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G. Abbas, H. Nazar, "Dynamics of Dissipative Viscous Cylindrical Collapse with Full Causal Approach in Gravity", Advances in High Energy Physics, vol. 2018, Article ID 9250786, 13 pages, 2018. https://doi.org/10.1155/2018/9250786
Dynamics of Dissipative Viscous Cylindrical Collapse with Full Causal Approach in Gravity
The idea of this article is to examine the effects on dynamics of dissipative gravitational collapse in nonstatic cylindrical symmetric geometry by using Misner-Sharp concept in framework of metric gravity theory. In this interest, we extended our study to the dissipative dark source case in both forms of heat flow and the free radiation streaming. Moreover, the role of different quantities such as heat flux, bulk, and shear viscosity in the dynamical equation is evaluated in thorough version. The dynamical equation is then coupled with full causal transportation equations in the context of Israel-Stewart formalism. The present scheme explains the physical consequences of the gravitational collapse and that is given in the decreasing form of inertial mass density which depends on thermodynamics viscous/heat coupling factors in background of theory of gravity. It is very interesting to tell us that the motives of this theory are reproduced for into general theory of relativity that has been done earlier.
A multiple work has been done in the significance of relativistic geometry of the collapsing star to the general relativity theory and higher order curvature theories. In this scenario, too many open challenges and curious issues still exist in modern physics, gravitational physics, cosmological physics, relativistic dense objects, group of galaxies, neutron star, and black holes. Currently, our aim is to discuss the gravitational collapse of the relativistic dense object in the presence of dissipative term.
In this arrangement, the formation of the dense star is highly explosive and exhaustible due to the highly inmost pull of gravity and this process is terminated with the external pressure of the interior fuel. However, once the object has wakened due to its thermodynamical fission reaction, there is no any extended thermodynamical burning and there will be an interminable gravitating collapse. The geometry of the object is made up by its explosive mass which is constantly gravitational and it is stimulating by the interior core of the object due to the gravitating interaction between their particles. Oppenheimer and Snyder  made an activity stride in the field of gravitating collapse and examined the nitty gritty work in regard to the gravitational collapse issues. In this order after the midway of the 20th century, a brief sensible evaluation was recognized by Misner and Sharp  with isotropic matter distribution for the interior of the imploding object as well as in vacuum of the outside object. Vaidya  explained the gravitational collapse with the radiating source.
Numerous models still exist to explain the enigmatic role of the dark energy in a system. In this inference, we use the higher order modified gravity theories such as , , and gravity theories; it can provide the well-established framework in the significance of rushing evolution of the universe. Capozziello  studied the notable issues for the accelerated growth of the universe and quintessence in the Friedmann Robertson Walker (FRW) model with higher curvature gravity theories. Sharif and Yousaf  explored the stability of the anisotropic fluid distribution in nonstatic spherically symmetric geometry with metric formalism. Mak and Harko  investigated the notional indications through many astrophysical estimations and found the significant natural dissimilarities in the pressure. Herrera and Santos  determined the impacts on imperfect fluid formation with astral spherical object for the behavior of restful rotation. Webber  examined the solutions of pressure anisotropy in dense models with electromagnetic field. Chakraborty et al.  analyzed the gears of tangential pressure and along with radial pressure in stellar structures by quasispherical paradigms in the form of collapse. Garattini  evaluated the formation of exposed singularity through mass limitation in metric theory of gravity. Sharif and his coresearchers [11–18] studied the various consequences of energy density inhomogeneity and Weyl tensor with gravitational collapse in dense objects with General relativity theory as well as in modified theories of gravity. Cognola et al.  introduced the results of spherical imploding object for the black hole in context by the parameter of positive Ricci invariant . Capozziello et al.  examined the modified Lané-Emden equation in context of metric gravity theory and studied the hydrostatic periods of astral models. Copeland et al.  analyzed the several concepts for the rushing growth of universe. Amendola et al.  discussed the feasible paradigms of metric gravity in the context of both Einstein and Jordan frames.
Chandrasekhar  introduced the impacts on the constancy of dynamics with perfect fluid collapsing geometry. Herrera et al.  discussed the results of dynamical disequilibrium with spherically symmetric nonadiabatic collapse and found the heat flow out from the gravitating source in the form of radial heat flux. Abbas et al.  presented the results of compact star models in anisotropic fluid distribution with cylindrical symmetric static spacetime geometry. Mak and Harko  studied the kind of exact solutions of field equations by considering the spherical symmetric structure and also conferred the energy density along with tangential and radial pressure that are limited and increased in the core of the imperfect fluid object. Rahaman et al.  prolonged the Krori-Barua solution with spherical symmetric static spacetime for the investigation of charge imperfect fluid distribution. Herrera and his collaborators [28–33] did work in different directions, such as self-gravitating dens models discussed with local anisotropic matter distribution in the cracking spherical symmetric structure in context of equation of state with perturbed approach. They established the various results in gravitational collapse for the dissipative case by using Misner and Sharp approach; such gravitating source undergoes in form of free radiation streaming and diffusion approximations. They investigated the expansion free conditions for the spherically symmetric anisotropic dissipative collapsing matter. Further, Herrera and his collaborators [34–38] have evaluated the Lake algorithm in static spherically symmetric anisotropic configuration for the determination of new formalism of two functions (instead of one) to generate all possible results. They studied the results of structure scalars for the charged dissipative anisotropic collapsing geometry in presence of cosmological constant. They have also done detailed work on the dynamical disequilibrium with spherically symmetric locally anisotropic fluid which collapses adiabatically under the expansion-free condition. In few investigations they have used the noncasual as well as casual approaches to discuss the dynamics of gravitating source. Nolan  examined the effects on gravitational collapse for counter rotating dust cloud with cylindrical paradigms and found the naked singularity. Hayward  produced the results of cylindrical geometry for black holes, gravitational waves, and cosmic strings.
