Conformal Mappings in Relativistic Astrophysics

Abstract

We describe the use of conformal mappings as a mathematical mechanism to obtain exact solutions of the Einstein field equations in general relativity. The behaviour of the spacetime geometry quantities is given under a conformal transformation, and the Einstein field equations are exhibited for a perfect fluid distribution matter configuration. The field equations are simplified and then exact static and nonstatic solutions are found. We investigate the solutions as candidates to represent realistic distributions of matter. In particular, we consider the positive definiteness of the energy density and pressure and the causality criterion, as well as the existence of a vanishing pressure hypersurface to mark the boundary of the astrophysical fluid.

1. Introduction

The gravitational evolution of celestial bodies may be modeled by the Einstein field equations. These are a system of ten highly coupled partial differential equations expressing an equivalence between matter and geometry. The equations are extremely difficult to solve in general and so simpler cases have to be treated to gain an understanding into how certain types of matter behave under the influence of the gravitational field. For example, the most studied configuration of a matter distribution is that of a static spherically symmetric perfect fluid. The assumption of spherical symmetry has the effect of reducing the field equations to a system of three equations in four unknowns if the matter is neutral. While this is a severe, but reasonable, restriction, even in this case not all solutions to the system of field equations have been found. Work on this problem has been ongoing since the first exact solution appeared in 1916 when Karl Schwarzschild published his solution for a vacuum (matter free), and, to the present, this exterior solution continues to be used to model phenomena such as black holes. This solution is unique, and, moreover, Birkhoff [1] showed that the solution is independent of whether the sphere is static or not. In other words, the Schwarzschild exterior solution [2] is simply a consequence of the spherical geometry. From this theorem also follows the conclusion that pulsating fluid spheres do not generate gravitational waves.

By considering the case of a uniform density sphere, Schwarzschild [3] found a unique interior solution. However, the problem of finding all possible solutions for a nonconstant energy density is still an open problem. Since the system of field equations is underdetermined, one of the geometric or dynamical variables must be specified at the outset. This freedom of choice renders it impractical to determine all possible solutions. A comprehensive review of the static spherically symmetric fluid sphere has been compiled by Delgaty and Lake [4]. Recent work by Fodor [5], Martin and Visser [6], Lake [7], Boonserm et al. [8], and Rahman and Visser [9], which resuscitated an idea first proposed by Wyman [10], reported algorithms for finding all possible exact solutions. However, each prescription required an integration or two which may be intractable in practice given that a certain variable had to be selected in some ad hoc fortuitous way. In any event, even if solutions to the field equations are found, they have to satisfy certain physical requirements to be considered candidates for realistic matter. Finch and Skea [11] studied over 100 exact solutions and have found that only about 16 satisfy the most elementary physical requirements. It should be remarked that the importance of some of the so called requirements for physical plausibility is debatable given that the gravitational processes inside a star may not be accurately determined.

If spherical symmetry is maintained and if, in addition, the matter distribution contained charge, then the Einstein field equations must be supplemented by Maxwell’s equations which incorporate the effects of the electromagnetic field. These Einstein-Maxwell equations constitute a set of six field equations in four unknowns. In this case, the problem is simpler as now two of the matter or geometrical quantities must be chosen at the beginning and the remaining four will follow from the integration of the field equations. A detailed collection of the two-variable choices that have been considered has been achieved by Ivanov [12]. The caveat is that, although there is an extra degree of freedom, finding physically palatable solutions is extremely rare. Additionally, it is required that the exact solution for the interior be matched with the unique exterior solution for charged spheres according to Reissner [13] and Nordstrom [14]. The solution of Hansraj and Maharaj [15] in the form of Bessel functions of half integer order was shown to satisfy elementary physical requirements. Other matter configurations include radiation and rotation. In each case a unique exterior solution has been found—the exterior metric for a radiating star is credited to Vaidya [16], while an exterior metric for a rotating sphere was constructed by Kerr [17]. While in the case of radiating spheres many interior solutions have been found, this is not the case for a rotating sphere. The problem of finding a solution to the Einstein field equations that incorporate rotations is still unsolved problem in classical general relativity.

The mathematical approach described earlier often relies on choosing functional forms for some of the variables that eventually allow for the integration of the entire system of differential equations. Fortunately, the analysis is simplified by the fact that a single master equation holds the key to unlocking the whole system. This master equation may be interpreted as a second-order linear differential equation (see Duorah and Ray [18], Durgapal and Bannerji [19], and Finch and Skea [20]) or as a first-order Ricatti equation (for example, see Lake [7] and Fodor [5]). Once a solution is found, then the physical and geometric quantities must be established and checked for physical plausibility. An alternative approach is to impose some physical constraints a priori, for example, to prescribe an equation of state relating the pressure and energy density. However, the drawback of this approach is that the field equations may not be solvable. Only a few solutions of this kind have been reported in the literature. Interestingly, the charged analogue of the Finch-Skea [20] stars reported by Hansraj and Maharaj [15] turned out to possess a barotropic equation of state. This is a desirable feature of physically reasonable perfect fluids.

The situation when considering nonstatic (that is, time dependent) matter configurations introduces another level of complexity into the problem. The partial differential equations are now in terms of the spatial and temporal coordinates. Finding exact solutions usually amounts to prescribing relationships between the geometric quantities in order to solve the field equations completely. Therefore, in view of this difficulty, we consider an alternative mathematical approach. We employ conformal mappings on existing (possibly defective) solutions possessing Killing algebras (see Table 1) with the intention of solving the now conformally related Einstein field equations. The reason for the potential success of this approach has to do with the fact that the existence of conformal Killing vectors is known to simplify the field equations—in other words, they involve a geometric constraint. This is in opposition to an algebraic constraint which may be imposed, for example, by demanding that the eigenvectors of the Weyl tensor have certain preferred alignments. This gives rise to the Petrov [21] classification scheme which in reality is a result in pure mathematics applicable to any Lorentzian manifold. In our approach, the procedure is aided by the Defrise-Carter [22] theorem which specifies how conformal Killing vectors become Killing vectors under conformal transformations. The theorem states the following: suppose that a manifold (𝑀,𝐠) is neither conformally flat nor conformally related to a generalised plane wave. Then, a Lie algebra of conformal Killing vectors on 𝑀 with respect to 𝐠 can be regarded as a Lie algebra of Killing vectors with regard to some metric on 𝑀 conformally related to 𝐠 [23].

