# Rapidly rotating neutron stars in $f(R,T)$ gravity

F. M. da Silva<EMAIL_ADDRESS>Núcleo Cosmo–ufes & Departamento de
Física, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 540, CEP
29.075-910, Vitória, ES, Brazil  L. C. N. Santos<EMAIL_ADDRESS>Departamento de Física, CCEN–Universidade Federal da Paraíba; C.P. 5008, CEP
58.051-970, João Pessoa, PB, Brazil  C. E. Mota<EMAIL_ADDRESS>Departamento de Física, CFM–Universidade Federal de Santa Catarina; C.P. 476,
CEP 88.040-900, Florianópolis, SC, Brazil  T. O. F. da Costa
<EMAIL_ADDRESS>Núcleo Cosmo–ufes & Departamento de Física,
Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 540, CEP
29.075-910, Vitória, ES, Brazil  J. C. Fabris<EMAIL_ADDRESS>Núcleo Cosmo–ufes & Departamento de Física, Universidade Federal do Espírito
Santo, Av. Fernando Ferrari, 540, CEP 29.075-910, Vitória, ES, Brazil
National Research Nuclear University MEPhI, Kashirskoe sh.31, Moscow 115409,
Russia

###### Abstract

In this work, we study the influence of $f(R,T)$ gravity on rapidly rotating
neutron stars. First we discuss the main aspects of this modified theory of
gravity where the gravitational Lagrangian is an arbitrary function of the
Ricci scalar $R$ and of the trace of the energy–momentum tensor $T$. Then we
present the basic equations for neutron stars including the equations of state
used in the present work to describe the hadronic matter. Some physical
quantities of interest are calculated such as mass–radius relations, moments
of inertia, angular momentum, and compactness. By considering four different
rotation regimes, we obtain results that indicate substantial modifications in
the physical properties of neutron stars in $f(R,T)$ gravity when compared to
those in the context of general relativity. In particular, the mass–radius
relation for sequences of stars indicates that $f(R,T)$ gravity increases the
mass and the equatorial radius of the neutron stars for stars rotating with an
angular velocity smaller than Kepler limit.

##  I Introduction

General relativity (GR) describes the space–time geometric structure based on
the content of energy and matter contained within it. On this regard, there
are many experimental tests in which GR theory has been confirmed considering
geometries of the space–time associated with different astrophysical systems.
More recently, measurements of gravitational waves by the collaboration Virgo
and LIGO (Laser Interferometer Gravitational–Wave Observatory)
abbott2016observation ; abbott2019gwtc  and the first images of black holes by
the project Event Horizon Telescope akiyama2019first ; akiyama2022first  have
confirmed predictions of the theory. On the other hand, there are still open
problems in the field of cosmology and astrophysics regarding the dark energy
and dark matter not yet well understood in the context of GR. In this way,
modifications of GR have been developed due to the aforementioned issues. In
addition, there are formal aspects associated to the quantization of
gravitational field that can be addressed in the framework of modified
theories of gravity. Considering the divergences that arise in the context of
renormalization at one loop, DeWitt observed that the action for gravity
should be constructed with higher–order curvature terms witt . In this way,
extensions of GR as $f(R,T)$ gravity Harko:2011kv  can provide an action for
gravity with this requirement since that the theory is formulated with
arbitrary functions of curvature terms.

In a cosmological context, the $f(R,T)$ theory was considered in the study of
the evolution of scalar cosmological perturbations alvarenga2013dynamics  and
cosmological models with different equations of state (EoS)
jamil2012reconstruction ; sharif2012thermodynamics ; houndjo2012reconstruction
; shabani2014cosmological ; moraes2017simplest . This modified theory of
gravity can be linked to noncommutative quantum theory by energy
nonconservation of the energy–momentum tensor in $f(R,T)$ context
lobato2019energy . The connection between Rastall gravity and $f(R,T)$ gravity
was studied in shabani2020connection , where the authors considered that the
matter content is a perfect fluid with linear EoS.

In an astrophysical context, neutron stars (NS) have been used for the study
of the effects of modified theories of gravity due to the very high mass
density of these compact objects. In the case of static stars, we can mention
some works that consider the influence of different modifications of GR paper2
; paper1 ; harada1998neutron ; orellana2013structure ; momeni2015tolman ;
oliveira2015neutron ; hendi2016modified ; singh2019einstein ;
maurya2020charged ; mota2019combined ; fabris2021stellar . In particular, the
theory of $f(R,T)$ gravity has been studied in several papers that explore
effects of this gravity theory on different types of stars moraes2016stellar ;
yousaf2016causes ; carvalho2017stellar ; deb2019study ; dos2019conservative ;
bhatti2019stability ; lobato2020neutron ; rahaman2020anisotropic .

Currently, almost all of the accurate measurements that we have of NS mass and
radius come from rotating stars in binary pulsar systems ozel2016masses .
There are evidences that NS can rotate at an angular velocity of up to $716$
Hz hessels2006radio . Besides, some theoretical models indicate that magnetars
can rotate even faster uso1992millisecond ; metzger2011protomagnetar ;
giacomazzo2013formation ; kumar2015physics . In this way, it is important to
explicitly include effects of fast rotation in models of NS. Indeed, rapidly
rotating NS have been approached in the context of modified theories of
gravity such as scalar–tensor theories of gravity doneva2013rapidly ;
doneva2018differentially ; astashenok2020rotating , $f(R)$ gravity
doneva2015iq ; yazadjiev2015rapidly , dilatonic Einstein–Gauss–Bonnet theory
kleihaus2016rapidly  and in Rastall’s gravity da2021rapidly .

In this paper, we consider the effects of $f(R,T)$ gravity on rapidly rotating
NS. To take into account a star with a rapid rotation, we use the method
developed by Komatsu, Eriguchi and Hachisu (KEH) komatsu1989rapidly ;
komatsu1989rapidlyii . The organization of the paper has the following
structure: In Sec. II, we discuss the main aspects of the $f(R,T)$ gravity,
and in Sec. III we obtain equations that describe a rapidly spinning star. In
Sec. IV, we present the EoS used to describe hadronic matter in this paper.
The results obtained in our work are in Sec. V and the conclusions are in Sec.
VI.

##  II $f(R,T)$ gravity

In order to study rapidly rotating NS in $f(R,T)$ gravity, we start by briefly
discussing the main aspects of this modified theory of gravity.

In 2011, T. Harko et al. Harko:2011kv  proposed the theory of gravity
$f(R,T)$, where the gravitational Lagrangian is an arbitrary function of the
Ricci scalar $R$ and of the trace of the energy–momentum tensor $T$. The
dependence on $T$ is justified by the presence of imperfect exotic fluids and
quantum effects. To obtain the field equations, we proceed with the
generalization $R\rightarrow f(R,T)$ in the Hilbert–Einstein Lagrangian and
thus we obtain

$S=\frac{1}{16\pi}\int d^{4}xf(R,T)\sqrt{-g}+\int
d^{4}x\mathcal{L}_{m}\sqrt{-g},$ (1)

where $g$ is the determinant of the metric tensor $g_{\mu\nu}$. We will work
with geometrized units, that is, $c=G=1$ and we use the metric signature
$(-,+,+,+)$. $\mathcal{L}_{m}$ is the Lagrangian density of matter such that
we define the energy–momentum tensor of matter as Landau:1975pou

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}}.$ (2)

Now, assuming that the Lagrangian density $\mathcal{L}_{m}$ depends only on
the components $g_{\mu\nu}$ and not on its derivatives, we have

$T_{\mu\nu}=g_{\mu\nu}\mathcal{L}_{m}-2\frac{\partial\mathcal{L}_{m}}{\partial
g^{\mu\nu}}.$ (3)

