# Impact of projective curvature tensor in $f\left(R,G\right)$,
$f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity

Young Jin Suh1, Krishnendu De * 2 and Uday Chand De3 1 Department of
Mathematics and RIRCM, Kyungpook National University, Daegu-41566, South Korea
<EMAIL_ADDRESS>2 Department of Mathematics, Kabi Sukanta Mahavidyalaya, The
University of Burdwan. Bhadreswar, P.O.-Angus, Hooghly, Pin 712221, West
Bengal, India. ORCID iD: https://orcid.org/0000-0001-6520-4520
<EMAIL_ADDRESS>3Department of Pure Mathematics, University of
Calcutta, 35 Ballygunge Circular Road, Kolkata 700019, West Bengal, India.
ORCID iD: https://orcid.org/0000-0002-8990-4609<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This article concerns with the characterization of a spacetime and modified
gravity, such as $f\left(R,G\right)$, $f\left(R,T\right)$ and
$f\left(R,L_{m}\right)$-gravity equipped with the projective curvature tensor.
We establish that a projectively flat perfect fluid spacetime represents dark
energy era. Also, we prove that a projectively flat perfect fluid spacetime is
either locally isometric to Minkowski spacetime or a de-Sitter spacetime.
Furthermore, it is shown that a perfect fluid spacetime permitting harmonic
projective curvature tensor becomes a generalized Robertson-Walker spacetime
and is of Petrov type $I$, $D$ or $O$. Lastly, we investigate the effect of
projectively flat perfect fluid spacetime solutions in $f\left(R,G\right)$,
$f\left(R,T\right)$ and $f\left(R,L_{m}\right)$-gravity, respectively. We also
investigate the spacetime as a $f\left(R,G\right)$-gravity solution of and use
the flat Friedmann-Robertson-Walker metric to establish a relation among jerk,
snap, and deceleration parameters. Numerous energy conditions are studied in
terms of Ricci scalar with the model
$f\left(R,G\right)=\exp(R)+\alpha\left(6G\right)^{\beta}$. For this model, the
strong energy condition is violated but the weak, dominant and null energy
conditions are fulfilled, which is in excellent accordance with current
observational investigations that show the universe is now accelerating.

###### Key words and phrases:

perfect fluid spacetime, Robertson-Walker spacetime, projective curvature
tensor, energy condition, modified gravity.
*Corresponding author
The first author was supported by the grant NRF-2018-R1D1A1B-05040381 from
National Research Foundation of Korea.

###### 2020 Mathematics Subject Classification:

83C05, 83C10, 83C40, 83C56.

##  1\. Introduction

In general relativity (briefly, GR) theory, a spacetime is a Lorentzian
manifold $M^{4}$ with the metric (Lorentzian) $g$ of signature
$\left(+,+,+,-\right)$ which admits a globally time-oriented vector. As a
spontaneous source of Einstein’s field equations (briefly, EFEs) that are
compatible with the Bianchi identities, perfect fluids (briefly, PFs) play an
outstanding role in the theory of GR. In several sectors of physics, including
nuclear physics, plasma physics, and astrophysics, relativistic PF models are
of immense interest. Numerous researchers have studied spacetimes in several
techniques in ([6], [17], [18], [23]).

A Lorentzian manifold $M^{n}$ $\left(n\geq 4\right)$ whose metric can be
expressed as

$ds^{2}=-\left(d\zeta\right)^{2}+\phi^{2}\left(\zeta\right)g^{\ast}_{v_{1}v_{2}}dx^{v_{1}}dx^{v_{2}},$
(1.1)

in which $\phi$ is a function of $\zeta$ and
$g^{\ast}_{v_{1}v_{2}}=g^{\ast}_{v_{1}v_{2}}\left(x^{v_{3}}\right)$ are only
functions of $x^{v_{3}}$ $(v_{1},v_{2},v_{3}$$=2,3,\ldots,n)$ is named a
generalized Robertson Walker (briefly, GRW) spacetime ([2], [14], [15]).
Equation (1.1) can also be shaped as the warped product
$-\mathcal{I}\times\phi^{2}\tilde{M}$, in which $\tilde{M}$ denotes an
$\left(n-1\right)$-dimensional Riemannian manifold and the open interval
$\mathcal{I}$ is contained in $\mathbb{R}$. If $M$ is of constant sectional
curvature and of dimension three, this GRW spacetime turns into a Robertson
Walker (briefly, RW) spacetime.

The spacetime $M^{4}$ is described as a PF- spacetime if the Ricci tensor
$R_{lk}$ obeys

$R_{lk}=cg_{lk}+du_{l}u_{k},$ (1.2)

in which $c$, $d$ indicate scalars and $u_{k}$ stands for a unit time-like
vector ($u_{k}u^{k}=-1$) also known as a flow vector or velocity vector.
According to GR theory, the matter field is indicated by $T_{lk}$ (symmetric
tensor field), named the energy-momentum tensor (briefly, EMT) and due to the
absence of the heat conduction term, the fluid is called perfect [28]. For a
PF-spacetime, the EMT [38] is of the shape

$T_{lk}=pg_{lk}+\left(p+\mu\right)u_{l}u_{k},$ (1.3)

in which $\mu$ and $p$ indicate energy density and isotropic pressure
respectively. Additionally, an equation of state (briefly, EoS) having the
shape $p=p\left(\mu\right)$ links $p$ and $\mu$, and the PF-spacetime is named
as isentropic. Furthermore, this spacetime is known as stiff matter for
$p=\mu$. According to [13], the PF-spacetime is represented the radiation era
if $p=\dfrac{\mu}{3}$, the dust matter fluid if $p=0$, and the dark energy
period if $p+\mu=0$.

