# Exact solutions of Cotton Gravity

Roberto A. Sussman<EMAIL_ADDRESS>Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543,
México D.F. 04510, México.  Sebastián Nájera
<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad
Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543, México D.F.
04510, México.

###### Abstract

We examine various exact solutions in “Cotton Gravity” (CG), a new gravity
theory that provides an extension of General Relativity (GR) based on the
Cotton tensor. Using an alternative formulation of the field equations in
terms of a Codazzi tensor, we obtain various non-trivial CG exact solutions
that generalize known GR solutions: FLRW cosmologies, Lemaître-Tolman-Bondi
(LTB) and Szekeres dust solutions, as well as static perfect fluid spheres and
solutions with a shear-free 4 velocity. We show that CG modifies the spatial
curvature of the nonstatic GR solutions, allowing for significant changes in
the dynamics, an evolution from initial deceleration to a late time
accelerated cosmic expansion driven by negative spatial curvature and without
assuming a dark energy source or imposing a cosmological constant. In fact,
the cosmological constant emerges naturally as an integration constant
incorporated into the modified spatial curvature and can be characterized as
the case with constant negative spatial curvature. Static fluid spheres in the
weak field regime of CG allow for modeling the flattening of rotation
velocities in galactic systems without assuming dark matter. The methods we
have presented can be improved to be able to obtain more general solutions
that will facilitate the application of CG to current open problems in
gravitational systems in general.

##  I Introduction

.-

Dark sector sources (dark matter and dark energy) are essential for the
compatibility between cosmological observations, the Copernican principle at
large scales and cosmic dynamics as determined by General Relativity (GR).
However, detection of these sources and the understanding of their fundamental
has remained elusive. This has motivated the search of gravity theories
alternative to GR.

In 2021 Harada harada2021emergence  proposed a new gravity theory that
generalizes GR through the rank-3 Cotton tensor cotton1899varietes ;
ellis2012relativistic . He showed that all solutions of GR are also solutions
of “Cotton Gravity” theory (henceforth CG theory), but there are non-trivial
CG solutions not reducible or equivalent to GR solutions (see Harada’s
response harada2021reply  to a critical comment bargueno2021comment ). As a
first non-trivial solution, Harada derived a Schwarzschild CG analogue that
follows as a spherically symmetric vacuum solution. In a follow up article
harada2022cotton  Harada showed that the Newtonian limit of CG allows for
fitting galactic rotation curves without assuming the existence of dark
matter. The main problem with Harada’s formulation of CG is its inability to
obtain non-trivial solutions for non-vacuum conformally flat spacetimes
characterized by a vanishing Weyl tensor, since the Cotton tensor
$C_{abc}=-4\nabla_{d}C_{abc}^{d}$ vanishes automatically if the Weyl tensor
vanishes garcia2004cotton .

An alternative formulation of the field equations of CG was found by Mantica
and Molinary mantica2023codazzi  in terms of the constraints for a symmetric
tensor formed with the Ricci and energy momentum tensors of GR to be a Codazzi
tensor. These authors further discussed the compatibility between Codazzi
tensors, CG and various assumptions of sources and their connection with known
GR solutions. Since in this formulation it is not necessary to compute
beforehand the Cotton tensor, it can be applied to all conformally flat
spacetimes.

In the present article we adapt (to make it more intuitive) the
parametrization of CG field equations proposed by Mantica and Molinary
mantica2023codazzi . Our aim is not to present realistic spacetimes, but to
illustrate the potential of CG as an extension of GR by generating non-trivial
CG solutions from solutions of GR. Evidently, looking at the feasibility of CG
requires much more research beyond the results we have presented. We use first
the parametrization of CG to reproduce the Schwarzchild CG analogue found in
harada2021emergence . We present then Friedman-Lemaître-Robertson-Walker
(FLRW) models reported in an accompanying paper sussman2023cottonletter , as
well as CG solutions that generalize inhomogeneous GR solutions: Lemaître-
Tolman-Bondi (LTB) and quasi-spherical Szekeres dust models, spherically
symmetric static perfect fluid spheres, as well as perfect fluid solutions
with a shear-free 4-velocity (which mantica2023codazzi  denote as “Generalized
Stephani” solutions).

The non-static CG solutions we found modify the spatial curvature of their GR
analogues, an important degree of freedom absent in GR that introduces a
significant modification of the dynamics. In particular, an accelerated
expansion is possible in FLRW and LTB dust models, without assuming a dark
energy source. The specific $\Lambda$CDM model can be identified as the unique
FLRW dust model in CG with constant negative spatial curvature (constant Ricci
scalar of hypersurfaces orthonormal to the 4-velocity). In the case of static
spheres, we show that rotation curves in the fluid region can exhibit a
flattened profile without assuming dark matter, complementing the results of
Harada in harada2022cotton  in fitting rotational velocity profiles of 84
galactic systems through the Schwarzschild CG vacuum analogue as the far field
of the galactic systems and in the quasi-Newtonian limit. We also reproduce
the results of Mantica and Molinari in mantica2023codazzi  on shear-free
solutions with a perfect fluid source.

The section by section content of this paper is as follows: sections II and
III present the field equations of CG, in the original formulation and in the
alternative one. Sections IV, V, VI, VII and VIII respectively present CG
solutions for FLRW, LTB and Szekeres models, perfect fluid spherical symmetric
solutions for static spacetimes and a shear-free 4-velocity. Our conclusions
are stated in section IX.

##  II Cotton gravity in its original formulation

In harada2021emergence  Harada introduced CG by constructing the following
rank $4$ tensors constructed with the metric, curvature objects and the energy
momentum tensor of GR

$\displaystyle G_{abcd}:$ $\displaystyle=$ $\displaystyle{\cal
R}_{abcd}-\frac{1}{6}\left(g_{ac}g_{bd}-g_{bc}g_{ad}\right){\cal R},$ (1)
$\displaystyle T^{abcd}:$ $\displaystyle=$
$\displaystyle\frac{1}{2}(g^{ac}T^{bd}-g^{bc}T^{ad}-g^{ad}T^{bc}+g^{bd}T^{ac})$
(2) $\displaystyle-\frac{1}{6}(g^{ac}g^{bd}-g^{bc}g^{ad})T.$

where ${\cal R}_{abcd}$ and ${\cal R}$ are the Riemann tensor and the Ricci
scalar.and $T^{ab}$ is the GR energy momentum tensor. The connection with GR
follows from the trace of (1)-(2)

$\displaystyle g^{cd}G_{acbd}=G_{ab},\qquad g^{cd}T_{acbd}=T_{ab}.$ (3)

while the CG field equations (in geometrical units $c=G=1$) follow from the
divergence of the 4th order tensors

$C_{abc}=\nabla_{d}\tensor{G}{{}^{d}_{a}{}_{b}{}_{c}}=16\pi\nabla_{d}\tensor{T}{{}^{d}_{a}{}_{b}{}_{c}},$
(4)

where $C_{abc}$ is the Cotton tensor cotton1899varietes ; stephani2009exact ;
garcia2004cotton  given explicitly as

$C_{abc}=\nabla_{b}{\cal R}_{ac}-\nabla_{c}{\cal
R}_{ab}-\frac{1}{6}\left(g_{ac}\nabla_{b}{\cal R}-g_{ab}\nabla_{c}{\cal
R}\right),$ (5)

with ${\cal R}_{ab}$ the Ricci tensor. The Cotton tensor is proportional to
the divergence of the Weyl tensor: $C_{abc}=-4\pi\nabla_{d}C_{abc}^{d}$, hence
$C_{abc}=0$ automatically follows if $C_{abcd}=0$ holds garcia2004cotton . .

