Research Article Conformal Mappings in Relativistic ...Journal of Applied Mathematics where is a -dimensional di erentiable manifold with respecttoasymmetric,nonsingularmetric eld

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 196385, 12 pageshttp://dx.doi.org/10.1155/2013/196385

Research ArticleConformal Mappings in Relativistic Astrophysics

S. Hansraj, K. S. Govinder, and N. Mewalal

Astrophysics and Cosmology Research Unit, School of Mathematics, University of KwaZulu-Natal, Private Bag X54001,Durban 4000, South Africa

Correspondence should be addressed to K. S. Govinder; [email protected]

Received 1 March 2013; Accepted 10 June 2013

Academic Editor: Md Sazzad Chowdhury

Copyright © 2013 S. Hansraj et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We describe the use of conformalmappings as amathematicalmechanism to obtain exact solutions of the Einstein field equations ingeneral relativity. The behaviour of the spacetime geometry quantities is given under a conformal transformation, and the Einsteinfield equations are exhibited for a perfect fluid distribution matter configuration. The field equations are simplified and then exactstatic 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 theexistence of a vanishing pressure hypersurface to mark the boundary of the astrophysical fluid.

1. Introduction

The gravitational evolution of celestial bodies may be mod-eled by the Einstein field equations. These are a system often highly coupled partial differential equations expressingan equivalence between matter and geometry. The equationsare extremely difficult to solve in general and so simpler caseshave to be treated to gain an understanding into how certaintypes ofmatter behave under the influence of the gravitationalfield. For example, the most studied configuration of a matterdistribution is that of a static spherically symmetric perfectfluid. The assumption of spherical symmetry has the effect ofreducing the field equations to a system of three equations infour unknowns if the matter is neutral. While this is a severe,but reasonable, restriction, even in this case not all solutionsto the system of field equations have been found. Work onthis problem has been ongoing since the first exact solutionappeared in 1916 when Karl Schwarzschild published hissolution for a vacuum (matter free), and, to the present, thisexterior solution continues to be used to model phenomenasuch as black holes. This solution is unique, and, moreover,Birkhoff [1] showed that the solution is independent ofwhether the sphere is static or not. In other words, theSchwarzschild exterior solution [2] is simply a consequenceof the spherical geometry. From this theorem also followsthe conclusion that pulsating fluid spheres do not generategravitational waves.

By considering the case of a uniform density sphere,Schwarzschild [3] found a unique interior solution. However,the problemof finding all possible solutions for a nonconstantenergy density is still an open problem. Since the system offield equations is underdetermined, one of the geometric ordynamical variables must be specified at the outset.This free-dom of choice renders it impractical to determine all possiblesolutions. A comprehensive review of the static sphericallysymmetric fluid sphere has been compiled by Delgaty andLake [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 twowhich may be intractable in practice given that a certainvariable had to be selected in some ad hoc fortuitous way.In any event, even if solutions to the field equations arefound, they have to satisfy certain physical requirements tobe considered candidates for realistic matter. Finch and Skea[11] studied over 100 exact solutions and have found that onlyabout 16 satisfy the most elementary physical requirements.It should be remarked that the importance of some of theso called requirements for physical plausibility is debatablegiven that the gravitational processes inside a star may not beaccurately determined.

If spherical symmetry is maintained and if, in addition,the matter distribution contained charge, then the Einstein

2 Journal of Applied Mathematics

field equationsmust be supplemented byMaxwell’s equationswhich incorporate the effects of the electromagnetic field.These Einstein-Maxwell equations constitute a set of six fieldequations in four unknowns. In this case, the problem issimpler as now two of the matter or geometrical quantitiesmust be chosen at the beginning and the remaining fourwill follow from the integration of the field equations. Adetailed collection of the two-variable choices that have beenconsidered has been achieved by Ivanov [12]. The caveat isthat, although there is an extra degree of freedom, findingphysically palatable solutions is extremely rare. Addition-ally, it is required that the exact solution for the interiorbe matched with the unique exterior solution for chargedspheres according to Reissner [13] and Nordstrom [14]. Thesolution of Hansraj and Maharaj [15] in the form of Besselfunctions of half integer order was shown to satisfy ele-mentary physical requirements. Other matter configurationsinclude radiation and rotation. In each case a unique exteriorsolution has been found—the exterior metric for a radiatingstar is credited to Vaidya [16], while an exterior metric fora rotating sphere was constructed by Kerr [17]. While in thecase of radiating spheres many interior solutions have beenfound, this is not the case for a rotating sphere. The problemof finding a solution to the Einstein field equations thatincorporate rotations is still unsolved problem in classicalgeneral relativity.

