Neutron stars structure in the context of massive gravity
aa r X i v : . [ g r- q c ] J u l Neutron stars structure in the context of massive gravity
S. H. Hendi , ∗ , G. H. Bordbar , † , B. Eslam Panah , ‡ and S. Panahiyan , § Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O. Box 55134-441, Maragha, Iran Center for Excellence in Astronomy and Astrophysics (CEAA-RIAAM)-Maragha, P. O. Box 55134-441, Maragha, Iran ICRANet, Piazza della Repubblica 10, I-65122 Pescara, Italy Physics Department, Shahid Beheshti University, Tehran 19839, Iran
Motivated by the recent interests in spin − I. INTRODUCTION
The Einstein theory of gravity has been a pioneering tool for understanding and describing the gravitational systems.Most of the results and validations of this theory have been confirmed by observations done on solar system level. Inaddition, the results of LIGO proved the existence of gravitational wave which was one of the challenging predictionsof general relativity (GR) [1]. It is expected to observe the advantages of GR for beyond the Newtonian theory regimes(high curvature regimes) such as near the compact objects. The existence of compact objects has been confirmed inthe Einstein theory. Due to the physical properties of these objects, it is necessary to take the curvature of spacetimeinto account in order to have reliable predictions. Therefore, we employ the Einstein theory of gravity to study theneutron star.Einstein theory predicts the existence of massless spin-2 gravitons with two degrees of freedom as intermediateparticles for signaling the gravitational interactions. But, there have been several arguments regarding the possibilityof existence of massive gravitons. These arguments are supported by studies that are conducted on quantum levelof gravity and brane-world gravity. Especially problems such as the hierarchy problem and their brane-world gravitysolutions have expressed on the possibility of existence of massive spin-2 gravitons [2, 3]. Therefore, it is naturalto relax massless constraint and consider the modifications and generalizations of the general relativity to includemassive graviton. In this paper, we generalize the Einstein theory of gravity to include the massive graviton andinvestigate its effects on the hydrostatic equilibrium equation of a typical neutron star.The first attempt for constructing the massive gravity was done by Fierz and Pauli [4]. This theory has thespecific problem known as vDVZ (van Dam-Veltman-Zakharov) discontinuity which indicates that the propagatorsof massless and massive in the limit of m →
0, are not the same [5–7]. One of the resolutions of this problem wasVainshtein mechanism which requires the system to be considered in the nonlinear framework [8] (it is notable thatin nonlinear dRGT, there are also vacua that are free from vDVZ discontinuity [9]). Such generalization to nonlinearcase introduces a ghost into the theory which is known as Boulware-Deser ghost [10]. There are various ghost freescenarios for considering the massive gravity in the nonlinear framework. One of the interesting ghost free theories ofmassive gravity is known as dRGT theory which was developed by de Rham, Gabadadze and Tolley [11, 12]. In thistheory, a reference metric is employed to build massive terms [11–14]. These massive terms are inserted in the actionto provide massive gravitons. Cosmological results, black hole solutions and their thermodynamical properties in thismassive gravity are investigated by many authors [15–30]. In addition, Katsuragawa et al, studied the neutron starsin the context of dRGT theory and showed that, the massive gravity leads to small deviation from the GR [31].It is worthwhile to mention that the reference metric plays a key role for constructing the massive theory of gravity ∗ email address: [email protected] † email address: [email protected] ‡ email address: [email protected] § email address: [email protected] [32]. One of the modifications in reference metric was done by Vegh which introduced a new massive gravity [33].This new massive theory has specific applications in the gauge/gravity duality especially in lattice physics whichmotivate one to use it in other frameworks as well. This theory was employed in the context of black holes and it wasshown that geometrical and thermodynamical structures of the black holes will be modified and new phenomena werereported for massive black holes [34–42]. Here, we use this massive theory to conduct our studies in the properties ofneutron star.The structure of stars and their phenomenological properties are described with hydrostatic equilibrium equation(HEE). This equation is based on the fact that a typical star will be in equilibrium when there is a balance betweenthe internal pressure and the gravitational force. Historically speaking, the first HEE equation for GR was introducedand employed by Tolman, Oppenheimer and Volkoff (TOV) [43–45]. After that, a series of studies were dedicated toobtain HEE of neutron star [46–53]. In addition, the compact objects and their TOV equations were investigated inthe presence of different modified gravities such as; gravity’s rainbow [54], vector-tensor-Horndeski theory of gravity[55], dilaton gravity [56], F ( R ) and F ( G ) gravities [57–61] (see [62–81] for more details).According to recent studies on the neutron stars and observations of the interesting properties of them [82–101], wewant to investigate these stars in the context of massive gravity. In other words, our main