According to the EFEs for a gravitational constant $\kappa$,

$R_{lk}-\dfrac{1}{2}\,g_{lk}R=\kappa T_{lk},$ (1.4)

in which $R$ denotes the Ricci scalar. The equation (1.2)can be obtained from
the equations (1.3) and (1.4) [33].

From the perspective of differential geometry, the projective curvature tensor
$P$ is a significant tensor and $P$ vanishes if and only if the manifold
$M^{n}$ ($n\geq 3$) is locally projectively flat. This time, $P$ is described
by [36]

$P_{lijk}=R_{lijk}-\dfrac{1}{\left(n-1\right)}\left\\{g_{lk}R_{ij}-g_{lj}R_{ik}\right\\},$
(1.5)

in which $R_{lijk}$ stands for the curvature tensor. According to (pp. 84-85
of [44]), a manifold is projectively flat if and only if it has constant
curvature.

On the other hand, Weyl tensor performs a significant role in both geometry
and relativity theory. Several researchers have characterized spacetimes with
Weyl tensor. The Weyl tensor $C$ is defined by

$\displaystyle C_{hijk}$ $\displaystyle=$ $\displaystyle
R_{hijk}+\frac{R}{(n-1)(n-2)}\\{g_{hk}g_{ij}-g_{hj}g_{ik}\\}$ (1.6)
$\displaystyle-\frac{1}{n-2}(g_{ij}R_{hk}-g_{ik}R_{hj}+g_{hk}R_{ij}-g_{hj}R_{ik}),$

where $R_{hijk}$ stands for the curvature tensor.

Moreover, we know that

$\displaystyle\nabla_{k}C^{k}_{lij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\\{\nabla_{j}R_{li}-\nabla_{i}R_{lj}\\}-\frac{1}{2(n-1)}\\{g_{li}\nabla_{j}R-g_{lj}\nabla_{i}R\\}].$
(1.7)

The Weyl tensor is called harmonic if $\nabla_{k}C^{k}_{lij}=0$. The
harmonicity of the tensor appears in conservation laws of physics.

It is widely circulated that in GR, energy conditions (briefly, ECs) are
essential tools to investigate black holes and wormholes in many modified
gravities ([4], [28]). In [41], the Raychaudhuri equations which throw back
the character of gravity through $R_{jk}v^{j}v^{k}\geq 0$ (the positivity
condition), in which $v^{j}$ indicates a null vector, were used to produce the
ECs. In geometry, the last stated condition is equivalent to the null energy
condition (briefly, NEC) $T_{jk}u^{j}u^{k}\geq 0$ in GR theoty. Moreover, the
weak energy condition (briefly, WEC) states that $T_{jk}u^{j}u^{k}\geq 0,$ for
all time-like vector $u^{j}$ and presumes a local energy density which is
positive. Also, we know that [20] a spacetime fulfills the strong energy
condition (briefly, SEC) if for all time-like vectors $u$,
$R_{hj}v^{h}v^{j}\geq 0$ holds. Various modifications to EFE have been
developed and thoroughly investigated for modified gravity theories in (see
[11, 37, 40]).

The theory of $f(R,G)$-gravity was one of these modified theories [21], that
was created by replacing the original Ricci scalar $R$ with a function of $R$
and $G$. In their investigation of the stability of the power-law and de-
Sitter solutions in the $f(R,G)$ theory, the authors of [19] found that both
depend on the structure of the $f(R,G)$-gravity and the parameters of the
model, and that gravitational action strongly contributes to the stability of
the solutions. The weak-field limit of $f(R,G)$ by choosing the parametrized
Post-Newtonian formalism was explored in [30]. The probability to get
inflation by taking into account a general $f(R,G)$ theory was demonstrated in
[31]. Also in [3], the authors considered the implementation of ECs for flat
Friedmann cosmological models and analysed them in relation to Hubble,
deceleration, snap, and jerk parameters in $f\left(R,G\right)$-gravity.

Moreover, $f\left(R\right)$-gravity is generalized by
$f\left(R,T\right)$-gravity. In [25], Harko et al. were the first to present
this modified theory of gravity. Ordines et al.[39] have mentioned the
modifications in Earth’s atmospheric models with the use of the
$f\left(R,T\right)$-gravity. Many authors have investigated
$f\left(R,T\right)$-gravity features from various angles (see [16], [42]).

Also, another modified theory was the $f\left(R,L_{m}\right)$-gravity,
introduced in [27] by Harko and Lobo. It is a spontaneous generalization of
$f\left(R\right)$-gravity ([5], [9], [10]) that directly links any arbitrary
function of $R$ with the matter-related Lagrangian density $L_{m}$. The
application of this gravity to the situation of arbitrary coupling among
matter and geometry was done in [26]. Some specific models, for instance,
$f\left(R,L_{m}\right)=\lambda+\dfrac{R}{2}+L_{m}$ ($\lambda>0$ is an
arbitrary constant), suggested by Harko and Lobo [27].

In [32], the authors have investigated the $f\left(R\right)$-gravity in a
projectively flat spacetime and analyze their outcomes utilizing two common
models of $f\left(R\right)$-gravity.

These findings served as an inspiration for the present article, which is
designed to examine ECs in term of Ricci scalar $R$ in a projectively flat PF-
spacetime solutions fulfilling $f\left(R,G\right)$, $f\left(R,T\right)$ and
$f\left(R,L_{m}\right)$-gravity respectively and we set a new model
$f\left(R,G\right)=\exp(R)+\alpha\left(6G\right)^{\beta}$ ($\alpha$ is scalar)
to explain ECs.