Table 1 

The benefit of utilising this approach is that we may utilise a known exact solution of the Einstein field equations and then solve the associated conformally related equations. This is because the conformal Einstein tensor splits neatly into the original Einstein tensor and a conformally related part. So, for example, if one begins with a vacuum seed solution, then it is known that the Einstein tensor is zero and so only the conformal part must now be considered in conjunction with a perfect fluid energy momentum tensor. Such solutions are referred to as conformally Ricci-flat spacetimes. This scheme has yielded useful results. A pioneering work on this method was conducted by van den Bergh [24, 25], and, recently Castejón-Amenedo and Coley [26] and Hansraj et al. [27] found new classes of exact solutions that were nonstatic. In fact, the fluid congruences were found to be accelerating, shearing, and expanding, which is a category of solution types that is rare. Hansraj [28] analysed the conjecture of van den Bergh [24] that perfect fluid spacetimes can be found by using the nonconformally flat Schwarzschild exterior solution. It was proved that all such spacetimes are necessarily static, and a class of exact solutions were reported. In Hansraj et al. [27], use was made of a nonvacuum seed solution however, no physical analysis of the resulting solutions was carried out. Govinder and Hansraj [29] completely analysed the most general set of field equations with the help of Lie group analysis methods and found new exact solutions. As the solutions were four dimensional, it was not possible to provide a full physical analysis. We consider these aspects in the current paper.

2. Differential Geometry

A brief consideration of the mathematical framework for this problem will be in order. We consider a spacetime (𝑀,𝑔), where 𝑀 is a 4-dimensional differentiable manifold with respect to a symmetric, nonsingular metric field 𝑔. Points in 𝑀 are labelled by real coordinates, three of which are spacelike and one is timelike. The points are represented as (𝑥^𝑎)=(𝑥⁰,𝑥¹,𝑥²,𝑥³), where 𝑥⁰ is timelike and 𝑥¹,  𝑥²,  and  𝑥³ are spacelike.

The invariant distance between neighbouring points of a curve in 𝑀 is defined by the fundamental metric form 𝑑𝑠²=𝑔_𝑎𝑏𝑑𝑥^𝑎𝑑𝑥^𝑏, (1) where 𝑔_𝑎𝑏 is referred to as the metric tensor. It is symmetric and possesses an inverse 𝑔^𝑎𝑏=1/𝑔_𝑎𝑏. The Christoffel symbol Γ is a metric connection that preserves inner products under parallel transport. The coefficients of Γ are calculated by Γ^𝑎_𝑏𝑐=12𝑔^𝑎𝑑𝑔_{𝑐𝑑,𝑏}+𝑔_{𝑑𝑏,𝑐}−𝑔_{𝑏𝑐,𝑑}, (2) where commas denote partial differentiation.

The Riemann curvature tensor (also known as Riemann-Christoffel tensor) is a (1,3) tensor field whose coordinate components are given in terms of the coordinate components of the connection as follows: 𝑅^𝑎_𝑏𝑐𝑑=Γ^𝑎_{𝑏𝑑,𝑐}−Γ^𝑎_{𝑏𝑐,𝑑}+Γ^𝑎_𝑒𝑐Γ^𝑒_𝑏𝑑−Γ^𝑎_𝑒𝑑Γ^𝑒_𝑏𝑐. (3) The Ricci tensor 𝑅_𝑎𝑏, obtained by contraction of the Riemann curvature tensor, is given by 𝑅_𝑎𝑏=Γ^𝑑_{𝑎𝑏,𝑑}−Γ^𝑑_{𝑎𝑑,𝑏}+Γ^𝑒_𝑎𝑏Γ^𝑑_𝑒𝑑−Γ^𝑒_𝑎𝑑Γ^𝑑_𝑒𝑏. (4) Upon contraction of the Ricci tensor, we obtain the Ricci scalar 𝑅, given by 𝑅=𝑔^𝑎𝑏𝑅_𝑎𝑏. (5) The Einstein tensor 𝐺_𝑎𝑏 is obtained from the Ricci tensor and the Ricci scalar in the following way: 𝐺_𝑎𝑏=𝑅_𝑎𝑏−12𝑅𝑔_𝑎𝑏. (6) For matter in its neutral state, the energy-momentum tensor 𝑇_𝑎𝑏 is given by 𝑇_𝑎𝑏=𝜇+𝑝𝑢_𝑎𝑢_𝑏+𝑝𝑔_𝑎𝑏+𝑞_𝑎𝑢_𝑏+𝑞_𝑏𝑢_𝑎+𝜋_𝑎𝑏, (7) where 𝜇 is the energy density, 𝑝 is the isotropic pressure, 𝑞_𝑎 is the heat flow vector, 𝜋_𝑎𝑏 is the stress tensor and 𝑢_𝑎 is the velocity field vector.

For a perfect fluid, the energy-momentum tensor, reduces to 𝑇_𝑎𝑏=𝜇+𝑝𝑢_𝑎𝑢_𝑏+𝑝𝑔_𝑎𝑏. (8) The energy-momentum tensor (8) and the Einstein tensor (6) are related via the Einstein field equations in the following way: 𝐺_𝑎𝑏=𝑇_𝑎𝑏. (9) This is a system of 10 partial differential equations which are highly nonlinear and hence difficult to integrate in general.

It is also of importance to our work to mention the Weyl conformal tensor which has physical relevance in expressing the tidal force a body experiences when moving along a geodesic. In fact, the Weyl tensor is the traceless component of the Riemann tensor and is given by 𝐶_{𝑎𝑏𝑐𝑑}=𝑅_{𝑎𝑏𝑐𝑑}+1()𝑅𝑔𝑛−1)(𝑛−2^𝑎𝑐𝑔_𝑏𝑑−𝑔_𝑎𝑑𝑔_𝑏𝑐−1𝑔𝑛−2^𝑎𝑐𝑅_𝑏𝑑−𝑔_𝑏𝑐𝑅_𝑎𝑑+𝑔_𝑏𝑑𝑅_𝑎𝑐−𝑔_𝑎𝑑𝑅_𝑏𝑐. (10)

3. Conformal Geometry

A variety of symmetries may be defined on the manifold by the action of ℒ_𝐗 on the metric tensor and associated quantities [30, 31]. Of the various symmetries that are possible, we are primarily concerned with conformal motions. A conformal Killing vector 𝐗 is defined by the action of ℒ_𝐗 on the metric tensor field 𝐠 so that ℒ_𝐗𝑔_𝑎𝑏=2𝜓𝑔_𝑎𝑏, (11) where 𝜓(𝑥^𝑎) is the conformal factor. There are four special cases associated with (11), namely.