motivation in this paperis studying the effects of considering the massive gravity on the structure of neutron stars. Previous studies in thecontext of other astrophysical objects have proven a wide variation in the properties of these objects comparing to themassless graviton case. Therefore, we are expecting to see the specific modifications in the properties of neutron staras well. Here, we would like to address how the structure of neutron star will be modified in the presence of massivegravity and which contributions this generalization has into properties of these objects. We regard the HEE equationin 4 and higher dimensions with a suitable equation of state (EoS), which satisfies stability, energy conditions and LeChatelier’s principle. We obtain the maximum mass and corresponding radius, Schwarzschild radius, compactness,gravitational redshift and dynamical stability of the neutron stars. We give details regarding the effects of massivegravity on these properties. These studies provide an insight into the structure of neutron stars and enable one tomake a comparison between the massive and massless gravity theories. Remembering that the neutron stars, similarto other massive objects, propagate the gravitational waves, one is urged to study the neutron stars in the presenceof massive gravity which is the aim of this paper.The outline of our paper is as follows. In Sec. II, we consider a spherical symmetric metric and obtain the modifiedTOV in Einstein-massive gravity in four dimensions. Next, we employ the specific many-body EoS and study itsproperties such as the Le Chatelier’s principle, stability and energy conditions. Then, considering the Einstein-massive gravity, we investigate the neutron star structure and obtain other properties of this star. In next section,we extract mass and radius of this star versus the Planck mass as a fundamental physical constant. Finally, we finishour paper with some closing remarks. II. MODIFIED TOV EQUATION IN THE MASSIVE GRAVITY
The action of Einstein-massive (EN-massive) gravity with the cosmological constant in d -dimensions is given by I = − π Z d d x √− g " R −
2Λ + m X i c i U i ( g, f ) + I matter , (1)where R and m are the Ricci scalar and the massive parameter, Λ is the negative cosmological constant, and f and g are a fixed symmetric tensor and metric tensor, respectively. In addition, c i ’s are constants and U i ’s are symmetricpolynomials of the eigenvalues of d × d matrix K µν = √ g µα f αν where they can be written in the following forms U = [ K ] , U = [ K ] − (cid:2) K (cid:3) , U = [ K ] − K ] (cid:2) K (cid:3) + 2 (cid:2) K (cid:3) , U = [ K ] − (cid:2) K (cid:3) [ K ] + 8 (cid:2) K (cid:3) [ K ] + 3 (cid:2) K (cid:3) − (cid:2) K (cid:3) . By variation of Eq. (1) with respect to the metric tensor g νµ , the equation of motion for EN-massive gravity can bewritten as G νµ + Λ g υµ + m χ υµ = K d T νµ , (2)where K d = πG d c , G d is d -dimensional gravitational constant, G µν is the Einstein tensor and c is the speed of lightin vacuum. Also, T νµ denotes the energy-momentum tensor which comes from the variation of I matter and χ µν is themassive term with the following explicit form χ µν = − c U g µν − K µν ) − c (cid:0) U g µν − U K µν + 2 K µν (cid:1) − c U g µν − U K µν + 6 U K µν − K µν ) − c (cid:0) U g µν − U K µν + 12 U K µν − U K µν + 24 K µν (cid:1) . (3) A. Modified TOV equation in (3+1)-dimensions
In this section, the static solutions of EN-massive gravity in (3 + 1)-dimensions are obtained. For this purpose, weconsider a spherical symmetric space-time in the following form g µν = diag (cid:0) f ( r ) , − g ( r ) − , − r , − r sin θ (cid:1) , (4)where f ( r ) and g ( r ) are unknown metric functions. Now, in order to obtain exact solutions, we should consider asuitable reference metric. Obeying the ansatz of Ref. [34], we consider the following relation for the reference metric f µν = diag (0 , , C r , C r sin θ ) , (5)in which C is a positive constant. Considering the metric ansatz (5), we can obtain the explicit forms of nonzero U i ’sas [34] U = 2 Cr , U = 2 C r . Here, we regard the neutron star as a perfect fluid with the following energy-momentum tensor T µν = (cid:0) c ρ + P (cid:1) U µ U ν − P g µν , (6)where P and ρ are the pressure and density of the fluid which are measured by the local observer, respectively, and U µ is the fluid four-velocity. Using Eqs. (2) and (6) with the metric introduced in Eq. (4), it is easy to obtain thecomponents of energy-momentum as T = ρc & T = T = T = − P. (7)In addition, taking into account Eqs. (4) and (7), it is straightforward to achieve the following nonzero componentsof field equation (2) Kc r ρ = Λ r + (1 − g ) − rg ′ − m C ( c r + c C ) , (8) Kf r P = − Λ r f − (1 − g ) f + rgf ′ + m f C ( c r + c C ) , (9)4 Kf rP = − rf + 2 ( gf ) ′ f − rgf ′ + r [ g ′ f ′ + 2 gf ′′ ] f + 2 m Cc f , (10)where K = πGc , and f , g , ρ and P are functions of r . Also, we note that the prime and double prime denote the firstand the second derivatives with respect to r , respectively.Using Eqs. (8)-(10) and after some calculations, we obtain dPdr + (cid:0) c ρ + P (cid:1) f ′ f = 0 . (11)In addition, for extracting g ( r ), we can use Eq. (8) which leads to g ( r ) = 1 + Λ3 r − m C (cid:16) c r c C (cid:17) − c KM ( r )4 πr , (12)in which M ( r ) = R πr ρ ( r ) dr . Now, we obtain f ′ from Eq. (9) and insert it with Eq. (12) in Eq. (11) to calculateHEE in the Einstein-massive gravity with the following form dPdr = (cid:0) c