After preliminaries in Section $3,$ the properties of PF-spacetime permitting
projective curvature tensor are explored. Finally, we provide projectively
flat PF-spacetime solutions in $f\left(R,G\right)$, $f\left(R,T\right)$ and
$f\left(R,L_{m}\right)$-gravity, respectively in the last three Sections.

##  2\. Preliminaries

We choose throughout the article a spacetime of dimension $4$. If $P=0$ at
each point of the spacetime, then the spacetime is named projectively flat. At
first we consider projectively flat spacetime. Then the equation (1.5) yields

$R_{lijk}=\dfrac{1}{3}\left\\{g_{lk}R_{ij}-g_{lj}R_{ik}\right\\},$ (2.1)

from which we can get

$R_{iljk}=\dfrac{1}{3}\left\\{g_{ik}R_{lj}-g_{ij}R_{lk}\right\\}.$ (2.2)

Since $R_{lijk}+R_{iljk}=0$, hence the foregoing two equations give

$\left\\{g_{lk}R_{ij}-g_{lj}R_{ik}+g_{ik}R_{lj}-g_{ij}R_{lk}\right\\}=0.$
(2.3)

Multiplying (2.3) by $g^{ij}$, we acquire

$R_{lk}=\dfrac{R}{4}\,g_{lk}.$ (2.4)

Therefore, we write:

######  Proposition 1.

A projectively flat spacetime represents an Einstein spacetime.

Making use of (2.4) in (2.1), we have

$R_{lijk}=\dfrac{R}{12}\left\\{g_{ij}g_{lk}-g_{ik}g_{lj}\right\\}.$ (2.5)

Hence, the spacetime is of constant sectional curvature.

Let the space be a space of constant curvature $\lambda$. Then, we acquire

$R_{lijk}=\lambda\left\\{g_{ij}g_{lk}-g_{ik}g_{lj}\right\\},$ (2.6)

which entails $R_{lk}=3\lambda g_{lk}$. Substituting this value in (1.5), we
infer $P_{lijk}=0$.

Hence, we state:

######  Proposition 2.

A spacetime is projectively flat if and only if the spacetime is of constant
sectional curvature.

######  Remark 1.

A space of constant curvature is an Einstein space, but generally, the
converse is not valid. Although, an Einstein space of dimension three is a
space of constant curvature. As for example ‘ a space with Schwarzschild
metric is an Einstein space, but not a space of constant curvature.’

######  Remark 2.

The foregoing spacetime is either a de-Sitter or anti de-Sitter spacetime.

For $n=4$, taking covariant derivative of (1.5), we acquire

$\nabla_{h}P^{h}_{ijk}=\nabla_{h}R^{h}_{ijk}-\dfrac{1}{3}\left\\{\delta^{h}_{k}\nabla_{h}R_{ij}-\delta^{h}_{j}\nabla_{h}R_{ik}\right\\}.$
(2.7)

It is well-known that

$\nabla_{h}R^{h}_{ijk}=\nabla_{k}R_{ij}-\nabla_{j}R_{ik}.$ (2.8)

Using (2.8) in (2.7) provides

$\nabla_{h}P^{h}_{ijk}=\dfrac{2}{3}\left\\{\nabla_{k}R_{ij}-\nabla_{j}R_{ik}\right\\}.$
(2.9)

If the tensor $P^{h}_{ijk}$ is harmonic, that is, $\nabla_{h}P^{h}_{ijk}=0$,
then (2.9) gives

$\nabla_{k}R_{ij}-\nabla_{j}R_{ik}=0,$ (2.10)

which means that $R_{lk}$ is of Codazzi type.
Conversely, if $R_{lk}$ is of Codazzi type, then

$\nabla_{k}R_{ij}-\nabla_{j}R_{ik}=0.$ (2.11)

Hence, the equation (2.9) turns into

$\nabla_{h}P^{h}_{ijk}=0.$ (2.12)

This means that projective curvature tensor is harmonic. So we have

######  Proposition 3.

In a semi-Riemannian space $P$ is harmonic if and only if $R_{lk}$ is of
Codazzi type.

##  3\. PF-spacetime permitting projective curvature tensor

Now consider a projectively flat PF-spacetime obeying EFE.

From (1.3), (1.4) and (2.4), we acquire

$\left(\kappa
p+\dfrac{R}{4}\right)g_{lk}+\kappa\left(p+\mu\right)u_{l}u_{k}=0.$ (3.1)

Multiplying (3.1) with $g^{lk}$ implies that

$3\kappa p+R-\kappa\mu=0.$ (3.2)

Also, multiplying (3.1) with $u^{l}$ gives

$R=4\kappa\mu.$ (3.3)

Combining the equations (3.2) and (3.3) yield

$p+\mu=0,$ (3.4)

which entails a dark energy era [13]. Therefore, we write:

######  Theorem 1.

A projectively flat PF-spacetime obeying EFE becomes a dark energy era.

Now, equations (1.3) and (1.4) together provide

$R_{lk}=\left(\kappa
p+\dfrac{R}{2}\right)g_{lk}+\kappa\left(p+\mu\right)u_{l}u_{k}.$ (3.5)

Multiplying (3.5) by $u^{l}u^{k}$, we obtain

$R_{lk}u^{l}u^{k}=-\dfrac{R}{2}+\kappa\mu.$ (3.6)

Hence from equations (3.3) and (3.6), we find

$R_{lk}u^{l}u^{k}=-\kappa\mu.$ (3.7)

Here, we consider the spacetime under consideration meets the SEC. Then

$\kappa\mu\leq 0.$ (3.8)

As $\kappa>0$ and $\mu$ is non-negative, the equations (3.3) and (3.8) provide
us

$R=0.$ (3.9)

Then (2.5) infers $R_{lijk}=0$, which means that the spacetime has zero
sectional curvature. Therefore a projectively flat PF-spacetime and Minkowski
spacetime are locally isometric ([20], p. 67).