In this paper, we analyze the self-gravitational collapse with cylindrical symmetric geometry in context of metric gravity theory. The main persistence of this study is to examine the viscous dissipative collapse in the form of radial heat flow and free radiation streaming in cylindrical nonstatic spacetime with full causal approach.
The proposal of this work is as follows. In Section 2, we express the cylindrical symmetric isotropic collapsing matter distribution and its related variables in context of theory of gravity. Section 3 consists of Einstein modified field equations; matching conditions are given for the interior and exterior manifolds and and the dynamical equation that was accompanied with physical parameters such as heat flux, bulk, and shear viscosity. Section 4 is devoted to the transportation equations attained in frame of -Israel-Stewart formalism [41–43]; then transportation equations are associated with dynamical equation. Finally, the conclusions of the study have been discussed in the last section.
2. Self-Gravitational Collapsing Geometry and Related Conventions
In this section, we consider a locally isotropic collapsing fluid with full causal description in context of Misner-Sharp approach, formed by cylindrically symmetric nonstatic interior and exterior geometry of the dissipative fluid which undergoes dissipation in the form of heat flow and free streaming radiation. In general relativity the Einstein-Hilbert (EH) action can be expressed as follows:The extended formulation of the Einstein-Hilbert (EH) in context iswhere is arbitrary function of Ricci scalar , is coupling constant, and the Lagrangian matter density of the action. The following modified field equations are obtained by the variation of (2) with respect to :where , is the D’ Alembert operator, denotes the covariant derivative, and is the isotropic energy-momentum tensor. The above equations can be expressed in Einstein tensor as given below:whererepresent the action energy-momentum tensor. By considering the comoving coordinates bounded by cylindrically symmetric nonstatic spacetime geometry, which contains the dissipative nature of the fluid defines by interior metric,Here , , , are the metric functions. We labeled the coordinates , , and .
The tensor of the dissipative collapsing matter distribution is where is the energy density, is the isotropic pressure, the bulk viscosity, the heat flux, is the shear viscosity, the radiation density, and the four velocity of the fluid. Furthermore, denotes radial null four vector. The above extents gratify the results as below.In the general nonreversible thermodynamics taken into [44, 45],wherever is the factor of shear viscosity and denotes the factor of bulk viscosity, and are the shear tensor and heat flow expansion, respectively.
The shear tensor is defined bywhere the acceleration and the expansion are recognized byand is the projector onto the hypersurface orthogonal to the four velocity. let us define the following quantities for the given metric:where depending on and . Also it trails from (8) so thatIn a more explicit formation we can writewhere is a unit four vector along the radial direction, fulfilling the conditionsand .
From the information of (13), we obtain the following nonnull components of shear tensor.wherewhereHere dot denotes derivative w.r.t. .
3. Modified Field Equations in Gravity
The modified field equation (4) for cylindrical symmetric spacetime in metric formalism gives the following system of equations.The energy of gravitating source per specific length defined by cylindrical symmetric space-time is given by [40, 46–48].For a cylindrical symmetric paradigms by killing vectors, the circumference radius , and specific length and, moreover, arial radius are described in [40, 46–48].
, so that .
In the whole interior region C-energy has the following form :
3.1. Matching Conditions
We consider an exterior spacetime over hypersurface that describes the 4-dimensional manifold in cylindrically symmetric geometry [16, 49] is given bywhere is the total mass and is the usual retarded time coordinate, while is a constant having the dimension of . Moreover, the interior spacetime of manifold over the boundary hypersurface isHere is constant. The interior metric of over bounding hypersurface becomeswhereindicates the time coordinate on hypersurface and means the estimation is applied on of both sides.
In a similar manner, we prescribed the exterior line-element on hypersurface takes the formThus, the exterior spacetime manifold on hypersurface is given byThe following conditions due to Darmois  are to be satisfied. (i)The continuity of the first fundamental form on becomes(ii)The continuity of the second fundamental form over boundary three-surface is as follows: where is the extrinsic curvature over the hypersurface and given into
From the above equation of the extrinsic curvature, the Christoffel symbols are calculated for the interior and exterior manifolds and accordingly and we labeled = for the intrinsic coordinates on boundary surface and are the components of the outward unit normal over in the coordinates , thus we haveThe following expressions are taken from the continuity of the first fundamental form:Hence, the nonzero components of extrinsic curvature areThus, continuity of the extrinsic curvature is organized with (38); we obtain the following results over hypersurface [16, 51]:Here and are the dark source components given in Appendices (A.1) and (A.2), respectively.