The spanning set {𝐗_𝐴}={𝐗₁,𝐗₂,…,𝐗_𝑟} of all the conformal Killing vectors of a spacetime generates a Lie algebra. The elements of this basis are related by 𝐗_𝐴,𝐗_𝐵=𝐗_𝐴𝐗_𝐵−𝐗_𝐵𝐗_𝐴=𝐶^𝐷_𝐴𝐵𝐗_𝐷, (12) where the quantities 𝐶^𝐷_𝐴𝐵 are the structure constants of the group. The structure constants have the property of being independent of the coordinate system but do depend on the choice of the basis. The structure constants are skew symmetric so that 𝐶^𝐷_𝐴𝐵=−𝐶^𝐷_𝐵𝐴 (13) and satisfy the identity 𝐶^𝐸_𝐴𝐷𝐶^𝐷_𝐵𝐶+𝐶^𝐸_𝐵𝐷𝐶^𝐷_𝐶𝐴+𝐶^𝐸_𝐶𝐷𝐶^𝐷_𝐴𝐵=0. (14) The integrability condition for the existence of a conformal symmetry is given by ℒ_𝐗𝐶^𝑎_𝑏𝑐𝑑=0. (15) Suppose that we are given a spacetime (𝑀,𝐠) with line element 𝑑𝑠²=𝑔_𝑎𝑏𝑑𝑥^𝑎𝑑𝑥^𝑏 (16) and a related spacetime (𝑀,𝐠) with the line element 𝑑𝑠²=𝑔_𝑎𝑏𝑑𝑥^𝑎𝑑𝑥^𝑏. (17) Then, the previous two line elements are said to be conformally related if 𝑔_𝑎𝑏=𝑒^2𝑈𝑔_𝑎𝑏,𝑔^𝑎𝑏=𝑒^−2𝑈𝑔^𝑎𝑏, (18) where 𝑈(𝑥^𝑐) is a nonzero, real-valued function of the coordinates on 𝑀. This transformation between 𝐠 and 𝐠 is called a conformal transformation and is a special type of mapping between metric spaces given by dilatation (or contraction) of all lengths by a common factor which varies from point to point. If 𝐗 is a conformal Killing vector in the (𝑀,𝐠) spacetime so that (11) holds, that is, ℒ_𝐗𝑔_𝑎𝑏=2𝜓𝑔_𝑎𝑏, (19) then 𝐗 is also a conformal Killing vector in the related spacetime (𝑀,𝐠), and we have ℒ_𝐗𝑔_𝑎𝑏=2𝜎𝑔_𝑎𝑏. (20) With the help of (11), (18), and (20), we can show that the factors 𝜎 and 𝜓 are related by ℒ𝜎=^𝐗𝑒^2𝑈2𝑒^2𝑈+𝜓. (21) From this, we can conclude that conformal transformations map conformal Killing vectors to conformal Killing vectors, with the conformal factors being different. However, conformal transformations do not map Killing vectors or homothetic Killing vectors to their respective counterparts in the conformally related spacetime. Note that, in the trivial case where 𝑒^2𝑈 is constant, we have 𝜎=𝜓. However, it is important to note that the converse does not hold. If 𝜎=𝜓, then (21) implies that ℒ_𝐗(𝑒^2𝑈)=0. This means that 𝑈 is constant along the integral curves of the vector 𝐗, but it may vary elsewhere on the manifold.

It should be noted that the connection coefficients, Riemann curvature tensor, Ricci tensor, and Ricci scalar for the metric 𝑔_𝑎𝑏 are related to those of the metric 𝑔_𝑎𝑏=𝑒^2𝑈𝑔_𝑎𝑏 by the following formulae which are given in de Felice and Clarke [32]: Γ^𝑎_𝑏𝑐=Γ^𝑎_𝑏𝑐+12𝛿^𝑎_𝑏𝜙_𝑐+𝛿^𝑎_𝑐𝜙_𝑏−𝑔_𝑏𝑐𝜙^𝑎,𝑅^𝑎_𝑏𝑐𝑑=𝑅^𝑎_𝑏𝑐𝑑+𝛿^𝑎_[𝑑∇_𝑐]𝜙_𝑏+𝑔_𝑏[𝑐∇_𝑑]𝜙^𝑎+12𝛿^𝑎_[𝑐𝜙_𝑑]𝜙_𝑏−12𝑔_𝑏[𝑐𝜙_𝑑]𝜙^𝑎−12𝛿^𝑎_[𝑐𝑔_𝑑]𝑏𝜙^𝑒𝜙_𝑒,𝑅_𝑏𝑑=𝑅_𝑏𝑑−122𝜙_𝑏;𝑑−𝜙_𝑎𝜙_𝑏+𝑔_𝑎𝑏𝜙^𝑒𝜙_𝑒−12𝑔_𝑏𝑑𝜙^𝑎_;𝑎,1𝑅=Ω𝑅−3𝜙^𝑒_;𝑒−32𝜙_𝑒𝜙^𝑒, (22) where we have defined 𝜙_𝑎=𝜕_𝑎(lnΩ) and Ω=𝑒^2𝑈. Additionally, the conformal Einstein tensor 𝐆 is given by 𝐺_𝑎𝑏=𝐺_𝑎𝑏+2𝑈_𝑎𝑈_𝑏−12𝑈^𝑐𝑈_𝑐𝑔_𝑎𝑏+2𝑈^𝑐_;𝑐+𝑈^𝑐𝑈_𝑐𝑔_𝑎𝑏−2𝑈_𝑎;𝑏, (23) where the covariant derivatives and contractions are calculated on the original metric 𝑔_𝑎𝑏. Note that we may also write the conformal Einstein tensor (23) as 𝐺_𝑎𝑏=𝐺_𝑎𝑏−2𝑈_𝑎𝑈_𝑏−2𝑈_𝑎;𝑏+2𝑈^𝑐_;𝑐−𝑈^𝑐𝑈_𝑐𝑔_𝑎𝑏, (24) where the geometric quantities are now evaluated using the conformally related metric 𝑔_𝑎𝑏 [33].