By varying the equation (1) with respect to $g^{\mu\nu}$, we obtain the field
equations of $f(R,T)$ gravity in the metric formalism as

$\begin{split}f_{R}(&R,T)R_{\mu\nu}-\frac{1}{2}f(R,T)g_{\mu\nu}+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{R}(R,T)\\\
&=8\pi T_{\mu\nu}-f_{T}(R,T)T_{\mu\nu}-f_{T}(R,T)\Theta_{\mu\nu},\end{split}$
(4)

where, as originally proposed by the authors, $f_{R}(R,T)\equiv\partial
f(R,T)/\partial R$, $f_{T}(R,T)\equiv\partial f(R,T)/\partial T$,
$\Box\equiv\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})/\sqrt{-g}$ and
the tensor $\Theta_{\mu\nu}$ is defined as Harko:2011kv

$\Theta_{\mu\nu}\equiv g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta
g^{\mu\nu}}=-2T_{\mu\nu}+g_{\mu\nu}\mathcal{L}_{m}-2g^{\alpha\beta}\frac{\partial^{2}\mathcal{L}_{m}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}.$ (5)

Note that when $f(R,T)=f(R)$, from equations (4) we obtain the field equations
in the context of $f(R)$ gravity. At this stage, taking the covariant
divergence of (4), with the use of mathematical identity Koivisto:2005yk

$\nabla^{\mu}\left[f_{R}(R,T)R_{\mu\nu}-\frac{1}{2}f(R,T)g_{\mu\nu}+(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu})f_{R}(R,T)\right]\equiv
0,$ (6)

we find the following modified equation for the four–divergence of the
energy–momentum tensor $T_{\mu\nu}$ BarrientosO:2014mys

$\begin{split}\nabla^{\mu}T_{\mu\nu}=&\frac{f_{T}(R,T)}{8\pi-
f_{T}(R,T)}\left[(T_{\mu\nu}+\Theta_{\mu\nu})\nabla^{\mu}\ln
f_{T}(R,T)\right.\\\
&\left.+\nabla^{\mu}\Theta_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\nabla^{\mu}T\right].\end{split}$
(7)

##  III Rapidly rotating star in $f(R,T)$ gravity

Now, we will obtain the equations that describe a stationary rapidly spinning
star in $f(R,T)$ gravity. We start by writing the line element that describes
the space–time geometry of a stationary axisymmetric rotating star, which can
be written in terms of spherical coordinates $\left(t,r,\theta,\phi\right)$ as
follows:

$ds^{2}=-e^{\gamma+\rho}dt^{2}+e^{2\alpha}\left(dr^{2}+r^{2}d\theta^{2}\right)+e^{\gamma-\rho}r^{2}\sin^{2}\theta\left(d\phi-\omega
dt\right)^{2},$ (8)

where $\alpha$, $\gamma$, $\rho$ and $\omega$ are the metric functions which
depend only on $r$ and $\theta$, and the function $\omega$ is associated to
the Lense–Thirring effect pfister2007history ; ciufolini2004confirmation .

We assume that the matter of which the star is made is a perfect fluid. In
this case, there is no unique definition of the matter Lagrangian density.
However, in the present study we assume that $\mathcal{L}_{m}=p$, with $p$
being the pressure of the fluid schutz1970perfect . In this case the tensor
$\Theta_{\mu\nu}$ is given by

$\Theta_{\mu\nu}=-2T_{\mu\nu}+pg_{\mu\nu}.$ (9)

And for the energy–momentum tensor, we use the general definition
Fisher:2019ekh  given by the following expression

$T_{\mu\nu}=pg_{\mu\nu}+(p+\varepsilon)U_{\mu}U_{\nu},$ (10)

where $\varepsilon$ is the energy density and $U_{\mu}$ represents the
4–velocity of the fluid, which obeys the conditions: $U_{\mu}U^{\mu}=-1$ and
$U^{\mu}\nabla_{\nu}U_{\mu}=0$. In the case of a stationary rotating star the
expression for $U^{\mu}$ is given by komatsu1989rapidly :

$U^{\mu}=\frac{dx^{\mu}}{d\tau}=\frac{e^{-\frac{\gamma+\rho}{2}}}{\sqrt{1-v^{2}}}\left(1,0,0,\Omega\right),$
(11)

where $\Omega$ represents the angular velocity of an element of mass of the
star with respect to a static observer at infinity, and $v$ is the 3–velocity
in the ZAMO (zero momentum angular observer), which is the referential frame
of an observer locally without rotation, given by

$v=\left(\Omega-\omega\right)r\sin\theta e^{-\rho}.$ (12)

For the functional form of the $f(R,T)$ function, we use one that was
originally suggested by T.Harko et al. in Harko:2011kv , i.e,
$f(R,T)=R+2\lambda^{\prime}T$, with $\lambda^{\prime}$ a constant. The
substitution of $f(R,T)=R+2\lambda^{\prime}T$ in equations (4) and (7) give us

$G_{\mu\nu}=(8\pi+2\lambda^{\prime})T_{\mu\nu}+\lambda^{\prime}g_{\mu\nu}(T-2p),$
(13)

and

$\nabla^{\mu}T_{\mu\nu}=\frac{2\lambda^{\prime}}{8\pi+2\lambda^{\prime}}\left[\nabla^{\mu}(pg_{\mu\nu})-\frac{1}{2}g_{\mu\nu}\nabla^{\mu}T\right],$
(14)

where $G_{\mu\nu}$ is the usual Einstein tensor, so that the left side of
equation (13) is equal to the left side of Einstein’s equations of GR, and in
the right side we have additional terms proportional to the parameter
$\lambda^{\prime}$ of the $f(R,T)$ theory.

Now, we can define an effective energy–momentum tensor given by

$\tau_{\mu\nu}=T_{\mu\nu}+\frac{\lambda^{\prime}}{8\pi+2\lambda^{\prime}}g_{\mu\nu}(T-2p),$
(15)

and using the definition above we rewrite the field equations (13) in a
compact form as follows

$G_{\mu\nu}=8\pi(1+2\lambda)\tau_{\mu\nu},$ (16)

where $\lambda=\lambda^{\prime}/(8\pi)$. With such formalism, the application
of the Bianchi identities in the above equation yields

$\nabla^{\mu}\tau_{\mu\nu}=0.$ (17)

It is interesting to note that for a perfect fluid the effective
energy–momentum tensor $\tau_{\mu\nu}$ can be written as

$\tau_{\mu\nu}=p^{eff}g_{\mu\nu}+(p^{eff}+\varepsilon^{eff})U_{\mu}U_{\nu},$
(18)

where $\varepsilon^{eff}$ is the effective energy density and $p^{eff}$ is the
effective pressure as follows

$\begin{split}&\varepsilon^{eff}=\frac{\varepsilon(1+3\lambda)-p\lambda}{1+2\lambda},\\\
&p^{eff}=\frac{p(1+3\lambda)-\varepsilon\lambda}{1+2\lambda}.\end{split}$ (19)

We can verify that equations (14) and (17) are equivalent.

