1 Introduction

## Abstract

We investigate in this paper the structures of neutron stars under the strong magnetic field in the framework of gravity where denotes the scalar torsion. The TOV equations in this theory of gravity have been considered and numerical resolution of these equations has been performed within perturbative approach taking into account the equation of state of neutron dense matter in magnetic field. We simplify the problem by considering the very strong magnetic field which affects considerably the dense matter; and for quadratic and cubic corrections to Teleparallel term, one finds that the mass of neutron stars can increase for different values of the perturbation parameter. The deviation from Teleparallel for different values of magnetic field is found out and this feature is very appreciable in the case of cubic correction. Our results are related to the hadronic particles description with very small hyperon contributions and the mass-radius evolution is consistency with the observational data.

Strong Magnetic field effects on Neutron Stars within theory of gravity

M. G. Ganiou 1, C. Aïnamon 2, M. J. S. Houndjo 3 and J. Tossa 4

Institut de Mathématiques et de Sciences Physiques (IMSP)

01 BP 613, Porto-Novo, Bénin

Faculté des Sciences et Techniques de Natitingou - Université de Parakou - Bénin

Keywords: TOV, magnetic field , Teleparallel, mass-radius

## 1 Introduction

The explanation of the present state of our Universe and the several structures appearing in it, has opened the gate to various reflexions. Indeed, the current acceleration of the Universe is widely accepted through several independent observational data, as supernovae Ia [1]-[2], the large scale structure of the Universe [3], cosmic shear through gravitational weak lensing surveys [4] and the Lyman alpha forest absorption lines [5]. Due to the need to understand the acceleration of the Universe, various theories have been introduced. The well known standard equivalent theories, the General Relativity (GR) and the Teleparallel Theory, are the first theories used for explaining the acceleration of the Universe, including the existence of the dark energy as a new component of the Universe [6]. We recall here that the Teleparallel Theory is an idea of Einstein in 1928 in his attempt to unify gravity and electromagnetism. This theory nowadays, is formulated as higher gauge theory [7]; furthermore it has already been established that GR can in fact be re-cast into Teleparallel language [8], known as the Teleparallel Equivalent of General Relativity (TEGR). The second attempt in theories of gravity for explaining this state of Universe consists to modify the GR and the Teleparallel Theory. Due to the fact that the Teleparallel theory does not yield consistent results according to the observational data, the immediate attempt to modify this theory of gravity is replacing the Teleparallel term, the torsion scalar by an algebraic function based on the Weitzenböck connection instead of Levi-Civita connection used in GR.

Furthermore, structure formations have also been investigated in this modified theory of gravity. The neutron stars structures have already been executed in several theories of gravity mainly in [24, 25, 42] and [23]. Indeed the structures of neutron and quark Stars have recently been investigated by [23] through the deviation of the mass-radius diagrams for power-law and exponential type correction from the Teleparallel Theory gravity by considering some equations of state. In this paper, we focus our attention on the study of these stars in particular the neutron stars by investigating the effects of strong magnetic fields in the framework of gravity. Note that such studies have been performed in the so-called theory of gravity where interesting result have been found [24, 32, 41, 42, 43, 44]). We emphasize here that neutron stars are observed as several classes of self-gravitating systems [24]. The structure of neutron stars and the relation between the mass and the radius are determined by equations of state (EOS) of dense matter. Moreover, the maximal mass of neutron star is still an open question. Recent observations allow to estimate this limit at least as . It exits some well-measured limits according to several observational data on the pulsars and massive neutron stars: there are for pulsars PSR J [9], for pulsars J [10], for Vela X- [11], for U[12], for JB [46]. Several works with different approaches show that the mass limit of neutron star can increase: larger hyperon-vector couplings with stiffness of the EOS [14], model with chiral quark-meson coupling [15],

the quark-meson coupling model which naturally incorporates hyperons [16] and others. However, it has been shown in the literature that ultrastrong magnetic field affected considerably the equation of state (EOS) of neutron-star matter and consequently could lead to the increase of the maximal mass limit of these stars [40]. In order to explain this fact, various models of dense nuclear matter in presence of strong magnetic fields have been considered such as models with hyperons and quarks, and model with interacting gas. It has also been performed that the Landau quantization leads to the softening of the EoS for matter but account for contributions of magnetic field into pressure and density.