An important characteristic of conformal mappings is that the Weyl tensor 𝐶 is invariant under the transformation; that is, 𝐶_{𝑎𝑏𝑐𝑑}=𝐶_{𝑎𝑏𝑐𝑑}, (25) and consequently a conformal transformation is sometimes referred to as a Weyl rescaling in the literature. A necessary and sufficient condition that a spacetime is conformally flat is that the Weyl tensor 𝐂 vanishes.

The covariant derivative of the timelike fluid 4-vector field 𝐮 can be decomposed as follows 𝑢_𝑎;𝑏=𝜎_𝑎𝑏+13Θℎ_𝑎𝑏−𝑢̇_𝑎𝑢_𝑏+𝜔_𝑎𝑏, (26) where ℎ_𝑎𝑏=𝑔_𝑎𝑏+𝑢_𝑎𝑢_𝑏 is the projection tensor. In the previous, we have defined 𝑢̇_𝑎=𝑢_𝑎;𝑏𝑢^𝑏,𝜔_𝑎𝑏=𝑢_[𝑎;𝑏]+𝑢̇_[𝑎𝑢_𝑏],𝜎_𝑎𝑏=𝑢_(𝑎;𝑏)+𝑢̇_(𝑎𝑢_𝑏)−13Θℎ_𝑎𝑏,Θ=𝑢^𝑎_;𝑎, (27) where 𝑢̇_𝑎 is the acceleration vector (𝑢̇^𝑎𝑢_𝑎=0),  𝜔_𝑎𝑏 is the skew-symmetric vorticity tensor (𝜔_𝑎𝑏𝑢^𝑎=0),  𝜎_𝑎𝑏 is the symmetric shear tensor (𝜎_𝑎𝑏𝑢^𝑏=0=𝜎^𝑎_𝑎), and Θ is the rate of expansion.

Under a conformal transformation 𝑔_𝑎𝑏=𝑒^2𝑈𝑔_𝑎𝑏, the world lines are the same and the velocity field transforms as 𝑢_𝑎=𝑒^𝑈𝑢_𝑎 (28) and we obtain 𝑢̇_𝑎=𝑒^𝑈𝑢̇_𝑎+𝑢_𝑎𝑢_𝑏𝑈^,𝑏+𝑈_,𝑎,Θ=𝑒^−𝑈Θ−3𝑢^𝑎𝑒^−𝑈_,𝑎,𝜔_𝑎𝑏=𝑒^𝑈𝜔_𝑎𝑏,𝜎_𝑎𝑏=𝑒^𝑈𝜎_𝑎𝑏 (29) for the transformed kinematical quantities listed in (27). The quantities, (29) will be useful in studying the physical behaviour of the models generated by a conformal transformation [27].

4. Einstein Field Equations

We consider the metric 𝑑𝑠²=−𝑑𝑡²+𝑑𝑥²+𝑒^{2𝜈(𝑦,𝑧)}𝑑𝑦²+𝑑𝑧², (30) which is of Petrov type D. Solving (11) for this metric reveals that (30) admits three Killing vectors, namely, 𝑋₁=𝜕_𝑡,𝑋₂=𝜕_𝑥,𝑋₃=𝑥𝜕_𝑡+𝑡𝜕_𝑥 (31) with the Lie bracket relations 𝑋₁,𝑋₂𝑋=0,₁,𝑋₃=𝑋₂,𝑋₂,𝑋₃=𝑋₁. (32) The conformally related analogue is given by 𝑑𝑠²=𝑒^{2𝑈(𝑡,𝑥,𝑦,𝑧)}−𝑑𝑡²+𝑑𝑥²+𝑒^{2𝜈(𝑦,𝑧)}𝑑𝑦²+𝑑𝑧². (33) By the Defrise-Carter [22] theorem, the previous Killing vectors are now conformal Killing vectors, given by 𝑌₁=𝑈_𝑡,𝑌₂=𝑈_𝑥,𝑌₃=𝑥𝑈_𝑡+𝑡𝑈_𝑥. (34)

To determine the perfect fluid energy-momentum tensor, we select a fluid 4-velocity vector 𝐮 that is noncomoving of the form 𝑢^𝑎=𝑒^−𝑈cosh𝑣𝛿^𝑎₀+sinh𝑣𝛿^𝑎₁, (35) where 𝑣=𝑣(𝑡,𝑥). Note that utilising a comoving velocity field would also lead to conformal flatness, and this case is not of immense interest as explained previously.

The coupling of the conformal Einstein tensor and the transformed energy momentum tensor yields the Einstein field equations given by 𝑈_𝑡𝑈_𝑦−𝑈_𝑡𝑦𝑈=0,_𝑡𝑈_𝑧−𝑈_𝑡𝑧𝑈=0,_𝑥𝑈_𝑦−𝑈_𝑥𝑦𝑈=0,_𝑥𝑈_𝑧−𝑈_𝑥𝑧𝑈=0,_𝑡𝑈_𝑥−𝑈_𝑡𝑥1=−4𝜇+𝑝𝑒^2𝑈𝑈sinh2𝑣,_𝑦𝑈_𝑧−𝑈_𝑦𝑧+𝜈_𝑧𝑈_𝑦+𝜈_𝑦𝑈_𝑧=0,−2𝑈_𝑥𝑥−𝑈²_𝑥+3𝑈²_𝑡−𝑒^−2𝜈2𝑈_𝑦𝑦+2𝑈_𝑧𝑧+𝑈²_𝑦+𝑈²_𝑧+𝜈_𝑦𝑦+𝜈_𝑧𝑧=𝜇+𝑝𝑒^2𝑈cosh²𝑣−𝑝𝑒^2𝑈,−2𝑈_𝑡𝑡−𝑈²_𝑡+3𝑈²_𝑥+𝑒^−2𝜈2𝑈_𝑦𝑦+2𝑈_𝑧𝑧+𝑈²_𝑦+𝑈²_𝑧+𝜈_𝑦𝑦+𝜈_𝑧𝑧=𝜇+𝑝𝑒^2𝑈sinh²𝑣+𝑝𝑒^2𝑈,2𝑈_𝑧𝑧+𝑈²_𝑧+3𝑈²_𝑦+2𝜈_𝑦𝑈_𝑦−2𝜈_𝑧𝑈_𝑧+𝑒^2𝜈2𝑈_𝑥𝑥−2𝑈_𝑡𝑡+𝑈²_𝑥−𝑈²_𝑡=𝑝𝑒^2𝜈+2𝑈,2𝑈_𝑦𝑦+𝑈²_𝑦+3𝑈²_𝑧−2𝜈_𝑦𝑈_𝑦+2𝜈_𝑧𝑈_𝑧+𝑒^2𝜈2𝑈_𝑥𝑥−2𝑈_𝑡𝑡+𝑈²_𝑥−𝑈²_𝑡=𝑝𝑒^2𝜈+2𝑈 (36)(37)(38)(39)(40)(41)(42)(43)(44)(45) for the line element (33).