The inability to handle conformally flat spacetimes is a serious problem with
the original formulation of CG in (4). Harada was aware of this fact and
proposed that condition $C_{abc}=0$ should characterize conformally flat
spacetimes together with vacuum solutions, as a generalization of the
condition ${\cal R}_{ab}=0$ characterizing vacuum solutions in in GR. However,
${\cal R}_{ab}=0$ and $C_{abc}=0$ are very different conditions, the former is
a non-trivial constraint that must be solved to get a vacuum solution, the
latter is a trivial condition that holds automatically for conformally flat
spacetimes regardless of their source or symmetries.

##  III CG in terms of a Codazzi tensor

An alternative formulation equivalent to the original CG field equations (4)
was derived by mantica2023codazzi  in terms of the condition that defines a
Codazzi tensor

$\nabla_{b}{\cal C}_{ac}-\nabla_{c}{\cal C}_{ab}=0,$ (6)

where ${\cal C}_{ab}$ is a symmetric tensor constructed with ${\cal
R}_{ab},\,\,T^{ab}$ and $g_{ab}$ that substituted in (6) yields (4). The form
of ${\cal C}_{ab}$ obtained in mantica2023codazzi  can be made more intuitive
and amenable for practical usage by rewriting this tensor in terms of the
Einstein tensors $G_{ab}$ and $T_{ab}$ as

$\displaystyle{\cal C}_{ab}$ $\displaystyle=$ $\displaystyle G_{ab}-8\pi
T_{ab}-\frac{1}{3}(G-8\pi T)\,g_{ab},$ (7) with
$\displaystyle\quad\nabla_{b}T^{ab}=0,$ (8)

where $G=g^{cd}G_{cd},\,\,T=g^{cd}T_{cd}$.

It is straightforward to show that (6)-(8) are equivalent to (4)-(5). Hence,
we consider as non-trivial CG solutions all spacetimes complying with ${\cal
C}_{ab}\neq 0$ in (7) that satisfy the constraints involved in (6) for an
energy momentum tensor $T^{ab}$ complying with the conservation law (8). We
will not consider the following options on (7)-(6):

  1. 1.

GR solutions that are trivial CG solutions. If Einstein equations $G_{ab}-8\pi
T_{ab}=0$ hold, ${\cal C}_{ab}=0$ holds by construction and (6) holds
trivially.

  2. 2.

Solutions with “effective” fluids. If ${\cal C}_{ab}\neq 0$ and (6) hold,
Mantica and Molinari mantica2023codazzi  show it is possible to rearrange
terms in (7) so that it reads as Einstein’s field equations with an
“effective” energy momentum tensor

$\displaystyle G^{ab}$ $\displaystyle=$ $\displaystyle 8\pi
T^{ab}_{(\hbox{\tiny{mod}})}=8\pi[T_{(\hbox{\tiny{gr}})}^{ab}+{\cal
C}^{ab}-{\cal C}\,g^{ab}],$ $\displaystyle\Rightarrow$
$\displaystyle\quad\nabla_{b}T^{ab}_{(\hbox{\tiny{mod}})}=0,$

where ${\cal C}=g^{cd}{\cal C}_{cd}$ and $T_{(\hbox{\tiny{gr}})}^{ab}=T^{ab}$.
It is common in alternative theories (see examples in $f({\cal R})$ gravity in
Clifton:2011jh ; de2010f ; sotiriou2010f ; chakraborty2022note  ) to rewrite
the field equations in the form (2), with geometric terms passed to the right
hand side of Einstein’s equations as “effective fluids”. While expressing CG
as (2) is mathematically self-consistent, the added energy momentum tensor
essentially describes geometric fluids not sustained by physical assumptions
and thus will not be considered. Unless stated otherwise, we will assume that
$T^{ab}\neq T^{ab}_{(\hbox{\tiny{mod}})}$.

Vacuum solutions in GR lead to vacuum solutions in CG, as can be appreciated
by substituting $G^{ab}={\cal R}^{ab}-(1/2){\cal R}g^{ab}$ and ${\cal
R}^{ab}=0$ in (7). The first solution found by Harada is the CG analogue of
the Schwarzschild solution. We obtain this solution from the spherically
symmetric metric

$ds^{2}=-\Phi(r)dt^{2}+\frac{dr^{2}}{\Phi(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (10)

leading to the tensor (7) with components ${\cal C}^{\theta}_{\theta}={\cal
C}^{\phi}_{\phi}=-2{\cal C}^{t}_{t}=-2{\cal
C}^{r}_{r}=(\Phi_{,rr}r^{2}-2\Phi_{,r}r-4\Phi)/(6r^{2})$, which inserted into
(6) yields

$\Phi(r)=1-\frac{2M_{s}}{r}+\gamma\,r+\frac{8\pi}{3}\,\Lambda\,r^{2},$ (11)

where we have identified the integration constants as the Schwarzschild mass
$2M_{s}$ and the cosmological constant (emerging naturally as an integration
constant), while $\gamma r$ is the linear term found by Harada in
harada2021emergence . It is straightforward to obtain the CG analogue of the
Reissner Nordstrom solution by considering the energy momentum tensor
$T^{a}_{b}=-Q/r^{4}\times\hbox{diag}[-1,1,1-1]$ of a point mass with electric
charge $Q$ which adds the term $8\pi Q/r^{2}$ to (11).

##  IV FLRW solutions in Cotton Gravity

FLRW models are non-vacuum conformally flat spacetimes, hence the Cotton
tensor vanishes identically and the field equations (4) become inconsistent.
However, these spacetimes can be handled by the equivalent formulation
(6)-(8). Consider the general spacetime compatible with the Robertson-Walker
metric in spherical coordinates

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ $\displaystyle T^{ab}$ $\displaystyle=$
$\displaystyle(\rho+p)u^{a}u^{b}+pg^{ab},\qquad u^{a}=\delta^{a}_{t},$ (13)

with $k=k_{0}H_{0}^{2}|\Omega_{0}^{(k)}|,\,\,k_{0}=0,\pm\,1$. The components
of the tensor (7) for (LABEL:flrw)-(13) are

$\displaystyle{\cal C}^{r}_{r}$ $\displaystyle=$ $\displaystyle{\cal
C}^{\theta}_{\theta}={\cal
C}^{\phi}_{\phi}=-\frac{8\pi}{3}\rho+\frac{\dot{a}^{2}+k}{a^{2}},$ (14)
$\displaystyle{\cal C}^{t}_{t}$ $\displaystyle=$ $\displaystyle
8\pi\left(\frac{2}{3}\rho+p\right)+\frac{2a\ddot{a}-\dot{a}^{2}-k}{a^{2}},$
(15)

Substitution of (14)-(15) in the condition (6) for (7) to be a Codazzi tensor
yields:

$a\,\dot{\rho}+3\dot{a}(\rho+p)=0,$ (16)

coinciding with the conservation law $\nabla_{b}T^{ab}=0$ of (13), which can
be integrated once an equation of state $p(\rho)$ is imposed, leading to
$\rho=\rho(a)$.

FLRW models of GR follow as trivial CG solutions, ${\cal C}_{ab}=0$. Einstein
equations hold for $T^{ab}$, leading to the Friedman equation of GR

$\dot{a}^{2}=\frac{8\pi}{3}\rho(a)\,a^{2}-k,$ (17)

which can be integrated knowing $\rho(a)$ from (16) to find $a=a(t)$. As
mentioned before, if ${\cal C}_{ab}\neq 0$ and satisfies (6) it is possible to
write the CG field equations as in (2), with an energy momentum tensor
$T^{ab}_{(\hbox{\tiny{mod}})}$ containing ‘effective” fluids that cannot be
justified on physical grounds. We omit this case.

