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G. Abbas, M. Tahir, "Models of Anisotropic Self-Gravitating Source in Einstein-Gauss-Bonnet Gravity", Advances in High Energy Physics, vol. 2018, Article ID 7420546, 12 pages, 2018. https://doi.org/10.1155/2018/7420546
Models of Anisotropic Self-Gravitating Source in Einstein-Gauss-Bonnet Gravity
In this paper, we have studied gravitational collapse and expansion of nonstatic anisotropic fluid in 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 (appearing in the form of metric functions) for which , 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 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.
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  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  described the ghost-free nontrivial gravitational interactions for greater than 4 dimensions in the study of -dimensional action in .
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  described that the 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 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  of super heterotic sting theory. The exact black hole (BH) solution in greater than or equal to five-dimensional gravitational theories is studied by . Dadhich  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  are generalization of -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  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  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 . 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.  studied a dynamical stability of cylindrical symmetric collapse of sphere filled with locally anisotropic fluid in theory of gravity. A lot of literatures are available about the gravitational collapse and BH in GB theory of gravity [50–58]. If the theory of gravity obeys the stress-energy tensor conservation, then unknown function can be obtained in the closed form . Abbas and Riaz  determined the exact solution of nonstatic anisotropic gravitational fluid in theory of gravity, which may lead to collapse and expansion of the star. Sharif and Aisha  studied the models for collapse and expansion of charged self-gravitating objects in theory of gravity.
Oppenheimer and Snyder  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  studied this work for positive cosmological constant, and Lake  discussed this for negative as well as positive cosmological constant. Sharif and Abbas  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  discussed the 5-dimenssional symmetric spherical gravitational collapse with positive cosmological constant in the existence of an electromagnetic field. Abbas and Zubair  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 . Jhingan and Ghosh  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  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  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 changes the structure of the singularities. Glass  generated the collapsing and expansion solutions of anisotropic fluid of Einstein field equations. In this paper, we extended the work of Glass  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 Section 2, 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 the action given aswhere is gravitational constant and is a Ricci scalar in , and is known as the coupling constant of the GB term. The GB Lagrangian is given as follows: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 . The action in (1) gives the following field equations:whereis the Einstein tensor andis the Lanczos tensor. We want to find the solution for collapse and expansion of a spherical anisotropic fluid in 5D-EGB gravity.where is a metric on three spheres and , , and . The energy-momentum tensor for anisotropic fluid iswhere , , , , and 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), and are given by  as follows:which satisfyThe expansion scalar isThe dimensionless measure of anisotropy is defined asEquation (3) for the metric in (6) with the help of (7) is given as follows:where the prime and dot denote the partial derivative with respect to r and t, respectively. The auxiliary solution of (15) isBy Using the auxiliary solution in (16) in (10), the expansion scalar becomesFor and , we obtained expansion and collapse regions. The matter components from (12), (13), and (14) with the help of (16) areThe Misner-Sharp mass function m(t,r) is given as follows:where comma represents partial differentiation and the surface of dimensional unit space is represented by . By using , with , and (16) in (21), we getThe specific values of and form an anisotropic configuration. When , there exist trapped surfaces at , provided that . Therefore, in this case, is trapped surface condition.
3. Generating Solutions
For the different values of , expansion scalar for collapse and for expansion solutions are discussed as follows.
4. Collapse with
The expansion scalar must be negative in case of the collapse, so for , from (17), we get . By assuming and the trapped condition and then integration, we obtainIt 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  and take the value of as posivite scalar () multiple of , such that . Hence the convenient form of the areal radius for the solution outside the trapped surface is given byThus (18)–(20) with (24) and yield the following set of equations:In this case, (22) is reduced to the following form:Equation (11), with the help of (26) and (27), becomeswhere , , , and = + +
The energy conditions for the curvature matter coupled gravity are(i)Null energy condition: ; (ii)Weak energy condition: , , and (iii)Strong energy condition:
In this case, we analyzed our results for time profile , , and the different values of . For , we obtained ; the energy density is decreasing with respect to radius r and time t and remains positive for different values of coupling constant 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 . 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 . The anisotropy is directed outward when ; this implies that and it is directed inward when ; this implies that . In this case for increasing r and t at various values of as shown in Figure 5. This represents that the anisotropy force allows the formation of more massive object and has attractive force for near the center. In the present case, the GB coupling term 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.
5. Expansion with
In this case, the expansion scalar must be positive. When , from (17), the expansion scalar is positive. We assume thatwhere is constant and we take .
For , (18), (19), and (20) take the formsAssuming that and , (31), (32), and (33) in this case areThe mass function (22) in this case reduces toWith the help of (35) and (36), (11) takes the formwhere = + − − − + + , = + + + − , = − , and = + − − .
In this case, we take , , and the various values of and analyze our results. For , we obtained ; Figure 7 shows that, for the variation of , the energy density increases for the time profile. The radial pressure is initially maximum and then decreases along ; the tangential pressure is decreasing near the center along at different values of as shown in Figures 8 and 9. The mass of the sphere is increasing along and the different values of as shown in Figure 10. In this case, ; this implies that . The anisotropy parameter increases in this case for the different values of 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 . The energy conditions for the expansion case are also satisfied as shown in Figure 12.
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  studied the effects of anisotropy on the late time expansion of inhomogeneous universe. Mahmood et al.  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 . The gravitational collapse is a highly dissipative process, in which a big amount of energy is released . The dynamic of an anisotropic fluid collapse is observed in 5-dimensional Einstein-Gauss-Bonnet gravity . Collin  modeled the inhomogeneous cosmological nonstatic expanding solutions. For the suitable values of and , Glass  observed anisotropic collapsing and expanding of spherical object. We have modeled the field equations of anisotropic fluid in EGB gravity.
The aim of this paper is to study the generating solutions for anisotropic spherical symmetric fluid in 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 has been depending on the range of free constant , for which being positive or negative leads to expansion and collapse of the fluid. We use the condition in (22), which leads to two trapped horizons at , provided that . The curvature singularity is hidden at the common center of the inner and outer horizon. The matter components like density, radial pressure, tangential pressure, anisotropic parameter, and mass functions have been determined in the EGB theory of gravity. Equation (17) implies that, for , , and for the expansion scalar is negative and for the expansion scalar is positive, which lead to bouncing, collapsing, and expanding, respectively. In other words, and ; and and . The solutions have been modeled by taking and for collapse and expansion of gravitating source, respectively.
We assume that the areal radius outside the sphere for . It is interesting that, for , and it is valid only for trapped surface, so we take for collapsing solutions outside the trapped surface. Further, 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 term . 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.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this manuscript.
One of the authors, G. Abbas, acknowledges the financial support from HEC, Islamabad, Pakistan, under NRPU project with Grant no. 20-4059/NRPU/R&D/HEC/14/1217.
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