Charged anisotropic white dwarfs in fR T gravity Zhe Feng Dated October 5 2022_2

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Charged anisotropic white dwarfs in f(R, T )gravity

Zhe Feng∗

(Dated: October 5, 2022)

In the context of f(R, T ) = R+ 2βT gravity, where Ris the Ricci scalar and Tis the trace of

the energy-momentum tensor, the equilibrium structure of charged anisotropic white dwarfs (WDs)

is studied. The stellar equations for the general case are derived and numerical solutions are found

for the Chandrasekhar equation of state (EoS) and a charge density distribution proportional to

the energy density ρch =αρ. By adjusting diﬀerent parameters, the properties of the solutions

under various conditions are compared. Most importantly, by going beyond the trivial WD in GR

in various ways, the solutions may exhibit super-Chandrasekhar behavior. This paper is a study of a

WD structure, and the results obtained may have a contrasting eﬀect on astronomical observations

such as superluminous type Ia supernovae.

I. INTRODUCTION

General relativity(GR) has withstood almost all obser-

vations and experimental tests from weak gravity (e.g.,

Mercury’s perihelion shift, gravitational lensing, gravi-

tational redshift, etc. at the scale of the solar system)

and strong gravity (e.g. pulsar binary systems), and

is therefore a beautiful and successful theory of grav-

ity. With the discovery of Type Ia supernovae [1], the

cosmic microwave background radiation (CWBR) [2,3],

and most importantly, the accelerating expansion of the

universe, Einstein’s theory have shown increasing limita-

tions. Even in a purely academic interest, there is a desire

to explore theories outside the standard GR framework

[4]. Hence many alternative models have been proposed

[5].

The simplest correction scheme is to replace the

Einstein-Hilbert action with a general function of the

Ricci scalar Rto obtain f(R) gravitation[6,7], where

Ris treated as a redundant degree of freedom. As a fa-

mous example, the Starobinsky theory[8] is used to deal

with many questions about stars. f(R) = R1+εgravity

is applied as a correction close to GR in the study of

compact astronomical objects[9].

Further, coupling in the trace T=gabTab of the matter

energy tensor gives the f(R, T ) theory[10], which further

explores the role of matter in the gravitational ﬁeld. The

f(R, T ) theory has special signiﬁcance for the static equi-

librium structure of compact stars, see [11–19]. This has

been an active research area in the last few years, and in

the present work, we adopt a similar path.

White dwarfs (WD), quark stars (QS), neutron stars

(NS), etc. are the compact astronomical objects of inter-

est. A white dwarf is a stellar core remnant composed

mostly of electron-degenerate matter. White dwarfs are

thought to be the ﬁnal evolutionary state of stars whose

mass is not high enough to become a neutron star or black

hole. This includes over 97% of the other stars in the

Milky Way. Chandrasekhar has long established a mass

∗2010020129@hhu.edu.cn; College of Science, Hohai University,

Nanjing, People’s Republic of China

limit for WD[20] that its mass cannot exceed 1.44M.

Type Ia supernova (SNIa) explosions may occur when

WD masses exceed this limit, with equal brightness con-

sidered standard candles. However, the discovery of a

group of super-bright supernovae[21,22] has driven the

hypothesis that their ancestors were WDs with super-

Chandrasekhar masses. In the scope of GR and modi-

ﬁed gravity (MG), various possible WD structures with

Chandrasekhar equation of state (EoS)[23] have been ex-

tensively studied, including uniformly charged WD under

GR [24], WD under f(R, T ) gravity[11], charged WD un-

der GR [25], charged WD under f(R, T ) gravity [12], the

charge of the latter two are mainly concentrated on the

surface of stars.

The idea of stellar structure discussed in this paper

mainly comes from the studies of several authors for QS.

A quark star is a hypothetical type of compact, exotic

star, where extremely high core temperature and pres-

sure have forced nuclear particles to form quark matter,

a continuous state of matter consisting of free quarks.

For QS, more abundant structures were considered[13–

19,26], including: surface or internal charge, isotropic

or anisotropic pressure, GR or f(R, T ) gravity. Usually,

when dealing with QS, the MIT Bag Model EoS is cho-

sen, although some authors [19] are happy to use EoS

with O(m4

s) correction term to obtain a more accurate

case. Although the structure of WD is solved in this pa-

per, a charge density proportional to the energy density

and anisotropic pressure distribution are introduced.

This paper is organized as follows. Following this intro-

duction(I), in Sec. II, we brieﬂy review f(R, T ) gravity

and derive the ﬁeld equations in the presence of mat-

ter and electromagnetic ﬁelds. In Sec. III, the modi-

ﬁed Tolman-Oppenheimer-Volkoﬀ (TOV) equation(Sec.