FLRW models that are non-trivial CG solutions and not GR solutions follow by
demanding first that Einstein equations do not hold: $G^{ab}-8\pi T^{ab}\neq
0$ so that ${\cal C}_{ab}\neq 0$. Since we do not wish to modify $T^{ab}$, the
only option is to modify the Friedman equation (17) as

$\dot{a}^{2}=\frac{8\pi}{3}\rho(a)\,a^{2}-k-\gamma{\cal K}(t),$ (18)

where ${\cal K}(t)$ is (for the time being) a dimensionless free function and
$\gamma$ is constant with inverse squared length units to be determined
further ahead. The components of the nonzero ${\cal C}_{ab}$ for (18) are

${\cal C}^{r}_{r}={\cal C}^{\theta}_{\theta}={\cal
C}^{\phi}_{\phi}=\frac{\gamma{\cal K}}{a^{2}},\quad{\cal
C}^{t}_{t}=\frac{\gamma}{a}\left(\frac{\dot{\cal K}}{2\dot{a}}-\frac{{\cal
K}}{a}\right),$ (19)

and (6) is automatically satisfied.

Since the function ${\cal K}(t)$ is not the density of an “effective” fluid
(as in (2)), it is necessarily a geometric degree of freedom brought by CG
that modifies the spatial curvature of the models. The modified Friedman
equation (18) changes the expansion scalar
$\Theta=\nabla_{a}u^{a}=3\dot{a}/a$, which in turn implies a modification of
the extrinsic curvature of the hypersurfaces of constant $t$ orthonormal to
the 4-velocity: $K_{ab}=h_{a}^{c}h_{b}^{d}\nabla_{b}u_{a}=(\Theta/3)h_{ab}$
that determines the embedding of these submanifolds in the enveloping
spacetime. The connection with spatial curvature follows readily from the
Gauss-Codazzi equations thorne2000gravitation ; poisson2004relativist ;
ellis2012relativistic , leading to

$\displaystyle{}^{3}{\cal R}$ $\displaystyle=$ $\displaystyle 16\pi
T_{ab}u^{a}u^{b}+K_{ab}K^{ab}-K^{2},\qquad K=K^{a}_{a},$ (20) $\displaystyle=$
$\displaystyle 16\pi\rho-\frac{2}{3}\Theta^{2}=\frac{6[k+\gamma{\cal
K}]}{a^{2}}.$

where ${}^{3}{\cal R}$ is the Ricci scalar of the hypersurfaces orthonormal to
$u^{a}$.

The modification of ${}^{3}{\cal R}$ produced by CG modifies the dynamics of
the models, as can be seen in the following system of 3 evolution equations
and the Friedman equation (18) as the Hamiltonian constraint:

$\displaystyle\dot{\rho}$ $\displaystyle=$ $\displaystyle-(\rho+p)\Theta,$
(21) $\displaystyle\dot{\Theta}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\Theta^{2}-4\pi(\rho+3p)-\frac{3\lambda}{2a}\frac{\textrm{d}{\cal
K}(a)}{\textrm{d}a},$ (22) $\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\Theta a,$ (23) $\displaystyle\frac{\Theta^{2}}{9}$
$\displaystyle=$ $\displaystyle\frac{8\pi}{3}\rho-\frac{k+\lambda{\cal
K}}{a^{2}},$ (24)

where $d{\cal K}(a)/da=\dot{\cal K}/\dot{a}$. The integration of (21)-(24)
requires a functional form ${\cal K}={\cal K}(a)$ to be specified, but there
is no evolution equation or conservation law for this term. Hence, for a
general ${\cal K}(a)$ the system (21)-(24) is incompatible with a well posed
initial value formulation that requires finding a unique solution given
initial data at an initial hypersurface $t=t_{0}$, which only determines
${\cal K}_{0}={\cal K}(a(t_{0}))={\cal K}(1)$, but without an evolution
equation does not determine ${\cal K}$ for $t\neq t_{0}$. While ${\cal K}(a)$
can be prescribed beforehand and could lead to interesting dynamics
unachievable by FLRW models in GR, this involves a priori determination of
future (even asymptotic) evolution disconnected from initial conditions.

However, there are three particular cases of FLRW models in CG in which
(21)-(24) are compatible with a well posed initial value formulation:

$\displaystyle{\cal K}$ $\displaystyle=$ $\displaystyle\hbox{constant},\qquad
H^{2}=\frac{8\pi}{3}\rho-\frac{k+\lambda}{a^{2}},$ (25) $\displaystyle k$
$\displaystyle=$ $\displaystyle 0\,\,\hbox{and}\,\,{\cal K}=a^{2},\quad
H^{2}=\frac{8\pi}{3}\rho-\lambda,$ (26) $\displaystyle k$ $\displaystyle\neq$
$\displaystyle 0\,\,\hbox{and}\,\,{\cal K}=a^{2},\quad
H^{2}=\frac{8\pi}{3}\rho-\frac{k}{a^{2}}-\lambda,$ (27)

Models (25) are FLRW models of GR with $\Lambda=0$ and $k$ rescaled, trivial
solutions of CG with ${\cal C}^{a}_{b}=0$. Models (26) are non-trivial CG
solutions with ${\cal C}^{a}_{b}\neq 0$, characterized by constant spatial
curvature: ${}^{3}{\cal R}=6\lambda$, whose sign is determined by the sign of
$\lambda$. Models (27) are models (26) with $k\neq 0$.

The models (26) and (27) above coincide with FLRW models of GR with nonzero
$\Lambda$ by identifying $\lambda=(8\pi/3)\Lambda$. While they operationally
coincide with FLRW models of GR, there is an important conceptual difference.
In GR, $\Lambda$ has units of energy density, so theoretically it is a source
(likely an empiric approximation of dark energy), hence it is normalized by
$8\pi G/c^{4}$ to put it in 1/$\hbox{length}^{2}$ units. In CG $\lambda$ has
the same dynamical effect as $(8\pi/3)\Lambda$, but it is a constant spatial
curvature, not a source. Hence, to emphasize the conceptual difference it is
better to use a different symbol $\lambda$ and since it already has
1/$\hbox{length}^{2}$ units, it can be fixed simply as a value proportional to
$H_{0}^{2}$ that provides a best fit for observations.

For cosmological applications we assume $\Lambda>0$ (hence $\lambda<0$) and
$\rho=\rho_{0}/a^{3}$ to represent cold dark matter (CDM) density (neglecting
other constituents like baryons, electrons, photons and neutrinos). We
consider then the models (26) and (27) with $\lambda<0$ and assigning its
numerical value as $\lambda=-(8\pi/3)\Lambda$ determined by observations. The
Hubble scalar is given only in terms of CDM and spatial curvature, without a
cosmological constant $\Lambda$ in energy density units taken as an
approximation of a dark energy source:

$\displaystyle\frac{H^{2}}{H_{0}^{2}}$ $\displaystyle=$
$\displaystyle\Omega^{(\hbox{\tiny{CDM}})}+\Omega^{(K)},$ (28)
$\displaystyle\Omega^{(\hbox{\tiny{CDM}})}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{0}^{(\hbox{\tiny{CDM}})}}{a^{3}},\quad\Omega^{(K)}=\frac{\Omega_{0}^{(k)}}{a^{2}}+\Omega_{0}^{(\lambda)},$
(29) $\displaystyle\Omega_{0}^{(m)}$ $\displaystyle=$
$\displaystyle\frac{8\pi\rho_{0}}{3H_{0}^{2}},\quad\Omega_{0}^{(k)}=\frac{|k|}{H_{0}^{2}},\quad\Omega_{0}^{(\lambda)}=\frac{|\lambda|}{H_{0}^{2}}$
(30)

with
$\Omega_{0}^{(K)}=\Omega_{0}^{(k)}+\Omega_{0}^{(\lambda)}=1-\Omega_{0}^{(\hbox{\tiny{CDM}})}$.
In particular, the $\Lambda$CDM is equivalent to the FLRW model (2) with
$\Omega_{0}^{(k)}=0$, which is the unique FLRW dust model in CG with constant
negative spatial curvature. This theoretical characterization of the
$\Lambda$CDM model is more appealing than that of GR, either as a spatially
flat FLRW model providing a late time description of an accelerated cosmic
expansion dominated by a dark energy source approximated by $\Lambda$, or
simply as an FLRW model empirically constructed to fit observations.