Applying the metric (8) and the effective energy–momentum tensor (18) into
(16) give us the following field equations:

$\nabla^{2}\left(\rho e^{\gamma/2}\right)=S_{\rho}\left(r,\mu\right),$ (20)
$\left(\nabla^{2}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\mu\frac{\partial}{\partial\mu}\right)\gamma
e^{\gamma/2}=S_{\gamma}\left(r,\mu\right),$ (21)
$\left(\nabla^{2}+\frac{2}{r}\frac{\partial}{\partial
r}-\frac{2}{r^{2}}\mu\frac{\partial}{\partial\mu}\right)\omega
e^{\left(\gamma-2\rho\right)/2}=S_{\omega}\left(r,\mu\right)$ (22)

and

$\begin{split}\alpha_{,\mu}=&-\nu_{,\mu}-\left\\{\left(1-\mu^{2}\right)\left(1+rB^{-1}B_{,r}\right)^{2}+\left[\mu-\left(1-\mu^{2}\right)B^{-1}B_{,\mu}\right]^{2}\right\\}^{-1}\\\
&\left[\frac{1}{2}B^{-1}\left\\{r^{2}B_{,rr}-\left[\left(1-\mu^{2}\right)B_{,\mu}\right]_{,\mu}-2\mu
B_{,\mu}\right\\}\left[-\mu+\left(1-\mu^{2}\right)B^{-1}B_{,\mu}\right]\right.\\\
&+rB^{-1}B_{,r}\left[\frac{1}{2}\mu+\mu
rB^{-1}B_{,r}+\frac{1}{2}\left(1-\mu^{2}\right)B^{-1}B_{,\mu}\right]\\\
&+\frac{3}{2}B^{-1}B_{,\mu}\left[-\mu^{2}+\mu\left(1-\mu^{2}\right)B^{-1}B_{,\mu}\right]-\left(1-\mu^{2}\right)rB^{-1}B_{,\mu
r}\\\ &(1+rB^{-1}B_{,r})-\mu
r^{2}\left(\nu_{,r}\right)^{2}-2\left(1-\mu^{2}\right)r\nu_{,\mu}\nu_{,r}+\mu\left(1-\mu^{2}\right)\left(\nu_{,\mu}\right)^{2}\\\
&-2\left(1-\mu^{2}\right)r^{2}B^{-1}B_{,r}\nu_{,\mu}\nu_{,r}+\left(1-\mu^{2}\right)B^{-1}B_{,\mu}\left[r^{2}\left(\nu_{,r}\right)^{2}\right.\\\
&\left.-\left(1-\mu^{2}\right)\left(\nu_{,\mu}\right)^{2}\right]+\left(1-\mu^{2}\right)B^{2}e^{-4\nu}\left\\{\frac{1}{4}\mu
r^{4}\left(\omega_{,r}\right)^{2}+\frac{1}{2}\left(1-\mu^{2}\right)r^{3}\omega_{,\mu}\omega_{,r}\right.\\\
&-\frac{1}{4}\mu\left(1-\mu^{2}\right)r^{2}\left(\omega_{,\mu}\right)^{2}+\frac{1}{2}\left(1-\mu^{2}\right)r^{4}B^{-1}B_{,r}\omega_{,\mu}\omega_{,r}\\\
&\left.\left.-\frac{1}{4}\left(1-\mu^{2}\right)r^{2}B^{-1}B_{,\mu}\left[r^{2}\left(\omega_{,r}\right)^{2}-\left(1-\mu^{2}\right)\left(\omega_{,\mu}\right)^{2}\right]\right\\}\right],\end{split}$
(23)

where $\mu$ is defined as

$\mu=\cos\theta,$ (24)

the function $B$ is as follows

$B=e^{\gamma}$ (25)

and the operator $\nabla^{2}$ is given by

$\nabla^{2}=\frac{\partial^{2}}{\partial
r^{2}}+\frac{2}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\frac{\partial}{\partial\mu}\left((1-\mu^{2})\frac{\partial}{\partial\mu}\right)+\frac{1}{r^{2}(1-\mu^{2})}\frac{\partial^{2}}{\partial\phi^{2}}.$
(26)

The “source” terms that appear in equations $(\ref{rs6})$, $(\ref{rs7})$ and
$(\ref{rs8})$ are given by

$\begin{split}S_{\rho}(r,\mu)=&\left[e^{\gamma/2}ke^{2\alpha}\left(\varepsilon+p\right)\frac{1+v^{2}}{1-v^{2}}\right.\\\
&+r^{2}(1-\mu^{2})e^{-2\rho}\left[\omega_{,r}^{2}+\frac{1}{r^{2}}\left(1-\mu^{2}\right)\omega_{,\mu}^{2}\right]+\frac{1}{r}\gamma_{,r}-\frac{1}{r^{2}}\mu\gamma_{,\mu}\\\
&\left.+\frac{\rho}{2}\left\\{2ke^{2\alpha}\frac{p(1+3\lambda)-\varepsilon\lambda}{1+2\lambda}-\gamma_{,r}\left(\frac{1}{2}\gamma_{,r}+\frac{1}{r}\right)-\frac{1}{r^{2}}\gamma_{,\mu}\left[\frac{1}{2}\gamma_{,\mu}\left(1-\mu^{2}\right)-\mu\right]\right\\}\right],\end{split}$
(27)
$S_{\gamma}\left(r,\mu\right)=e^{\gamma/2}\left\\{2ke^{2\alpha}\frac{p(1+3\lambda)-\varepsilon\lambda}{1+2\lambda}+\frac{\gamma}{2}\left[2ke^{2\alpha}\frac{p(1+3\lambda)-\varepsilon\lambda}{1+2\lambda}-\frac{1}{2}\gamma_{,r}^{2}-\frac{1}{2r^{2}}\left(1-\mu^{2}\right)\frac{1}{2}\gamma_{,\mu}^{2}\right]\right\\}$
(28)

and

$\begin{split}S_{\omega}\left(r,\mu\right)=&e^{\left(\gamma-2\rho\right)/2}\left[-2ke^{2\alpha}\frac{\left(\Omega-\omega\right)\left(\varepsilon+p\right)}{1-v^{2}}\right.\\\
&+\omega\left\\{-ke^{2\alpha}\left[\frac{\left(1+v^{2}\right)\left(\varepsilon+p\right)}{1-v^{2}}-\frac{p(1+3\lambda)-\varepsilon\lambda}{1+2\lambda}\right]-\frac{1}{r}\left(2\rho_{,r}+\frac{1}{2}\gamma_{,r}\right)\right.\\\
&+\frac{1}{r^{2}}\mu\left(2\rho_{,\mu}+\frac{1}{2}\gamma_{,\mu}\right)+\frac{1}{4}\left(4\rho_{,r}^{2}-\gamma_{,r}^{2}\right)+\frac{1}{4r^{2}}\left(1-\mu^{2}\right)\left(4\rho_{,\mu}^{2}-\gamma_{,\mu}^{2}\right)\\\
&\left.\left.-r^{2}\left(1-\mu^{2}\right)e^{-2\rho}\left[\omega_{,r}^{2}+\frac{1}{r^{2}}\left(1-\mu^{2}\right)\omega_{,\mu}^{2}\right]\right\\}\right].\end{split}$
(29)

respectively, with $k=8\pi(1+2\lambda)$.

In this work we use the scheme developed by KEH komatsu1989rapidly ;
komatsu1989rapidlyii , which is an iterative numerical method that uses
suitable Green functions to obtain the following integral representation of
the equations $(\ref{rs6})$, $(\ref{rs7})$ and $(\ref{rs8})$:

$\rho=-e^{-\gamma/2}\sum_{n=0}^{\infty}P_{2n}\left(\mu\right)\int_{0}^{\infty}r^{\prime
2}f_{2n}^{2}\left(r,r^{\prime}\right)\int_{0}^{1}P_{2n}\left(\mu^{\prime}\right)S_{\rho}\left(r^{\prime},\mu^{\prime}\right)d\mu^{\prime}dr^{\prime},$
(30) $\gamma=-\frac{2e^{-\gamma/2}}{\pi
r\sin\theta}\sum_{n=1}^{\infty}\frac{\sin{\left(2n-1\right)\theta}}{2n-1}\int_{0}^{\infty}r^{\prime
2}f_{2n-1}^{1}\left(r,r^{\prime}\right)\int_{0}^{1}\sin{\left(2n-1\right)\theta^{\prime}}S_{\gamma}\left(r^{\prime},\mu^{\prime}\right)d\mu^{\prime}dr^{\prime},$
(31)

and

$\begin{split}\omega&=-\frac{e^{\left(2\rho-\gamma\right)/2}}{r\sin\theta}\sum_{n=1}^{\infty}\frac{P_{2n-1}^{1}\left(\mu\right)}{2n\left(2n-1\right)}\int_{0}^{\infty}r^{\prime
3}f_{2n-1}^{2}\left(r,r^{\prime}\right)\int_{0}^{1}P_{2n-1}^{1}\left(\mu^{\prime}\right)S_{\omega}\left(r^{\prime},\mu^{\prime}\right)d\mu^{\prime}dr^{\prime},\end{split}$
(32)

where $P_{n}$ are the Legendre polynomials, $P^{m}_{n}$ are the Associated
Legendre functions and the functions $f_{n}^{1}\left(r,r^{\prime}\right)$ and
$f_{n}^{2}\left(r,r^{\prime}\right)$ are given by

$f_{n}^{1}\left(r,r^{\prime}\right)=\left\\{\begin{tabular}[]{l}${\left(r^{\prime}/r\right)^{n}\text{,
if }r^{\prime}\leq r,}$\\\ ${\left(r/r^{\prime}\right)^{n}\text{, if
}r^{\prime}>r,}$\end{tabular}\right.$ (33)
$f_{n}^{2}\left(r,r^{\prime}\right)=\left\\{\begin{tabular}[]{l}${r^{\prime
n}/r^{n+1}\text{, if }r^{\prime}\leq r,}$\\\ ${r^{n}/r^{\prime n+1}\text{, if
}r^{\prime}>r}$.\end{tabular}\right.$ (34)

The equation (23) for the function $\alpha$ is the only one that is not
transformed to a integral equation, it can be integrated using the following
condition,

$\alpha(r,1)=\frac{\gamma(r,1)-\rho(r,1)}{2}.$ (35)

A convenient feature of the KEH method is that the requirement that the metric
functions $\rho$, $\gamma$ and $\omega$ be asymptotically flat at infinity is
promptly satisfied on the condition that the “source” terms $S_{\rho}$,
$S_{\gamma}$ and $S_{\omega}$ are well behaved. As for the function $\alpha$,
its flatness condition is automatically satisfied because the other functions
already satisfy their respective conditions.

Computing equation (14), or equivalently equation (17), for divergence of the
energy–momentum tensor lead us to the following expression

$\frac{1}{(1+2\lambda)(p+\varepsilon)}\left[\left(1+3\lambda\right)\nabla
p-\lambda\nabla\varepsilon\right]=\nabla\ln U^{t},$ (36)

that can be integrated to obtain

$h^{eff}=\log(c)-\frac{\gamma+\rho}{2}-\frac{1}{2}\log(1-v^{2}),$ (37)

where $c$ is a constant of integration and $h^{eff}$ can be referred as
effective specific enthalpy and is given by the equation bellow

$\frac{dh^{eff}}{dp^{eff}}=\frac{1}{p^{eff}+\varepsilon^{eff}},$ (38)

where $p^{eff}$ and $\varepsilon^{eff}$ are defined in equation (19). To be
able to solve equation (37), (38) we need an EoS, and so in the next section
we present the EoS we use in this work.

##  IV Equations of state

In this section, we outline the EoS used in the present work to describe
hadronic matter. These EoS will be the input to the stellar structure
equations that govern rapidly rotating NS in $f(R,T)$ gravity. For a didactic
and more extensive explanation on EoS, we refer the reader to ref. Menezes2021
and references therein.

###  IV.1 The Nonlinear Walecka Model

The first relativistic model used here to describe the hadronic matter is a
rather generalized version of the quantum hadrodynamics (QHD) livrowalecka ;
walecka ; bogutabodmer , which is based on a relativistic mean–field theory
and describes the baryon interaction through the exchange of scalar and vector
mesons, known as nonlinear Walecka model (NLWM).

Although the first version of the model Walecka-74  reproduced
well–established properties of infinite nuclear matter, such as, the binding
energy and the nuclear saturation density ($\rho_{0}\sim$ 0.15 fm-3), other
important properties such as incompressibility and the effective mass of
nucleons are not obtained with satisfactory values. This problem was
circumvented with the introduction of self–interacting terms, cubic and
quartic, in the scalar field by Boguta and Bodmer bogutabodmer . Likewise, to
deal with asymmetric systems with respect to the numbers of protons and
neutrons, the vector–isovector meson $\rho$ (not to confound with the notation
for the density) was introduced. And, to adjust other properties such as
symmetry energy and the fact that protons and neutrons have slightly different
masses, other mesons and interactions were included, leading to extensive
generalizations and parameterization of this model Dutra2014 . Therefore, the
more general Lagrangian density of the NLWM model is given by Menezes2021 ;
Agrawal2010 ; Dutra2014 :

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm
nm}+\mathcal{L}_{\sigma}+\mathcal{L}_{\omega}+\mathcal{L}_{\rho}+\mathcal{L}_{\delta}+\mathcal{L}_{\sigma\omega\rho},$
(39)

where

$\displaystyle\mathcal{L}_{\rm nm}$ $\displaystyle=$
$\displaystyle\overline{\psi}(i\gamma^{\mu}\partial_{\mu}-M)\psi+g_{\sigma}\sigma\overline{\psi}\psi-
g_{\omega}\overline{\psi}\gamma^{\mu}\omega_{\mu}\psi-\frac{g_{\rho}}{2}\overline{\psi}\gamma^{\mu}\vec{\rho}_{\mu}\cdot\vec{\tau}\psi+g_{\delta}\overline{\psi}\vec{\delta}\cdot\vec{\tau}\psi,$
(40) $\displaystyle\mathcal{L}_{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial^{\mu}\sigma\partial_{\mu}\sigma-m^{2}_{\sigma}\sigma^{2})-\frac{A}{3}\sigma^{3}-\frac{B}{4}\sigma^{4},$
(41) $\displaystyle\mathcal{L}_{\omega}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}m^{2}_{\omega}\omega_{\mu}\omega^{\mu}+\frac{C}{4}(g_{\omega}^{2}\omega_{\mu}\omega^{\mu})^{2},$
(42) $\displaystyle\mathcal{L}_{\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\vec{B}^{\mu\nu}\vec{B}_{\mu\nu}+\frac{1}{2}m^{2}_{\rho}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\mu},$
(43) $\displaystyle\mathcal{L}_{\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial^{\mu}\vec{\delta}\partial_{\mu}\vec{\delta}-m^{2}_{\delta}\vec{\delta}^{2}),$
(44) $\displaystyle\mathcal{L}_{\sigma\omega\rho}$ $\displaystyle=$
$\displaystyle
g_{\sigma}g_{\omega}^{2}\sigma\omega_{\mu}\omega^{\mu}\left(\alpha_{1}+\frac{1}{2}{\alpha_{1}}^{\prime}g_{\sigma}\sigma\right)+g_{\sigma}g_{\rho}^{2}\sigma\vec{\rho}_{\mu}\cdot\vec{\rho}^{\mu}\left(\alpha_{2}+\frac{1}{2}{\alpha_{2}}^{\prime}g_{\sigma}\sigma\right)$
(45) $\displaystyle+$
$\displaystyle\frac{1}{2}{\alpha_{3}}^{\prime}g_{\omega}^{2}g_{\rho}^{2}\omega_{\mu}\omega^{\mu}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\mu}.$

In this Lagrangian density: $\mathcal{L}_{\rm nm}$ represents the kinetic part
of the nucleons plus the terms standing for the interaction between them and
mesons $\sigma$, $\delta$, $\omega$, and $\rho$; the terms $\mathcal{L}_{j}$
represents the free and self–interacting terms of the meson $j$, where
$j=\sigma,\delta,\omega,$ and $\rho$. The term
$\mathcal{L}_{\sigma\omega\rho}$, accounts for crossing interactions between
the meson fields. The antisymmetric field tensors $F_{\mu\nu}$ and
$\vec{B}_{\mu\nu}$ are given by
$F_{\mu\nu}=\partial_{\nu}\omega_{\mu}-\partial_{\mu}\omega_{\nu}$ and
$\vec{B}_{\mu\nu}=\partial_{\nu}\vec{\rho}_{\mu}-\partial_{\mu}\vec{\rho}_{\nu}-g_{\rho}(\vec{\rho}_{\mu}\times\vec{\rho}_{\nu})$.
Finally, $M$ and $m_{j}$ are respectively the nucleon mass and the meson
masses 111Observe that the notation used here is valid within Section IV
only..

