A family of exact models for radiating matter

In this paper, the cosmological constant and electric charge are incorporated in the Einstein–Maxwell field equations. Two approaches are used to investigate the problem. First, the boundary condition is expressed as a generalized Riccati equation in one of the gravitational potentials. New classes of exact solutions are found by writing the Riccati equation in linear, Bernoulli, and inhomogeneous forms. Our solutions contain previous results in the absence of the cosmological constant and charge. Second, it is possible to preserve the form of the generalized Riccati equation by introducing a transformation called the horizon function. This transformation simplifies the generalized Riccati equation. We generate new solutions to the transformed Riccati equation when one of the metric functions serves as a generating function. We also obtain other families of new classes of exact solutions, where the horizon function serves as a generating function. Interestingly, new uncharged solutions, not contained in previous studies, arise as special cases of the inhomogeneous Riccati equation in both approaches. All otherwise


I. INTRODUCTION
Radiating stellar models have been studied in many different physical situations over the years. The junction conditions at the stellar surface were completed by Santos, 1 who showed that the presence of heat flow must be taken into account so that the interior can match to the exterior Vaidya metric 2 at the boundary. Particular solutions of the boundary condition have been studied in astrophysical settings including recent investigations of Naidu et al., 3 Govender et al., 4 Sharma et al., 5 and Pretel and da Silva. 6 A systematic approach to investigating this problem is to use the Lie analysis of infinitesimal symmetry generators. Abebe et al. 7,8 used the Lie approach to study conformally flat and geodesic stars, respectively. Stars with shear-free matter distributions were analyzed by Abebe et al. 9 The interesting physical case of Euclidean stars, containing Newtonian stars in the limit, using the Lie analysis was studied by Govinder and Govender 10 and Abebe et al. 11 The results for matter, which are expanding, accelerating, and shearing, are contained in the works of Mohanlal et al. 12 using the Lie method of symmetry generators.
A second approach to solving the boundary condition at the stellar surface is to exploit physical properties, such as the formation of horizons. This approach was first suggested by Ivanov 13 for anisotropic spherical collapse for geodesic particles with shear. Later, Ivanov 14 extended this method to accelerating matter with shear and radiation. Recently, generating functions in the presence of the electromagnetic field with the horizon function were considered by Ivanov. 15, 16 Maharaj et al. 17 and Mohanlal et al. 18 showed that Riccati equations arise in the presence of the horizon function using the Lie approach. Several new families of exact solutions were obtained with the horizon function. These solutions are helpful in describing gravitational collapse scenarios in radiating stars. In this process, the interior heat conducting matter loses energy across the stellar boundary in the form of null radiation to the exterior. The gravitational mass of the radiating star decreases with time.
In this paper, we consider the boundary condition at the stellar surface for a general spherically symmetric line element. The horizon function of Ivanov 13 is utilized to reduce the Einstein-Maxwell system to generate a single evolution equation. The matter field is generalized to include the cosmological constant and the electromagnetic field. Our intention is to investigate the qualitative differences that arise from these additions. The boundary condition that arises is a generalization of a Riccati equation first derived by Thirukkanesh et al. 19 We find that the addition of charge and cosmological constant to the Einstein field equations leads to several families of exact solutions to the Riccati equations that arise. Our analysis indicates that Riccati equations are the fundamental equations that determine the evolution of the boundary for a bounded radiating system, such as a star. The presence of charge and the cosmological constant lead to several new linear, Bernoulli, and Riccati equations.
We make a number of points relating to the regularity of the fields, the energy conditions at the stellar boundary, the physical relevance of exact solutions, and stability. First, the study of the junction conditions in general relativity has a long history starting with the treatment of Darmois; 20 see the treatment of Bonnor and Vickers. 21 A comprehensive analysis of the junction conditions for stellar surfaces and surface layers in a general relativistic setting was completed by Lake. 22 In these treatments, there is the requirement of a maximal atlas, in which the metric is continuous. Mars and Senovilla 23 considered the problem of matching two spacetimes across a general hypersurface. They found the conditions on the Riemann, Ricci, and Weyl tensors across the junction hypersurface of two matching spacetimes (see their Theorem 12). The particular case of matching two spherically symmetric spacetimes (containing our model) was completed by Fayos et al. 24 In the case when the exterior spacetime is the Vaidya radiating metric, the junction conditions were explicitly generated by Santos. 1 The spherically symmetric metric functions in the interior of the star that arise in specific models have to satisfy the Einstein field equations. Second, the energy conditions for an imperfect fluid with heat conduction were found by Kolassis et al. 25 These conditions can be applied to a collapsing sphere, which radiates energy to the exterior. Particular examples where the energy conditions are satisfied have been contained in the treatments of Govinder and Govender 10 and Abebe et al. 11 Third, exact solutions to the boundary condition at the stellar surface were found which have desirable physical features. Kolassis et al. 26 and Rajah and Maharaj 27 analyzed a model of a dissipating star, which is regular and contains the Friedmann dust model in the limit. Conformally flat models, which match to the Vaidya exterior, have been obtained by Herrera et al., 28 Misthry et al., 29 and Herrera et al. 30 Euclidean stars in general relativity, with equal areal and proper radii, containing Newtonian stars have been studied by Herrera et al. 31 and Govender et al. 32 Models with barotropic equations of state of the form p = p(μ), relating the pressure to the energy density, have been generated by Abebe and Maharaj 33 using the Lie analysis of differential equations; this class of models contains many earlier results. Fourth, the stability properties are difficult to study, in particular radiating stellar models. The general treatment for stability of locally anisotropic selfgravitating systems was completed by Herrera and Santos. 34 Stability is related to the Weyl tensor in shear-free fluids as established by Herrera et al. 35 Another approach by Reddy et al. 36 to study stability is to use a perturbative scheme on an initially static configuration for a body undergoing collapse because of heat dissipation and anisotropic stresses. The temperature profiles in the interior of the star can be found in the Eckart theory and also in causal thermo-dynamics (see, for example, Naidu et al. 37 ). Clearly, several models exist with desirable physical features which point to them being realistic.
In Sec. II, we present the geodesic model of a radiating star, the corresponding Einstein field equations with the cosmological constant, electric charge, and the boundary conditions at the surface. The boundary condition is expressed as a Riccati equation in one of the metric potentials. Section III introduces three cases for the