The mathematical approach described earlier often relieson choosing functional forms for some of the variables thateventually allow for the integration of the entire system ofdifferential equations. Fortunately, the analysis is simplifiedby the fact that a single master equation holds the key tounlocking the whole system. This master equation may beinterpreted as a second-order linear differential equation(see Duorah and Ray [18], Durgapal and Bannerji [19], andFinch and Skea [20]) or as a first-order Ricatti equation(for example, see Lake [7] and Fodor [5]). Once a solutionis found, then the physical and geometric quantities mustbe established and checked for physical plausibility. Analternative approach is to impose some physical constraints apriori, for example, to prescribe an equation of state relatingthe pressure and energy density. However, the drawback ofthis approach is that the field equations may not be solvable.Only a few solutions of this kind have been reported in theliterature. Interestingly, the charged analogue of the Finch-Skea [20] stars reported by Hansraj and Maharaj [15] turnedout to possess a barotropic equation of state.This is a desirablefeature of physically reasonable perfect fluids.

The situation when considering nonstatic (that is, timedependent) matter configurations introduces another level ofcomplexity into the problem. The partial differential equa-tions are now in terms of the spatial and temporal coordi-nates. Finding exact solutions usually amounts to prescribingrelationships between the geometric quantities in order tosolve the field equations completely.Therefore, in view of thisdifficulty, we consider an alternative mathematical approach.We employ conformal mappings on existing (possibly defec-tive) solutions possessing Killing algebras (see Table 1) withthe intention of solving the now conformally related Einsteinfield equations. The reason for the potential success of this

Table 1

(1) 𝜓 = 0 X is a Killing vector(2) 𝜓;𝑎= 0 ̸= 𝜓 X is a homothetic Killing vector

(3) 𝜓;𝑎𝑏

= 0 ̸= 𝜓;𝑎

X is a special conformal Killing vector(4) 𝜓;𝑎𝑏

̸= 0 X is a nonspecial conformal Killing vector

approach has to do with the fact that the existence of confor-mal Killing vectors is known to simplify the field equations—in other words, they involve a geometric constraint.This is inopposition to an algebraic constraint which may be imposed,for example, by demanding that the eigenvectors of the Weyltensor have certain preferred alignments.This gives rise to thePetrov [21] classification scheme which in reality is a result inpure mathematics applicable to any Lorentzian manifold. Inour approach, the procedure is aided by the Defrise-Carter[22] theorem which specifies how conformal Killing vectorsbecome Killing vectors under conformal transformations.The theorem states the following: suppose that a manifold(𝑀, g) is neither conformally flat nor conformally related toa generalised plane wave. Then, a Lie algebra of conformalKilling vectors on 𝑀 with respect to g can be regarded as aLie algebra of Killing vectors with regard to some metric on𝑀 conformally related to g [23].

The benefit of utilising this approach is that wemay utilisea known exact solution of the Einstein field equations andthen solve the associated conformally related equations. Thisis because the conformal Einstein tensor splits neatly into theoriginal Einstein tensor and a conformally related part. So,for example, if one begins with a vacuum seed solution, thenit is known that the Einstein tensor is zero and so only theconformal part must now be considered in conjunction witha perfect fluid energy momentum tensor. Such solutions arereferred to as conformally Ricci-flat spacetimes. This schemehas yielded useful results. A pioneering work on this methodwas conducted by van den Bergh [24, 25], and, recentlyCastejón-Amenedo and Coley [26] and Hansraj et al. [27]found new classes of exact solutions that were nonstatic. Infact, the fluid congruences were found to be accelerating,shearing, and expanding, which is a category of solution typesthat is rare. Hansraj [28] analysed the conjecture of van denBergh [24] that perfect fluid spacetimes can be found by usingthe nonconformally flat Schwarzschild exterior solution. Itwas proved that all such spacetimes are necessarily static,and a class of exact solutions were reported. In Hansrajet al. [27], use was made of a nonvacuum seed solutionhowever, no physical analysis of the resulting solutions wascarried out. Govinder and Hansraj [29] completely analysedthe most general set of field equations with the help of Liegroup analysis methods and found new exact solutions. Asthe solutions were four dimensional, it was not possible toprovide a full physical analysis. We consider these aspects inthe current paper.