ρ + P (cid:1) h c KM ( r )2 + 2 πr (cid:0) + KP (cid:1) − m πr c C i r (cid:2) m πr c C + c KM ( r ) + 4 πr (cid:0) m c C − Λ3 r − (cid:1)(cid:3) , (13)which is modified TOV equation due to the presence of massive graviton. As one expects, for m = 0, Eq. (13) isreduced to the following TOV equation obtained in Einstein-Λ gravity [102, 103] dPdr = (cid:2) c GM ( r ) + r (cid:0) Λ c + 12 πGP (cid:1)(cid:3) c r [6 GM ( r ) − c r (Λ r + 3)] (cid:0) c ρ + P (cid:1) . (14)In addition, in the absence of both massive term and cosmological constant ( m = Λ = 0), Eq. (13) leads to theusual TOV equation of Einstein gravity (see [43–45] for more details). It is notable that, the generalization to higherdimensions is done in the appendix A.Before applying the mentioned gravitational framework on the neutron star structure, we should point out somecomments for the mentioned massive gravity. As we mentioned before, the massive gravity employed in this paperis essentially a dRGT like [32]. It was shown that all dRGT like theories of the massive gravity in d dimensionswith N scalar fields (St¨uckelberg fields) enjoy at most d ( d −
3) + N number of degrees of freedom. The referencemetric employed in this paper is spatial reference metric which in its simpler form in the appropriate orthonormalcoordinate, it will be (0 , , , m tt , m ij , etc. These are the mass fluctuations of the modes which are depending on the non-trivial contributionsof the St¨uckelberg function. In other words, such mass fluctuations are depending on the free parameters of themassive theory such as c and c , etc. So, it would be useful to calculate the physical mass of fluctuations and imposeappropriate conditions to avoid tachyon-like instabilities. Taking into account the point of Ref. [32], we can regardthat the mass parameter is of the order of the Hubble parameter today, and therefore, such an instability would notbe problematic. However, since the tachyon-like instabilities are very important in some gravitational framework,such as black holes, we will address such substantial point in an independent paper. III. STRUCTURE PROPERTIES OF NEUTRON STARA. Equation of state of neutron star matter
The interior region of a typical neutron star is a mixed soup of neutrons, protons, electrons and muons in chargeneutrality and beta equilibrium conditions (beta-stable matter) [105]. This balanced mixture is governed by unknownEoS. One of the EoS which could be employed to study the neutron star is the microscopic constrained variationalcalculations based on the cluster expansion. This EoS has been employed to study the structure of neutron starmatter before [56, 106]. Fundamentally, the mentioned model is based on two-nucleon potentials which are themodern Argonne AV18 [107] and charged dependent Reid-93 [108]. It is notable that this method requires no freeparameter, has a good convergence and is more accurate comparing to other semi-empirical parabolic approximationmethods. These advantages come from a microscopic computation of asymmetry energy which is carried on for theasymmetric nuclear matter calculations. The necessity of microscopic calculations with the modern nucleon-nucleonpotentials which is isospin projection ( T z ) dependent was pointed out in Ref. [109]. Here, we employ the lowest orderconstrained variational (LOCV) method with the AV18 potential [106] for obtaining the modern EoS for neutron starmatter and investigating some physical properties of neutron star structure.As we mentioned, the energy of the system under study is obtained by the LOCV method which is a fully self-consistent formalism. Through a normalization constraint, this method keeps the higher order terms as small aspossible [110]. In addition, this method has been employed to calculate the properties of neutron, nuclear and asym-metric nuclear matters at zero and finite temperatures [110–112]. The functional minimization procedure representsan enormous computational simplification over the unconstrained methods which attempt to go beyond the lowestorder.A trial many-body wave function is ψ = F φ, (15)where φ is the uncorrelated ground-state wave function of N independent neutrons, and F is a proper N -bodycorrelation function. Here, we apply Jastrow approximation [113] to replace F as F = S Y i>j f ( ij ) , (16)where S and f ( ij ) are a symmetrizing operator and the two-body correlation function, respectively. Besides, weconsider a cluster expansion of the energy functional up to the two-body term E ([ f ]) = 1 N h ψ | H | ψ ih ψ | ψ i = E + E , (17)in which ψ and H are wave function and Hamiltonian system, respectively. In other words, the energy per particleup to the two-body term is E ([ f ]) = E + E , (18)where E = P i =+ , − ~ k ( i )2 F m ρ ( i ) ρ and E = N P ij h ij | ν (12) | ij − ji i are one-body and two-body energy terms, re-spectively. It is notable that, k ( i ) F = (cid:0) π ρ ( i ) (cid:1) / is the Fermi momentum of a neutron with spin projection i . Theoperator ν (12) is nuclear potential and it has been given in Ref. [114] (see Refs. [109] for more details). The behaviorof obtained EoS of neutron star matter is shown in Fig. 1. We extract the mathematical forms for the EoS presentedin Fig. 1 as P = X i =1 A i ρ − i , (19)in which A i are A = − . × − , A = 3 . × − , A = − . × − , A = 3 . × − , A = − . × , A = 2 . × , A = − . × . In order to investigate the properties of such EoS with more details, we study Le Chatelier’s principle condition inthe following subsection.