In this paper, we present the models of neutron star for simple EOS in the ultrastrong magnetic field via the gravity. In Ref.[40], it is clearly shown that the structure of a magnetized neutron star will be mostly affected by contributions from the magnetic field stress Mev f which greatly exceeds the matter pressure at all relevant densities for . Having in hand such fundamental result and through a perturbative approach, one obtains that the maximal mass limit of neutron stars can increase under an ultrastrong field magnetic. We consider two corrections of the Teleparallel Theory (quadratic and cubic corrections) and find that the parameter of correction plays an important role in this analysis. The deviation of mass-radius from Teleparallel Theory appears clearly in the case of cubic corrections. Our investigation through perturbative approach shows that in the case of null magnetic field, the mass of neutron stars can increase as it was predicted by [24]. The paper is organized as follows. In Sec.II, we make some reviews on the Teleparallel Theory via its equivalence to GR and its modified version, the so-called gravity. In Sec.III, we present the TOV equations and its perturbative versions. Strong magnetic field effect on the dense matter in the framework of the relativistic mean field has been described in the Sec.IV. Our main results have been presented in the Sec.V and the conclusions in Sec.VI.

## 2 From Teleparallel equivalent of General Relativity to f(T)

In Teleparall, equivalent of General Relativity, as in General Relativity, the structure of space-time is represented by a manifold . At every point in the local coordinate chart , the tangent space at is spanned by the coordinate vector fields . The corresponding dual space is denoted by and generated by . In addition, the tangent space is a dimension space described by the Lorentzian metric with signature . We will label all space-time coordinates by Greek subscripts that run from to , with denoting the time dimension, while all spatial coordinates will be labeled by , , , . . . that run from to .

Let us assume as an arbitrary base of . We can express the total derivative covariant as :

 ∇eA(x)=ΓBA(x)eB(x), (1)

with , the 1-form connection satisfying

 ΓBA(x)=⟨zB(x),∇eA(x)⟩=ΓBνA(x)dxν. (2)

In this last expression, represents the dual of where as stayed for base dual of local base . Then, it comes that and . In this paper, the capital letters , , ,… take the values running from to and , , … take the values running from to .

Assuming that the spacetime is parallelisable (i.e. there exist vector fields such that at any point the tangent vectors provide a basis of the tangent space at ), the mapping between the bases in coordinate frame to that of non-coordinate frame is an isomorphism . This also comes from the fact that all dimensional parallelisable manifold has a tangent space which can be decomposed in the direct product of and . Note that the frame field depends only on the affine structure of the manifold and hence a priori has any relation with the metric. Actually, we describe the frames with the metric by equipping the space with the minkowskian metric such as

 gμν=ηABeAμeBν. (3)

This also means that on the manifold , we arbitrary choose a frame at each point namely locally on some open chart . This approach can be extended by parallelisability. One can define the metric on the open chart by

 η(eA,eB)=ηAB, (4)

which shows the orthonormality on the tetrad. It is important to notify here that for the dimension gravity, the previous results hold according to the Steenrod theorem: all dimensional and orientable manifold is parallelisable [34, 35]. Thus a -dimensional spacetime with orientable spatial section is also parallelisable. To put it slightly differently, if any spatial slice of spacetime is an orientable -manifold (and as such parallelizable) with initial data that can be propagated uniquely in time, in the manner of decomposition of [36], then the entire spacetime is parallelizable. We also emphasize here from Geroch theorem [37] that a non-compact 4-dimensional Lorentzian manifold M admits a spin structure if and only if it is parallelisable.
Let introduce , the Weitzenböck connection [38] defined by

 \lx@stackrel∙∇XY:=(XYA)eA, (5)

with . The teleparallelism condition on the tetrads imposes which allows to define the coefficients of the connection as

 \lx@stackrel∙Γλμν=eλA∂νeAμ=−eAμ∂νeλA. (6)

The coefficients defined in (6) is for an unique connection [39] which, at each vector field , gives rise to parallelisation on , assuming of course that is parallelisable.

 \lx@stackrel∙T(X,Y)=\lx@stackrel∙∇XY−\lx@stackrel∙∇YX−[X,Y]. (7)

And then

 \lx@stackrel∙T(X,Y)=XAYB[eA,eB]. (8)

The equation (8) shows that the torsion tensor associated to Weitzenböck connection is generically non zero because in general, the basis vectors are not integrable. In the local coordinates, the torsion tensor components can be expressed by

 \lx@stackrel∙Tλμν=\lx@stackrel∙Γλνμ−\lx@stackrel∙Γλμν=eλA(∂μeAν−∂νeAμ)≠0. (9)

We also define here the curvature tensor associated to the Weitzenböck connection through the Riemann curvature by

 \lx@stackrel∙R(X,Y)Z=(\lx@stackrel∙∇X\lx@stackrel∙∇Y−\lx@stackrel∙∇Y\lx@stackrel∙∇X−\lx@stackrel∙∇[X,Y])Z. (10)

From the fact that , one gets

 \lx@stackrel∙R(eA,eB)eC=\lx@stackrel∙∇eA(\lx@stackrel∙∇eBeC)−\lx@stackrel∙∇eB(\lx@stackrel∙∇eAeC)−\lx@stackrel∙∇[eA,eB]eC=0. (11)

It follows that the curvature tensor associated to the Weitzenböck connection is equal to zero contrary to the case of the Levi-Civita connection in GR. We can thus say that the curvature is an intrinsic property of spacetime but it also only depends on the connection that is put on spacetime.