In fact, the CDM filled FLRW models of CG with ${\cal K}=-a^{2}$ and ${\cal
K}=$ constant can fit observations and account for an accelerated expansion
driven by spatial curvature, without the need to invoke a dark energy source.
We can also identify as particular cases of models (2) the constant curvature
vacuum cases setting $\rho=0$, lading to de Sitter
($\lambda=-(8\pi/3)\Lambda$), anti-de Sitter ($\lambda=(8\pi/3)\Lambda$) and
Minkwski ($\lambda=0$).

##  V LTB dust solutions in Cotton Gravity

The spherically symmetric Lemaître-Tolman-Bondi (LTB) dust solutions of GR
provide a simple inhomogeneous and anisotropic generalization of FLRW dust
models. As we show in this section, CG can generate non-trivial dust solutions
based on the LTB metric

$ds^{2}=-dt^{2}+\frac{R^{\prime
2}}{1-K}\,dr^{2}+R^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$ (31)

where $R=R(t,r),\,K=K(r)$ and a prime denotes $\partial_{r}$. For a dust
source $T^{ab}=\rho u^{a}u^{b}$ in a comoving frame $u^{a}=\delta^{a}_{t}$,
the field equations of GR reduce to a Friedman-like equation that determines
the evolution of dust layers and an equation for the dust density

$\displaystyle\dot{R}^{2}$ $\displaystyle=$ $\displaystyle\frac{2M}{R}-K,$
(32) $\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle\frac{2M^{\prime}}{R^{2}R^{\prime}},$ (33)

where $M=M(r)$. The GR solution for (31)-(33) yields ${\cal C}^{a}_{b}=0$.

Hence, from our approach followed for FLRW models, we avoid casting CG field
equations (6)-(7) as Einstein’s equation with “effective” fluids. Instead, we
obtain a non-trivial dust CG solution with metric (31) by modifying the
Friedman-like equation (32) as

$\dot{R}^{2}=\frac{2M}{R}-K-{\cal K}(R),$ (34)

where ${\cal K}(R)$ is a dimensionless free function to be determined by (6)
(${\cal K}={\cal K}(R)$ is not the most general choice, but it facilitates the
integration of (34)). From (34) we get the nonzero components of (7)

$\displaystyle{\cal C}^{\theta}_{\theta}$ $\displaystyle=$ $\displaystyle{\cal
C}^{\phi}_{\phi}=-\frac{1}{6}\frac{d^{2}{\cal
K}}{dR^{2}}+\frac{1}{3R}\frac{d{\cal K}}{dR}+\frac{2{\cal K}}{3R},$ (35)
$\displaystyle{\cal C}^{r}_{r}$ $\displaystyle=$ $\displaystyle{\cal
C}^{t}_{t}=\frac{1}{3}\frac{d^{2}{\cal K}}{dR^{2}}+\frac{1}{3R}\frac{d{\cal
K}}{dR}-\frac{{\cal K}}{3R^{2}},$ (36)

which inserted in (6) yields the following constraint

$\frac{d^{3}{\cal K}}{dR^{3}}+\frac{1}{R}\frac{d^{2}{\cal
K}}{dR^{2}}-\frac{2}{R^{2}}\frac{d{\cal K}}{dR}+\frac{2{\cal K}}{R^{3}}=0,$
(37)

whose solution is very similar to the CG Schwarzschild-like solution (11)

${\cal K}(R)=\gamma(r)R+\frac{\beta(r)}{R}+\lambda(r)R^{2},$ (38)

where we can identify $\beta(r)=2M(r)$ as the effective mass of comoving dust
layers, while $\gamma(r)$ and $\lambda(r)$ are arbitrary functions with
inverse length and inverse square length units that appear as integration
constants. Equation (34) becomes now

$\frac{\dot{R}^{2}}{R^{2}}=\frac{2M}{R^{3}}-\frac{K+\gamma R+\lambda
R^{2}}{R^{2}},$ (39)

Notice that the quadratic term leads to the cosmological constant, which need
not be imposed, but can be incorporated directly in the modification of the
Friedman-like equation (34) by setting $\lambda(r)=(8\pi/3)\Lambda$. The term
$\gamma(r)R$ is analogous to the extra linear term in the Schwarzschild CG
solution (11).

We emphasize that we have derived a non-trivial CG solution for the LTB metric
(31) with a dust source given by (33). While LTB models in GR with a nonzero
$\Lambda$ (LTB de Sitter) are also characterized by (31) and (33), the metric
functions $R,\,\,R^{\prime}$ must be now computed with the solution $R(t,r)$
obtained with the CG Friedman-like equation (39), which contains the term
$\gamma R$ and thus are necessarily different from those obtained in GR from
(32) (with $\Lambda$ zero or nonzero).

As with FLRW models, the modification of the Friedman-like equation leading to
(39) also involves a change of the spatial curvature. To illustrate this fact,
we stress first that a modified Friedman-like equation modifies the kinematic
parameters (expansion scalar $\Theta$ and shear tensor $\sigma_{ab}$) in the
extrinsic curvature of the hypersurfaces orthogonal to $u^{a}$

$\displaystyle K_{ab}$ $\displaystyle=$ $\displaystyle
h_{a}^{c}h_{b}^{d}\nabla_{b}u_{a}=\frac{\Theta}{3}h_{ab}+\sigma_{ab},$
$\displaystyle K^{r}_{r}$ $\displaystyle=$
$\displaystyle\frac{\dot{R}^{\prime}}{R^{\prime}},\qquad
K^{\theta}_{\theta}=K^{\phi}_{\phi}=\frac{\dot{R}}{R}$ (40)
$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\nabla_{a}u^{a}=\frac{2\dot{R}}{R}+\frac{\dot{R}^{\prime}}{R^{\prime}},$
(41) $\displaystyle\sigma^{a}_{b}$ $\displaystyle=$
$\displaystyle\hbox{diag}[0,-2\Sigma,\Sigma,\Sigma],\quad\Sigma=-\frac{1}{3}\left(\frac{\dot{R}^{\prime}}{R^{\prime}}-\frac{\dot{R}}{R}\right),$

Substitution of (39) into (40)-(LABEL:LTBshear) into the Gauss-Codazzi
equations (20) and the LTB dust density (33) we get the Ricci scalar of the
hypersurfaces orthogonal to $u^{a}$

$\displaystyle{}^{3}{\cal
R}=16\pi\rho-\frac{2}{3}\Theta^{2}+\sigma_{ab}\sigma^{ab}=\frac{2\left[(K+\gamma
R+\lambda R^{2})\,R\right]^{\prime}}{R^{2}\,R^{\prime}},$ (43)

which contains the term $K+\gamma R+\lambda R^{2}$ in the modified Friedman-
like equation (39).