1. Le Chatelier’s principle
The matter of star satisfies dP/dρ ≥ ≤ v = (cid:16) dPdρ (cid:17) ≤ c ) and energy conditions for this EoS are investigated in Ref. [54], and it is shownthat this EoS satisfied these conditions. FIG. 1: Equation of state of neutron star matter (pressure, P (10 erg/ cm ) versus density, ρ (10 g/ cm )). B. Mass-radius relation and other properties of neutron star in massive gravity
Considering the maximum gravitational mass of a neutron star for dynamical stability against gravitational collapseinto a black hole, one is able to make differences between neutron star and black holes. In other words, there is acritical maximum mass for the massive object in which for masses larger than the maximum value, the massive objectbecomes a black hole [105]. The value of maximum mass originated from the nucleons degeneracy pressure is evidentlythe possible maximum mass of neutron star. Therefore, obtaining the maximum gravitational mass of neutron staris of a great interest, and important in astrophysics. Unfortunately, the advanced observational technologies formeasuring the mass of neutron star by investigating the X-ray pulsars and X-ray bursters were not able to produceaccurate results. Nevertheless the measurements that are done with the binary radio pulsars [116–119], provided highlyaccurate results for the mass of neutron star. In Ref. [49], the Einstein gravity has been investigated, and maximummass of neutron star has been obtained using the modern equations of state of neutron star matter obtained from themicroscopic calculations. It was shown that the maximum mass of neutron star is about 1 . M ⊙ . In addition, theEoS with dilaton gravity was employed and the properties of neutron star were investigated [56]. The results showedthat by increasing the effects of dilaton gravity, the maximum mass of this star decreases ( M max ≤ . M ⊙ ). Here, weintend to obtain the maximum mass of neutron star by considering the obtained TOV equation for Einstein-massivegravity (Eq. (13)) and investigate the properties of neutron star.Now, by employing the EoS of neutron star matter presented in Fig. 1 and numerical approach for integrating theHEE obtained in Eq. (13), we can calculate the maximum mass and other properties of the neutron star. To do so,one can consider the boundary conditions P ( r = 0) = P c and m ( r = 0) = 0, and integrates Eq. (13) outwards to aradius r = R in which P vanishes for selecting a ρ c . This leads to the neutron star radius R and mass M = m ( R ).We present the results in different figures and tables (see Figs. 2 and 3, and tables I, II and III for more details).It is notable that, here, we ignore the effects of cosmological constant on the structure of neutron star. Forinvestigating its effects, we refer the interested reader to Ref. [103], in which, it was shown that, this constant has noeffect on the structure of this star when the cosmological constant is about 10 − m − . Now, we are in a position tostudy the properties of neutron star in massive gravity. First, we consider the mass of graviton about 1 . × − g ,which was obtained in Ref. [120]. Next, we use obtained results by A. W. Steiner et al [121] in which an empiricaldense matter EoS from a heterogeneous data set of six neutron stars was obtained. Their results showed that theradius of a neutron star must be in the range of R ≤ (11 ∼ km . In the present paper, we consider the maximumradius of neutron star in the range of R ≤ km and investigate the maximum mass for neutron star in the massivegravity by employing the modern EoS of neutron star matter derived from microscopic calculations. According tothe table I, considering the spacial values for the parameters of the modified TOV equation, the maximum mass ofneutron star is an increasing function of m c . Calculations show that the maximum mass of neutron star can bemore than 3 M ⊙ ( M max ≈ . M ⊙ ), whereas in the Einstein gravity and by using this EoS, the maximum mass wasin the range of M max ≤ . M ⊙ . A mass measurement for PSR J1614-2230 [122] showed that the mass for neutronstar was about 2 M ⊙ . In other words, our results cover the mass measurement of massive neutron star, and also,predict that the mass of neutron star in massive gravity can be in the range upper than 3 M ⊙ (see the table I for more TABLE I: Structure properties of neutron star in massive gravity for C = 2 and m c = 3 . × − . m c M max ( M ⊙ ) R ( km ) R Sch ( km ) ρ (10 g cm − ) σ (10 − ) z (10 − ) − . × − .
68 8 .
42 4 .
95 13 .
36 5 .
88 5 . − . × − .
68 8 .
42 4 .
95 13 .
36 5 .
88 5 . − . × − .
71 8 .
47 4 .
97 13 .
36 5 .
87 5 . − . × − .
84 8 .
68 5 .
10 13 .
36 5 .
87 5 . − . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . − . × − .
36 9 .
43 5 .
56 13 .
36 5 .
89 5 . − . × − .
73 9 .