Therefore, we state:

######  Theorem 2.

A projectively flat $\mathrm{PF-spacetime}$ fulfilling the $\mathrm{SEC}$ and
a Minkowski spacetime are locally isometric.

As $\mu$ is non-negative, (3.3) reflects that

$R\geq 0,$ (3.10)

which implies $R>0$ or, $R=0$.

Case (i). For $R=0,$ (2.5) infers $R_{lijk}=0$. Hence, this spacetime and the
Minkowski spacetime are locally isometric.

Case (ii). For $R>0,$ (2.5) infers that the space is of positive constant
curvature. Therefore, the space is a de-Sitter spacetime [20].

Thus, we write:

######  Theorem 3.

A projectively flat PF-spacetime fulfilling the $\mathrm{SEC}$ is either
locally isometric to Minkowski spacetime or a de-Sitter spacetime.

It is well circulated that the de-Sitter spacetime is always conformally flat.
Hence, the spacetime belongs to Petrov classification $O.$

Thus, we have

######  Corollary 1.

A projectively flat PF-spacetime fulfilling the $\mathrm{SEC}$ belongs to
Petrov classification $O$ or locally isometric to Minkowski spacetime.

Next we consider the harmonic projective curvature, that is,
$\nabla_{h}P^{h}_{ijk}=0$.

A Yang pure space [22] is a Lorentzian manifold whose metric obeys the Yang’s
equation:

$\nabla_{h}R_{lk}=\nabla_{k}R_{lh}.$ (3.11)

Therefore by Proposition $3$, we can say that a spacetime permitting
$\nabla_{h}P^{h}_{ijk}=0$ is a Yang pure space.

Thus, we state:

######  Theorem 4.

A PF-spacetime permitting harmonic projective curvature tensor is a Yang pure
space.

Again $divP=0$ implies $divC=0$, since $\nabla_{l}R_{ij}=\nabla_{j}R_{il}$ and
hence $R=$ constant.

In [35], Mantica et al established the subsequent:

Theorem A. If $\nabla_{m}C_{jkl}^{m}=0$ and $R=$constant in a PF-spacetime,
then it is a $GRW$ spacetime.

Therefore, by Theorem A, we write:

######  Theorem 5.

A PF-spacetime with $\nabla_{h}P^{h}_{ijk}=0$ represents a GRW spacetime.

In a GRW spacetime ( p. 14, [34]), we have

$\nabla_{h}C^{h}_{ijk}=0\Longleftrightarrow u^{h}C_{hijk}=0.$

In dimension 4, $u^{h}C_{hijk}=0$ implies $C_{hijk}=0$ and hence the spacetime
represents a RW spacetime.

######  Theorem 6.

A PF-spacetime permitting harmonic projective curvature tensor represents a RW
spacetime.

Also, $C_{hijk}=0$ implies $C$ is purely electric[29]. We know ([43], p. 73)
that the spacetime is of Petrov type $I$, $D$ or $O$, since $C$ is purely
electric.

######  Theorem 7.

A PF-spacetime permitting harmonic projective curvature tensor is of Petrov
type $I$, $D$ or $O$.

##  4\. Projectively flat PF-spacetime solutions fulfilling
$f\left(R,G\right)$-gravity

Now, we concentrate on a particular subclass of modified gravity known as
$f(R,G)$-gravity. The expression for gravitational force is

$S=\dfrac{1}{2\kappa}\int(-g)^{\frac{1}{2}}f\left(R,G\right)d^{4}x+S_{\mathrm{mat}},$
(4.1)

$S_{\mathrm{mat}}$ indicates the matter action. The Gauss-Bonnet invariant $G$
is described by

$G=R^{2}+R_{lijk}R^{lijk}-4R_{lk}R^{lk}.$ (4.2)

The action term of (4.1) provides the field equations as:

$R_{ij}-\dfrac{R}{2}\,g_{ij}=\kappa T_{ij}+\Omega_{ij}=\kappa
T_{ij}^{\mathrm{\,eff}},$ (4.3)

where

$\displaystyle\Omega_{ij}$ $\displaystyle=\nabla_{i}\nabla_{j}f_{R}-g_{ij}\Box
f_{R}+2R\nabla_{i}\nabla_{j}f_{G}-2g_{ij}R\Box
f_{G}-4R^{l}_{i}\nabla_{l}\nabla_{j}f_{G}$
$\displaystyle-4R^{l}_{j}\nabla_{l}\nabla_{i}f_{G}+4R_{ij}\Box
f_{G}+4g_{ij}R^{lk}\nabla_{l}\nabla_{k}f_{G}+4R_{ilkj}\nabla^{l}\nabla^{k}f_{G}$
$\displaystyle-\dfrac{1}{2}\,g_{ij}\left(Rf_{R}+Gf_{G}-f\right)+\left(1-f_{R}\right)\left(R_{ij}-\dfrac{1}{2}\,g_{ij}R\right)$
(4.4)

and $T_{ij}^{\mathrm{\,eff}}$ denotes the effective EMT. Notice that
$f_{R}\equiv\dfrac{\partial f}{\partial R}$, $f_{G}\equiv\dfrac{\partial
f}{\partial G}$ and $\Box$ represent the d’Alembert operator.