The spatial curvature term that appears in the right hand side of (39) is the
quasilocal scalar (“q-scalar”) associated to ${}^{3}{\cal R}$ by the average
distribution sussman2013weighed1 ; sussman2013weighed2

${}^{3}{\cal R}_{q}=\frac{\int_{0}^{r}{{}^{3}{\cal
R}\,R^{2}\,R^{\prime}d\bar{r}}}{\int_{0}^{r}{R^{2}\,R^{\prime}d\bar{r}}}=\frac{K+\gamma
R+\lambda R^{2}}{R^{2}},$ (44)

which applied to the density $8\pi\rho R^{2}R^{\prime}=2M^{\prime}$ and the
expansion scalar $\Theta\,R^{2}R^{\prime}=\partial_{t}(R^{2}R^{\prime})$,
leads to

$\frac{4\pi}{3}\rho_{q}=\frac{M}{R^{3}},\qquad\frac{\Theta_{q}}{3}=\frac{\dot{R}}{R},$
(45)

transforming (39) into a relation between q-scalars that is formally identical
to the Friedman equation of FLRW dust models:

$\left(\frac{\Theta_{q}}{3}\right)^{2}=\frac{8\pi}{3}\rho_{q}-\frac{1}{6}\,{}^{3}{\cal
R}_{q},$ (46)

with ${}^{3}{\cal R}_{q}$ given by (44).

Notice that CG allows to incorporate into the spatial curvature a position
dependent term $\lambda(r)$ analogous to the cosmological constant. In fact,
as in FLRW models, by setting $K=\gamma=0$ and ${}^{3}{\cal
R}=(8\pi/3)\Lambda=-\lambda(r)$ in (43), a positive cosmological constant can
be characterized as the particular case of LTB models in CG with constant
negative spatial curvature.

As a consequence of the modification imposed by CG that results in the
Friedman-like equation (39), the evolution of comoving dust layers of this CG
solution in general is found by integrating the quadrature

$t-t_{bb}(r)=\int_{0}^{R}{\frac{\sqrt{\bar{R}}\,d\bar{R}}{\sqrt{2M-K\bar{R}-\gamma\,\bar{R}^{2}-\lambda\,\bar{R}^{3}}}},$
(47)

where $t_{bb}(r)$ is an integration constant denoting the “big bang time”. As
opposed to FLRW models for which CG does not provide an evolution equation for
the term ${\cal K}(a)$, in the case of LTB models of CG the field equations
provide a fully determined form of ${\cal K}(R)$ in (38). Hence, the initial
value formulation is well posed, with the evolution of the models fully
determined at all times by initial conditions given by specifying at an
initial hypersurface orthonormal to the 4-velocity the functions
$K(r),\,M(r),\,t_{bb}(r),\,\gamma(r)$ and $\lambda(r)$ (though one of these
functions can be eliminated by a choice of radial coordinate, for example
$R_{0}=R(t_{0},r)=r$).

The presence of the term $\gamma R^{2}$ means that solutions of (47) are
manifestly distinct from those of the equivalent quadrature for the LTB dust
solution of GR with $\Lambda$ zero or nonzero (the case $\gamma=0$). For the
usual LTB models in GR ($\Lambda=0$) the spatially flat sub-case ($K=0$) dust
layers are ever expanding, while the equivalent CG solution
($K=0,\,\,\gamma\neq 0$) allows for collapsing layers if $\gamma>0$. Another
interesting feature is the radial dependence of $\lambda(r)$ analogous to the
cosmological constant $\Lambda$. Hence LTB models can model inhomogeneities in
which the effect of $\Lambda$ is negligible ($\lambda=0$ near $r=0$) and
becomes dominant ($\lambda\to 8\pi\Lambda/3$ as $r\to\infty$).

The FLRW limit of LTB dust models in CG is analogous to the same limit in GR:
it follows as $R$ becomes a separable function $R=a(t)f(r)$, with $a(t)$
satisfying the modified Friedman equation (18) with $\gamma{\cal K}=\gamma
a+\lambda a^{2}$. While the linear term $\gamma a$ of ${\cal K}$ is not
compatible with a well posed initial value formulation when deriving the FLRW
models, in this case it would follow as a limit of LTB models complying with
this formulation.

If $M$ is a constant the LTB solution from CG also reduces to the
Schwarzschild CG solution (11) (but described in comoving coordinates
constructed with radial timelike geodesics). Also, as with the GC solution for
FLRW models, accelerated expansion of dust layers is possible (even with
$\Lambda=0$) in many cases in which the layers decelerate in the GR solutions.
Since LTB dust solutions are simple but useful models of structure formation,
this difference of evolution due to modifications from CG might have
interesting consequences.

Finally, the description of the LTB dust solution in CG in terms of an
“effective” energy momentum tensor $T_{(\hbox{\tiny{mod}})}$, as described in
(2), leads to an unphysical fluid with anisotropic pressure whose components
are

$\displaystyle-T^{t}_{t}{}_{(\hbox{\tiny{mod}})}$ $\displaystyle=$
$\displaystyle\rho+c_{3}+\frac{2c_{1}}{R},$ (48) $\displaystyle
T^{r}_{r}{}_{(\hbox{\tiny{mod}})}$ $\displaystyle=$ $\displaystyle-
c_{3}-\frac{2c_{1}}{R},$ (49) $\displaystyle
T^{\theta}_{\theta}{}_{(\hbox{\tiny{mod}})}$ $\displaystyle=$ $\displaystyle
T^{\phi}_{\phi}{}_{(\hbox{\tiny{mod}})}=-c_{3}-\frac{c_{1}}{R},$ (50)

with $\rho$ given by (33). The “effective” fluid satisfies
$T^{ab}_{(\hbox{\tiny{mod}})}\,{}_{;b}=0$ as long as
$c_{1}^{\prime}=c_{3}^{\prime}=0$. However, it is not compatible with a
regular center worldline ($\rho,\,p$ bounded at $r=0$ with
$R(t,0)=\dot{R}(t,0)=0$), hence it cannot describe a model of a localized
spherical inhomogeneity. Also, this fluid has no FLRW limit as $R$ becomes a
separable function $R=a(t)f(r)$.

##  VI Szekeres solutions in Cotton Gravity

Szekeres models are a large class of exact solutions, typically with a
comoving dust source ($T^{ab}=\rho u^{a}u^{b},\,\,u^{a}=\delta^{a}_{t}$),
admitting (in general) no isometries. They are expressible in the line element

$ds^{2}=-dt^{2}+e^{2\alpha}dr^{2}+e^{2\beta}[dx^{2}+dy^{2}]$ (51)

where $\alpha,\,\beta$ depend on all four coordinates. We consider class I
models ($\beta^{\prime}=\beta_{,r}\neq 0$) and the quasi-spherical sub-class
of solutions whose spherically symmetric limit are LTB models. For this
solution (51) is expressible in the following LTB-like parametrization

$\displaystyle.ds^{2}$ $\displaystyle=$ $\displaystyle-dt^{2}+\frac{{\cal
E}^{2}Y^{\prime}{}^{2}}{1-K}\,dr^{2}+Y^{2}[dx^{2}+dy^{2}],$ (52)
$\displaystyle Y$ $\displaystyle=$ $\displaystyle\frac{R(t,r)}{{\cal
E}(r,x,y)},\quad K=K(r),$ (53)

with ${\cal E}$ and $Y$ satisfying

$\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle\frac{S}{2}\left[1+\left(\frac{x-P(r)}{S}\right)^{2}+\left(\frac{y-Q(r)}{S}\right)^{2}\right],$
(54) $\displaystyle\dot{Y}^{2}$ $\displaystyle=$
$\displaystyle\frac{2\tilde{M}}{Y}-\tilde{K}\quad\Rightarrow\quad\dot{R}^{2}=\frac{2M}{R}-K,$
(55)

with $\tilde{M}=M(r)/{\cal E}^{3}$ and $\tilde{K}=K/{\cal E}^{2}$ and we
assume $\Lambda=0$. Einstein’s field equations reduce to the single nonzero
component:

$\ G_{tt}=8\pi\rho=\frac{2\tilde{M}^{\prime}}{Y^{2}\,Y^{\prime}}.$ (56)

In this parametrization the Friedman-like equation and the nonzero component
of the field equations of the Szekeres solution are formally identical to
their LTB equivalents (32) and (33). Therefore, the CG Szekeres solution
follows the same pattern as the LTB solution in equations, with the Friedman-
like equation

$\displaystyle\dot{Y}^{2}=\frac{2\tilde{M}}{Y}-\tilde{K}-\tilde{\cal
K},\qquad\tilde{\cal K}=\frac{{\cal K}(R)}{{\cal E}^{2}}$
$\displaystyle\Rightarrow\quad\dot{R}^{2}=\frac{2M}{R}-K-{\cal K}$ (57)

leading to the same components of the prospective Codazzi tensor (7) for the
LTB metric in (35)-(36), with quantities normalized by ${\cal E}^{2}$ and the
constraint (37) by ${\cal E}^{3}$. All comments and results on degress of
liberty that CG affords to LTB models apply also to the much more general
quasi-spherical Szekeres models, which provide more elements for modeling
cosmological inhomogeneities.

##  VII A spherically symmetric static solution

Consider the spherically symmetric line element for static GR solutions

$ds^{2}=-\exp[2\Phi(r)]dt^{2}+\frac{dr^{2}}{1-\frac{2M(r)}{r}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (58)

where the source is a perfect fluid $T^{ab}=\rho u^{a}u^{b}+ph^{ab}$ with
$u^{a}=\exp(-\Phi)\delta^{a}_{t}$. Einstein’s field equations are

$\displaystyle M^{\prime}$ $\displaystyle=$ $\displaystyle 4\pi\rho r^{2},$
(59) $\displaystyle\Phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{4\pi\,p\,r^{3}+M}{r(r-2M)},$ (60)

where a prime denotes radial derivative. Auxiliary conditions follow from the
momentum balance $T^{ab}\,_{;b}=0$ and the isotropy of pressure
$T^{r}_{r}-T^{\theta}_{\theta}=0$

$\displaystyle p^{\prime}=-(\rho+p)\Phi^{\prime},$ (61)
$\displaystyle\left(1-\frac{2M}{r}\right)\left[\Phi^{\prime\prime}+\Phi^{\prime}{}^{2}-\frac{2\Phi^{\prime}}{r}\right]-\frac{[rM^{\prime}+M-r]\Phi^{\prime}}{r^{2}}$
$\displaystyle+\frac{3M-rM^{\prime}}{r^{3}}=0.$ (62)

Using the equations above, the Codazzi tensor (7) for GR

$\displaystyle{\cal C}^{r}_{r}$ $\displaystyle=$ $\displaystyle{\cal
C}^{\theta}_{\theta}={\cal
C}^{\phi}_{\phi}=-\frac{8\pi}{3}\rho+\frac{2M^{\prime}}{3r^{2}}=0,$ (63)
$\displaystyle{\cal C}^{t}_{t}$ $\displaystyle=$ $\displaystyle
8\pi\left(\frac{2}{3}\rho+p\right)-2\left(1-\frac{2M}{r}\right)\frac{\Phi^{\prime}}{2}$
(64) $\displaystyle+$ $\displaystyle\frac{2(3M-2rM^{\prime})}{3r^{3}}=0.$

where we used (59)-(60). The GR equations (63)-(64) yield ${\cal
C}^{a}_{b}=0$.

The modification of rotation velocities of circular geodesics provides a
motivation to obtain a nonzero Codazzi tensor and thus a CG solution. From the
particle Lagrangean $-2{\cal
L}=-\exp(2\Phi)\dot{t}^{2}+r_{0}^{2}\dot{\phi}^{2}$, the velocity of observers
along timelike circular geodesics ($\dot{r}=dr/d\tau=0,\theta=\pi/2$)
associated with (58) are

$v^{2}(r)=r^{2}\,\dot{\phi}^{2}=r\left[e^{2\Phi(r)}-1\right]\approx
r\Phi^{\prime}\approx\frac{M(r)}{r},$ (65)

where we used the weak field limit $2M/r\ll 1,\,\,\Phi\ll 1$ also applied to
(60) ($p/\rho\sim\sigma^{2}\ll c^{2}$, with $\sigma^{2}$ the dispersion
velocity). Equation (65) shows that in GR the only way to account for the
observed increase of $v^{2}(r)$ in galactic rotation curves is an increase of
$M$ (and of density $\rho$ from (59)) and a modification of $\Phi^{\prime}$
produced by the assumption of an extra “invisible” mass.

Thus, CG, as an alternative gravity theory, can be useful to increase
$v^{2}(r)$ without modifying the matter-energy source taken as visible mass
and density $M$ and $\rho$. For this purpose we need to generate a nonzero
${\cal C}^{a}_{b}$ from (7), thus implying that Einstein’s equations are not
satisfied for $T^{ab}$. Hence, we replace the metric function $\Phi(r)$ by
$\Phi(r)+\Psi(r)$, where $\Psi(r)$ becomes determined by inserting (7) into
(6). The result is

${\cal
C}^{t}_{t}=-2\left(1-\frac{2M}{r}\right)\frac{\Psi^{\prime}}{r},\quad{\cal
C}^{r}_{r}={\cal C}^{\theta}_{\theta}={\cal C}^{\phi}_{\phi}=0,$ (66)

which inserted into (6) leads to the following quadrature

$\Psi^{\prime}=C_{0}\exp\left[-\Phi+\int_{0}^{r}{\frac{1-4M/\bar{r}+2M^{\prime}}{\bar{r}(1-2M/\bar{r})}d\bar{r}}\right],$
(67)

where $C_{0}$ is a constant with inverse length units. The quadrature (67) can
be integrated for any given GR solution with $\rho,\,\Phi$ and $M$ computed
only with visible sources. The integral in (67) must be evaluated up to the
boundary $r=r_{b}$ of the galactic system, where the static solution should
match with the equivalent Schwarzschild CG solution (10)-(11). Harada assumed
this matching in harada2022cotton , but matching conditions in CG still need
to be examined.

The matter-energy source of the CG solution we discuss in this section is
$T^{ab}$, but it can also be expressed in terms of an “effective” energy
momentum tensor $T_{(\hbox{\tiny{mod}})}$, as described in (2), leading to a
perfect fluid $T^{ab}=\rho u^{a}u^{b}+p_{(\hbox{\tiny{mod}})}h^{ab}$, with the
modified pressure given by

$p_{(\hbox{\tiny{mod}})}=p-2\left(1-\frac{2M}{r}\right)\frac{\Psi^{\prime}}{r}.$
(68)

and the modified equilibrium equation (60) taking the form

$(\Phi+\Psi)^{\prime}=\frac{4\pi[p+p_{(\hbox{\tiny{mod}})}]r^{3}+M}{r(r-2M)},$
(69)

with $p$ given by (68). However, the usage of this “effective” fluid is not
necessary nor useful.