89 5 .
83 13 .
40 5 .
89 5 . − . × − .
11 10 .
34 6 .
08 13 .
36 5 .
88 5 . − . × − .
52 10 .
76 6 .
35 13 .
41 5 .
90 5 . − . × − .
76 11 .
00 6 .
48 13 .
40 5 .
89 5 . m c = 3 . × − and m c = − . × − . C M max ( M ⊙ ) R ( km ) R Sch ( km ) ρ (10 g cm − ) σ (10 − ) z (10 − )0 .
01 1 .
68 8 .
42 4 .
20 13 .
36 5 .
00 5 . .
10 1 .
68 8 .
42 4 .
84 13 .
36 5 .
75 5 . .
50 1 .
70 8 .
45 4 .
96 13 .
37 5 .
87 5 . .
00 1 .
76 8 .
55 5 .
03 13 .
37 5 .
88 5 . .
00 2 .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . .
00 2 .
45 9 .
55 5 .
62 13 .
36 5 .
88 5 . .
00 3 .
11 10 .
34 6 .
08 13 .
36 5 .
88 5 . .
73 3 .
76 11 .
00 6 .
49 13 .
40 5 .
90 5 . C = 2 and m c = − . × − . m c M max ( M ⊙ ) R ( km ) R Sch ( km ) ρ (10 g cm − ) σ (10 − ) z (10 − )3 . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . − . × − .
01 8 .
94 5 .
36 13 .
36 5 .
99 5 . − . × − .
01 8 .
94 5 .
27 13 .
36 5 .
89 5 . − . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . − . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . − . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . − . × − .
01 8 .
94 5 .
26 13 .
36 5 .
88 5 . FIG. 2: Gravitational mass versus central mass density (radius), ρ c (10 gr/ cm ), for C = 2 and m c = 3 . × − .Left diagrams: gravitational mass versus central mass density for m c = − . × − (bold line), m c = − . × − (doted line), m c = − . × − (dashed line), m c = − . × − (dashed-dotted line) and m c = − . × − (continuous line).Right diagrams: gravitational mass versus radius for m c = − . × − (bold line), m c = − . × − (doted line), m c = − . × − (dashed line), m c = − . × − (dashed-dotted line) and m c = − . × − (continuous line). details). Also, by decreasing the value of m c less than (10 − ), the maximum mass and radius of neutron star arenot affected. In other words, considering the value of m c about − − , the maximum mass and radius of neutronstar reduce to the obtained results of massless Einstein gravity [49].On the other hand, the average density ( ρ ) of the neutron star calculated in the tables I, II and III shows that thecentral density may exceed a few times as 10 g cm − . In other words, it is larger than the normal nuclear density, ρ = 2 . × g cm − [107].For further investigation, we plot the mass of neutron star versus the central mass density ( ρ c ) in left panels ofFigs. 2 and 3. As one can see, the maximum mass of this star increases as m c increases. On the other hand, thevariation of maximum mass versus radius is also shown in right panels of Figs. 2 and 3.Now, we complete our discussion by considering the gravitational mass equal to 1 . × − g , with various valuesfor different parameters of modified TOV equation (see Eq. (14)) and obtain the maximum mass of neutron starin the massive gravity. The results are presented in the tables II and III. According to the table II, the maximummass of neutron star is an increasing function of C . It is notable that considering the values less than 0 . C , themaximum mass and corresponding radius of neutron star are not affected. In other words, these results reduce to theobtained results of the maximum mass and radius of neutron star in the Einstein gravity [49]. The variation of m c has very interesting effects. In this case, the maximum mass and radius of this star are constant and by variation of m c , these quantities are not affected (see the table III).For completeness, in the following, we investigate other properties of neutron star in this gravity such as theSchwarzschild radius, average density, compactness, the gravitational redshift and dynamical stability.
1. modified Schwarzschild Radius
It is clear that by applying the massive term to the Einstein gravity, the Schwarzschild radius is modified. Consid-ering Eq. (12) and using the horizon radius constraint ( g ( r ) = 0), we can obtain the Schwarzschild radius ( R Sch ) forthe EN-massive gravity. After some calculations, the Schwarzschild radius for this gravity without the cosmologicalconstant is obtained as R Sch = c (cid:0) − m c C (cid:1) m cc C − q c ( m c C − − m c CGMm cc C . (20)
FIG. 3: Gravitational mass versus central mass density (radius), ρ c (10 gr/ cm ), for m c = 3 . × − and m c = − . × − .Left diagrams: gravitational mass versus central mass density for C = 1 .
20 (bold line), C = 2 .
32 (doted line), C = 3 .
08 (dashedline), C = 3 .
65 (dashed-dotted line) and C = 3 .
85 (continuous line).Right diagrams: gravitational mass versus radius for C = 1 .
20 (bold line), C = 2 .
32 (doted line), C = 3 .
08 (dashed line), C = 3 .
65 (dashed-dotted line) and C = 3 .
85 (continuous line).
