# Models of Anisotropic Self-Gravitating Source in Einstein-Gauss-Bonnet Gravity

Authors: G. Abbas; M. Tahir
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

In this paper, we have studied gravitational collapse and expansion of nonstatic anisotropic fluid in5D Einstein-Gauss-Bonnet gravity. For this purpose, the field equations have been modeled and evaluated for the given source and geometry. The two metric functions have been expressed in terms of parametric form of third metric function. We have examined the range of parameter β (appearing in the form of metric functions) for which Θ, the expansion scalar, becoming positive/negative leads to expansion/collapse of the source. The trapped surface condition has been explored by using definition of Misner-Sharp mass and auxiliary solutions. The auxiliary solutions of the field equations involve a single function that generates two types of anisotropic solutions. Each solution can be represented in term of arbitrary function of time; this function has been chosen arbitrarily to fit the different astrophysical time profiles. The existing solutions forecast gravitational expansion and collapse depending on the choice of initial data. In this case, wall to wall collapse of spherical source has been investigated. The dynamics of the spherical source have been observed graphically with the effects of Gauss-Bonnet coupling term α in the case of collapse and expansion. The energy conditions are satisfied for the specific values of parameters in both solutions; this implies that the solutions are physically acceptable.


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## Body

## 1. Introduction

In the gravitational study of more than four dimensions, the unification problem of gravity with electromagnetism and other basic connections are discussed by [1–3]. The study on supergravity by Witten [4] has strongly sported the work on unification problem of gravity. The problem of gravitational unification study is entirely based on the string theory [5, 6]. The ten-dimensional gravity that arises from string theory contains a quadratic term in its action [7, 8]; in low energy limit. Zwiebach [9] described the ghost-free nontrivial gravitational interactions for greater than 4 dimensions in the study of n-dimensional action in [7].In recent years, the higher-order gravity included higher derivative curvature terms, which is an interesting and developing study. The most widely studied theory in the higher curvature gravities is known as Einstein-Gauss-Bonnet (EGB) theory of gravity. The EGB theory of gravity is a special case of the Lovelocks gravity. The Lagrangian of EGB gravity was obtained from the first three terms of Lagrangian of the Lovelock theory. Pedro [10] described that the 2nd Euler density is the Gauss-Bonnet combination and is topological invariant in four dimensions. He also pointed out that, to make dynamical GB combination in four-dimensional theory, couple it to dynamical scalar field. The dynamical stability and adiabatic and anisotropic fluid collapse of stars in 5D EGB theory of gravity have been studied in [11, 12]. Gross and Sloan [13, 14] investigated that EGB theory of gravity occurs in the low energy effectual action [8] of super heterotic sting theory. The exact black hole (BH) solution in greater than or equal to five-dimensional gravitational theories is studied by [8]. Dadhich [15] examined in Newtonian theory that gravity is independent of space-time dimensions with constant density of static sphere, and this result is valid for Einstein and higher-order EGB theory of gravity. The conditions for universality of Schwarzschild interior solution describe sphere with uniform density for the dimensions greater than or equal to four. The authors in [16–20] described that the existence of EGB term in the string theory leads to singularity-free solutions in cosmology and hairy black holes. The solutions described in [8] are generalization of n-dimensional spherically symmetric BH solution determined in [21, 22]. In literatures [23–25], the authors examined the other spherical symmetric BH solutions