Therefore, (44) shows the total mass of the interior boundary surface , whereas (45) gives the smooth association between the energy for the cylindrical symmetric geometry and the active mass on hypersurface . Moreover, (46) describes the linear association among the fluid quantities over the boundary three-surface . Thus effective pressure is in usual nonvanishing over due to dissipative nature of the dark source and this dissipation through viscosity parameters in the form of heat flow and undergoes free radiation streaming on the boundary surface. The extra components on the right hand side defines the dark source and appeared due to higher curvature of the geometry. It should be very interesting to mention that if the fluid has no dark source components then effective pressure and heat flux are equal on the boundary surface .
Furthermore, other important results of the collapsing fluid are performed on the bounding three-surface for the matching conditions of the Ricci invariant and its normal derivative. The detailed description of matching conditions in gravity theory has been given in [50, 52–56]. We require the matching conditions for the continuity of both spacetimes and over the hypersurface ; the following matching conditions theory are given byThe above expressions are required for the continuity of Ricci invariant over the boundary surface in the cylindrical collapsing fluid.
3.2. Dynamical Equations
The proper time and radial derivatives are given; we use Misner and Sharp technique .where is the areal radius of a spherical surface inside the limit. The velocity of the collapsing matter is described by the proper time derivative of and ; i.e.,which is always negative. Applying these results and (27), it turns out intoThe rate of change of mass is w.r.t. proper time in (27); we use (21), (22), and (23)-(25) as follows:The overhead result leads to telling us that the variation rate of the total energy interior of the cylinder of radius C and the R.H.S of (51), reflects that the energy is increasing in the interior surface of radius (in case of collapsing situation ) through the rate of work being done by the active radial pressure, the energy density , and the radiation pressure . The usual thermodynamical result is used in the relaxation phase. The second value of the right-hand side illustrates the matter energy of the fluid, which is leaving the cylindrical surface. Moreover, all other extra terms play the natural role of the dark energy dissipative source under the formation of modified theory of gravity and decreasing the pressure in the core of the star due to continuous collapsing phase of the dark energy. Similarly, we can calculateThis solution of (52) explains how variational quantities affect the matter distribution between the neighboring exteriors in the object of radius C. The picture of the first two quantities on the R.H.S of (52) is related with energy density of the fluid and plus the addiction of the null fluid conferring dissipation in the outgoing streaming approximation. The last factor plays negative role in the collapsing situation and estimates the dissipation in terms of heat flow and radiation streaming. The extra values signify the input of the dark energy (DE) due to its higher order of the curvature matter. We Apply integration on (52) with ; we getThe above equation gives the total mass in terms of C-energy inside the cylinder with the contribution of dark source term. To observe the dynamical equation by assuming Misner and Sharp approach[2, 57], the contracted Bianchi identities take the form,which produceHere and are the dark source components shown in Appendix (A.3) and (A.4), respectively.
From (23) and (48)-(50), it follows thatThe acceleration of the dissipative viscous collapsing source is attained by using (23) and (48)-(50); it becomesAfter, inserting from (59) into (58), we getIn this interpretation, to analyze the factor , that comes on the L.H.S and on the R.H.S, this is effective inertial mass and defines through equivalence principle. It can also be recognized as passive gravitational mass. On the R.H.S, the first term of square bracket illustrates the impacts of the collapsing variables on the active gravitational mass for the cylindrical collapsing object with the dissipative dark source in f(R) metric theory; this datum was early pointed by Herrera et al.  in the context of GR. Moreover, in the second square bracket there is the gradient of the whole active pressure which is influenced by collapsing variables and radiating density. The final bracket includes unalike quantities that define the collapsing nature of the source. The second value of this bracket is positive concluding that dissipation is continuous in the form of heat flow and free streaming radiation and finally decreases the total energy of the system, which reduces the degree of collapse. The last term of this bracket expresses the dark energy source of the dissipative gravitating collapse.
4. Transport Equations
The purpose of this profile is to discuss the full causal technique for the viscous dissipative gravitating collapse in self-gravitating system accompanied by heat transference. This study tells us that all collapsing variables should satisfy the transportation equations attained from causal thermodynamics. Therefore, we take the transportation equations for heat, bulk, and shear viscosity from -Israel-Stewart formalism [41–43] for dissipative source. Herrera et al.  discussed transportation equations for heat, bulk, and shear viscosity. The entropy flux is given bywhere and are thermodynamic factors for unalike additions to entropy density, and are thermodynamics heat coupling factors, and is temperature.
Furthermore, from the Gibbs equation and Bianchi identities, it follows that Finally, by the standard procedure, the constitutive transport equations follow from the requirement :withand where the relaxational times are given bywhere and are the factors of bulk and shear viscosity. We use the interior metric of the cylindrical structure from (63)-(65), to get the following set of equations:where we have to analyze the effect on several dissipative variables for the cylindrical interior surface. For this persistence, we use (69) and (60) and getHere and are introduced into the formand is defined asFrom (68) and (71) subsequently it becomes