2. Differential Geometry

A brief consideration of the mathematical framework for thisproblem will be in order. We consider a spacetime (𝑀, 𝑔),

Journal of Applied Mathematics 3

where 𝑀 is a 4-dimensional differentiable manifold withrespect to a symmetric, nonsingular metric field 𝑔. Points in𝑀 are labelled by real coordinates, three of which are space-like and one is timelike. The points are represented as (𝑥𝑎) =(𝑥0, 𝑥1, 𝑥2, 𝑥3), where 𝑥0 is timelike and 𝑥1, 𝑥2, and 𝑥3 are

spacelike.The invariant distance between neighbouring points of a

curve in𝑀 is defined by the fundamental metric form

𝑑𝑠2

= 𝑔𝑎𝑏𝑑𝑥𝑎

𝑑𝑥𝑏

, (1)

where 𝑔𝑎𝑏

is referred to as the metric tensor. It is symmetricand possesses an inverse 𝑔𝑎𝑏 = 1/𝑔

𝑎𝑏. The Christoffel symbol

Γ is a metric connection that preserves inner products underparallel transport. The coefficients of Γ are calculated by

Γ𝑎

𝑏𝑐=1

2𝑔𝑎𝑑

(𝑔𝑐𝑑,𝑏

+ 𝑔𝑑𝑏,𝑐

− 𝑔𝑏𝑐,𝑑

) , (2)

where commas denote partial differentiation.The Riemann curvature tensor (also known as Riemann-

Christoffel tensor) is a (1, 3) tensor field whose coordinatecomponents are given in terms of the coordinate componentsof the connection as follows:

𝑅𝑎

𝑏𝑐𝑑= Γ𝑎

𝑏𝑑,𝑐− Γ𝑎

𝑏𝑐,𝑑+ Γ𝑎

𝑒𝑐Γ𝑒

𝑏𝑑− Γ𝑎

𝑒𝑑Γ𝑒

𝑏𝑐. (3)

The Ricci tensor 𝑅𝑎𝑏, obtained by contraction of the Riemann

curvature tensor, is given by

𝑅𝑎𝑏= Γ𝑑

𝑎𝑏,𝑑− Γ𝑑

𝑎𝑑,𝑏+ Γ𝑒

𝑎𝑏Γ𝑑

𝑒𝑑− Γ𝑒

𝑎𝑑Γ𝑑

𝑒𝑏. (4)

Upon contraction of the Ricci tensor, we obtain the Ricciscalar 𝑅, given by

𝑅 = 𝑔𝑎𝑏

𝑅𝑎𝑏. (5)

The Einstein tensor𝐺𝑎𝑏is obtained from the Ricci tensor and

the Ricci scalar in the following way:

𝐺𝑎𝑏= 𝑅𝑎𝑏−1

2𝑅𝑔𝑎𝑏. (6)

For matter in its neutral state, the energy-momentum tensor𝑇𝑎𝑏is given by

𝑇𝑎𝑏= (𝜇 + 𝑝) 𝑢

𝑎𝑢𝑏+ 𝑝𝑔𝑎𝑏+ 𝑞𝑎𝑢𝑏+ 𝑞𝑏𝑢𝑎+ 𝜋𝑎𝑏, (7)

where 𝜇 is the energy density, 𝑝 is the isotropic pressure, 𝑞𝑎

is the heat flow vector, 𝜋𝑎𝑏

is the stress tensor and 𝑢𝑎is the

velocity field vector.For a perfect fluid, the energy-momentum tensor, reduces

to

𝑇𝑎𝑏= (𝜇 + 𝑝) 𝑢

𝑎𝑢𝑏+ 𝑝𝑔𝑎𝑏. (8)

The energy-momentum tensor (8) and the Einstein tensor (6)are related via the Einstein field equations in the followingway:

𝐺𝑎𝑏= 𝑇𝑎𝑏. (9)

This is a system of 10 partial differential equations which arehighly nonlinear and hence difficult to integrate in general.