in GB theory of gravity. Cai [26, 27] discussed the structure of topologically nontrivial black holes. The effects of GB term on the Vaidya solutions have been studied in [28–32]. Wheeler [23] discussed the spherical symmetric BH solutions and their physical properties. The GB term has no effect on the existence of local naked singularity, but the strength of the curvature is affected.In the composition of stars, the nuclear matter is enclosed inside the stars. The stars are gravitated and attracted continuously to the direction of their center because of gravitational interaction of their matter particles; this phenomenon is known as gravitational contraction of stars, which leads to gravitational collapse. It is studied in [33] that, during the gravitational collapse, the space-time singularities are generated. When the massive stars collapse due to their own gravity, the end state of this collapse may be a neutron star, white dwarf, a BH, or a naked singularity [34]. The spherical symmetric collapse of perfect fluid is discussed in [35, 36]. The dissipative and viscous fluid gravitational collapse in GR is discussed in literatures [37–48].Zubair et al. [49] studied a dynamical stability of cylindrical symmetric collapse of sphere filled with locally anisotropic fluid in f(R,T) theory of gravity. A lot of literatures are available about the gravitational collapse and BH in GB theory of gravity [50–58]. If the f(R,T) theory of gravity obeys the stress-energy tensor conservation, then unknown f(R,T) function can be obtained in the closed form [59]. Abbas and Riaz [60] determined the exact solution of nonstatic anisotropic gravitational fluid in f(R,T) theory of gravity, which may lead to collapse and expansion of the star. Sharif and Aisha [61] studied the models for collapse and expansion of charged self-gravitating objects in f(R,T) theory of gravity.Oppenheimer and Snyder [62] observed the gravitational contraction of inhomogeneous spherically symmetric dust collapse, and according to this end state of the gravitational collapse is BH. Markovic and Shapiro [63] studied this work for positive cosmological constant, and Lake [64] discussed this for negative as well as positive cosmological constant. Sharif and Abbas [65] studied the gravitational perfect fluid charged collapse with cosmological constant in the Friedmann universe model with weak electromagnetic field. Sharif and Ahmad [66–69] worked on the spherical symmetric gravitational collapse of perfect fluid withe positive cosmological constant. Sharif and Abbas [70] discussed the 5-dimenssional symmetric spherical gravitational collapse with positive cosmological constant in the existence of an electromagnetic field. Abbas and Zubair [71] investigated the dynamical anisotropic gravitational collapse in EGB theory of gravity. A homogeneous spherical cloud collapse with zero rotation and disappearing internal pressure leads to a singularity covered by an event horizon [72]. Jhingan and Ghosh [73] discussed the five or greater than five-dimensional gravitational inhomogeneous dust collapse in EGB theory of gravity. They investigated the exact solution in closed form. Sunil et. al [74] investigated the exact solution to the field equations for five-dimensional spherical symmetric and static distribution of the prefect fluid in EGB modified gravity. Abbas and Tahir [75] studied the exact solution of motion during gravitational collapse of prefect fluid in EGB theory of gravity. It should be observed in [73, 75–78] that coupling term α changes the structure of the singularities. Glass [79] generated the collapsing and expansion solutions of anisotropic fluid of Einstein field equations. In this paper, we extended the work of Glass [79] to model the solutions for collapse and expansion of anisotropic fluid in the EGB theory of gravity. The paper has been arranged as follows.In Section2, we present interior matter distribution and field equations. We discuss the generation of solutions for the gravitational collapse and expansion in Section 3. The last section presents the summary of the results of this paper.