Despite of the complexity of the above expression, in [7] Capozziello et al.
established that in a Friedmann-Robertson-Walker space-time of dimension n,
for any analytical $f\left(R,G\right)$ model of gravity, the tensor
$\Omega_{ij}$ is a perfect fluid form. In [24], it is proved that the field
equations of the general $f\left(R,G\right)$ gravity theory are of the perfect
fluid type. Geometric perfect fluids in $f\left(R,G\right)$ gravity is also
studied in [8].

These field equations are utilized to acquire the ECs of $f(R,G)$ -gravity,
and get the following:

$\displaystyle\mathrm{NEC}$ $\displaystyle\Longleftrightarrow p+\mu\geq 0,$
(4.5) $\displaystyle\mathrm{WEC}$ $\displaystyle\Longleftrightarrow\mu\geq
0\quad\mathrm{and}\quad p+\mu\geq 0,$ (4.6) $\displaystyle\mathrm{DEC}$
$\displaystyle\Longleftrightarrow\mu\geq 0\quad\mathrm{and}\quad\mu\pm p\geq
0,$ (4.7) $\displaystyle\mathrm{SEC}$ $\displaystyle\Longleftrightarrow
3p+\mu\geq 0\quad\mathrm{and}\quad\mu+p\geq 0,$ (4.8)

where DEC indicates the dominant energy condition.
From (2.4), it follows that

$R^{lk}=\dfrac{R}{4}\,g^{lk}.$ (4.9)

Equations (2.4) and (4.9) together imply

$R_{lk}R^{lk}=\dfrac{R^{2}}{4}\,.$ (4.10)

From (2.5), it follows that

$R^{lijk}=\dfrac{R}{12}\left\\{g^{ij}g^{lk}-g^{ik}g^{lj}\right\\}.$ (4.11)

Multiplying (2.5) and (4.11), one infers

$R_{lijk}R^{lijk}=\dfrac{R^{2}}{6}\,.$ (4.12)

Equations (4.2), (4.10) and (4.12) reflect that

$G=\dfrac{R^{2}}{6}\,.$ (4.13)

Since $R$ is constant for a projectively flat spacetime, the equation (4)
becomes

$\Omega_{ij}=R_{ij}+\left(\dfrac{f}{2}-\dfrac{R}{2}\right)g_{ij}.$ (4.14)

For a PF-spacetime the EMT is given by

$T_{lk}=pg_{lk}+\left(\mu+p\right)u_{l}u_{k}$ (4.15)

and

$T_{lk}^{\mathrm{\,eff}}=p^{\mathrm{\,eff}}g_{lk}+\left(\mu^{\mathrm{\,eff}}+p^{\mathrm{\,eff}}\right)u_{l}u_{k},$
(4.16)

in which $p^{\mathrm{\,eff}}$ and $\mu^{\mathrm{\,eff}}$ stand for the
effective isotropic pressure and energy density, respectively.

Using (4.14) and (1.3) in (4.3), we obtain

$\left(\kappa
p+\dfrac{f}{2}\right)g_{ij}+\kappa\left(p+\mu\right)u_{i}u_{j}=0.$ (4.17)

Multiplying (4.17) with $u^{i}$, we have

$\mu=\dfrac{f}{2\kappa}\,.$ (4.18)

Again multiplying (4.17) with $g^{ij}$ and using (4.18), we arrive at

$p=-\dfrac{f}{2\kappa}\,.$ (4.19)

Hence, we provide:

######  Theorem 8.

For a projectively flat $\mathrm{PF-spacetime}$ solutions satisfying
$f\left(R,G\right)$-gravity, $\mu$ and $p$ are described by (4.18) and (4.19),
respectively.

The combination of (4.18) and (4.19) give

$p+\mu=0,$

which means that NEC is satisfied. When acting on neighbouring particles that
are also travelling null geodesics, locally gravity is typically appealing
(preferably not repulsive), which is the physical reason for NEC [32].

Utilizing (4.14)–(4.16) in (4.3), one infers

$R_{lk}+\left(\kappa
p+\dfrac{f}{2}-\dfrac{R}{2}\right)g_{lk}+\kappa\left(\mu+p\right)u_{l}u_{k}=\kappa
p^{\mathrm{\,eff}}g_{lk}+\kappa\left(\mu^{\mathrm{\,eff}}+p^{\mathrm{\,eff}}\right)u_{l}u_{k}.$
(4.20)

In light of (2.4) and (4.18)–(4.20), we acquire

$\left(\kappa
p^{\mathrm{\,eff}}+\dfrac{R}{4}\right)g_{lk}+\kappa\left(\mu^{\mathrm{\,eff}}+p^{\mathrm{\,eff}}\right)u_{l}u_{k}=0.$
(4.21)

Multiplying (4.21) with $u^{l}$, we have

$\mu^{\mathrm{\,eff}}=\dfrac{R}{4\kappa}\,.$ (4.22)

Again multiplying (4.21) with $g^{lk}$ and using (4.22), we arrive at

$p^{\mathrm{\,eff}}=-\dfrac{R}{4\kappa}\,.$ (4.23)

Now, we choose

$ds^{2}=a^{2}\left(t\right)\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)-dt^{2},$
(4.24)

in which $a\left(t\right)$ indicates the scale factor of the universe. The
preceding metric is commonly known as the flat Friedmann Robertson Walker
(briefly, FRW) metric. The field equations for $f\left(R,G\right)$-gravity
follow from the FRW background and a PF equation of state for ordinary matter
are described by