Using the series expansion of R Sch for the limit m →
0, we find that R Sch ≈ GMc + 2 GM C (cid:0) c c C + c GM (cid:1) c m + O ( m ) , (21)where the first term is the Schwarzschild radius in Einstein gravity [123], as expected, and the second term indicatesthe massive correction.In order to investigate the effects of various parameters on the modified Schwarzschild radius, one can look at thetables I, II and III. As one can see in tables I and II, by increasing the maximum mass and radius of neutron star,the Schwarzschild radius increases and these stars are out of the Schwarzschild radius. Also, considering the negativevalue of m c and increasing m c , the Schwarzschild radius increases (see table I). On the other hand, by increasing C , the Schwarzschild radius increases (see table II). In addition, considering the positive (negative) values of m c and increasing (decreasing) m c , the Schwarzschild radius almost does not change (see table III).
2. Average Density
Now, using the maximum mass and radius obtained in the massive gravity, we can calculate the average density ofneutron star in 4 − dimensions as ρ = 3 M πR , (22)where the results for variation of the massive parameters are presented in the tables I, II and III. Considering differentparameters introduced in this theory, the average density of this star is almost the same. In other words, by variationsof the different parameters, the average density remains fixed.
3. Compactness
The compactness of a spherical object may be defined by the ratio of Schwarzschild radius to radius of that object σ = R Sch
R , (23)0 r(cm) a d i a b a t i c i nd ex m c =−3.168 · −3 m c =−3.168 · −2 m c =−1.584 · −1 r(cm) a d i a b a t i c i nd ex C=1C=3C=4
FIG. 4: Adiabatic index versus radius for m c = 3 . × − .Left diagrams: for C = 2, m c = − . × − (doted line), m c = − . × − (continuous line) and m c = − . × − (dashed line).Right diagrams: for m c = − . × − , C = 1 . C = 3 . C = 4 . which may be indicated as the strength of gravity. For the massive gravity, we obtain the values of σ in the tables I, IIand III. For different values of m c and C , the results show that the strength of gravity is almost the same (see tablesI and II). But, for different values of m c , there are two interesting behaviors. A) considering the positive value of m c and increasing m c , the strength of gravity do not change. B) considering the negative values of m c andincreasing m c , the strength of gravity decreases and the strength of gravity is not affected for m c > − . × − (see the table III).
4. Gravitational redshift
Considering Eq. (12) for vanishing Λ and by using definition of the gravitational redshift, we obtain this quantityin the massive gravity as z = 1 q − m C (cid:0) c r + c C (cid:1) − GMc r − , (24)in which it reduces to the gravitational redshift in the Einstein gravity when m = 0. The results show that, thegravitational redshift of neutron star is almost independent of different parameters. The gravitational redshift of eachcompact object depends on its average density, so as one can see, the average density for these stars are almost thesame, therefore the gravitational redshift of them must be the same.
5. Dynamical Stability
The dynamical stability of stellar model against infinitesimal radial adiabatic perturbation was introduced byChandrasekhar in Ref. [124]. This stability condition was developed and applied to astrophysical cases by manyauthors [125–128]. The adiabatic index ( γ ) is defined as γ = ρc + Pc P dPdρ . (25)It is notable that, in order to have the dynamical stability, γ should be more than ( γ > = 1 .
33) everywherewithin the isotropic star. Therefore, we plot two diagrams related to γ versus radius for different values of m c and C in Fig. 4. As one can see, these stellar models in massive gravity are stable against the radial adiabatic infinitesimalperturbations.Also, we plot the density (pressure) versus distance from the center of neutron star. As one can see, the densityand pressure are maximum at the center and they decrease monotonically towards the boundary (see Figs 5 and 6).1 r(cm) d e n s i t y m c =−3.168 · −3 m c =−3.168 · −2 m c =−1.584 · −1 r(cm) d e n s i t y C=1C=3C=4
FIG. 5: Density versus radius for m c = 3 . × − .Left diagrams: for C = 2, m c = − . × − (doted line), m c = − . × − (continuous line) and m c = − . × − (dashed line).Right diagrams: for m c = − . × − , C = 1 . C = 3 . C = 4 . r(cm) p r ess u r e m c =−3.168 · −3 m c =−3.168 · −2 m c =−1.584 · −1 r(cm) p r ess u r e C=1C=3C=4
FIG. 6: Pressure versus radius for m c = 3 . × − .Left diagrams: for C = 2, m c = − . × − (doted line), m c = − . × − (continuous line) and m c = − . × − (dashed line).Right diagrams: for m c = − . × − , C = 1 . C = 3 . C = 4 . IV. NEUTRON STAR PROPERTIES VIA PLANCK MASS
Here, our aim is to obtain the mass of neutron star according to the Planck mass. The neutron stars are supportedagainst the gravitational force by degeneracy pressure of nucleons which is mainly related to the strong repulsiveinter-nucleons force. It is notable that the nucleon-nucleon interaction is so strong, and it is taken place through thepion exchange. Therefore, we can consider the average density of a neutron star in term of the nucleus density usingthe following form (see Refs. [54, 129] for more details) ρ nuc ∼ m p πλ π , (26)where m p and λ π = ~ m π c are, respectively, the proton mass and Compton wavelength ( m π is the pion mass). Now,we are going to use an analogy for obtaining a relation between the mass of neutron star in massive gravity and thePlanck mass. Using the equations (20) and (22) and by considering ρ nuc , one can derive the following corresponding2mass as 3 m p πλ π ∼ M πR Sch ,M ∼ m c Cm p m π G (cid:8)(cid:2) A + 8 cm p m π G (cid:3) s m c C ~ (cid:18) m c C ~ − A (cid:19) +24 m c C ~ (cid:2) m C A − cm p m π G (cid:3)(cid:9) , (27)where A = m C (cid:0) m c ~ − cc m p m π G (cid:1) , A = cGm p m π (cid:0) m c C − (cid:1) , A = cc m p m π G − m c ~ . Now, we use the relation between the proton (pion) mass and the Planck mass [129] to obtain the mass of neutronstar with respect to the Planck mass m p = m pl η p & m π = m pl η π ,M ∼ η p η π m m pl c CG (cid:26)(cid:20) m c C ~ η p η π − B (cid:21) s m c C ~ η p η π (cid:18) m c C ~ η p η π − B (cid:19) + m c C ~ (cid:20) − m c C ~ η p η π