## 2. Matter Distribution Inside the Star and the Field Equations

We start with the5D action given as(1)S=∫d5x-g12κ52R+αLGB+Smatter,where κ5≡8πG5 is gravitational constant and R is a Ricci scalar in 5D, and α is known as the coupling constant of the GB term. The GB Lagrangian is given as follows:(2)LGB=R2-4RabRab+RabcdRabcd.This kind of action is discussed in the low energy limit of supersting theory [13, 14]. In this paper, we consider only the case with α>0. The action in (1) gives the following field equations:(3)Gab+αHab=Tab,where(4)Gab=Rab-12gabRis the Einstein tensor and(5)Hab=2RRab-2RaαRbα-2RαβRaαbβ+RaαβγRbαβγ-12gabLGBis the Lanczos tensor. We want to find the solution for collapse and expansion of a spherical anisotropic fluid in 5D-EGB gravity.(6)ds2=-At,r2dt2+Bt,r2dr2+Rt,r2dΩ32,where dΩ32=(dθ2+sin2θ(dϕ2+sin2ϕdψ2)) is a metric on three spheres and R=R(t,r)≥0, A=A(t,r), and B=B(t,r). The energy-momentum tensor for anisotropic fluid is(7)Tab=μ+p⊥VaVb+p⊥gab+pr-p⊥χaχb,where μ, pr, p⊥, χa, and Va are the energy density, radial pressure, tangential pressure, unit four vector along the radial direction, and four velocity of the fluid, respectively. For the metric in (6), Va and χa are given by [60] as follows:(8)Va=A-1δ0a,χa=B-1δ1awhich satisfy(9)VaVa=-1,χaχa=1,χaVa=0.The expansion scalar is(10)Θ=1AB˙B+3R˙R.The dimensionless measure of anisotropy is defined as(11)▵a=pr-p⊥pr.Equation (3) for the metric in (6) with the help of (7) is given as follows:(12)μ=12αA4B5R3R˙R˙A′2R-B2B˙R˙+A2BR′′-B′R′B2+A2B2B˙R′2-B2+2AA′B2RR˙B˙R′-BR˙′+A4B2-R′2BR′′-B′R′-3A2B3R2B3A2+R˙2+A2R′B′R-BR′+BRBB˙R˙-A2R′′,(13)pr=12αA5B4R3B2R˙2A˙R˙B2-A2A′2R+B2R¨+A2A2A′R′+B2A˙R˙-AR¨R′2-B2+B2A′R′R˙2+3A3B2R2A2A′R+AR′R′-AB2+B2A˙R˙-AR¨R-AR˙2,(14)p⊥=-4αA5B5R2B3B˙3BA˙R˙2+AA′2R˙-BB¨R˙+2B˙R¨+AB2R˙BA′′R˙+A′4B˙R′-2BR˙′-B′R˙+A˙B2-R′2B˙+2R˙B′R′-BR′′+A3BB˙2R′2-B3B¨+BR′B¨R′-2B′R¨-2B˙R˙′+B2R˙′2+2R¨R′′+A4A′′BB2-R′2-A′BBB′+2R′R′′-3B′R′2-1AB3R2R2AB¨-A˙B˙B2+A2A′B′-A′′B+2RABB˙R˙+BR¨B-AA′R′+AR′′+A3B′R′-B3A˙R˙+ABB2A2+R˙2-A2R′2,(15)0=-12αA4B4R3A′BR˙-ABR˙′+AB˙R′2A2A′B′-A′′BR2-B2ABR˙2+2A˙B˙R2+A3BR′2-B2+3ABRA′BR˙-ABR˙′+AB˙R′where the prime and dot denote the partial derivative with respect to r and t, respectively. The auxiliary solution of (15) is(16)A=R˙Rβ,B=Rβ.By Using the auxiliary solution in (16) in (10), the expansion scalar becomes(17)Θ=Rβ-13+β.For β>-3 and β<-3, we obtained expansion and collapse regions. The matter components from (12), (13), and (14) with the help of (16) are(18)
μ
=
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(19)
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(20)
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.The Misner-Sharp mass function m(t,r) is given as follows:(21)mt,r=n-22kn2Vn-2kRn-3k-gabR,aR,b+n-3n-4αk-gabR,aR,b2where comma represents partial differentiation and the surface of (n-2) dimensional unit space is represented by Vn-2k. By using Vn-21=(2π(n-1)/2)/Γ((n-1)2), k=1 with n=5, and (16) in (21), we get(22)mr,t=32R21-R′2R2β+R2β+2α1-R′2R2β+R2β2.The specific values of β and R(r,t) form an anisotropic configuration. When R′=R2β, there exist trapped surfaces at R=±2/3m-2α, provided that m≥3α. Therefore, in this case, R′=R2β is trapped surface condition.


## 3. Generating Solutions

For the different values ofβ, expansion scalar Θ<0 for collapse and Θ>0 for expansion solutions are discussed as follows.