II. THE MODEL
The metric for a general spherically symmetric matter distribution, in our case describing a radiating star, can be written as where A = A(r, t), B = B(r, t), and Y = Y(r, t) are functions that describe gravity. The spacetime (1) is a connected, time-oriented, four-dimensional Lorentzian manifold. The metric (1) is invariant under rotations and admits three Killing vectors given by SO (3).
Conditions on A, B, and Y will arise from dynamical considerations. The energy momentum tensor for the matter distribution is where μ is the energy density, p denotes the isotropic pressure, qa is the heat flow, ε ab is the anisotropic stress tensor, and E ab denotes the electromagnetic field. The radial and tangential pressures, p ∥ and p , respectively, can be written in terms of the isotropic pressure by p = 1 3 (p ∥ + 2p⊥). The electromagnetic energy tensor is defined as where F ab is the Faraday tensor. For charged matter with a cosmological constant λ, the Einstein-Maxwell equations are given by where J a = κu a is the four-current and κ is the proper charge density.
We can write the Einstein-Maxwell equations as a system of coupled equations for the metric (1) and the energy momentum ARTICLE scitation.org/journal/adv tensor (2). The field equations [Eq. (4)] then reduce to where l = l(r) is the total charge within the stellar boundary. When l = 0 and λ = 0, we regain the equations of Thirukkanesh et al. 19 The presence of cosmological constant λ ≠ 0 and the electromagnetic charge l ≠ 0 affects the dynamics of the gravitating model. Exact solutions of system (II) will be very different from the results in earlier investigations. Equation (5e) relates the proper charge density κ to the total charge l. The remaining equations in the system comprise four equations in the seven unknowns μ, p ∥ , p , q, A, B, and Y. Once functional forms for potentials A, B, and Y are found, the expressions for μ, p ∥ , p , and q follow. The forms for A, B, and Y are obtained by solving the boundary condition at the stellar surface.
The surface of spherically bound radiating matter is the boundary that separates the interior matter from the exterior spacetime. The spherically symmetric spacetime (1) must match, at the boundary, to the exterior radiating metric. We take the exterior metric to be where m(v) denotes the mass of the star and Q is the total charge measured by an observer at infinity. The metric (1) is the Vaidya spacetime 2 generalized to include the cosmological constant λ and charge l. We must match the line elements (1) and (6), and also the extrinsic curvature at the surface Σ of the star. This gives the junction conditions at the hypersurface Σ. Null radiation is emitted by a body with heat flow and generates a gravitational redshift given by In Eq. (8), L Σ is the luminosity at the surface and L∞ = − dm dv is the luminosity as seen by an observer at infinity. Note the important condition in (7) that the radial pressure p ∥ is nonvanishing at the surface Σ. Boundary condition (7e) can be written as the nonlinear differential equation We can interpret (9)  Thirukkanesh et al. 19 found new classes of exact solutions for λ = 0 and l = 0 by transforming (9) to simpler forms. Ivanov 13,15 also studied (9) using a particular transformation related to the formation of horizons, the so called horizon function, to obtain a generating function without the presence of the cosmological constant. For our investigation, we will consider (9) in general with λ ≠ 0 and l ≠ 0.