It is also of importance to our work to mention the Weylconformal tensor which has physical relevance in expressingthe tidal force a body experiences when moving along ageodesic. In fact, the Weyl tensor is the traceless componentof the Riemann tensor and is given by

𝐶𝑎𝑏𝑐𝑑

= 𝑅𝑎𝑏𝑐𝑑

+1

(𝑛 − 1) (𝑛 − 2)𝑅 (𝑔𝑎𝑐𝑔𝑏𝑑− 𝑔𝑎𝑑𝑔𝑏𝑐)

−1

𝑛 − 2(𝑔𝑎𝑐𝑅𝑏𝑑− 𝑔𝑏𝑐𝑅𝑎𝑑+ 𝑔𝑏𝑑𝑅𝑎𝑐− 𝑔𝑎𝑑𝑅𝑏𝑐) .

(10)

3. Conformal Geometry

A variety of symmetries may be defined on the manifoldby the action of LX on the metric tensor and associatedquantities [30, 31]. Of the various symmetries that are pos-sible, we are primarily concerned with conformal motions. Aconformal Killing vector X is defined by the action ofLX onthe metric tensor field g so that

LX𝑔𝑎𝑏 = 2𝜓𝑔𝑎𝑏, (11)

where 𝜓(𝑥𝑎) is the conformal factor. There are four specialcases associated with (11), namely.

The spanning set {X𝐴} = {X

1,X2, . . . ,X

𝑟} of all the

conformal Killing vectors of a spacetime generates a Liealgebra. The elements of this basis are related by

[X𝐴,X𝐵] = X

𝐴X𝐵− X𝐵X𝐴

= 𝐶𝐷

𝐴𝐵X𝐷,

(12)

where the quantities 𝐶𝐷𝐴𝐵

are the structure constants of thegroup. The structure constants have the property of beingindependent of the coordinate system but do depend onthe choice of the basis. The structure constants are skewsymmetric so that

𝐶𝐷

𝐴𝐵= −𝐶𝐷

𝐵𝐴(13)

and satisfy the identity

𝐶𝐸

𝐴𝐷𝐶𝐷

𝐵𝐶+ 𝐶𝐸

𝐵𝐷𝐶𝐷

𝐶𝐴+ 𝐶𝐸

𝐶𝐷𝐶𝐷

𝐴𝐵= 0. (14)

The integrability condition for the existence of a conformalsymmetry is given by

LX𝐶𝑎

𝑏𝑐𝑑= 0. (15)

Suppose that we are given a spacetime (𝑀, g) with lineelement

𝑑𝑠2


𝑑𝑥𝑏 (16)

and a related spacetime (𝑀, g) with the line element

𝑑𝑠2


𝑑𝑥𝑏

. (17)


Then, the previous two line elements are said to be confor-mally related if

𝑔𝑎𝑏= 𝑒2𝑈

𝑔𝑎𝑏, 𝑔

𝑎𝑏

= 𝑒−2𝑈

𝑔𝑎𝑏

, (18)

where𝑈(𝑥𝑐) is a nonzero, real-valued function of the coordi-nates on𝑀. This transformation between g and g is called aconformal transformation and is a special type of mappingbetween metric spaces given by dilatation (or contraction)of all lengths by a common factor which varies from pointto point. If X is a conformal Killing vector in the (𝑀, g)spacetime so that (11) holds, that is,

LX𝑔𝑎𝑏 = 2𝜓𝑔𝑎𝑏, (19)

then X is also a conformal Killing vector in the relatedspacetime (𝑀, g), and we have