## 4. Collapse withβ=-7/2

The expansion scalar must be negative in case of the collapse, so forβ<-3, from (17), we get Θ<0. By assuming β=-7/2 and the trapped condition R′=R2β and then integration, we obtain(23)Rtrap=8r+z1t1/8.It is to noted that (23) is only valid for the trapped surface, and it can not be used everywhere. In order to discuss the solutions outside the trapped surface, we follow Glass [79] and take the value of R(r,t) as posivite scalar (k>1) multiple of Rtrap, such that R(r,t)>Rtrap. Hence the convenient form of the areal radius R(r,t) for the solution outside the trapped surface is given by(24)R=k8r+z1t1/8,z11=1+t2.Thus (18)–(20) with (24) and β=-7/2 yield the following set of equations:(25)μ=-21αk188r+z19/4-2+k711-2k16k9-28r+z17/81-k25+32k98r+z19/85k16-1+2k78r+z17/8,(26)pr=-42αk188r+z19/4-1+k79-k16k9-8r+z17/81-k25-31024k28r+z15/4k9896k7z˙1+8r+z11/81024+96z˙12-k8z˙1128+7z˙12+128r+16z18r+z11/8k8z˙1-8k9z¨1-64,(27)p⊥=αk188r+z19/4210-k7375-165k16k9-8r+z17/8112-67k25+14k98r+z19/860-k7315k9-48r+z17/8.In this case, (22) is reduced to the following form:(28)mr,t=32k28r+z11/4-k118r+z15/8+8r+z11/8k5+2α1-k98r+z17/8+1k78r+z17/8.Equation (11), with the help of (26) and (27), becomes(29)Δa=1-C1+C2C3+C4where C1=45α/8r+z111/8, C2=1/8r+z11/845/48r+z1+1, C3=-294α8r+z1-9/4, and C4 = -3/1024[896z˙1/8r+z15/4 + 1/8r+z19/81024-128z˙1+96z˙12-7z˙13 + 128r+16z1/8r+z15/48r+z11/8z˙1-8z¨1-64].The energy conditions for the curvature matter coupled gravity are(i)
Null energy condition:μ+pr≥0; μ+1/2pt≥0(ii)
Weak energy condition:μ≥0, μ+pr≥0, and μ+1/2pt≥0(iii)
Strong energy condition:μ+1/4pr+1/2pt≥0In this case, we analyzed our results for time profilez1=1+t2, k=2.5, and the different values of α. For β=-7/2, we obtained Θ<0; the energy density is decreasing with respect to radius r and time t and remains positive for different values of coupling constant α as shown in Figure 1. As the density is decreasing, spherical object goes to collapse outward the point. It is observed in Figure 2 that the radial pressure is increasing outward the center, and it is also noted in Figure 3 that the tangential pressure is increasing with respect to radius r and time t at various values of α. Due to this increase of radial and tangential pressure on the surface, the sphere loses the equilibrium state, which may cause the collapse of the sphere outward the center. Figure 4 shows that the mass is decreasing function of r and t during the collapse at different values of α. The anisotropy is directed outward when pt<pr; this implies that Δa>0 and it is directed inward when pr<pt; this implies that Δa<0. In this case Δa>0 for increasing r and t at various values of α as shown in Figure 5. This represents that the anisotropy force allows the formation of more massive object and has attractive force for Δa<0 near the center. In the present case, the GB coupling term α affects the anisotropy and the homogeneity of the collapsing sphere. All the energy conditions are plotted in Figure 6; these plots represent that all the energy conditions are satisfied for considered parameters in collapse solutions.Figure 1
Plot of density alongr and t for k=2.5 and the different values of α.Figure 2
Behavior of radial pressure alongr and t for k=2.5 and the different values of α.Figure 3
Tangential pressure alongr and t for k=2.5 and the different values of α.Figure 4
Mass plot alongr and t for k=2.5 and the different values of α.Figure 5
Plot of anisotropy parameter alongr and t for k=2.5 and the different values of α.Figure 6
Plot of energy conditions alongr and t for k=2.5.