In the weak field limit $2M/r\ll 1,\,\,\Phi\ll 1$ (67) becomes

$\displaystyle\Psi^{\prime}$ $\displaystyle\approx$ $\displaystyle
C_{0}\exp\left\\{-\Phi+\int_{0}^{r}{\left[\frac{1}{\bar{r}}+8\pi\rho\bar{r}\right]d\bar{r}}\right\\},$
(70) $\displaystyle\approx$
$\displaystyle\frac{C_{0}r}{r_{b}}\exp\left[-\Phi+8\pi\int_{0}^{r}{\rho\bar{r}d\bar{r}}\right],$

where we used (59) and $r_{b}$ is a second integration constant that can be
identified with the radius of a galactic system. Velocities of circular
geodesics follow from the Lagrangean associated with the combined potentials
$v^{2}\approx r(\Phi+\Psi)^{\prime}$

$\displaystyle v^{2}(r)\approx
r\Phi^{\prime}+\frac{C_{0}r^{2}}{r_{b}}\exp\left[-\Phi+8\pi\int_{0}^{r}{\rho\bar{r}d\bar{r}}\right],$
(71)

identifying the Keplerian term $r\Phi^{\prime}\sim M(r)/r$ that is dominant at
small scales $r\approx 0$ and the CG contribution that grows and becomes
dominant as $r$ increases, since the arguments of the exponential rapidly
decay in this limit.

To probe this approximation, we use the parameters of the Plummer sphere, a
toy model of a spherical galaxy binney2011galactic

$\Phi(r)=-\frac{GM_{0}}{\sqrt{b^{2}+r^{2}}}\quad\frac{4\pi}{3}\rho(r)=\frac{4\pi
M_{0}b^{2}}{\left(b^{2}+r^{2}\right)^{5/2}},$ (72)

where $M_{0},\,b$ are constants with mass and length units. Assuming a
spherical galaxy of total mass $M_{0}=10^{11}\hbox{M}_{\odot}$ with radius
$r_{b}=20$ kpc, we can define a dimensionless radius and relate the length
parameters $b$ and $C_{0}$ as

$x=\frac{r}{b},\quad b=\frac{1}{C_{0}}=\frac{4\pi G\,\rho_{c}}{3v_{e}^{2}},$
(73)

where $\rho_{c}=\rho(0)=0.4\,\hbox{M}_{\odot}/\hbox{pc}^{3}$ and $v_{e}=500\,$
km/s is the escape velocity daod2019density . Using these parameters we plot
the rotation curves $v(r)$ from (71) in figure 1, separating the purely
Keplerian curve (blue), the CG contribution (red) and their sum (black).
Notice that the CG contribution compensates the decay of the Keplerian curve,
with the sum of both producing a roughly flat profile as $r$ increases.

The increase of rotation velocities caused by CG occurs without modifying the
GR density and mass in each spherical layer (as determined by the mass-
luminosity relation). While rotation velocities grow too fast $v\sim r$ as $r$
increases, the radial range is bounded by $0\leq r\leq r_{b}$, the boundary
radius at which the fluid CG solution should match the Schwarzschild analogue
(11). Evidently, this toy model is unrealistic, though it serves to illustrate
the possibility the extra degrees of freedom of CG can be useful to address
the rotation velocities of galactic systems, either without assuming dark
matter or assuming lesser quantity. Harada has already proven that CG can fit
galactic rotation velocities without dark matter through the Schwarzschils
analogue, but he did not actually derive the CG solution in the inner fluid
region.

Figure 1: Rotational curves for a Plummer sphere. The curves respectively
correspond to the Keplerian contribution (blue), the CG contribution (red) and
their sum (black). They were obtained from (71) with the parameters defined in
(72)-(73). Notice how the CG contribution lifts and flattens the decaying
Keplerian curve that would result in the weak field of GR without sark matter.

##  VIII Spherically symmetric shear-free solutions

A large class of solutions describing inhomogeneous and anisotropic perfect
fluid sources $T^{ab}=\rho u^{a}u^{b}+ph^{ab}$ with a shear-free 4-velocity
($\sigma_{ab}=0$, the spherically symmetric Stephani-Barnes solutions in
Krasinski ) is given by the metric

$ds^{2}=-N^{2}dt^{2}+B^{2}\left[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (74)

where $N$ and $B$ depend on $(t,r)$ and the comoving 4-velocity is
$u^{a}=N^{-1}\delta^{a}_{t}$. The constraint of Einstein equations
$h_{a}^{c}\nabla_{b}\sigma^{b}_{c}-h_{a}^{b}\nabla_{b}\Theta=0$ implies
$\Theta=\Theta(t)$. The FLRW limit of the solutions follow from $B=a(t)$ and
$N=1$, hence the time coordinate can be fixed as

$N=\frac{B_{,t}/B}{\Theta/3},\qquad\frac{\Theta}{3}=\frac{S_{,t}}{S},$ (75)

where $S(t)$ is an arbitrary function analogous to the FLRW scale factor. The
condition of pressure isotropy $G^{r}_{r}-G^{\theta}_{\theta}=0$ leads to the
constraint

$L_{yy}=F(y)L^{2},\qquad L=\frac{1}{B},\qquad y=\frac{r^{2}}{2},$ (76)

where $F(y)$ is related to the conformal invariant $\Psi_{2}$ that determines
the Weyl tensor for Petrov type D spacetimes sussman1987spherically  and has
only an electric part: $C_{ab}^{cd}=4(h_{[a}^{[c}+u_{]a}u^{[c})E_{b]}^{d]}$
ellis2012relativistic . The components of the electric Weyl tensor and the
radial derivative of the density are given by

$\displaystyle E^{r}_{r}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\,y\,L^{3}\,F=-2E^{\theta}_{\theta}=-2E^{\phi}_{\phi},$
(77) $\displaystyle 8\pi\rho_{,y}$ $\displaystyle=$
$\displaystyle(2yF_{,y}+5F)\,L^{3},$ (78)

Individual solutions follow by choosing the functional form of $F(y)$ and will
produce two arbitrary functions of time ($S(t)$ and another one) as
integration constants. In particular, the spherically symmetric conformally
flat Stephani solution is case with $L_{yy}=F=0$ (so that $C_{ab}^{cd}=0$ and
$\rho=\rho(t)$) , while the McVittie solution corresponds to $F(y)=y^{-5/2}$.

The conservation equations $\nabla_{b}T^{ab}=0$ are

$\displaystyle\rho_{,t}$ $\displaystyle=$
$\displaystyle-\frac{3L_{,t}}{L}(\rho+p),\quad
h_{a}^{b}p_{,b}=-\dot{u}_{a}(\rho+p),$ (79) $\displaystyle\dot{u}_{a}$
$\displaystyle=$ $\displaystyle
h_{a}^{b}\frac{N_{,b}}{N}=\left(\frac{L_{,yt}}{L_{,t}}-\frac{L_{,y}}{L}\right)\delta_{a}^{y},$
(80)

while the GR density and pressure are given by

$\displaystyle\frac{8\pi}{3}\rho$ $\displaystyle=$
$\displaystyle\frac{S_{,t}^{2}}{S^{2}}+\left(\frac{2}{3}L^{2}F-L_{,y}^{2}\right)y+LL_{,y},$
(81) $\displaystyle p$ $\displaystyle=$
$\displaystyle-\rho+\frac{3L}{L_{,t}}\rho_{,t},$ (82)

where (82) follows from (79). The spatial components of the tensor (7) ${\cal
C}^{i}_{i},\,\,\,i=y,\theta,\phi$ (no sum) take the form (81) and ${\cal
C}^{t}_{t}$ takes the form (81) . To obtain a nonzero Codazzi tensor we modify
${\cal C}^{a}_{b}$ adding to ${\cal C}^{i}_{i}$ a function $\alpha(t,y)$ and
to ${\cal C}^{t}_{t}$ a function $\beta(t,y)$. Substitution into (6) yields
three constraints whose solution is

$\alpha_{,y}=0,\qquad\beta=\alpha(t)+\alpha_{,t}\frac{L}{L_{,t}},$ (83)

which suggests identifying $\alpha$ with a modification of spatial curvature
following the steps of (20) (since $\Theta=\Theta(t)$), while the function
$\beta$ can be identified with a modification of $L/L_{,t}$ that appears in
the lapse function $N$ and alters the relation between proper and coordinate
time: $\tau=\int{dt/N}$. In terms of an “effective” fluid, the functions
$\alpha,\,\beta$ involve a modified energy momentum tensor
$T^{ab}_{(\hbox{\tiny{mod}})}$, with $\rho_{(\hbox{\tiny{mod}})}=\rho-3\alpha$
and $p_{(\hbox{\tiny{mod}})}=p-2\alpha-\beta$. This corroborates the result of
mantica2023codazzi  that these solutions admit an “effective” energy momentum
tensor of the form of a perfect fluid.