III. NEW SOLUTIONS
Equation (9) has been studied before without the presence of the cosmological constant and electric charge. In our investigation, we consider how these parameters change the nature of the boundary condition by comparing previous results with new classes of exact solutions. We find that the parameters l and λ change the nature of the families of exact solutions that are possible. These parameters affect the gravitational dynamics of the collapsing star. The various solutions to the Riccati equation found are listed in Table I.

A. Linear equation
We start by setting the coefficient of B 2 in (9) to zero, which gives which is a Bernoulli equation in A. Solving (10) gives where f (r) is the function of integration. Substituting (11) into (9) yields ARTICLE scitation.org/journal/adv

Cases Gravitational potentials Features
Explicit solution which makes this a linear equation in B. The solution to (12) is given by where g(r) is the integration constant, and we have as an arbitrary function. When l = λ = 0, we regain the results of Thirukkanesh et al. 19 A particular choice for potential Y will lead to explicit forms for potentials A and B.

B. Bernoulli equation
By setting the inhomogeneous term in (9) to zero, we get which yields where f (t) is the integration constant. With the help of (16), Eq. (9) becomes Equation (17) is a Bernoulli equation in B, which can be solved to give where

ARTICLE scitation.org/journal/adv and the constant of integration is g(r). For this class of solutions, we have
as an arbitrary function. When l = λ = 0, we regain the results of Thirukkanesh et al. 19 as a special case. If A is specified, then we can obtain explicit forms for potentials B and Y.

C. Inhomogeneous Riccati equation
Particular solutions to the Riccati equation [Eq. (9)] do exist, but the presence of the parameters l and λ leads to complications. Setting the coefficient of B to zero in (9) gives which can be integrated to give where f (t) is the constant of integration. By substituting (21) into (9), we obtain (22) Equation (22) is not integrable as it is presented. If we set λ = l = 0 in Eq. (22), we can regain the equations of Thirukkanesh et al. 19 They presented an exact solution to Eq. (22) by assuming that the potential Y(r, t) is a separable function. Unfortunately, this approach does not work in (22). It is not possible to integrate (22) in general; particular solutions exist under certain assumptions. We demonstrate this below.
An interesting possibility arises if Yt = 0. Then, (20) is identically satisfied but the relation in (21) does not hold. If we let Yt = 0 in Eq. (9), then the boundary condition becomes which is a simpler Riccati equation in B. We observe that on setting A = A(r) in Eq. (23), we can rewrite (23) as where An explicit solution to (24) can be found if we let where α is the proportionality constant. Equation (26) implies which is a separable equation in A. Integrating (27) gives where C 1 is the integration constant. By using Eq. (26), we can integrate (24), which yields two solutions for B. The first solution is given by where C 2 (r) is the constant of integration. The second solution is where C 3 (r) is the constant of integration. For this class of solutions, which is an arbitrary function.
In the above solution, potential A is given in (28); potential B contains A and Y through the function f (r) and g(r), and Y(r) is arbitrary. This class of models obtained by integrating the Riccati equation is new. Observe that if λ = l = 0, then potential A becomes We observe that this uncharged model is also a new solution to the Riccati equation and is not contained in earlier treatments.

IV. NEW SOLUTIONS: TRANSFORMED EQUATIONS
Particular transformations reduce the boundary condition (9) to simpler forms. The horizon function was introduced by Ivanov, 13 which gives a relation between the gravitational potentials and horizon formation. The transformation is given by where H = H(r, t). The horizon function is used as a transformation to simplify Eq. (9), which becomes This is also a Riccati equation in H. We employ a similar method used to solve (9) in Sec. III to integrate (34) for which we find that three cases arise. The various solutions are listed in Table II.