LX𝑔𝑎𝑏 = 2𝜎𝑔𝑎𝑏. (20)

With the help of (11), (18), and (20), we can show that thefactors 𝜎 and 𝜓 are related by

𝜎 = [LX (𝑒

2𝑈)

2𝑒2𝑈+ 𝜓] . (21)

From this, we can conclude that conformal transformationsmap conformal Killing vectors to conformal Killing vec-tors, with the conformal factors being different. However,conformal transformations do not map Killing vectors orhomothetic Killing vectors to their respective counterpartsin the conformally related spacetime. Note that, in the trivialcase where 𝑒2𝑈 is constant, we have 𝜎 = 𝜓. However, it isimportant to note that the converse does not hold. If 𝜎 = 𝜓,then (21) implies that LX(𝑒

2𝑈) = 0. This means that 𝑈 is

constant along the integral curves of the vector X, but it mayvary elsewhere on the manifold.

It should be noted that the connection coefficients,Riemann curvature tensor, Ricci tensor, and Ricci scalar forthe metric 𝑔

𝑎𝑏are related to those of the metric 𝑔

𝑎𝑏= 𝑒2𝑈𝑔𝑎𝑏

by the following formulae which are given in de Felice andClarke [32]:

Γ𝑎

𝑏𝑐= Γ𝑎

𝑏𝑐+1

2[𝛿𝑎

𝑏𝜙𝑐+ 𝛿𝑎

𝑐𝜙𝑏− 𝑔𝑏𝑐𝜙𝑎

] ,

𝑅𝑎

𝑏𝑐𝑑= 𝑅𝑎

𝑏𝑐𝑑+ 𝛿𝑎

[𝑑∇𝑐]𝜙𝑏+ 𝑔𝑏[𝑐∇𝑑]𝜙𝑎

+1

2𝛿𝑎

[𝑐𝜙𝑑]𝜙𝑏−1

2𝑔𝑏[𝑐𝜙𝑑]𝜙𝑎

−1

2𝛿𝑎

[𝑐𝑔𝑑]𝑏𝜙𝑒

𝜙𝑒,

𝑅𝑏𝑑= 𝑅𝑏𝑑−1

2[2𝜙𝑏;𝑑

− 𝜙𝑎𝜙𝑏+ 𝑔𝑎𝑏𝜙𝑒

𝜙𝑒]

−1

2𝑔𝑏𝑑𝜙𝑎

;𝑎,

𝑅 =1

Ω[𝑅 − 3𝜙

𝑒

;𝑒−3

2𝜙𝑒𝜙𝑒

] ,

(22)

where we have defined 𝜙𝑎= 𝜕𝑎(lnΩ) and Ω = 𝑒2𝑈. Addi-

tionally, the conformal Einstein tensor G is given by

𝐺𝑎𝑏= 𝐺𝑎𝑏+ 2 (𝑈

𝑎𝑈𝑏−1

2𝑈𝑐

𝑈𝑐𝑔𝑎𝑏)

+ 2 (𝑈𝑐

;𝑐+ 𝑈𝑐

𝑈𝑐) 𝑔𝑎𝑏− 2𝑈𝑎;𝑏,

(23)

where the covariant derivatives and contractions are calcu-lated on the original metric 𝑔

𝑎𝑏. Note that we may also write

the conformal Einstein tensor (23) as

𝐺𝑎𝑏= 𝐺𝑎𝑏− 2𝑈𝑎𝑈𝑏− 2𝑈𝑎;𝑏

+ (2𝑈𝑐

;𝑐− 𝑈𝑐

𝑈𝑐) 𝑔𝑎𝑏,

(24)

where the geometric quantities are now evaluated using theconformally related metric 𝑔

𝑎𝑏[33].

An important characteristic of conformal mappings isthat the Weyl tensor 𝐶 is invariant under the transformation;that is,

𝐶𝑎𝑏𝑐𝑑

= 𝐶𝑎𝑏𝑐𝑑

, (25)

and consequently a conformal transformation is sometimesreferred to as a Weyl rescaling in the literature. A necessaryand sufficient condition that a spacetime is conformally flat isthat the Weyl tensor C vanishes.