In the relativistic mean field (RMF) approximation, the meson fields are
treated as classical fields, and the equations of motion are obtained using
the Euler–Lagrange equations assuming rotational and translational invariance.
Therefore, the RMF consists of the application of substitutions

$\displaystyle\sigma\rightarrow\left<\sigma\right>\equiv\sigma_{0},$
$\displaystyle\omega_{\mu}\rightarrow\left<\omega_{0}\right>\equiv\omega_{0},$
$\displaystyle\vec{\rho}_{\mu}\rightarrow\left<\vec{\rho}_{0}\right>\equiv\bar{\rho}_{0(3)},$
$\displaystyle\vec{\delta}\rightarrow\,\,<\vec{\delta}>\equiv\delta_{(3)}.$

and the equations of motion:

$\displaystyle
m^{2}_{\sigma}\sigma_{0}=g_{\sigma}\rho_{s}-A\sigma_{0}^{2}-B\sigma_{0}^{3}+g_{\sigma}g_{\omega}^{2}\omega_{0}^{2}(\alpha_{1}+{\alpha_{1}}^{\prime}g_{\sigma}\sigma)+g_{\sigma}g_{\rho}^{2}\bar{\rho}_{0(3)}^{2}(\alpha_{2}+{\alpha_{2}}^{\prime}g_{\sigma}\sigma)\,\mbox{,}\quad$
(46) $\displaystyle m_{\omega}^{2}\omega_{0}=g_{\omega}\rho-
Cg_{\omega}(g_{\omega}\omega_{0})^{3}-g_{\sigma}g_{\omega}^{2}\sigma_{0}\omega_{0}(2\alpha_{1}+{\alpha_{1}}^{\prime}g_{\sigma}\sigma_{0})-{\alpha_{3}}^{\prime}g_{\omega}^{2}g_{\rho}^{2}\bar{\rho}_{0(3)}^{2}\omega_{0},$
(47) $\displaystyle
m_{\rho}^{2}\bar{\rho}_{0(3)}=\frac{g_{\rho}}{2}\rho_{3}-g_{\sigma}g_{\rho}^{2}\sigma_{0}\bar{\rho}_{0(3)}(2\alpha_{2}+{\alpha_{2}}^{\prime}g_{\sigma}\sigma_{0})-{\alpha_{3}}^{\prime}g_{\omega}^{2}g_{\rho}^{2}\bar{\rho}_{0(3)}\omega_{0}^{2},$
(48) $\displaystyle m_{\delta}^{2}\delta_{(3)}=g_{\delta}\rho_{s3},$ (49)
$\displaystyle\left[i\gamma^{\mu}\partial_{\mu}-\gamma^{0}V_{\tau}-(M+S_{\tau})\right]\psi=0,$
(50)

where

$\rho_{s}=\left<\overline{\psi}\psi\right>={\rho_{s}}_{p}+{\rho_{s}}_{n},$
(51)

$\rho_{s3}=\left<\overline{\psi}{\tau}_{3}\psi\right>={\rho_{s}}_{p}-{\rho_{s}}_{n},$
(52)

$\rho=\left<\overline{\psi}\gamma^{0}\psi\right>=\rho_{p}+\rho_{n},$ (53)

$\rho_{3}=\left<\overline{\psi}\gamma^{0}{\tau}_{3}\psi\right>=\rho_{p}-\rho_{n}=(2y_{p}-1)\rho,$
(54)

with

${\rho_{s}}_{p,n}=\frac{\gamma
M^{*}_{p,n}}{2\pi^{2}}\int_{0}^{{k_{F}}_{p,n}}\frac{k^{2}dk}{\sqrt{k^{2}+M^{*2}_{p,n}}},$
(55)

$\rho_{p,n}=\frac{\gamma}{2\pi^{2}}\int_{0}^{{k_{F}}_{p,n}}k^{2}dk=\frac{\gamma}{6\pi^{2}}{k_{F}^{3}}_{p,n},$
(56)

$V_{\tau}=g_{\omega}\omega_{0}+\frac{g_{\rho}}{2}\bar{\rho}_{0(3)}\tau_{3},\qquad
S_{\tau}=-g_{\sigma}\sigma_{0}-g_{\delta}\delta_{(3)}\tau_{3}.$ (57)

where $\gamma$ is the spin degeneracy and ${k_{F}}_{p,n}$ is the Fermi
momentum. The indices $p,n$ correspond to protons and neutrons respectively.

The proton and neutron effective masses are given by:

$M^{*}_{p}=M-g_{\sigma}\sigma_{0}-g_{\delta}\delta_{(3)},\quad
M^{*}_{n}=M-g_{\sigma}\sigma_{0}+g_{\delta}\delta_{(3)}.$ (58)

Finally, after some analytical calculations it is possible to obtain the
energy density and pressure in the NLWM model. These quantities are given as
follows Dutra2014 ; Menezes2021 :

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}_{\sigma}\sigma_{0}^{2}+\frac{A}{3}\sigma_{0}^{3}+\frac{B}{4}\sigma_{0}^{4}-\frac{1}{2}m^{2}_{\omega}\omega_{0}^{2}-\frac{C}{4}(g_{\omega}^{2}\omega_{0}^{2})^{2}-\frac{1}{2}m^{2}_{\rho}\bar{\rho}_{0(3)}^{2}+g_{\omega}\omega_{0}\rho+\frac{g_{\rho}}{2}\bar{\rho}_{0(3)}\rho_{3}$
(59) $\displaystyle+$
$\displaystyle\frac{1}{2}m^{2}_{\delta}\delta^{2}_{(3)}-g_{\sigma}g_{\omega}^{2}\sigma\omega_{0}^{2}\left(\alpha_{1}+\frac{1}{2}{\alpha_{1}}^{\prime}g_{\sigma}\sigma_{0}\right)-g_{\sigma}g_{\rho}^{2}\sigma\bar{\rho}_{0(3)}^{2}\left(\alpha_{2}+\frac{1}{2}{\alpha_{2}}^{\prime}g_{\sigma}\sigma_{0}\right)$
$\displaystyle-$
$\displaystyle\frac{1}{2}{\alpha_{3}}^{\prime}g_{\omega}^{2}g_{\rho}^{2}\omega_{0}^{2}\bar{\rho}_{0(3)}^{2}+\mathcal{E}_{\mbox{\tiny
kin}}^{p}+\mathcal{E}_{\mbox{\tiny kin}}^{n},$

with

$\displaystyle\mathcal{E}_{\mbox{\tiny kin}}^{p,n}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2\pi^{2}}\int_{0}^{{k_{F}}_{p,n}}k^{2}(k^{2}+M^{*2}_{p,n})^{1/2}dk,$
(60)

and pressure:

$\displaystyle P$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m^{2}_{\sigma}\sigma_{0}^{2}-\frac{A}{3}\sigma_{0}^{3}-\frac{B}{4}\sigma_{0}^{4}+\frac{1}{2}m^{2}_{\omega}\omega_{0}^{2}+\frac{C}{4}(g_{\omega}^{2}\omega_{0}^{2})^{2}+\frac{1}{2}m^{2}_{\rho}\bar{\rho}_{0(3)}^{2}+\frac{1}{2}{\alpha_{3}}^{\prime}g_{\omega}^{2}g_{\rho}^{2}\omega_{0}^{2}\bar{\rho}_{0(3)}^{2}$
(61) $\displaystyle-$
$\displaystyle\frac{1}{2}m^{2}_{\delta}\delta^{2}_{(3)}+g_{\sigma}g_{\omega}^{2}\sigma_{0}\omega_{0}^{2}\left(\alpha_{1}+\frac{1}{2}{\alpha_{1}}^{\prime}g_{\sigma}\sigma_{0}\right)+g_{\sigma}g_{\rho}^{2}\sigma\bar{\rho}_{0(3)}^{2}\left(\alpha_{2}+\frac{1}{2}{\alpha_{2}}^{\prime}g_{\sigma}\sigma\right)$
$\displaystyle+$ $\displaystyle P_{\mbox{\tiny kin}}^{p}+P_{\mbox{\tiny
kin}}^{n},\qquad$

with

$P_{\mbox{\tiny
kin}}^{p,n}=\frac{\gamma}{6\pi^{2}}\int_{0}^{{k_{F}}_{p,n}}\frac{k^{4}dk}{(k^{2}+M^{*2}_{p,n})^{1/2}}.$
(62)

When stellar matter is considered, charge neutrality and chemical $\beta$
equilibrium equations have to be imposed. To implement them, leptons
(generally electrons and muons) have to be present and they enter the system
as free gases. Therefore, charge neutrality and $\beta$-equilibrium conditions
require that:

$\mu_{p}=\mu_{n}-\mu_{e},\quad\mu_{e}=\mu_{\mu},\quad\rho_{p}=\rho_{e}+\rho_{\mu}.$
(63)

The energy density and pressure for the leptons are given by:

$\varepsilon_{l}=\frac{3}{\pi^{2}}\int^{K_{F_{l}}}_{0}k^{2}(m_{l}^{2}+k^{2})^{1/2}dk,$
(64)

and

$p=\frac{1}{3\pi^{2}}\int^{K_{F_{l}}}_{0}\frac{k^{4}}{(m_{l}^{2}+k^{2})^{1/2}}dk,$
(65)

where $K_{F_{l}}$ is the Fermi momentum for leptons and the electron and muon
mass values are 0.511 MeV and 105.66 MeV, respectively.

The electron and muon densities read:

$\rho_{l}={K_{F_{l}}^{3}}/{3\pi^{2}}.$ (66)

There are many possible parameterizations of the QHD model. We chose the
IU–FSU parameterization proposed in iufsu . Besides the tests performed in
Dutra2014 ; Dutra2015 , IU–FSU is also successful in explaining the recent
constraint that comes from the GW170817 observation Lourenco2018 .

For the outer crust of the neutron star, it is necessary to use a model that
describes the nuclear matter for the low density region. For this, we use the
full EoS BPS bps .

###  IV.2 The Quark–Meson Coupling Model

Now, we present the second relativistic EoS used in the present work to model
the nuclear matter in NS. For this we use the quark–meson coupling (QMC) model
guichon .

In the QMC model, nucleons in nuclear medium are considered as a system of
non–overlapping MIT bags mitbag , where quarks within one bag interact with
quarks in another bag through the exchange of scalar $(\sigma)$ and vector
$(\omega,\rho)$ mesons. The usual RMF approximation is used to treat the meson
fields.

The quark field, $\psi_{q}$, inside the bag then satisfies the equation of
motion:

$\left[i{\partial\mkern-7.5mu/}-(m_{q}^{0}-g_{\sigma}^{q})-g_{\omega}^{q}\omega\gamma^{0}+\frac{1}{2}g_{\rho}^{q}\tau_{z}\rho_{03}\gamma^{0}\right]\psi_{qB}(x)=0,$
(67)

where $q$ are the quarks (q = u, d) of mass $m_{q}^{0}$, $\tau_{z}$ is the
spin projection, and $g_{\sigma}^{q}$, $g_{\omega}^{q}$ and $g_{\rho}^{q}$
denote the quark–meson coupling constants. The energy of the static bag
describing a nucleon ($p$ or $n$) consisting of three quarks, in the ground
state, is expressed as

$E^{bag}_{p,n}=\sum_{q}n_{q}\frac{\Omega_{q_{p,n}}}{R_{p,n}}-\frac{Z_{p,n}}{R_{p,n}}+\frac{4}{3}\pi
R_{p,n}^{3}B_{p,n},$ (68)

where $Z_{p,n}$ is a parameter containing information about zero–point motion
of nucleon and $B_{p,n}$ is the bag constant of radius $R_{p,n}$. The
effective mass of a nucleon is defined as $M^{*}_{p,n}=E^{bag}_{p,n}$.

The equilibrium condition for the bag is obtained by minimizing the effective
mass $M^{*}_{p,n}$ with respect to the bag radius

$\frac{dM^{*}_{p,n}}{dR^{*}_{p,n}}=0.$ (69)

In our calculations, we consider $Z_{p,n}=4.0050668$ and
$E^{1/4}_{p,n}=210.85$ MeV. These values are obtained by fixing the bag radius
$R_{p,n}=0.6$ fm and considering the bare nucleon mass $M=939$ MeV. For more
details see ref. Grams:2015inf .

After some analytical calculations that can be found in Grams:2015inf ;
Grams18  the following expressions for energy density and pressure are
obtained, namely:

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}_{\sigma}\sigma+\frac{1}{2}m^{2}_{\omega}\omega_{0}^{2}+\frac{1}{2}m^{2}_{\rho}\rho_{03}^{2}+\sum_{p,n}\frac{1}{\pi^{2}}\int_{0}^{{k_{F}}_{p,n}}k^{2}(k^{2}+M^{*2}_{p,n})^{1/2}dk,$
(70)

and pressure,

$\displaystyle P$ $\displaystyle=$
$\displaystyle-\frac{1}{2}m^{2}_{\sigma}\sigma+\frac{1}{2}m^{2}_{\omega}\omega_{0}^{2}+\frac{1}{2}m^{2}_{\rho}\rho_{03}^{2}+\sum_{p,n}\frac{1}{3\pi^{2}}\int_{0}^{{k_{F}}_{p,n}}\frac{k^{4}dk}{(k^{2}+M^{*2}_{p,n})^{1/2}}.$
(71)

The mesonic fields $\omega_{0}$ e $\rho_{03}$ are determined through the
following relations

$\displaystyle\omega_{0}=\frac{g_{\omega}(\rho_{p}+\rho_{n})}{m^{2}_{\omega}}\
,\ \ \rho_{03}=\frac{g_{\rho}(\rho_{p}-\rho_{n})}{m^{2}_{\rho}},$ (72)

where

$\rho_{p,n}=\sum_{p,n}\frac{\gamma}{6\pi^{2}}{k_{F}^{3}}_{p,n},$ (73)

is the baryonic density.

Again, charge neutrality and chemical equilibrium conditions need to be
implemented, which depend on the inclusion of leptons. The leptonic
expressions for energy density, pressure and density are the same as given in
the last subsection.

Once again, we use the full BPS EoS to describe the star outer crust bps .