$2\dot{H_{1}}f_{R}+8H_{1}\dot{H_{1}}\dot{f_{G}}=H_{1}\dot{f_{R}}-\ddot{f_{R}}+4H_{1}^{3}\dot{f_{G}}-4H_{1}^{2}\ddot{f_{G}},$
(4.25)
$6H_{1}^{2}f_{R}+24H_{1}^{3}\dot{f_{G}}=f_{R}R-f\left(R,G\right)-6H_{1}\dot{f_{R}}+Gf_{G},$
(4.26)

in which $H_{1}=\dfrac{\dot{a}}{a}$ indicates the Hubble parameter and overdot
$\equiv\frac{d}{dt}$. Moreover, we have

$R=6\left(2H_{1}^{2}+\dot{H_{1}}\right)$ (4.27)

and

$G=24H_{1}^{2}\left(H_{1}^{2}+\dot{H_{1}}\right).$ (4.28)

From (4.13), (4.27) and (4.28), we get

$H_{1}^{2}=\dfrac{R}{12}\qquad\mathrm{and}\qquad\dot{H_{1}}=0.$ (4.29)

As $H_{1}=\dfrac{\dot{a}}{a}$ , $\dfrac{\dot{a}}{a}=\sqrt{\dfrac{R}{12}}$ .
Thus

$\ddot{a}=\dfrac{\dot{a}^{2}}{a}\,,\qquad\dddot{a}=\dfrac{\dot{a}^{3}}{a^{2}}\qquad\mathrm{and}\qquad\ddddot{a}=\dfrac{\dot{a}^{4}}{a^{3}}\,.$
(4.30)

Then, using an analogue with classical mechanics, we explain velocity,
acceleration, snap and jerk in the context of cosmology. The jerk,
deceleration, and snap parameters must be defined as

$j=\dfrac{1}{H_{1}^{3}}\dfrac{\dddot{a}}{a}\,,\qquad
q=-\dfrac{1}{H_{1}^{2}}\dfrac{\ddot{a}}{a}\qquad\mathrm{and}\qquad
s=\dfrac{1}{H_{1}^{4}}\dfrac{\ddddot{a}}{a},$ (4.31)

respectively. Using (4.30) in (4.31), we obtain

$s=j=-q.$ (4.32)

Hence, for a projectively flat PF-spacetime fulfilling
$f\left(R,G\right)$-gravity, the jerk, deceleration, and snap parameters are
linked by (4.32).

Now we concentrate on the ECs of a $f\left(R,G\right)$-gravity model.

## A. $f\left(R,G\right)=\exp(R)+\alpha\left(6G\right)^{\beta}$

Here, making use of (4.13), (4.18) and (4.19), the energy density and pressure
are expressed as

$\mu=\dfrac{\exp(R)+\alpha\left(6G\right)^{\beta}}{2\kappa}\,,$ (4.33)
$p=-\dfrac{\exp(R)+\alpha\left(6G\right)^{\beta}}{2\kappa}\,.$ (4.34)

Using (4.13) the foregoing equations reduce to

$\mu=\dfrac{\exp(R)+\alpha\left(R\right)^{2\beta}}{2\kappa}\,,$ (4.35)
$p=-\dfrac{\exp(R)+\alpha\left(R\right)^{2\beta}}{2\kappa}\,.$ (4.36)

The ECs for this setup can now be discussed using (4.35) and (4.36). In this
model the EoS reduces to $\omega=-1$. Therefore, the chosen model is
consistent with the $\Lambda$CDM model. Obviously, in this model NEC is
staisfied. Since WEC is the amalgamation of NEC and positive density, we
examine the behavior of density parameter, DEC and SEC.
Fig. 1: Development of $\mu$ with reference to $R\in[1,1.5]$ and
$\beta\in[1,2]$ for $\alpha=1$ and $\kappa=2.077\times 10^{-43}$. Fig. 2:
Development of $\mu-p$ with reference to $R\in[1,1.5]$ and $\beta\in[1,2]$ for
$\alpha=1$ and $\kappa=2.077\times 10^{-43}$.

Fig. 3: Development of $\mu+3p$ with reference to $R\in[1,1.5]$ and
$\beta\in[1,2]$ for $\alpha=1$ and $\kappa=2.077\times 10^{-43}$.
---

One can see from Fig. $1$, the energy density cannot be negative for the
parameter ranges $R\in[1,1.5]$ and $\beta\in[1,2]$ and for greater values of
$R$ and $\alpha$, it is high. In this situation, $\mu+p$ becomes zero. As NEC
belongs to WEC, as a consequence NEC and WEC are verified. Fig. $2$ gives the
$\mu-p$ profile, which yields a positive value range.. Using Fig. $1$, Fig.
$2$ and $\mu+p=0$, we see that DEC is verified. From Fig. $3$, we see that SEC
is violated. Therefore, it describes the Universe’s late-time acceleration
[32].

##  5\. Projectively flat PF-spacetime solutions fulfilling
$f\left(R,T\right)$-gravity

The field equations of $f\left(R,T\right)$-gravity have been investigated in
metric formalism for a number of special instances. Here, we set [25]

$f\left(R,T\right)=2f\left(T\right)+R.$ (5.1)

Here, modified Einstein-Hilbert action term is described by

$E=\int(-g)^{\frac{1}{2}}\left[\dfrac{16\pi
L_{m}+f\left(R,T\right)}{16\pi}\right]\,d^{4}x,$ (5.2)

in which $L_{m}$ stands for the scalar field’s matter Lagrangian. Here, the
stress energy tensor is described by

$T_{ij}=\dfrac{-2\delta\left(\sqrt{-g}\right)L_{m}}{\sqrt{-g}\,\delta^{ij}}\,,$
(5.3)

in which $L_{m}$ solely depends on $g$.