Another important tensor emerging from the use of the Weitzenböck connection is the contortion which shows the difference between the Weitzenböck connection and the Levi-Civita connection [18] according to

 \lx@stackrel∙Kλμν:=\lx@stackrel∙Γλμν−Γλμν=12(\lx@stackrel∙Tνλμ+\lx@stackrel∙Tμλν−\lx@stackrel∙Tλμν), (12)

or in the equivalent form as

 Missing or unrecognized delimiter for \Big (13)

where represents the coefficient of Levi-Civita connection. In order to show that the both connections make the same description of spacetime namely, prove that the Teleparallel Theory based on the Weitzenböck connection is equivalent to GR using the Levi-Civita connection, we recall here the action of Einstein-Hilbert GR as

 S=12κ2∫d4x√−gR, (14)

With , and is the Ricci scalar curvature coming from the Levi-Civita connection. The action (14) can be written with scalar torsion as

 S=12κ2∫d4xeT, (15)

where is equivalent to in GR and the scalar torsion defined by

 \lx@stackrel∙T:=\lx@stackrel∙Sβμν\lx@stackrel∙Tβμν, (16)

with

 \lx@stackrel∙Sβμν=12(\lx@stackrel∙Kμνβ+δμβ\lx@stackrel∙Tανα−δνβ\lx@stackrel∙Tαμα). (17)

By using the relations (12) and (17), the scalar torsion defined in (16) can be explicitly put in the following relations

 \lx@stackrel∙T=14\lx@stackrel∙Tβμν\lx@stackrel∙Tβμν+12\lx@stackrel∙Tβμν\lx@stackrel∙Tνμβ−\lx@stackrel∙Tβνβ\lx@stackrel∙Tμνμ. (18)

In the local base , the components of Riemann tensor given in (10) can be obtained in the framework of Levi-Civita connection by

 Rρμλν=∂λΓρμν−∂νΓρμλ+ΓρσλΓσμν−ΓρσνΓσμλ. (19)

By making using the relation (12) and after some contractions, one obtains the associated Ricci tensor as

 Rμν=∇ν\lx@stackrel∙Kρμρ−∇ρ\lx@stackrel∙Kρμν+\lx@stackrel∙Kρσν\lx@stackrel∙Kσμρ−\lx@stackrel∙Kρσρ\lx@stackrel∙Kσμν, (20)

where represents the covariant derivative in GR. By combining the relation (12) with the following relations and by taking into consideration
, we have

 Rμν=−∇ρ\lx@stackrel∙Sνρμ−gμν∇ρ\lx@stackrel∙Tσρσ−\lx@stackrel∙Sρσμ\lx@stackrel∙Kσρν, (21)

whose total contraction gives

 R=−\lx@stackrel∙T−2∇μ\lx@stackrel∙Tνμν. (22)

This last relation finds out the trivial equivalence of GR and Teleparallel under the actions defined in (14) and (15). It follows that these two relations are the Einstein-Hilbert action respectively in G and Teleparallel. In the rest of this work, we will leave the strackrel bullet () on all quantities resulting from the Weitzenböck connection.

The action of the modified versions of TEGR (Teleparallel Equivalent of GR) is obtained by substituting the scalar torsion of the action (15) by an arbitrary function of scalar torsion giving then to modified theory . This approach is similar in spirit to the generalisation of Ricci scalar curvature of Einstein-Hilbert action (14) ) by a function of this scalar leading to the well known theory. The action of theory can be defined as

 Missing or unrecognized delimiter for \right (23)

with the scalar torsion and the matter density Lagrangian which only depends from the tetrads without their derivative. The variation of the action with respect to tetrads gives [22, 23]

 1e∂μ(eSAμν)fT(T)−eAλTρμλSρμνfT(T)+SAμν∂μ(T)fTT(T)+14eAνf(T)=ΘνA, (24)

with , and the energy-momentum tensor naturally related to the matter.

