more exotic three- and
four-forms to specify a new measure for the action integral. Moreover, in our
formulation the Lagrange multiplier fields from these setups look like axions
and can even have normal kinetic terms, see actions (27), (76) and (81). For
the Newton constant to be a global degree of freedom the axion should either
non-minimally couple to curvature as in (76) or to couple universally to all
matter fields, as in (81). To describe sequester one requires to have at least
two different gauge fields and two different axions. It is rather intriguing
to understand quantum properties of such systems and their potential UV
completions. In particular, such formulation can be also useful for
supersymmetric generalizations, see e.g. (Nagy:2019ywi, ; Baulieu:2020obv, ;
Bansal:2020krz, ). Even though QCD seems to possess a composite three-form
(Luscher:1978rn, ; Aurilia:1980xj, ; Dvali:2005an, ; Dvali:2005ws, ;
Dvali:2005zk, ), we think it is more convenient to see usual Yang-Mills gauge
fields in the action. This brings the field content of such theories closer to
elementary fields present already in the Standard Model. This may be a hint
that it is the IR dynamics of the Yang-Mills vacuum (maybe even QCD or some
“dark” QCD) and its topological properties which secretly define both
gravitational constants $G_{N}$ and $\Lambda$.

###### Acknowledgements.

It is a pleasure to thank I. Saltas and I. Sawicki, for valuable discussions
and criticisms, while we are indebted to N. Kaloper and T. Padilla for very
useful email exchange. K. S. would like to thank the Central European
Institute for Cosmology and Fundamental Physics for their warm and welcoming
hospitality during early stages of the project. K. S. is supported by JSPS
KAKENHI Grant Number JP20J12585. M. Y. is supported in part by JSPS Grant-in-
Aid for Scientific Research Numbers 18K18764 and JSPS Bilateral Open
Partnership Joint Research Projects. This project originated when P. J. and A.
V. were enjoying very warm hospitality of the cosmology group at the Tokyo
Institute of Technology. This productive visit was possible thanks to the JSPS
Invitational Fellowships for Research in Japan (Fellowship ID:S19062) received
by A. V. The work of P. J. is supported by the Grant Agency of the Czech
Republic GAČR grant 20-28525S. The work of A. V. is supported by the J. E.
Purkyně Fellowship of the Czech Academy of Sciences and by the funds from the
European Regional Development Fund and the Czech Ministry of Education, Youth
and Sports (MŠMT): Project CoGraDS - CZ.02.1.01/0.0/0.0/15_003/0000437.

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