The conformally flat case (Stephani solution $L_{,yy}=F=0$) is particularly
interesting, as it is impossible to consider it in the original CG formulation
of Harada (equations (4)-(5)), since the Cotton tensor vanish automatically.
However, there is no problem in considering it in the formulation we used
based on mantica2023codazzi . For this case the GR solution from (75) and
(81)-(82) lead to $\rho=\rho(t)$ and a Friedman-like equation analogous to
that of FLRW models (though $p=p(t,y)$ through $1/N$).

$\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{1}{L}=\frac{S}{1+Ky},$
(84) $\displaystyle N$ $\displaystyle=$
$\displaystyle-\frac{L_{,t}/L}{S_{,t}/S}=\frac{1+(K-K_{,t}S/S_{,t})y}{1+Ky},$
(85) $\displaystyle\frac{S_{,t}^{2}}{S^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi}{3}\rho(t)-\frac{K}{S^{2}},$ (86) $\displaystyle 8\pi
p$ $\displaystyle=$
$\displaystyle-\frac{S_{,t}^{2}}{S^{2}}-\frac{K}{S^{2}}\frac{2SS_{,tt}-2S_{,t}^{2}-K_{,t}+2K}{N}.$
(87)

The Hamiltonian constraint (20) for the Stephani solution is similar to that
of FLRW: ${}^{3}{\cal R}=6K/S^{2}$, hence (86) suggest that the function
$\alpha(t)$ in (83) can be a modification spatial curvature, while $\beta$
involves a modification of $1/N$ in (87). However, the evolution of the models
might not follow a well posed initial value problem, since there are no
constraints on the function $K(t)$, but this also occurs in the GR solutions.

##  IX Conclusion

We have have discussed and shown how non-trivial exact solutions can be found
in “Cotton Gravity” CG, a gravity theory alternative to General Relativity
(GR) recently proposed by Harada harada2021emergence  based on the Cotton
tensor. Instead of the original formulation of CG, we adapted an alternative
formulation in terms of a Codazzi tensor derived by Mantica and Molinari
mantica2023codazzi . In particular, this alternative formulation allows to
apply CG to non-vacuum conformally flat spacetimes (an impossible task in the
original formulation because the Cotton tensor vanishes identically). After
re-deriving the Schwarzschild analogue found in harada2021emergence , we
obtained various non-trivial CG solutions that generalize well known solutions
of GR: FLRW, LTB and Szekeres models, as well as perfect fluid spherically
symmetric solutions: static and non-static with a shear-free 4-velocity.

We found that CG brings an interesting geometric degree of freedom in FLRW,
LTB and Szkeres models: it modifies the spatial curvature in the Hamiltonian
constraint, without modifying the energy momentum tensor (i.e. without adding
extra sources). In FLRW models this modification of spatial curvature must be
necessarily restricted to describe the evolution as a well posed initial value
problem, leading to FLRW models that are operationally the same as GR models,
but with an important conceptual difference: the cosmological constant now
interpreted as spatial curvature. In particular, the $\Lambda$CDM model
becomes defined in a precise covariant manner as the unique FLRW dust model in
CG with constant negative curvature, an appealing theoretical interpretation
of the $\Lambda$ term.

In LTB and Szekeres models the modification of spatial curvature is compatible
with a well posed initial value problem, leading to manifestly distinct
dynamical evolution of the models with respect to their GR equivalents. In
particular, it is easy and natural to obtain an spatial and temporal dependent
transition from a decelerated expansion to an accelerated expansion of dust
layers, an evolution produced only by dust sources and spatial curvature,
without assuming the presence of dark energy fluids with negative pressure or
imposing a cosmological constant.

In static fluid spheres, we show that CG qualitatively modifies the radial
profile of rotation curves of circular geodesics. Considering as source only
visible matter, without assuming dark matter, these curves can mimick the
flattening behavior with increasing radius that is associated with galactic
dark matter. We show this effect in a toy model of a spherical galaxy treated
in the weak field limit. Finally, we apply CG to spherically symmetric
solutions with a shear-free 4-velocity, corroborating the results of
mantica2023codazzi .

We believe that CG has a significant potential as an alternative gravity
theory, as it is closer to GR (and less technically complicated) in comparison
with other alternative gravity theories. The solutions we have presented are
highly idealized, but out aim has been to show how CG solutions can be
generated. In future research we will attempt to find more general models.

##  Appendix A Equivalence between the formulations of Harada and
Mantica/Molinary’s

We begin considering the definition of the potentially Codazzi tensor proposed
by Mantica and Molinari mantica2023codazzi

$\displaystyle{\cal C}_{ab}=R_{ab}-8\pi T_{ab}-\frac{1}{6}(R-16\pi
T)g_{ab},\,$ (88)

with $\nabla_{b}T^{ab}=0$, leads to (7)-(8) by substitution of ${\cal
R}_{ab}=G_{ab}+(1/2){\cal R}g_{ab}$. Using condition (6) for ${\cal C}_{ab}$
to be a Codazzi tensor

$\displaystyle\nabla_{b}\mathcal{C}_{ac}-\nabla_{c}\mathcal{C}_{ab}$
$\displaystyle=$ $\displaystyle 0,$

leading to

$\displaystyle\Rightarrow\nabla_{b}R_{ac}-\nabla_{c}R_{ab}-8\pi(\nabla_{b}T_{ac}-\nabla_{c}T_{ab})$
$\displaystyle-\frac{1}{6}\left[(g_{ac}\nabla_{b}R-g_{ab}\nabla_{c}R)+16\pi(g_{ac}\nabla_{b}T-g_{ab}\nabla_{c}T)\right]=0.$
(89)

Using the definition of the Cotton tensor in (5), equation (89) can be
rewritten as

$\displaystyle C_{abc}-8\pi(\nabla_{b}T_{ac}-\nabla_{c}T_{ab})$
$\displaystyle-\frac{8\pi}{3}(g_{ac}\nabla_{b}T-g_{ab}\nabla_{c}T)=0.$ (90)

It is sufficient to calculate the right hand side of the field equations (4)
to prove the equivalence between both formulations. Considering (2) and (8) a
direct calculation yields

$\displaystyle 16\pi\nabla_{d}\tensor{T}{{}^{d}_{a}{}_{b}{}_{c}}$
$\displaystyle=$ $\displaystyle 8\pi(\nabla_{b}T_{ac}-\nabla_{c}T_{ab})$ (91)
$\displaystyle-$
$\displaystyle\frac{8\pi}{3}(g_{ac}\nabla_{b}T-g_{ab}\nabla_{c}T).$

Hence, (90) and (4) are equivalent.

###### Acknowledgements.

. SN acknowledges financial support from SEP–CONACYT postgraduate grants
program

## References

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  * [3] George FR Ellis, Roy Maartens, and Malcolm AH MacCallum.  Relativistic cosmology.  Cambridge University Press, 2012.
  * [4] Junpei Harada.  Reply to “comment on ‘emergence of the cotton tensor for describing gravity”’.  Physical Review D, 104(8):088502, 2021.
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