24 + 3 B (cid:21)(cid:27) , (28)where B = cm pl G (cid:0) m c C − (cid:1) . It is notable that in the absence of massive term ( m → M ∼ (cid:18) ~ cG (cid:19) / m p (cid:18) η π η p (cid:19) / ∼ m pl η p (cid:18) η π η p (cid:19) / . (29)Now, we are in a position to obtain a constraint on the neutron star radius. Regarding the fact that the radius ofneutron star should be greater than the Schwarzschild radius, one can extract a limitation for the radius of neutronstar via the Planck mass as a fundamental physical constant. Using Eq. (28) with R NS > R Sch , we obtain R NS > m cc m pl C n √ cGm pl (cid:0) m c C − (cid:1) (cid:0) − m c C ~ η π (cid:2) m c C ~ η p η π − B (cid:3) × q ~ η p η π [ m c C ~ η p η π − B ] + (cid:0) m c C ~ η p η π (cid:1) − m c C ~ η p η π B +32 c G m pl (cid:0) m c C − (cid:1) (cid:17) / (cid:27) . (30)As a final comment, we should note that in the absence of the massive term, Eq. (30) reduces to R NS > G (cid:18) η p η π (cid:19) / (cid:18) η π cm pl (cid:19) (cid:18) ~ cG (cid:19) / , (31)which may indicate a minimum value for neutron star radius in usual general relativity.3 V. CLOSING REMARKS
In this paper, we considered the spherically symmetric metric and extracted a modified TOV equation of starsin the Einstein-massive gravity in 4 − dimensions. Then, we showed that for m → d − dimensions was also done (see appendix A). Furthermore, we have considered an EoS, which was derived frommicroscopic calculations and investigated Le Chatelier’s principle for the mentioned EoS. It was shown that thisequation is suitable for investigating the structure of neutron star.Considering the modified TOV obtained in this paper, the structure of neutron star was investigated. The resultsshowed that, the maximum mass of these stars increases when m c and C increase (the results represented in varioustables numerically). In addition, it was shown that by considering the constant values of C and m c , the maximummass of neutron star is independent of m c .Then, we showed that by increasing the maximum mass of neutron star, the radius and the Schwarzschild radiusincrease as well. It is notable that, by regarding massive graviton, the Schwarzschild radius is modified. In order toconduct more investigations, we plotted some diagrams related to the mass-radius and mass-central mass density. Wefound that these figures are similar to the diagrams related to the mass-radius and mass-central mass density in usualGR. In addition, these diagrams confirmed the validity of obtained results in massive gravity.After that the adiabatic index was investigated. It was shown that this star is dynamically stable. It is notable thatthe density and pressure are maximum at the center of the star and decrease monotonically towards the boundary.Jacoby et al [130] and Verbiest et al [131] used the detection of Shapiro delay to measure the masses of both theneutron star and its binary component. Also, using the same approach, the masses of compact objects were obtainedfor Vela X-1 (about 1 . M ⊙ ) [132], PSR J1614-2230 (about 1 . M ⊙ ) [122], PSR J0348+0432 (about 2 . M ⊙ ) [133],4U 1700-377 (about 2 . M ⊙ ) [134] and J1748-2021B (about 2 . M ⊙ ) [135]. It is notable that, in this paper, we showedthat the obtained maximum mass of neutron star in massive gravity can cover all the measured masses of pulsars andneutron stars. Also, we predicted the existence of possible mass of more than 3 M ⊙ .Briefly, we obtained the quite interesting results from massive gravity for the neutron star such as:I) Obtaining the modified TOV equation. II) Prediction of maximum mass for neutron star more than 3 M ⊙ ( M max ≈ . M ⊙ ), due to the existence of massive gravitons. III) Dynamically stable neutron star in the massivegravity. IV) EoS derived from microscopic calculations satisfied the energy, stability conditions and Le Chatelier’sprinciple, simultaneously. V) The Schwarzschild radius was modified in the presence of massive gravity. VI) Dueto the considering massive graviton, the gravitational redshift was modified. VII) The relations between the massand the radius of neutron star versus the Planck mass as a fundamental physical constant were extracted. VIII) Ourconsequences covered previous results and reduce to the Einstein gravity for massless graviton ( m = 0), as expected.Finally, it is notable that the investigation of other compact objects such as quark star and white dwarf in thecontext of massive gravity and its modified TOV equation are interesting subjects. Moreover, it is worth studyingthe effects of higher dimensions and other equation of states on the structure of compact objects. Also, anisotropiccompact objects [136–142], rotating, slowly rotating [143–149], rapidly rotating [150–155] neutron stars and obtainthe Buchdahl limit [156–162] in the context of massive gravity are interesting topics. Furthermore, regarding theconsiderable effects of free parameters on the existence of tachyon-like instabilities, it will be useful to address thementioned substantial instability. We leave these issues for the future works. VI. ACKNOWLEDGEMENTS
The author wish to thank Shiraz University Research Council. This work has been supported financially by theResearch Institute for Astronomy and Astrophysics of Maragha, Iran.