The covariant derivative of the timelike fluid 4-vector fieldu can be decomposed as follows

𝑢𝑎;𝑏

= 𝜎𝑎𝑏+1

3Θℎ𝑎𝑏− �̇�𝑎𝑢𝑏+ 𝜔𝑎𝑏, (26)

where ℎ𝑎𝑏

= 𝑔𝑎𝑏

+ 𝑢𝑎𝑢𝑏is the projection tensor. In the

previous, we have defined

�̇�𝑎= 𝑢𝑎;𝑏𝑢𝑏

,

𝜔𝑎𝑏= 𝑢[𝑎;𝑏]

+ �̇�[𝑎𝑢𝑏],

𝜎𝑎𝑏= 𝑢(𝑎;𝑏)

+ �̇�(𝑎𝑢𝑏)−1

3Θℎ𝑎𝑏,

Θ = 𝑢𝑎

;𝑎,

(27)

where �̇�𝑎is the acceleration vector (�̇�𝑎𝑢

𝑎= 0), 𝜔

𝑎𝑏is the

skew-symmetric vorticity tensor (𝜔𝑎𝑏𝑢𝑎

= 0), 𝜎𝑎𝑏

is thesymmetric shear tensor (𝜎

𝑎𝑏𝑢𝑏= 0 = 𝜎

𝑎

𝑎), and Θ is the rate

of expansion.Under a conformal transformation 𝑔

𝑎𝑏= 𝑒2𝑈𝑔𝑎𝑏, the

world lines are the same and the velocity field transforms as

𝑢𝑎= 𝑒𝑈

𝑢𝑎

(28)

and we obtain

�̇�𝑎= 𝑒𝑈

(�̇�𝑎+ 𝑢𝑎𝑢𝑏𝑈,𝑏

+ 𝑈,𝑎) ,

Θ = 𝑒−𝑈

Θ − 3𝑢𝑎

(𝑒−𝑈

),𝑎

,

𝜔𝑎𝑏= 𝑒𝑈

𝜔𝑎𝑏,

𝜎𝑎𝑏= 𝑒𝑈

𝜎𝑎𝑏

(29)


for the transformed kinematical quantities listed in (27).The quantities, (29) will be useful in studying the physicalbehaviour of the models generated by a conformal transfor-mation [27].

4. Einstein Field Equations

We consider the metric

𝑑𝑠2

= −𝑑𝑡2

+ 𝑑𝑥2

+ 𝑒2](𝑦,𝑧)

(𝑑𝑦2

+ 𝑑𝑧2

) , (30)

which is of Petrov type D. Solving (11) for this metric revealsthat (30) admits three Killing vectors, namely,

𝑋1= 𝜕𝑡,

𝑋2= 𝜕𝑥,

𝑋3= 𝑥𝜕𝑡+ 𝑡𝜕𝑥

(31)

with the Lie bracket relations

[𝑋1, 𝑋2] = 0,

[𝑋1, 𝑋3] = 𝑋

2,

[𝑋2, 𝑋3] = 𝑋

1.

(32)

The conformally related analogue is given by

𝑑𝑠2

= 𝑒2𝑈(𝑡,𝑥,𝑦,𝑧)

[−𝑑𝑡2

+ 𝑑𝑥2

+ 𝑒2](𝑦,𝑧)

(𝑑𝑦2

+ 𝑑𝑧2

)] . (33)

By the Defrise-Carter [22] theorem, the previous Killingvectors are now conformal Killing vectors, given by

𝑌1= 𝑈𝑡,

𝑌2= 𝑈𝑥,

𝑌3= 𝑥𝑈𝑡+ 𝑡𝑈𝑥.

(34)

To determine the perfect fluid energy-momentum tensor,we select a fluid 4-velocity vectoru that is noncomoving of theform

𝑢𝑎

= 𝑒−𝑈

(cosh V𝛿𝑎0+ sinh V𝛿𝑎

1) , (35)

where V = V(𝑡, 𝑥). Note that utilising a comoving velocity fieldwould also lead to conformal flatness, and this case is not ofimmense interest as explained previously.