A. Linear equation
We start by setting the coefficient of H 2 to zero, so that This can be solved to give

Cases Gravitational potentials Features
Generating function H Explicit solution Explicit solution where f (t) is the constant of integration. By substituting (36) into (34), we get which is a linear equation in H. We can solve this easily to give where g(r) is the integration constant. By using Eq. (33), we can find the metric function B to be For this class of solutions, is arbitrary. Note that this class of exact solutions is different from the linear solutions given in Sec. III. The horizon function ARTICLE scitation.org/journal/adv transformation (33) leads to a new class of exact models to the boundary condition (9).

B. Bernoulli equation
We can transform (34) into a Bernoulli equation in H. The condition for obtaining a Bernoulli equation is that which is a quadratic equation in Y. This condition is satisfied only if Y = Y(r). Unfortunately, this condition is inconsistent with transformation (33), which requires Yt ≠ 0. Hence, it is not possible to generate Bernoulli equations with the horizon function. This is in contrast with the results of Sec. III. We conclude that transformation (33) restricts the number of solutions that are admissible.

C. Inhomogeneous Riccati equation
The Riccati equation [Eq. (34)] in H cannot be solved in general. Setting the coefficient of H to zero gives This is a separable equation that can be solved to yield where f (t) is the integration constant. Substituting (43) into (34), we get which is a simpler Riccati equation in H. It is difficult to solve (44) to obtain an expression for H. However, it is possible to express Eq. (44) in the form We can treat (45) as an algebraic equation in variable Y. Four cases arise in the solution of (45). We consider these in turn. The various cases are listed in Table II. 1. Case 1: λ ≠ 0, l ≠ 0 In this case, we can solve (45) to get In this class, we observe that potentials A and Y are given in terms of the horizon function H. The function H(r, t) is arbitrary. Here, the quantity H plays the role of a generating function; a particular choice of H will lead to analytical forms for A and Y.
Despite the complicated form of potential Y in (46), it is possible to regain potential B explicitly. By substituting (46) into (43) and (33), we obtain In these expressions, the role of the horizon function H as a generating function is highlighted.

V. DISCUSSION
This paper consists of two important approaches that discuss the boundary condition for a radiating star in general relativity with cosmological constant and electric charge. We first studied the boundary condition as a Riccati equation in potential B. This Riccati equation is very difficult to solve in general. However, we obtained exact solutions for three cases: Linear, Bernoulli, and inhomogeneous Riccati. The linear and Bernoulli cases are extensions of the work of Thirukkanesh et al. 19 The inhomogeneous Riccati case is a special case that allows the equation to become a separable equation after making certain assumptions for potentials A and B. This allows for a simple new class of exact solutions. We next applied a transformation to the boundary condition, which was first introduced by Ivanov, 13 called the horizon function. This transformation resulted in a simplification of the boundary condition, which was then expressed as a Riccati equation in variable H. Three cases arose ARTICLE scitation.org/journal/adv when we solved the transformed Riccati equation: Linear, Bernoulli, and inhomogeneous Riccati. The linear case yields new classes of exact solutions in which potential Y can be treated as a generating function. The Bernoulli case is disregarded because the horizon function breaks down due to potential Y being a function of r only. The inhomogeneous Riccati case is characterized by four subcases when λ ≠ 0, λ = 0, l ≠ 0, and l = 0. Each subcase can be solved explicitly to yield a family of new exact solutions. These solutions are expressed in terms of potential H, which acts as a generating function. Tables I and II express the results collectively for both the Riccati and transformed Riccati equations. The solutions obtained in this paper from the various Riccati equations can be used to study the physical features of the astrophysical model. It is possible to express stellar quantities, such as the mass of the star from the junction condition (7d) as Note that (54) becomes Eq. (38) in Ivanov 15 when λ = l = 0. We also find the time derivative to be These expressions show how the cosmological constant and electric charge play a role in the mass function and its derivative. The compactness parameter 2M Y can also be obtained using (54). These physical quantities are given in terms of potential H, which serves as a generating function. Specific choices of the generating function will permit a detailed physical analysis of the evolution of the star. This is the object of further research. An expression or the mass function can be found after eliminating r, r, andv. This produces the result We can also establish the relationship

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This result shows that the pressure at the boundary of a radiating star is proportional to the heat flux. In terms of the metric potentials, we have