III A) is obtained by considering the spherical symmetry

metric. The selected Chandrasekhar EoS(Sec. III B) and

charge distribution(Sec. III C) are also presented in two

subsections in Sec. III. In the Sec. IV, the numerical so-

lution of the equation is obtained. The results are brieﬂy

analyzed. Finally, in Sec. V, conclusions are drawn from

the results.

arXiv:2210.01574v1 [gr-qc] 30 Sep 2022

II. f(R, T )GRAVITY FORMALISM

Proposed by Harko et al.[10], f(R, T ) gravity is a gen-

eralization based on f(R) [6,7]. where the action of the

gravitational ﬁeld contains an arbitrary function with re-

spect to the Ricci scalar Rand the trace Tof the energy-

momentum tensor. When there is a matter ﬁeld and a

gravitational ﬁeld, consider the following actions

S=Zd4x√−g[Lg+Lm+Le]

=Zd4x√−g1

16πf(R, T ) + Lm+Le,

(1)

where gis the determinant of the space-time metric gab

and Lmis the Lagrangian density of the matter ﬁeld.

Leis the Lagrangian density of the electromagnetic ﬁeld

with the following form

Le=jaAa−1

16πFabFab,(2)

where ja=ρchuais the four-current density with ρch

being the electric charge density and uabeing the four-

velocity, respectively. Aais the electromagnetic four-

potential, and Fab =∇aAb− ∇bAais the electromag-

netic ﬁeld strength tensor naturally. Additionally, ∇ais

a covariant derivative operator adapted to the metric gab.

The ﬁeld equation can be obtained by the variation

of action respect to metric. Firstly, the action of elec-

tromagnetic ﬁeld and matter are varied to obtain the

electromagnetic energy-momentum tensor and the mat-

ter energy-momentum tensor, respectively

Eab =−2

√−g

δSe

δgab =1

4πgcdFacFbd −1

4gabFcdFcd,(3)

Mab =−2

√−g

δSm

δgab =gabLm−2∂Lm

∂gab .(4)

The total energy-momentum tensor is the sum of the two,

namely Tab =Mab +Eab. Considering that the electro-

magnetic energy-momentum tensor is traceless, further

T≡gabTab =gabMab ≡ M.

The variation of the action of the gravitational ﬁeld is

continued with a view to obtaining the equations of the

gravitational ﬁeld.

16π

√−g

δSg

δgab =fRRab −1

2gabf+ (gab− ∇a∇b)fR

+fT(Mab + Θab),

(5)

where fR≡∂f(R, T )/∂R,fT≡∂f(R, T )/∂T ,=

∇a∇ais the d’Alembert operator, and

Θab ≡gcd δMcd

δgab =−2Mab +gabLm−2gcd ∂2Lm

∂gab∂gcd .(6)

According to the variational principle δS = 0, we can

get the equation of motion of the gravitational ﬁeld

fRRab −1

2gabf+ (gab− ∇a∇b)fR

= 8π(Mab +Eab)−fT(Mab + Θab).

(7)

As in the case of f(R) theory, the Ricci scalar is treated as

a redundant degree of freedom whose equation of motion

can be obtained by taking the trace of the tensor equation

of motion

3fR+RfR−2f= 8πM − fT(M+ Θ) .(8)

Similar to the case in GR, taking the covariant derivative

of the tensor equation of motion eq.7yields an equation

for the energy-momentum tensor

∇aMab =fT

8π−fT

[(Mab + Θab)∇aln fT+∇aΘab

−1

2gab∇aT−8π

fT∇aEab].

(9)

It can be veriﬁed that if f(R, T ) = Ris taken, the theory

will return to the standard case of GR: eq.7will return

to Einstein’s gravitational ﬁeld equation, while eq.9will

return to energy-momentum conservation.

Next, I will take f(R, T ) = R+ 2βT , following the

examples in [11–19,27–29].

III. STELLAR STRUCTURE EQUATIONS

A. modiﬁed TOV equations

In order to further expand the equation into a compo-

nent form, a static spherically symmetric space-time line

element is chosen

ds2=−e2ψ(r)dt2+ e2λ(r)dr2+r2dθ2+ sin2θdφ2.

(10)

At this point, the Maxwell equation obeyed by the elec-

tromagnetic ﬁeld can be written more explicitly

∇aFab =−4πjb,∇[aFbc]= 0.(11)

It has only two non-zero components, namely

F01 =−F10 =q(r)

r2e−ψ(r)−λ(r),(12)

where the charge function has the following form

q(r) = 4πZr

¯r2ρch(¯r)eλ(¯r)d¯r. (13)

where ρch is the charge density, see III C.

Following the approach in [14–18], we choose the

anisotropic matter energy-momentum density, namely

Mab = (ρ+pt)uaub+ptgab −σkakb,(14)

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Chargedanisotropicwhitedwarfsinf(R;T)gravityZheFeng(Dated:October5,2022)Inthecontextoff(R;T)=R+2Tgravity,whereRistheRicciscalarandTisthetraceoftheenergy-momentumtensor,theequilibriumstructureofchargedanisotropicwhitedwarfs(WDs)isstudied.Thestellarequationsforthegeneralcasearederivedandnumericalsol...

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Charged anisotropic white dwarfs in fR T gravity Zhe Feng Dated October 5 2022_2

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