The coupling of the conformal Einstein tensor and thetransformed energy momentum tensor yields the Einsteinfield equations given by

𝑈𝑡𝑈𝑦− 𝑈𝑡𝑦= 0, (36)

𝑈𝑡𝑈𝑧− 𝑈𝑡𝑧= 0, (37)

𝑈𝑥𝑈𝑦− 𝑈𝑥𝑦= 0, (38)

𝑈𝑥𝑈𝑧− 𝑈𝑥𝑧= 0, (39)

𝑈𝑡𝑈𝑥− 𝑈𝑡𝑥= −

1

4(𝜇 + 𝑝) 𝑒

2𝑈 sinh 2V, (40)

𝑈𝑦𝑈𝑧− 𝑈𝑦𝑧+ ]𝑧𝑈𝑦+ ]𝑦𝑈𝑧= 0, (41)

− 2𝑈𝑥𝑥− 𝑈2

𝑥+ 3𝑈2

𝑡

− 𝑒−2]

(2𝑈𝑦𝑦+ 2𝑈𝑧𝑧+ 𝑈2

𝑦+ 𝑈2

𝑧+ ]𝑦𝑦+ ]𝑧𝑧)

= (𝜇 + 𝑝) 𝑒2𝑈cosh2V − 𝑝𝑒2𝑈,

(42)

− 2𝑈𝑡𝑡− 𝑈2

𝑡+ 3𝑈2

𝑥

+ 𝑒−2]

(2𝑈𝑦𝑦+ 2𝑈𝑧𝑧+ 𝑈2

𝑦+ 𝑈2

𝑧+ ]𝑦𝑦+ ]𝑧𝑧)

= (𝜇 + 𝑝) 𝑒2𝑈sinh2V + 𝑝𝑒2𝑈,

(43)

2𝑈𝑧𝑧+ 𝑈2

𝑧+ 3𝑈2

𝑦+ 2]𝑦𝑈𝑦− 2]𝑧𝑈𝑧

+ 𝑒2](2𝑈𝑥𝑥− 2𝑈𝑡𝑡+ 𝑈2

𝑥− 𝑈2

𝑡) = 𝑝𝑒

2]+2𝑈,

(44)

2𝑈𝑦𝑦+ 𝑈2

𝑦+ 3𝑈2

𝑧− 2]𝑦𝑈𝑦+ 2]𝑧𝑈𝑧

+ 𝑒2](2𝑈𝑥𝑥− 2𝑈𝑡𝑡+ 𝑈2

𝑥− 𝑈2

𝑡) = 𝑝𝑒

2]+2𝑈(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 thefunctional form

𝑒−𝑈

= 𝑓 (𝑡, 𝑥) + ℎ (𝑦, 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 = 3 (𝑓2

𝑡− 𝑓2

𝑥) + (𝑓 + ℎ) (4𝑘𝑓 − 2𝑘ℎ + 3𝛼)

− 3𝑒−2]

(ℎ2

𝑦+ ℎ2

𝑧) ,

(47)

𝑝 = −3 (𝑓2

𝑡− 𝑓2

𝑥) + (𝑓 + ℎ) (2𝑓

𝑡𝑡− 2𝑓𝑥𝑥+ 2𝑘ℎ − 𝛼)

+ 3𝑒−2]

(ℎ2

𝑦+ ℎ2

𝑧) ,

(48)

tanh2V =2𝑓𝑥𝑥− 2𝑘𝑓 − 𝛼

2𝑓𝑡𝑡+ 2𝑘𝑓 + 𝛼

, (49)

𝑓2

𝑡𝑥=1

4(2𝑓𝑥𝑥− 2𝑘𝑓 − 𝛼) (2𝑓

𝑡𝑡+ 2𝑘𝑓 + 𝛼) , (50)


ℎ𝑦𝑧= ]𝑧ℎ𝑦+ ]𝑦ℎ𝑧, (51)