## 3 The Generalised Tolman-Oppenheimer-Volkoff (TOV) equations in f(T)

TOV equations are often used in gravitational theories to study the structure of relativistic stars. Their determination requires the choice of a given theory of gravity, the precision on the type of spacetime (metric) and the matter source in the Universe. In the framework of our present work, based on the theory, we express the geometric Lagrangian density as: with a real constant such that matches to the Teleparallel theory. The equations of motion (24) become [23]

 [1eeAσ∂μ(eeAρSρνμ)+TρμσSρμν](1+ξgT(T))+ξSσνμ∂μ(T)gTT(T)+14δνσ(T+ξg(T))=κ22Θνσ, (25)

In order to obtain solutions that describe the stellar objects, we consider a spherical symmetric metric having two functions and depending on the radial coordinate by

 ds2=−e2φdt2+e2λdr2+r2(dθ2+sin2θdϕ2), (26)

which can be generated by the following table of tetrads

 eAμ=diag(eφ,eλ,r,rsinθ). (27)

Let recall here that the most successful metric in the stellar objets description is the one of Schwarschild [23, 24, 26, 27] which is a particular case of metric described in (26) with the following fundamental relation

 e−2λ=1−2GMc2r. (28)

We also consider that the Universe described by this metric has as content, an isotropic fluid such that the associated energy-momentum tensor can be expressed by

 Θνσ=−(ρ+P)uσuν+Pδνσ, (29)

with the four-velocity, and the energy density and the pressure of the matter respectively. Having the metric (26) and the energy-momentum (29) in hand, we establish the field equations from the general equation (25) via the following relations

 14[T+ξg(T)]−12[T−1r2−2e−2λr(φ′+λ′)](1+ξgT(T)) = −4πρc4, (30) −14[T+ξg(T)]−14[3T2−1r2+e−2λ[φ′′+(φ′+1r)(φ′−λ′)]](1+ξgT(T)) = 4πPc4, (31) cotθ2r2ξT′gTT(T) = 0. (32)

The “prime” in these relations means the derivative with respect to the radial coordinate . The scalar torsion can also be calculated as

 T=−2e−2λr2(2rφ′+1). (33)

However, from the conservation of the energy-momentum tensor, , the quantity can be expressed in term of the energy density and the pressure as

 φ′=−(ρ+P)−1dPdr. (34)

In order to obtain the TOV equations, we introduce the following dimensionless variables , , , , , , with . Indeed, the relation (35 becomes

 e−2λ=1−2mr, (35)

and

 Missing or unrecognized delimiter for \right (36)

The scalar torsion becomes (33)

 T(r)=−2r2gr2(1−2mr)[1−2rrg(ρ+p)−1dpdr]. (37)

By using the field equations (30), (31) and the relations (34), (36) and (37), one gets the TOV equations as

 −4πρ = −(1−2mr)[12r2gr2(1+2ξgT(T))+12rgr(1−r2g+2ξgT(T)−r2gξgT(T))(ρ+p)−1dpdr+ (38) +rg2r2(1+2ξgT(T))]−12r2(1+ξgT(T)−rg−rgξgT(T))+14ξg(T)+rgr2(1+ξgT(T))dmdr, 4πp = −(1−2mr){−12r2gr2+12rgr(ρ+p)−1dpdr+[−32r2gr2 (39) +12rgr(ρ+p)−1dpdr+r2gddr[(ρ+p)−1dpdr]−r2g(ρ+p)−2(dpdr)2 +rg2r(ρ+p)−1dpdr−r2g2r(ρ+p)−1dpdr+rg4r2](1+ξgT(T)4)}−(1+ξgT(T)4) ×[−r2g2r(ρ+p)−1dpdr+rg4r2]−14ξg(T)+14r2(1+ξgT(T))+ +r2gr[−(ρ+p)−1dpdr+12rgr](1+ξgT(T)4)dmdr, 4π(ρ−3p) = −(1−2mr){14r2gr2(17+13ξgT(T))+1rgr(−134−r2g−52ξgT(T)−r2gξgT(T)) (40) ×(ρ+p)−1dpdr+rg2r2(1+2ξgT(T))+[3r2g(ρ+p)−2(dpdr)2−3r2gddr[(ρ+p)−1dpdr] +3rg2r(rg−1)(ρ+p)−1dpdr−3rg4r2](1+ξgT(T)4)}−(11rg+44r2+3r2g2r(ρ+p)−1dpdr) ×(1+ξgT(T)4)+ξg(T)+[5rg2r2+3r2gr(ρ+p)−1dpdr](1+ξgT(T)4)dmdr.