The field equations may be reduced to a simpler form by observing that an immediate consequence of (36)–(39) is the functional form 𝑒^−𝑈=𝑓(𝑡,𝑥)+ℎ𝑦,𝑧, (46) where 𝑓 and ℎ are arbitrary functions and 𝑈=𝑈(𝑡,𝑥,𝑦,𝑧). After a series of arduous calculations, it may be shown that (36)–(45) reduce to the simpler set of equations 𝜇=3𝑓²_𝑡−𝑓²_𝑥+𝑓+ℎ4𝑘𝑓−2𝑘ℎ+3𝛼−3𝑒^−2𝜈ℎ²_𝑦+ℎ²_𝑧,𝑝=−3𝑓²_𝑡−𝑓²_𝑥+𝑓+ℎ2𝑓_𝑡𝑡−2𝑓_𝑥𝑥+2𝑘ℎ−𝛼+3𝑒^−2𝜈ℎ²_𝑦+ℎ²_𝑧,tanh²𝑣=2𝑓_𝑥𝑥−2𝑘𝑓−𝛼2𝑓_𝑡𝑡,𝑓+2𝑘𝑓+𝛼_2𝑡𝑥=142𝑓_𝑥𝑥−2𝑘𝑓−𝛼2𝑓_𝑡𝑡,ℎ+2𝑘𝑓+𝛼_𝑦𝑧=𝜈_𝑧ℎ_𝑦+𝜈_𝑦ℎ_𝑧,ℎ_𝑦𝑦−ℎ_𝑧𝑧=2𝜈_𝑦ℎ_𝑦+2𝜈_𝑧ℎ_𝑧, (47)(48)(49)(50)(51)(52) where 𝛼 is a separation constant and we have set 𝜈_𝑦𝑦+𝜈_𝑧𝑧=−2𝑘𝑒^2𝜈,  𝑘 being a constant. Note that the expression 𝜈_𝑦𝑦+𝜈_𝑧𝑧 is nonzero. If this expression was zero, the Weyl tensor would vanish leading to conformally flat solutions. Such solutions have already been fully determined as Stephani [34] stars in the case of expansion or generalised Schwarzschild metrics for no expansion. The set of (47)–(52) has been discussed in general by Govinder and Hansraj [29] with the help of the Lie group analysis approach. New exact solutions were reported; however, the difficulty of working with such solutions is that they are 4 dimensional and therefore it is not transparent whether the solutions could represent physically acceptable matter configurations. For this reason, we elect to study some physical properties by neglecting the 𝑦 and 𝑧 directional contributions—that is, we set ℎ=0 ala Castejón-Amenedo and Coley [26].

5. The Static Case

As a first step, it is of interest to analyse the behaviour of our model when it is time independent. That is, we consider the conformal factor in the form 𝑈=𝑈(𝑥). The system of field equations (47)–(52) reduces to 𝜇=−3𝑓²_𝑥+𝑓4𝑘𝑓+3𝛼,𝑝=3𝑓²_𝑥−𝑓2𝑓_𝑥𝑥+𝛼,tanh²𝑣=2𝑓_𝑥𝑥−2𝑘𝑓−𝛼,2𝑘𝑓+𝛼0=2𝑓_𝑥𝑥−2𝑘𝑓−𝛼2𝑘𝑓+𝛼, (53)(54)(55)(56) where it should be recalled that 𝛼 is a separation constant (effectively an integration constant) and 𝑘 is an arbitrary constant. Equation (56) leads to 𝑓=−𝛼/2𝑘 which in turn implies a constant conformal factor. This is trivial. However, another consequence of (56) is the equation 2𝑓_𝑥𝑥−2𝑘𝑓−𝛼=0. (57) Equation (57) is a standard second-order differential equation with constant coefficients and its solution may be categorised as follows (note in what follows that we take the coupling constant 𝛼 to be positive without loss of generality: 𝑓(𝑥)=𝛼4𝑥²+𝐶₁𝑥+𝐶₂−𝛼,𝑘=0,2𝑘+𝐶₁cosh√𝑘𝑥+𝐶₂sinh√𝛼𝑘𝑥,𝑘>0,2𝑘+𝐶₁cos√−𝑘𝑥+𝐶₂sin√−𝑘𝑥,𝑘<0, (58) where 𝐶₁ and 𝐶₂ are integration constants.

5.1. The Case 𝑘=0

This case is not particularly interesting because 𝑘=0 results in a metric that is conformally flat. Such metrics have been found up to integration in general. However, in this case, we are able to provide the explicit forms for the kinematic and dynamical variables. When 𝑘=0, the complete solution of the conformally related Einstein field equations is given by 𝑒^{−𝑈(𝑥)}=𝛼4𝑥²+𝐶₁𝑥+𝐶₂,1𝜌=4𝑘𝛼²𝑥⁴+2𝑘𝛼𝐶₁𝑥³+2𝑘𝛼𝐶₂+2𝐶¹₁𝑥²+8𝑘𝐶₁𝐶₂𝑥+4𝑘𝐶²₂+3𝐶₂−3𝐶²₁,1𝑝=4𝛼²𝑥²−𝛼𝐶₁𝑥+3𝐶²₁−2𝛼𝐶₂,tanh𝑣=0. (59)

Note that the vanishing of the tilting angle 𝑣 in the velocity vector also corroborates the conformal flatness of the spacetime geometry for 𝑘=0.

5.2. The Case 𝑘>0

The energy density, pressure, and sound-speed parameter have the forms 𝜇=−3𝑘𝑔²+3𝛼−𝛼2𝑘+𝑔+4𝑘−𝛼2𝑘+𝑔²,𝑝=3𝑘𝑔²−−𝛼,2𝑘+𝑔𝛼+2𝑘𝑔𝑑𝑝𝛼𝑑𝜇=1−,2𝑘𝑔 (60) respectively, and we have made the redefinitions 𝑔=𝐶₁cosh√𝑘𝑥+𝐶₂sinh√𝑘𝑥 and 𝑔=𝐶₂cosh√𝑘𝑥+𝐶₁sinh√𝑘𝑥. In order to satisfy the weak, strong, and dominant energy conditions, we require that each of 𝜇−𝑝=−6𝑘𝑔²+2−𝛼,2𝑘+𝑔𝛼+4𝑔𝜇+𝑝=2𝑘𝑔−𝛼,𝑔2𝑘+𝑔𝜇+3𝑝=6𝑘²+𝛼+𝑘𝑔𝛼−2𝑘𝑔𝑘 (61) be positive.