The subsequent field equations are acquired from (5.2)

$\displaystyle f_{R}\left(R,T\right)R_{ji}$
$\displaystyle-\dfrac{1}{2}f\left(R,T\right)g_{ji}-\left[\nabla_{j}\nabla_{i}-g_{ji}\Box\right]f_{R}\left(R,T\right)$
$\displaystyle=8\pi
T_{ji}-\left[T_{ji}+\Theta_{ji}\right]f_{T}\left(R,T\right),$ (5.4)

$f_{R}\left(R,T\right)$ and $f_{T}\left(R,T\right)$ are the partial derivative
with regard to $R$ and $T$ respectively, $\Box$ stands for the d’Alembert
operator and

$\Theta_{ji}=-2T_{ji}+g_{ji}L_{m}-2g^{lk}\dfrac{\partial^{2}L_{m}}{\partial
g^{ab}\partial g^{lk}}\,.$ (5.5)

We presume that $L_{m}$ =$-p$ and utilizing (1.3), we infer that

$T_{lk}=-pg_{lk}+\left(p+\mu\right)u_{l}u_{k}.$ (5.6)

Using (5.6), we infer the variation of stress energy as

$\Theta_{lk}=-2T_{lk}-pg_{lk}.$ (5.7)

Equations (5.1) and (5) together produce

$\displaystyle R_{lk}=\dfrac{R}{2}g_{lk}$ $\displaystyle+8\pi
T_{lk}+f\left(T\right)g_{lk}$
$\displaystyle-2\left[T_{lk}+\Theta_{lk}\right]f^{\prime}\left(T\right).$
(5.8)

The conservation of the EMT was not taken into account when the field
equations were derived by Harko et al. [25]. But the author of [12], presumed
the conservation of the EMT. Here we assume that the EMT is conserved in the
PF-spacetime solution to the $f\left(R,T\right)$-gravity equation.

Making use of (5.6), (5.7) and (5) provide the Ricci tensor as

$R_{lk}=\left[\dfrac{R}{2}-8p\pi+f\left(T\right)\right]g_{lk}+\left(p+\mu\right)\left\\{2f^{\prime}\left(T\right)+8\pi\right\\}u_{l}u_{k}.$
(5.9)

Equations (2.4) and (5.9) reveal that

$\left[\dfrac{R}{4}-8p\pi+f\left(T\right)\right]g_{lk}+\left(p+\mu\right)\left\\{2f^{\prime}\left(T\right)+8\pi\right\\}u_{l}u_{k}=0.$
(5.10)

Contracting the foregoing equation reflects

$R-32p\pi+4f\left(T\right)-\left(p+\mu\right)\left\\{2f^{\prime}\left(T\right)+8\pi\right\\}=0.$
(5.11)

Multiplying (5.10) by $g^{lk}$, we acquire

$-R+32p\pi-4f\left(T\right)+4\left(p+\mu\right)\left\\{2f^{\prime}\left(T\right)+8\pi\right\\}=0.$
(5.12)

Adding (5.11) and (5.12), we obtain

$\left\\{4\pi+f^{\prime}\left(T\right)\right\\}\left(p+\mu\right)=0.$ (5.13)

From the above we conclude that

either $p+\mu=0$ or, $p+\mu\neq 0.$ Case i. If $p+\mu=0,$ then the spacetime
represents dark energy era. Case ii. If $p+\mu\neq 0,$ then
$f^{\prime}\left(T\right)+4\pi=0.$ Hence, (5.9) reveals that it is an Einstein
spacetime. Therefore we state:

######  Theorem 9.

A projectively flat spacetime satisfying $f\left(R,T\right)$-gravity
represents either Einstein spacetime, or dark energy era.

Equations (5.11) and (5.12) reflects

$p=\dfrac{4f\left(T\right)+R}{32\pi}\,.$ (5.14)

For dust matter era ($p=0$), the previous equation turns into
$f\left(T\right)=-\dfrac{R}{4}$ . Thus, we state:

######  Corollary 2.

For every viable $f\left(R,T\right)$ a projectively flat spacetime is
incapable to demonstrate dust matter era.

######  Remark 3.

When $f\left(T\right)$ is equal to zero, $f\left(R,T\right)$-gravity
transforms into $f\left(R\right)$-gravity. According to the aforementioned
Theorem, in projectively flat spacetime for $f\left(R\right)$-gravity
represents dark energy era. By virtue of energy density’s impossibility of
being negative, the EoS is $p+\mu=0,$ which means that
$\lvert\mu\rvert=\lvert-p\rvert,$ that is, $\mu=\lvert p\rvert$ . Thus, a
projectively flat spacetime obeys the DEC in $f\left(R\right)$-gravity. Hence,
the speed of light is the fastest that matter cannot travel in a projectively
flat spacetime obeying $f\left(R\right)$-gravity [20].