## 5. Expansion withβ=-5/2

In this case, the expansion scalarΘ must be positive. When β>-3, from (17), the expansion scalar Θ is positive. We assume that(30)R=r2+r02-1+z2t,where r0 is constant and we take z2=z2(t)=1+t2.Forβ=-5/2, (18), (19), and (20) take the forms(31)μ=α30R91R5+1+6R5R′2+2RR′′1-R5R′2+1R312R′′-75R′2R-12R′˙R2R˙1R˙+5R′R-321R23R5-2+R37R′2+2RR′′,(32)pr=-6α2R′˙R2R˙2R′˙R˙+9R′R+RR′2RR′˙R˙+5R˙3R′R5R′2-1+5R˙3R14R55R5R′2-1-1+342R3R′2RR′˙R˙+7R′-R¨R6R˙25R7/2+2-1R225R17/2+4R5+4,(33)p⊥=α1R4110R10+60R5+21R′2+RR′269R5R′2-19+10R′′1R3+R23R5R′2-1+2R2R˙R˙′′R5R5R′2-1-1+R′R˙′RR54R6R′′+25R5R′2-15-5+1R26R5+1-92R3RR′′+72R′2-R4R˙RR˙′′-192R′R˙′.Assuming that F(r,t)=1+z2(t)(r2+r02) and R=F/(r2+r02), (31), (32), and (33) in this case are(34)μ=α10r2+r02F91+r2+r02F5+8F2r2+r0254r2F5r2+r029-1F5r2+r02-z2r2+r02r02-3r2+4r220Fr2+r025-25F4-40r2F6r2+r0214-85r2+r02F2+8z2F3r2+r02r02-3r2-32r2+r0223r2+r025F7-2F2+4F3r2+r026r02-3r2z2-F25r2+r02r2+r02+7r2r2+r02,(35)pr=6α5r2+r02F91+r2+r02F5-4r225F4-5Fr2+r0251-4r2F5r2+r029+34-2z2¨z2˙r2+r02F19/25+2Fr2+r027/2+56r2F3r2+r027-25r2+r02F21/2-4r2+r02F7-4r2+r02F2,(36)p⊥=α60r2+r02F9+110r2+r028F14+10F2r2+r025-10F3-60r2F7r2+r02142F5r2+r02-2z2r2+r02r02-3r2+69-16F216r4F6r2+r0214+r2Fr2+r02564F2-76+r2F284F2+64+r2+r02F2+6r2+r02F7+F3r2+r027r216F2-63+9F22F5r2+r02-2z2r2+r02r02-3r2.The mass function (22) in this case reduces to(37)mr,t=32F2r2+r0221-F′r2+r02-2rFr2+r0222Fr2+r025+r2+r025F5+2α1-F′r2+r02-2rFr2+r0222Fr2+r025+r2+r025F52.With the help of (35) and (36), (11) takes the form(38)Δa=1-E1+E2E3+E4where E1 = α[60r2+r02/F9+110r2+r028/F14 + (10F2/r2+r025 − 10/F3 − 60r2F7/r2+r0214)(2F5r2+r02 − 2z2r2+r02r02-3r2) + 69-16F216r4F6/r2+r0214 + r2F/r2+r02564F2-76+r2/F284/F2+64], E2 = r2+r02/F2 + 6r2+r02/F7 + F3/r2+r027(r216F2-63 + 9F/2(2F5r2+r02 − 2z2r2+r02r02-3r2)), E3 = 6α[5r2+r02/F91+r2+r02/F5 − 4r225/F4-5F/r2+r0251-4r2F5/r2+r029], and E4 = 3/4[-2z2¨/z2˙r2+r02/F19/25+2F/r2+r027/2 + 56r2F3/r2+r027 − 25r2+r02/F21/2 − 4r2+r02/F7-4r2+r02/F2].In this case, we taker0=0.5, z2=1+t2, and the various values of α and analyze our results. For β=-5/2, we obtained Θ>0; Figure 7 shows that, for the variation of α, the energy density increases for the time profile. The radial pressure is initially maximum and then decreases along t; the tangential pressure is decreasing near the center along t at different values of α as shown in Figures 8 and 9. The mass of the sphere is increasing along t and the different values of α as shown in Figure 10. In this case, pt>pr; this implies that Δa>0. The anisotropy parameter increases in this case for the different values of α as shown in Figure 11. It is shown that, firstly, the anisotropy decreases and then increases with radial increase of star for various values of α. The energy conditions for the expansion case are also satisfied as shown in Figure 12.Figure 7
Density behavior alongr and t for r0=0.5 and the different values of α.Figure 8
Radial pressure behavior alongr and t for r0=0.5 and the different values of α.Figure 9
Tangential pressure behavior alongr and t for r0=0.5 and the different values of α.Figure 10
Mass behavior alongr and t for r0=0.5 and the different values of α.Figure 11
Anisotropy parameter behavior forr0=0.5 and the different values of α.Figure 12
Plot of energy conditions alongr and t for r0=0.5 at.