The last equation, namely the equation (40), is the trace of the equations (30) and (31). We emphasize here that in order to obtain the stellar structures, it is important to require some asymptotic flatness as the radial coordinate evolves [23].

 limr→+∞T(r)=0,limr→+∞m(r)=cst (41)

The relations (38), (39) and (40) can be numerically solved by having the algebraic function . But in this work, one can use the perturbative approach. In this approach, the terms containing must be of first order of the small parameter .

#### Perturbation TOV equations

In order to solve numerically the equations (38), (39) and (40), one proceeds by a perturbative approach . In the framework of a perturbative solution, the density, pressure, mass and scalar torsion can be expanded as [41, 42, 43]

 p=p(0)+ξp(0),ρ=ρ(0)+ξρ(0), (42) m=m(0)+ξm(0),T=T(0)+ξT(0), (43)

where , , and satisfy the standard TOV equations (see [41] for the zeroth order mass). The scalar torsion at zeroth order can be specified as . Finally, perturbative TOV equations can be established by [43]

 rgr2dmdr = ξg(0)T(T)[12r2(1−rg)+1r2r2g(1−2m(0)r)−8πrgρ(0)]+12r2(1−rg)+ (44) +12r2r2g(1−2mr)−1rgr(1−r2g)(ρ+p)−1dpdr−14ξg(0)(T)−4πρ,
 (ρ+p)−1dpdr[−r2g4rdmdr+r2g8r−(1−2mr)(58rgr−3r2r2g+14r2g+rg8r−r2g8r)]− (45) −14(1−2mr)[r2gddr[(ρ+p)−1dpdr]−r2g(ρ+p)−2(dpdr)2+rg4r2−2r2r2g]+ +ξ[−14(1−2m(0)r)(−32r2r2g+rg4r2)+14r2+πρ(0)rg]g(0)T(T)−14ξg(0)(T)−rg16r2−14r2−4πp=0.

## 4 Strong magnetic field effect on the dense matter in the framework of the relativistic mean field: brief reviews

Our goal in this work consists to study the effect of strong magnetic field on the neutron stars in the framework of theory. In general, for nuclear matter containing baryon octet interacting with the following elements: a magnetic field with quadripotentiel and a scalar , isoscalar-vector and isovector-vector , meson fields and leptons , the Lagrangian density is expressed as [28]:

 L=∑b¯ψb[γμ(i∂μ−qbAμ−gωbωμ−12gρbτ.ρμ)]ψb+∑l¯ψl[γμ(i∂μ−qlAμ)−ml]ψl+ (46) +12((∂μσ)2−m2σσ2)−V(σ)−14FσλFσλ+12m2ωω2−14ωσλωσλ−14ρσλρσλ+12m2ρρ2μ,

where the strong interaction couplings , and depend on the density. Furthermore, the following relations , define the mesonic and electromagnetic field strength, respectively. We also assume the frozen-field configurations of electromagnetic field and neglect the anomalous magnetic moments of baryons and lepton because their effect is very small. The terms of strong interaction are parametrized by [24]

 gj(ρ)=ajgj01+bj(x+dj)21+cj(x+dj)2, (47)

where and , , , are constants (see [28] ). The correspondent isovecteur field is also given by

 gbρ=gb0exp[−aρ(x−1)]. (48)

The mean field approximation constrains the mesonic fields in the following equations [24]

 m2σσ+dVdσ=∑bgbρnsb,m2ωω0=∑bgbρnb,m2ωρ03=∑bgρbnb. (49)

With , , , the expectation values of meson fields in the uniform matter, and the associated scalar and vector baryon number densities, respectively. One recalls the very used scalar field potential

 V(σ)=13pmN(gσNσ)3+14q(gσNσ)4, (50)

where and are dimensionless constants and the values of nucleon-meson couplings and parameters and for different models are given in [24].

The energy spectra of charged and neutral leptons and baryons with effective mass results from Dirac equation and can be expressed by

 EbΥ=(k2z+m∗2b+2Υ|qb|B)12+gωbω0+τ3bgρbρ0+ΣR0, (51) Eb=(k2+m2b)12+gωbω0+ΣR0, (52) ElΥ=(k2z+m2l+2Υ|ql|B)12. (53)

Here, is isospin projection, the number represents the Landau levels of the fermions with charge and spin number . Furthermore, correspond to the spin degeneracy for lowest and others levels of Landau, respectively while the last term stays for the rearrangement self-energy term and is defined by

 ΣR0=−∂lngσN∂nm2σσ2+∂lngωN∂nm2ωω20+∂lngρN∂nm2ρρ20, (54)

with . We define the scalar densities for neutral and charged baryons [25] by

 ns,nb = m∗2b2π2(Ebfkbf−