##  V Results

In this work we obtained sequences of solutions for four rotation regimes:
non–rotating stars which are represented by dashed curves in the plots, stars
rotating at $300$ Hz represented by dash double–dot curves, stars rotating at
$716$ Hz shown in the dash–dot curves and stars rotating at the Kepler limit
shown in the solid line curves. The Keplerian limit is reached when the
angular velocity of the star at the equator $\Omega(R_{e},\theta=\pi/2)$ is
the same as the angular velocity $\Omega_{K}$ of a free particle in circular
orbit

$\Omega_{K}=\left.\left(\omega+\frac{r\partial_{r}\omega}{2+r\partial_{r}\gamma-r\partial_{r}\rho}+\sqrt{\left(\frac{r\partial_{r}\omega}{2+r\partial_{r}\gamma-r\partial_{r}\rho}\right)^{2}+\frac{e^{2\rho}(\partial_{r}\gamma+\partial_{r}\rho)}{r(2+r\partial_{r}\gamma-r\partial_{r}\rho)}}\;\right)\right|_{r=R_{e},\theta=\frac{\pi}{2}}.$
(74)

If we increase the rotation of the star beyond $\Omega_{K}$ the star starts to
lose mass at the equator, so this limit is also known as the mass–shedding
limit. Besides, we obtained solutions for four values of the parameter
$\lambda^{\prime}$ of the $f(R,T)$ theory: $0$, which recovers the GR case and
is represented by the black color curves, $-0.02$ shown in the blue curves,
$-0.04$ shown in the green curves and $-0.06$ represented by the red curves.
We do not used values of $\lambda^{\prime}>0$ because this leads to negative
values for $p^{eff}$ and $\varepsilon^{eff}$, which violates the energy
conditions.

Once we have obtained solutions for the field equations (23), (30), (31) and
(32) together with the equation (37), which comes from the divergence of the
energy momentum tensor, we are interested in analysing relevant physical
quantities associated to the NS. The first analysis we do is for the
mass–radius relation. For that, we need to calculate the gravitational mass of
the star, also called total mass or tensor mass komatsu1989rapidly ;
cook1992spin ; doneva2013rapidly . This mass is obtained by the Komar integral
komar1959covariant ; wald2010general , which is equal to the ADM mass
arnowitt1959dynamical ; arnowitt1960canonical ; friedman2013rotating  in the
case of stationary, asymptotically flat space–times. In the $f(R,T)$ theory
such integral is given by

$\begin{split}M=&4\pi(1+2\lambda)\int_{0}^{\pi/2}\int_{0}^{R_{e}}e^{2\alpha+\frac{\gamma-\rho}{2}}\left[e^{\frac{\gamma+\rho}{2}}\left(\frac{\left(\varepsilon+p\right)\left(1+v^{2}\right)}{1-v^{2}}+2\frac{p(1+3\lambda)-\varepsilon\lambda}{1+2\lambda}\right)+\right.\\\
&\left.2r\sin\theta\;\omega
e^{\frac{\gamma-\rho}{2}}\frac{\left(\varepsilon+p\right)v}{1-v^{2}}\right]r^{2}\sin\theta\;drd\theta.\end{split}$
(75)

The mass–radius relation is shown in Figure 1 and, from these plots, we can
conclude that the effect of the rotation is of increase the mass and the
equatorial radius of the NS, in both GR and $f(R,T)$ gravity. We can also
observe that the influence of the rotation on the mass is more intense for
stars with the EoS IU–FSU than for those with QMC. For example, for the
solutions with $\lambda^{\prime}=-0.02$, when we go from non–rotating
solutions to the ones rotating at $716$ Hz, there is an increase of $3.0\%$ in
the maximum mass of the stars with EoS IU–FSU while for the stars with EoS QMC
the increase is of $2.6\%$. The effect of the parameter $\lambda^{\prime}$ on
the mass–radius relation is that as we increase the absolute value of this
parameter there is an increase in mass and equatorial radius for stars that
are static and for those rotating at $716$ Hz. However, for the stars at the
Kepler limit, the mass decreases as we increase the absolute value of
$\lambda^{\prime}$. This behavior is similar to that observed for Rastall
gravity da2021rapidly  and for the dilatonic Einstein–Gauss–Bonnet theory
kleihaus2016rapidly . Another effect of $\lambda^{\prime}$ that we can observe
on the mass–radius relation is that as we increase the absolute value of this
parameter, the curves gradually lose the characteristic of having several NS
with almost the same radius in the astrophysically pertinent mass range.

Figure 1: The mass–radius relation for sequences of non-rotating stars (dashed
curves), stars rotating at $716$ Hz (dash–dot curves) and for stars rotating
at the mass–shedding limit (solid line curves). The curves for
$\lambda^{\prime}=0$ correspond to the GR case.

In Figure 1 we show the masses of the two NS in the event GW170817
abbott2017gw170817 ; abbott2019gwtc , respectively
$M_{1}=1.46_{-0.10}^{+0.12}M_{\odot}$ (pink–dotted horizontal line) and
$M_{2}=1.27_{-0.09}^{+0.09}M_{\odot}$ (purple–dotted horizontal line)222In
Figure 1, the measurements for GW170817 and GW190814 are with $90\%$ credible
intervals and the ones for PSR J0030+0451 are with $68\%$. The shaded bands in
the plots indicate the intervals of each measurement.. Using data from LIGO
and Virgo abbott2018gw170817  it was estimated that the stars in this event
have radii $R_{1}=10.8_{-1.7}^{+2.0}$ km (pink–dotted vertical line) and
$R_{2}=10.7_{-1.5}^{+2.1}$ km (purple–dotted vertical line), respectively,
with shaded regions to indicate the uncertainty in the measurements. We can
observe that the static solutions in GR with EoS IU–FSU are in agreement with
these data. However, for other sequences of solutions shown in Figure 1 the
radii for stars with $M_{1}$ and $M_{2}$ are larger than those estimated in
abbott2018gw170817 . For example, the radii for NS with $M_{1}$ and $M_{2}$
which are rotating at $716$ Hz in $f(R,T)$ with $\lambda^{\prime}=-0.02$ and
EoS IU–FSU are $R_{1}\approx 15.6$ km and $R_{2}\approx 17.2$ km,
respectively.

Figure 1 also displays the estimates for the mass and the radius of the
isolated $205.53$ Hz millisecond pulsar PSR J0030+0451 made in miller2019psr ,
which are $M_{3}=1.44_{-0.14}^{+0.15}M_{\odot}$ (yellow–dotted horizontal
line) and $R_{3}=13.02_{-1.06}^{+1.24}$ km (yellow–dotted vertical line).
These estimates were made using data from NASA’s Neutron Star Interior
Composition Explorer (NICER) mission, installed on the International Space
Station (ISS). NICER uses X–ray timing and spectroscopy instrument, to
investigate soft X–ray emissions from hot spots on the surface of NS. Using
the same data, independent estimates for PSR J0030+0451, which are compatible
with those presented here, were also made in riley2019nicer . We can observe
that both EoS produce curves that have mass and radius values compatible with
those estimated for PSR J0030+0451. However, for QMC EoS only the curve for
static solutions in GR agrees with these data whereas for IU–FSU EoS we also
have solutions in $f(R,T)$ with $\lambda^{\prime}=-0.02$ the agree with
$M_{3}$ and $R_{3}$.

Lastly, in the left panel of Figure 1 we can see in orange–dotted horizontal
line the mass of the compact object detected in the event GW190814
abbott2020gw190814  ($M_{4}=2.59_{-0.09}^{+0.08}M_{\odot}$). We can observe
that stars rotating at the Kepler limit in GR with EoS QMC can achieve masses
in the range of values of $M_{4}$, with a radius around $R_{4}\approx 15.6$
km.

Figure 2: The mass as a function of the central energy density for sequences
of non–rotating stars (dashed curves), stars rotating at $716$ Hz (dash–dot
curves) and for stars rotating at the mass–shedding limit (solid line curves).
The curves for $\lambda^{\prime}=0$ correspond to the GR case.