## 6. Summary

There has been a great interest in studying the relativistic anisotropic systems due to the existence of such systems in the astronomical objects. The exact solutions of anisotropic sphere are helpful to determine the anisotropy of the universe during any era. Barrow and Maartens [80] studied the effects of anisotropy on the late time expansion of inhomogeneous universe. Mahmood et al. [81] modeled the exact solution for the gravitational collapse and expansion of charged anisotropic cylindrical source.The EGB gravity theory is the low energy limits of super-symmetric string theory of gravity [82]. The gravitational collapse is a highly dissipative process, in which a big amount of energy is released [37]. The dynamic of an anisotropic fluid collapse is observed in 5-dimensional Einstein-Gauss-Bonnet gravity [71]. Collin [83] modeled the inhomogeneous cosmological nonstatic expanding solutions. For the suitable values of R(t,r) and β, Glass [79] observed anisotropic collapsing and expanding of spherical object. We have modeled the field equations of anisotropic fluid in 5D EGB gravity.The aim of this paper is to study the generating solutions for anisotropic spherical symmetric fluid in5D EGB theory of gravity. We have used auxiliary solution of one field equation to obtain the solutions for remaining field equations. The solution for expansion scalar Θ has been depending on the range of free constant β, for which Θ being positive or negative leads to expansion and collapse of the fluid. We use the condition R′=R2β in (22), which leads to two trapped horizons at R=±2/3m-2α, provided that m≥3α. The curvature singularity is hidden at the common center of the inner (R+) and outer (R-) horizon. The matter components like density, radial pressure, tangential pressure, anisotropic parameter, and mass functions have been determined in the 5D EGB theory of gravity. Equation (17) implies that, for β=-3, Θ=0, and for β<-3 the expansion scalar is negative and for β>-3 the expansion scalar is positive, which lead to bouncing, collapsing, and expanding, respectively. In other words, β>0 and β∈(-3,∞); and β<0 and β∈(-∞,-3). The solutions have been modeled by taking β=-7/2 and β=-5/2 for collapse and expansion of gravitating source, respectively.We assume that the areal radiusR(r,t)=k(8r+z(t))1/8 outside the sphere for k>1. It is interesting that, for k=1, R(r,t)=Rtrap and it is valid only for trapped surface, so we take k>1 for collapsing solutions outside the trapped surface. Further, k<0 should not be considered because this leads to the solutions corresponding to the inner surface of the trapped region, which is not the case of interest in the present discussion.The dynamics of the spherical fluid are discussed in both cases. The density of the matter is decreasing/increasing in collapse/expansion with the arbitrary choice of constant/parameter, time profile, and different values of GB termα. The radial pressure, tangential pressure, and mass have different behaviors in both cases. The anisotropy is increasing in both cases; in other words, this nonvanishing pressure anisotropy in both cases leads to collapse and expansion of the fluid. The energy conditions are satisfied for collapse and expansion; this shows that our solutions are physically acceptable.



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*Source: 7420546-2018-12-27.xml*