##  6\. Projectively flat PF-spacetime solutions fulfilling
$f\left(R,L_{m}\right)$-gravity

Here, we describe the projectively flat PF-spacetime fulfilling
$f\left(R,L_{m}\right)$-gravity. According to our hypothesis, the shape of the
action term is as follows:

$S=\int\sqrt{-g}f\left(R,L_{m}\right)d^{4}x,$ (6.1)

The EMT of the matter is described as

$T_{lk}=-\dfrac{2}{\sqrt{-g}}\dfrac{\delta\left(\sqrt{-g}L_{m}\right)}{\delta
g^{lk}}\,.$ (6.2)

Supposing that $L_{m}$ is independent of the derivatives of the metric tensor
$g$. From the variation of action of (6.1), with regard to $g$, the field
equations of $f\left(R,L_{m}\right)$ theory are given in their modified form
[27]:

$\displaystyle f_{R}\left(R,L_{m}\right)$
$\displaystyle\left\\{R_{lk}-\dfrac{R}{3}\,g_{lk}\right\\}-\dfrac{1}{6}\left\\{f_{L_{m}}\left(R,L_{m}\right)L_{m}-f\left(R,L_{m}\right)\right\\}g_{lk}$
$\displaystyle=\dfrac{1}{2}\left\\{T_{lk}-\dfrac{t}{3}\,g_{lk}\right\\}f_{L_{m}}\left(R,L_{m}\right)+\nabla_{l}\nabla_{k}f_{R}\left(R,L_{m}\right),$
(6.3)

in which $t$ stands for the trace of the EMT. For our investigations, we take
into account the model described below [27]:

$f\left(R,L_{m}\right)=\lambda+\dfrac{R}{2}+L_{m},$ (6.4)

$\lambda>0$ being an arbitrary constant. Here, assuming that the EMT has the
form (1.3), we investigate PF-spacetime solutions to the
$f\left(R,L_{m}\right)$-gravity.
Contracting (1.3), we obtain

$t=3p-\mu.$ (6.5)

Equations (1.3), (2.4) and (6)–(6.5) reflect that

$\left(\dfrac{\lambda}{3}+\dfrac{R}{12}-\dfrac{\mu}{3}\right)g_{lk}-\left(p+\mu\right)u_{l}u_{k}=0.$
(6.6)

Multiplying (6.6) with $g^{lk}$ and $u^{l}$ separately, we get

$\dfrac{4\lambda+R}{3}+p-\dfrac{\mu}{3}=0$ (6.7)

and

$\dfrac{4\lambda+R}{12}+p+\dfrac{2\mu}{3}=0.$ (6.8)

Equations (6.7) and (6.8) together imply

$\mu=\dfrac{4\lambda+R}{4}$ (6.9)

and

$p=-\dfrac{4\lambda+R}{4}\,.$ (6.10)

The combination of (6.9) and (6.10) give $\mu+p=0$. Thus, we write:

######  Theorem 10.

A projectively flat $\mathrm{PF-spacetime}$ solutions obeying
$f\left(R,L_{m}\right)=\lambda+\dfrac{R}{2}+L_{m}$ becomes a dark energy era.

Now, we examine the ECs for the model (6.4). Using (6.9) and (6.10), one can
now discuss about the ECs for this configuration.
Fig. 4: Development of $\mu$ with reference to $R\in[1,1.5]$ and
$\lambda\in[1,1.5]$ Fig. 5: Development of $\mu-p$ with reference to
$R\in[1,1.5]$ and $\lambda\in[1,1.5]$ Fig. 6: Development of $\mu+3p$ with
reference to $R\in[1,1.5]$ and $\lambda\in[1,1.5]$ Figs. $4$, $5$, and $6$
denote the profiles of $\mu$, $\mu-p$, and $\mu+3p$. Also, for this
construction, as $\mu+p=0$ , WEC and NEC are also satisfied. From the
foregoing figures, we notice that DEC is verified for the parameter ranges
$R\in[1,1.5]$ and $\lambda\in[1,1.5]$ but the SEC is not valid.

##  7\. Discussion

Spacetime which is a time-oriented torsion-free Lorentzian manifold, serves as
the foundation for the present modelling of the physical universe. According
to GR theory, the appropriate EMT may be used to determine the matter content
of the universe, which is agreed to behave like a PF-spacetime in cosmological
models.

In this current investigation, we study a projectively flat PF-spacetime and
show that a projectively flat PF-spacetime is either locally isometric to
Minkowski spacetime or a de-Sitter spacetime. We also illustrate that a
projectively flat PF-spacetime fulfilling the SEC is locally isometric to
Minkowski spacetime. Furthermore, we notice that a PF-spacetime with
$\nabla_{h}P^{h}_{ijk}=0$ is of Petrov type $I$, $D$ or $O$.

The prime focus of this article has been the exploration of projectively flat
PF-spacetime solutions in relation to different modified gravity. In this
article, our outcomes have been evaluated analytically and graphically. We
used the analytic technique in order to construct our formulation and to
assess the stability of two cosmological models, like
$f\left(R,G\right)=\exp(R)+\alpha\left(6G\right)^{\beta}$ and
$f\left(R,L_{m}\right)=\lambda+\dfrac{R}{2}+L_{m}$. For the first model, Figs.
$1$, $2$ and $3$ show the profiles of ECs. Here we see thst although NEC, WEC
and DEC were satisfied, SEC violated for this agreement. In addition, the EoS
is $\dfrac{p}{\mu}=-1$, which denotes the dark energy era. However, these
outcomes are consistent with the $\Lambda$CDM model. Similar to the first
model, Figs. $4$, $5$ and $6$ show every ECs for the second model. The
outcomes we found for the second model are also consistent with those of the
first model.

##  8\. Declarations

###  8.1. Funding

NA.

###  8.2. Conflicts of interest/Competing interests

The authors have no conflicts to disclose and all authors contributed equally
to this work.

###  8.3. Availability of data and material

NA.

###  8.4. Code availability

NA.

##  9\. Acknowledgement

We would like to thank the Referees and the Editor for reviewing the paper
carefully and their valuable comments to improve the quality of the paper.

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