Now, it remains to select suitable values of the four constants in the problem in order to generate an astrophysical model that satisfies elementary conditions for physical reality. In order to reduce the arbitrariness in selecting these constants, we may consider the behaviour of the dynamical variables at the central axis corresponding to 𝑥=0—with the idea being that the constants should be compatible with a regular centre in the first place. Using the subscript 0 to denote central values, we obtain 𝜌₀=6𝑘²𝐶²₂−8𝑘²𝐶²₁+2𝛼𝑘𝐶₁+𝛼²𝑘𝑝≤0,₀=6𝑘²𝐶²₂−4𝑘²𝐶²₁−𝛼²𝑘≥0,0<𝑑𝑝𝑑𝜌₀𝛼=1−2𝑘𝐶₁<1, (62)(63)(64) where it must be recalled that 𝛼 and 𝑘 are positive. The right-most inequality of (64) implies that 𝐶₁>0, and the left inequality requires that 𝐶₁>𝛼/2𝑘. In addition, (62) and (63) lead to 𝛼²+4𝑘²𝐶²₁≤6𝑘²𝐶²₂≤8𝑘²𝐶²₁+2𝛼𝑘𝐶₁+𝛼², which places a constraint on 𝐶₂. Empirical testing with these constraints in mind results in the prescription 𝑘=1, 𝛼=2, 𝐶₁=12/5, and 𝐶₂=−2 which produces pleasing physical properties. Graphical representations of the dynamical and energy profiles are shown in Figures 1 and 2, respectively.

Figure 1 

Graph of energy density (𝜇), pressure (𝑝) and sound-speed index (𝑑𝑝/𝑑𝜇).

Figure 2 

Graph of 𝜇−𝑝,  𝜇+𝑝, and 𝜇+3𝑝.

From Figure 1, we observe that both the energy and pressure are positive definite and monotonically decreasing functions of the space variable 𝑥. Importantly, there exists a pressure-free hypersurface at 𝑥=0,11458 which identifies a boundary for the distribution of the perfect fluid. Therefore, this model may succeed in modeling astrophysical phenomena. In addition, it can be seen that the adiabatic sound-speed index 𝑑𝑝/𝑑𝜇 remains constrained between 0 and 1 throughout the interior of this fluid configuration. This is an indication that the sound speed is subluminal; that is, the sound speed never exceeds the speed of light, which is a fundamental postulate of the Einstein theory of the gravitational field. Figure 2 reveals that the weak, strong, and dominant energy conditions (𝜇−𝑝>0,  𝜇+𝑝>0  and  𝜇+3𝑝>0, resp.) are all satisfied throughout the fluid.

5.3. The Case 𝑘<0

In this case, the energy density, pressure and sound-speed index are given by 𝜇=3𝑘𝑗²̃+−𝛼,𝑗̃2𝑘+𝑗𝛼+4𝑘𝑗𝑝=−3𝑘²−−𝛼,2𝑘+𝑗𝛼+2𝑘𝑗𝑑𝑝𝛼𝑑𝜇=1−,2𝑘𝑗 (65) where we have defined 𝑗=𝐶₁cos√−𝑘𝑥+𝐶₂sin√−𝑘𝑥 and 𝑗̃=𝐶₂cos√−𝑘𝑥−𝐶₁√−𝑘𝑥. The quantities 𝜇−𝑝=6𝑘𝑗²̃+−𝛼+2𝑘𝑗𝛼+3𝑘𝑗𝑘,𝜇+𝑝=𝑘𝑗−𝛼+𝑘𝑗𝑘,𝜇+3𝑝=−6𝑘𝑗²̃−−𝛼+2𝑘𝑗𝛼+𝑘𝑗𝑘 (66)(67)(68) will assist in studying the energy conditions.

As above we now need to find suitable constants in order to create a viable fluid model. Again we examine the above quantities at the central axis 𝑥=0. We obtain 𝜌₀=6𝑘²𝐶²₂+−𝛼+2𝑘𝐶₁𝛼+4𝑘𝐶₁𝑝2𝑘>0,₀=−6𝑘²𝐶²₂−−𝛼+2𝑘𝐶₁𝛼+2𝑘𝐶₁2𝑘≥0,0≤𝑑𝑝𝛼𝑑𝜇=1−2𝑘𝐶₁<1 (69)(70)(71) as constraints on the constants. Now, from (71), we infer that 𝑟<0 and also that 𝛼<2𝑘𝐶₁ recalling that 𝛼>0 and 𝑘<0 in this case. Furthermore, reconciling (69) and (70), we obtain the relations 𝛼²−4𝑘²𝐶²₁<6𝑘²𝐶²₂<𝛼²+2𝛼𝑘𝐶₁−8𝑘²𝐶²₁. Disregarding the middle term in this inequality, we derive that 𝛼>2𝑘𝐶₁ which contradicts our earlier conclusion. Therefore, it is impossible to satisfy the three most important physical requirements 𝜌>0, 𝑝≥0, and 0<𝑑𝑝/𝑑𝜇<1 with any combination of the constants in the problem. In other words, the case 𝑘<0 cannot represent a realistic fluid. Therefore, we do not pursue any further consideration of this case.

6. Nonstatic Solutions

Finding nonstatic exact solutions of the Einstein equations for a particular configuration of matter is nontrivial. The method of conformal transformations gives us the ability to construct non-static solutions from known static metrics. In the case under investigation, we suppose that the conformal factor has the simple form 𝑈=𝑈(𝑡). The field equations then have the familiar form 𝜇=3𝑓²_𝑡+𝑓4𝑘𝑓+3𝛼,𝑝=−3𝑓²_𝑡+𝑓2𝑓_𝑡𝑡,−−𝛼tanh𝑣=2𝑘𝑓+𝛼2𝑓_𝑡𝑡,+2𝑘𝑓+𝛼0=2𝑘𝑓+𝛼2𝑓_𝑡𝑡+2𝑘𝑓+𝛼, (72) where now 𝑓=𝑓(𝑡) only. Once again, the conclusion 𝑓(𝑡)=𝛼/2𝑘 leads to a trivial constant conformal factor, so we disregard it. We examine the other possibility 2𝑓_𝑡𝑡+2𝑘𝑓+𝛼=0 (73) more closely. Equation (73) is a simple second-order differential equation and is solved by 𝑓(𝑥)=−𝛼4𝑡²+𝐶₁𝑡+𝐶₂−𝛼,𝑘=0,2𝑘+𝐶₁cos√𝑘𝑡+𝐶₂sin√𝛼𝑘𝑡,𝑘>0,2𝑘+𝐶₁cosh√−𝑘𝑡+𝐶₂sinh√−𝑘𝑡,𝑘<0, (74) where 𝐶₁ and 𝐶₂ are integration constants. Let us examine each case in turn.