In Figure 2 we show the relation between the mass and the central energy
density. We can see that stars with higher angular velocity have higher masses
for the same central energy density. As for the effect of the $f(R,T)$ gravity
we can observe that as we increase the absolute value of $\lambda^{\prime}$
there is an increase in the mass for the same value of $\varepsilon_{c}/c^{2}$
for stars that have $\Omega=0$ Hz and $\Omega=716$ Hz. But, for the solutions
in the mass–shedding limit the effect of the parameter is of decrease the
mass. We can observe that, in the Kepler limit, the maximum value of the
central energy density decreases as we increase the absolute value of
$\lambda^{\prime}$. And, comparing the two plots we can note that for the EoS
IU–FSU we can reach higher maximum central energy density than for EoS QMC.

Next, we are interested in the relation between the Kepler rotation and the
angular momentum of the NS. The angular momentum is giver by the Komar
integral as follows:

$J=4\pi(1+2\lambda)\int_{0}^{\pi/2}\int_{0}^{R_{e}}\frac{\left(\varepsilon+p\right)v}{1-v^{2}}e^{2\alpha+\gamma-\rho}r^{3}\sin^{2}\theta\;drd\theta.$
(76)

Figure 3: The angular velocity as a function of the angular momentum for
sequences of stars rotating at the mass–shedding limit. The curves for
$\lambda^{\prime}=0$ correspond to the GR case.

In Figure 3 we plot the Kepler rotation versus the angular momentum. In this
figure we can observe that the maximum angular momentum occurs to stars in GR
and as we increase the value of the $f(R,T)$ parameter the maximum angular
momentum decreases. The same occurs for the angular velocity, that is, as we
increase the parameter $\lambda^{\prime}$ the maximum velocity the star can
rotate is decreased. We can also verify that NS with the EoS QMC reach higher
values of $\Omega_{K}$ and $J$ than those with EoS IU–FSU.

Another quantity of interest is the moment of inertia, which can be found by
calculating the ratio of the angular momentum $J$ to the angular velocity
$\Omega$

$I=\frac{J}{\Omega}.$ (77)

Figure 4: The Moment of inertia as a function of the mass for sequences of
stars rotating at $300$ Hz (dash double–dot curves), stars rotating at $716$
Hz (dash–dot curves) and for stars rotating at the mass–shedding limit (solid
line curves). The curves for $\lambda^{\prime}=0$ correspond to the GR case.

This global parameter of the NS is very sensitive to the dense matter EoS, so
that its determination has relevant implications to the constraining of the
EoS models bejger2002moments ; lattimer2005constraining ; ozel2016masses . And
it is interesting to note that approximations for the moment of inertia can be
constructed, especially for the more stiff EoS, also for non-rotating stars
lattimer2000nuclear ; tello2019anisotropic ; singh2019minimally ;
singh2020static . Figure 4 shows the moment of inertia as a function of the
mass. We can observe that for stars with $\Omega=300$ Hz and $\Omega=716$ Hz
the moment of inertia increases as we increase the absolute value of
$\lambda^{\prime}$. However, for stars at the mass–shedding limit the maximum
value for $I$ is attained for the GR case and, as we increase the absolute
value of the parameter $\lambda^{\prime}$ of the $f(R,T)$ gravity the values
of the moment of inertia decrease. We can also see that NS with the EoS QMC
achieve higher values for $I$ than those with EoS IU–FSU.

Figure 5: The compactness $C$ for non-rotating stars (dashed curves), stars
rotating at $716$ Hz (dash–dot curves) and for stars rotating at the
mass–shedding limit (solid line curves). The curves for $\lambda^{\prime}=0$
correspond to the GR case.

The last physical quantity of interest we analyse is the compactness $C$ of
the NS, it can be defined in terms of the mass $M$ and the equatorial radius
$R_{e}$ as

$C=\frac{2M}{R_{e}}.$ (78)

This definition is the same one used in kleihaus2016rapidly , and is
normalized so that for black holes with mass $M$ and horizon radius $R_{e}$,
we have $C=1$. In Figure 5 we plot the compactness versus the mass for three
different rotation regimes. We can observe that the effect of the $f(R,T)$
theory is of decrease the values of $C$. The plots show that the static NS are
the ones with the highest values of compactness. And we can also note that
stars with EoS IU–FSU achieve smaller values of compactness than the ones with
EoS QMC.

##  VI Conclusions

In this work, we studied the influence of $f(R,T)$ theory on NS with realistic
EoS in a fast rotation regime. We obtained results that indicate substantial
modifications in the physical properties of NS in $f(R,T)$ gravity when
compared to those in the context of GR. In particular, we have considered
stars rotating at $0$ Hz, $300$ Hz, $716$ Hz and stars rotating at the Kepler
limit. Regarding the $f(R,T)$ theory we considered four values of the
parameter $\lambda^{\prime}$: $0$, $-0.02$, $-0.04$ and $-0.06$.

We obtained the mass–radius relation for sequences of stars and conclude that
the presence of rotation increases the mass and the equatorial radius of the
NS, in both GR and $f(R,T$) gravity. Besides, the effect of rotation is more
intense in IU-–FSU model. Concerning the consequence of $f(R,T)$ gravity to
the mass-radius relation, we have shown that the effect of increasing the
values of the parameter $\lambda^{\prime}$ is of increase the mass and
equatorial radius for stars that are static and those rotating at $716$ Hz.
But, for stars at the Kepler limit the mass decreases as we increase the value
of the parameter $\lambda^{\prime}$, similarly to what was observed in
da2021rapidly  and kleihaus2016rapidly . Using the mass and radius estimates
for GW170817 and PSR J0030+0451 as constraints, we conclude that only values
of $|\lambda^{\prime}|\leq 0.02$ produce mass and radius values compatible
with the experimental data considering the EoS used here. This bound for the
parameter of $f(R,T)$ gravity is in agreement with lobato2020neutron . As a
complementary analysis of the mass–radius relation, we studied the relation
between the mass and the central energy density. We were able to obtain
solutions with higher values for the central energy density in the IU–FSU
model than in the QMC model. Furthermore, in the mass shedding limit the
maximum value for the central energy density decreases with the increasing of
the parameter of the $f(R,T)$ gravity.

We also examined the effect of the $f(R,T)$ theory in the relation between the
Keplerian angular velocity and the angular momentum and concluded that this
theory produces stars with smaller Kepler limit and smaller angular momentum
than GR. Another physical quantity of great interest is the moment of inertia,
which was investigated as a function of the mass in Figure 4. We found that
for stars rotating at $300$ Hz and $716$ Hz the moment of inertia increases
with the increasing of the parameter $\lambda^{\prime}$, but for stars at the
mass–shedding limit the opposite occurs. We also saw that the QMC model
produces stars with higher values for the moment of inertia than the IU–FSU
model.

Lastly, we studied the consequence of the $f(R,T)$ gravity to the compactness
of the NS. We concluded that the effect of increasing the value of the
parameter of this theory is of decrease the compactness of the stars both in
the static and in the rotating case.

In general, we could observe that our results for rapidly rotating stars in
$f(R,T)$ gravity have some similarities with the ones obtained for Rastall’s
gravity da2021rapidly  and for dilatonic Einstein–Gauss–Bonnet theory
kleihaus2016rapidly .

##  VII Acknowledgements

We are very grateful to D.P. Menezes for providing the EoS tables used in this
work. C.E.M. and T.O.F.C. are supported by Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES) scholarship, L.C.N.S. would like to thank
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for
partial financial support through the research Project No. 164762/2020-5 and
F.M.S. would like to thank CNPq for financial support through the research
Project No. 165604/2020-4. J.C.F. thanks CNPq and FAPES for their financial
support.

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