Appendix: Modified TOV equation in higher dimension
Here, we are interested in obtaining the modified TOV equation in Einstein-massive gravity in higher dimensions.So, we consider a spherical symmetric space-time in higher dimensions as ds = f ( r ) dt − g − ( r ) dr − r h ij dx i dx j , (32)where i, j = 1 , , , ..., d −
2, and also h ij dx i dx j is the line element of a ( d − − dimensional unit sphere h ij dx i dx j = dθ + d − X i =2 i − Y j =1 sin θ j dθ i . (33)4We also use the following ansatz for the reference metric which is introduced in Ref. [34] f µν = diag (0 , , C r h ij ) . (34)Using the mentioned information and ansatz, we can find the explicit functional forms of U i ’s as U = d Cr , U = d d C r , U = d d d C r , U = d d d d C r , (35)where we denoted d i = d − i . We can also obtain the nonzero components of the energy-momentum for d -dimensionalperfect fluid as T = ρc & T = T = T = ... = T d − d − = − P. (36)Considering the metric (32) with Eq. (36), we can find the components of Eq. (2) are calculated as K d c r ρ = Λ r + d d − g ) − d rg ′ − m d C r d (cid:2) c r d + d c Cr d + d d (cid:0) c C r d + d c C r d (cid:1)(cid:3) , (37) K d f r P = − Λ r f − (1 − g ) f + rgf ′ + m d C r d (cid:2) c r d + d c Cr d + d d (cid:0) c C r d + d c C r d (cid:1)(cid:3) , (38)4 K d f rP = − rf + 2 ( gf ) ′ f − rgf ′ + r [ g ′ f ′ + 2 gf ′′ ] f + 2 m d Cf r d (cid:2) c r d + d c Cr d + d d (cid:0) c C r d + c C r d (cid:1)(cid:3) . (39)Considering Eqs. (37)-(39) and after some calculations, one can find a relation which is the same as Eq. (11). Inaddition, we can obtain the functional form of g ( r ) by using Eq. (37) as g ( r ) = 1 + 2Λ d d r − c K d M ( r )Γ (cid:0) d (cid:1) d π d / r d − m (cid:20) c Cd r + c C + d c C r + d d c C r (cid:21) , (40)where M ( r ) = R π d / Γ( d / r d ρ ( r ) dr and Γ is the gamma function, which satisfies some conditions such as Γ(1 /
2) = √ π ,Γ(1) = 1 and Γ( x + 1) = x Γ( x ).Now, we obtain f ′ from Eq. (38) and insert it with Eq. (40) into Eq. (11) to obtain the following higher dimensionalHEE in Einstein-massive gravity dPdr = (cid:0) c ρ + P (cid:1) h d [ g ( r ) − − r d (Λ + K d P ) + i rg + m C gr d (cid:2)(cid:0) c r d + d c Cr d + d d C (cid:0) c r d + d c Cr d (cid:1)(cid:1)(cid:3)(cid:9) , (41)where g ( r ) is presented in Eq. (40).As a special case, it is notable that for m = 0, Eq. (41) reduces to the following d -dimensional TOV equationobtained in Einstein-Λ gravity [103] dPdr = (cid:0) c ρ + P (cid:1) h ( d − d − d − )4 π ( d − / c K d M ( r ) + r d − (cid:0) Λ + d − K d P (cid:1)i r h − Λ r d − + ( d − (cid:16) Γ( d − )2 π ( d − / c K d M ( r ) − d − r d − (cid:17)i . (42)Regarding a suitable EoS for higher dimensional spacetime, with the obtained modified d -dimensional TOV equa-tion, one can investigate the neutron stars in higher dimensional massive gravity. We leave the mentioned problemfor the future works.5 Appendix: B brief dimensional analysis of massive parameters and its values
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