6.1. The Case 𝑘=0

This case degenerates into a conformal flat metric and so it is not worthy of further attention as such metrics have already been discovered.

6.2. The Case 𝑘>0

If we make the change of variables 𝑙=𝐶₁cos√𝑘𝑡+𝐶₂sin√𝑘𝑡, and 𝑙̃=𝐶₂cos√𝑘𝑡−𝐶₁sin√𝑘𝑡 then the relevant dynamical quantities have the following forms: 𝜇=−3𝑘𝑙²̃−1(),𝑙̃2𝑘𝛼−2𝑘𝑙)(𝛼+4𝑘𝑙𝑝=3𝑘²+1()2𝑘𝛼−2𝑘𝑙²,𝑑𝑝=𝑑𝜇𝛼−14𝑘𝑙2().𝛼+𝑘𝑙 (75) It is pleasing to note that the positive definiteness of pressure criterion is valid everywhere for this case 𝑘>0. We also expect all physical constraints to be satisfied at the initial condition 𝑡=0. Denoting initial values with the subscript 0, we have that the following inequalities must be satisfied: 𝜇₀=−6𝑘²𝐶²₂−𝛼²−2𝛼𝑘𝐶₁+8𝑘²𝐶²₁𝑝2𝑘>0,₀=6𝑘²𝐶²₂+𝛼²−4𝛼𝑘𝐶₁+4𝑘²𝐶²₁2𝑘≥0,0<𝑑𝑝𝑑𝜇₀=𝛼−14𝑘𝐶₁2𝛼+𝑘𝐶₁<1. (76)(77)(78) The first two relationships constrain 𝑘 as −𝛼²+4𝛼𝑘𝐶₁−4𝑘²𝐶²₁≤6𝑘²𝐶²₂<−𝛼²−2𝛼𝑘𝐶₁+8𝑘²𝐶²₁, (79) and the outer two expressions must satisfy 𝐶₁<0 or 𝐶₁>𝛼/2𝑘. The adiabatic sound-speed criterion (78) forces the value of 𝐶₁ as −𝛼/16𝑘<𝐶₁<𝛼/14𝑘 since 𝛼>0 and 𝑘>0. An additional constraint on 𝐶₁ is implied by requiring the positivity of the right hand side of (79). This is satisfied provided that 𝐶₁<−𝛼/4𝑘 or 𝐶₁>𝛼/2𝑘. Now, it can immediately be seen that these inequalities are never satisfied simultaneously for any value of 𝐶₁. Accordingly, we deduce that no physical reasonable model will result from this case. That is, positivity of pressure and energy density as well as a subluminal sound speed cannot all be achieved for any value of the constant 𝐶₁. There is little value in considering the value of 𝐶₂ in view of the nonsatisfaction of these requirements.

6.3. The Case 𝑘<0

In this case, the energy density, pressure, and sound-speed index are given, respectively, by 𝑣̃𝜇=6𝑘∗²+5𝛼²+14𝛼𝑘𝑣+8𝑘²𝑣²,𝑣̃2𝑘𝑝=−6𝑘²−𝛼²+4𝑘²𝑣²,2𝑘𝑑𝑝𝛼𝑑𝜇=−7,2𝑘𝑣+1 (80) where we have again redefined 𝑣=𝐶₁cosh√−𝑘𝑡+𝐶₂sinh√−𝑘𝑡 and 𝑣𝐶̃₂cosh√−𝑘𝑡+𝐶₁sinh√−𝑘𝑡. Reasoning as before together with empirical testing corroborates that it is not possible to achieve simultaneously a positive pressure and energy density. Additionally, the sound-speed index is always found to be negative. Therefore, this model cannot model a realistic distribution of stellar fluid.

7. A Spacetime Foliation 𝑈=𝑈(𝑡,𝑥)

The Lie group analysis method has proved successful in generating broader classes of solutions than the ones discussed previously. The functional forms for 𝑓(𝑡,𝑥) for ℎ=0 were previously obtained by Hansraj et al. [27] using group analysis techniques. We now take these solutions and substitute into expressions (47) and (48) to establish the dynamical quantities, energy density (𝜇) and pressure (𝑝). Thereafter, we verify that the pressure and density profiles satisfy conditions for physical acceptability. A rigorous analytical treatment is prohibitive for general 𝑓(𝑡,𝑥) and ℎ(𝑦,𝑧), in view of the complexity of the resultant expressions. Therefore, for graphical purposes, it is necessary to consider a simplified situation such as foliations of the distribution in terms of the temporal and one space variable. It is not possible to visually represent the complete 4-dimensional solution spacetime. We accordingly select the particular solution 𝑓(𝑥,𝑡)=𝐴sinh𝑘𝑡²−𝑥²+𝐵cosh𝑘𝑡²−𝑥² (81) presented by Hansraj et al. [27] together with that obtained from 𝑍₁. The conformally related metric (33) may now be given by 𝑑𝑠²=−𝑑𝑡²+𝑑𝑥²+𝑐₁𝑒^𝑧2/8𝑦2𝑧𝑦^3/4×𝑧²𝑦²−1^1/2𝑑𝑦²+𝑑𝑧²×𝑒^{𝑐1+𝑧2/8𝑦2}𝑦^3/4𝑧^1/4+𝐵cosh𝑘𝑡²−𝑥²+𝐴sinh𝑘𝑡²−𝑥²²⁻¹, (82) and the energy density takes the form 𝜇=−3𝑐₁𝑒^𝑧2/8𝑦2𝑧𝑦^3/4𝑧²𝑦²−1^1/2−1×𝑒^{𝑐1+𝑧2/8𝑦2}𝑦^3/44𝑧^3/4+𝑒^{𝑐1+𝑧2/8𝑦2}𝑧^5/44𝑦^5/42+3𝑒^{𝑐1+𝑧2/8𝑦2}𝑧^1/44𝑦^1/4−𝑒^{𝑐1+𝑧2/8𝑦2}𝑧^9/44𝑦^9/42+𝑒^{𝑐1+𝑧2/8𝑦2}𝑦^3/4𝑧^1/4+𝐵cosh𝑘𝑡²−𝑥²+𝐴sinh𝑘𝑡²−𝑥²×−2𝑒^{𝑐1+𝑧2/8𝑦2}𝑘𝑦^3/4𝑧^1/4−3𝛼+4𝑘𝐵cosh𝑘𝑡²−𝑥²+𝐴sinh𝑘𝑡²−𝑥²+3𝐴𝑘𝑡cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝐵𝑘𝑡sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²²−−𝐴𝑘𝑥cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²−𝐵𝑘𝑥sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²², (83) while the pressure is given by 𝑝=3𝑐₁𝑒^𝑧2/8𝑦2𝑧𝑦^3/4𝑧²𝑦²−1^1/2−1×𝑒^{𝑐1+𝑧2/8𝑦2}𝑦^3/44𝑧^3/4+𝑒^{𝑐1+𝑧2/8𝑦2}𝑧^5/44𝑦^5/42+3𝑒^{𝑐1+𝑧2/8𝑦2}𝑧^1/44𝑦^1/4−𝑒^{𝑐1+𝑧2/8𝑦2}𝑧^9/44𝑦^9/42−𝑘3𝑡²+𝑥²𝑡²−𝑥²𝐴cosh𝑘𝑡²−𝑥²+𝐵sinh𝑘𝑡²−𝑥²²+𝑒^{𝑐1+𝑧2/8𝑦2}𝑦^3/4𝑧^1/4+𝐵cosh𝑘𝑡²−𝑥²+𝐴sinh𝑘𝑡²−𝑥²××𝑒−𝛼+2_{𝑐1+𝑧2/8𝑦2}𝑘𝑦^3/4𝑧^1/4−𝑘+𝐴²𝑡²cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2+𝑘cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝑘𝑡²sinh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝑘−𝐴²𝑥²cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2−𝑘cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝑘𝑥²sinh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝐵𝑘𝑥²cosh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝑘²𝑥²sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2−𝑘sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝐵−𝑘𝑡²cosh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝑘²𝑡²sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2+𝑘sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥². (84)

In order to obtain an indication of the model’s feasibility to represent a realistic distribution, we elect to make a graphical study of this solution. The energy density and pressure are given by 𝜇=4𝑘𝐵cosh𝑘𝑡²−𝑥²+𝐴sinh𝑘𝑡²−𝑥²²−3𝐴𝑘𝑡cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝐵𝑘𝑡sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²²−−𝐴𝑘𝑥cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²−𝐵𝑘𝑥sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²²,𝑝=2𝐵cosh𝑘𝑡²−𝑥²+𝐴sinh𝑘𝑡²−𝑥²×𝐴−𝑘²𝑡²cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2+𝑘cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝑘𝑡²sinh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝐴−𝑘²𝑥²cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2−𝑘cosh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝑘𝑥²sinh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝐵𝑘𝑥²cosh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝑘²𝑥²sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2−𝑘sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²+𝐵𝑘𝑡²cosh𝑘𝑡²−𝑥²𝑡²−𝑥²−𝑘²𝑡²sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥^23/2+𝑘sinh𝑘𝑡²−𝑥²𝑘𝑡²−𝑥²−𝑘3𝑡²+𝑥²𝑡²−𝑥²×𝐴cosh𝑘𝑡²−𝑥²+𝐵sinh𝑘𝑡²−𝑥²², (85) We need to select appropriate values for the parameters 𝐴,𝐵,  and  𝑘 to finalise the model. It should be observed at this stage that, in many situations, it is possible to obtain bounds for the constants by examining the central conditions 𝑝(𝑡,0,0,0) and 𝜇(𝑡,0,0,0). We then require that all the physical conditions are satisfied at the centre. This is not fruitful in the present context. Furthermore, the approach in fixing integration constants in most problems in general relativity involves matching of the perfect fluid solution with its corresponding vacuum solution. For example, if we were devising a model of a static neutral sphere, then integration constants can be fixed by demanding continuity of the potentials at the boundary hypersurface. In addition, for such solutions, the pressure vanishes at the boundary, and this determines the size of the sphere. In the case of radiating spheres, this pressure-free interface does not exist, and this condition is modified to include the effects of radiation, and the junction conditions must be established via other means. The difficulty in our analysis is that we have no known vacuum solution. Consequently, our selection of the constants is completely random. We are fortunate that the available computing technology is able to quickly check the efficacy of our choices for the integration constants (see Figures 3 and 4).

Figure 3 

Plot of pressure versus 𝑡 and 𝑥 with 𝐴=1,  𝐵=10,  and  𝑘=0.1.

Figure 4 

Plot of energy density versus 𝑡 and 𝑥 with 𝐴=1,  𝐵=10,  and  𝑘=0.1.

These plots, generated via Mathematica [35], demonstrate many pleasing features. For example, it is evident that, in the region chosen, the pressure and energy density are both positive. These are the most basic requirements for models to serve as candidates for realistic celestial phenomena. Therefore, it is possible for this solution to model a realistic configuration of perfect fluid.

8. Conclusion

We have demonstrated that new static and nonstatic solutions of the Einstein field equations may be constructed from known solutions through the use of conformal mappings. Even solutions that are considered as defective may serve as seeds for new solutions which are physically reasonable. The reason for the success of this approach is that the seed solution is already a solution of the possibly vacuum Einstein field equations. The Einstein tensor splits up neatly into a part containing the original Einstein tensor and an expression involving the conformal factor. It is this latter part that must be coupled to an appropriate perfect fluid. The mathematical reason for this behaviour is that the conformal Killing vectors of the original seed metric become Killing vectors of the new metric. This generates symmetries which have the effect of simplifying the resultant field equations. While 4-dimensional solutions to the field equations have already been reported, solutions had not been analysed for physical plausibility. We have endeavoured to construct explicit models by considering cases where the conformal factor is a function of a space variable 𝑈=𝑈(𝑥), a function of the temporal coordinate 𝑈=𝑈(𝑡) only and finally the case of the conformal factor as a function of space and time 𝑈=𝑈(𝑡,𝑥). Analyses of each case show that it is possible to generate physically reasonable models through this method of conformal transformations.