ℎ𝑦𝑦− ℎ𝑧𝑧= 2]𝑦ℎ𝑦+ 2]𝑧ℎ𝑧, (52)

where 𝛼 is a separation constant and we have set ]𝑦𝑦+ ]𝑧𝑧=

−2𝑘𝑒2], 𝑘 being a constant. Note that the expression ]

𝑦𝑦+ ]𝑧𝑧

is nonzero. If this expression was zero, theWeyl tensor wouldvanish leading to conformally flat solutions. Such solutionshave already been fully determined as Stephani [34] stars inthe case of expansion or generalised Schwarzschild metricsfor no expansion. The set of (47)–(52) has been discussed ingeneral by Govinder andHansraj [29] with the help of the Liegroup analysis approach. New exact solutions were reported;however, the difficulty of working with such solutions is thatthey are 4 dimensional and therefore it is not transparentwhether the solutions could represent physically acceptablematter configurations. For this reason, we elect to study somephysical properties by neglecting the 𝑦 and 𝑧 directionalcontributions—that is, we set ℎ = 0 ala Castejón-Amenedoand Coley [26].

5. The Static Case

As a first step, it is of interest to analyse the behaviour of ourmodel when it is time independent. That is, we consider theconformal factor in the form 𝑈 = 𝑈(𝑥). The system of fieldequations (47)–(52) reduces to

𝜇 = −3𝑓2

𝑥+ 𝑓 (4𝑘𝑓 + 3𝛼) , (53)

𝑝 = 3𝑓2

𝑥− 𝑓 (2𝑓

𝑥𝑥+ 𝛼) , (54)

tanh2V =2𝑓𝑥𝑥− 2𝑘𝑓 − 𝛼

2𝑘𝑓 + 𝛼, (55)

0 = (2𝑓𝑥𝑥− 2𝑘𝑓 − 𝛼) (2𝑘𝑓 + 𝛼) , (56)

where it should be recalled that 𝛼 is a separation constant(effectively an integration constant) and 𝑘 is an arbitraryconstant. Equation (56) leads to 𝑓 = −𝛼/2𝑘 which in turnimplies a constant conformal factor. This is trivial. However,another consequence of (56) is the equation

2𝑓𝑥𝑥− 2𝑘𝑓 − 𝛼 = 0. (57)

Equation (57) is a standard second-order differential equationwith constant coefficients and its solution may be categorisedas follows (note in what follows that we take the couplingconstant 𝛼 to be positive without loss of generality:

𝑓 (𝑥) =

{{{{{

{{{{{

{

𝛼

4𝑥2+ 𝐶1𝑥 + 𝐶

2, 𝑘 = 0,

−𝛼

2𝑘+ 𝐶1cosh√𝑘𝑥 + 𝐶

2sinh√𝑘𝑥, 𝑘 > 0,

𝛼

2𝑘+ 𝐶1cos√−𝑘𝑥 + 𝐶

2sin√−𝑘𝑥, 𝑘 < 0,

(58)

where 𝐶1and 𝐶

2are integration constants.

5.1. The Case 𝑘 = 0. This case is not particularly interestingbecause 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 formsfor the kinematic and dynamical variables. When 𝑘 = 0, thecomplete solution of the conformally related Einstein fieldequations is given by

𝑒−𝑈(𝑥)

=𝛼

4𝑥2

+ 𝐶1𝑥 + 𝐶

2,

𝜌 =1

4𝑘𝛼2

𝑥4

+ 2𝑘𝛼𝐶1𝑥3

+ 2𝑘 (𝛼𝐶2+ 2𝐶1

1) 𝑥2

+ 8𝑘𝐶1𝐶2𝑥 + (4𝑘𝐶

2

2+ 3𝐶2− 3𝐶2

1) ,

𝑝 =1

4𝛼2

𝑥2

− 𝛼𝐶1𝑥 + (3𝐶

2

1− 2𝛼𝐶

2) ,

tanh V = 0.

(59)

Note that the vanishing of the tilting angle V in thevelocity vector also corroborates the conformal flatness of thespacetime geometry for 𝑘 = 0.

5.2.The Case 𝑘 > 0. The energy density, pressure, and sound-speed parameter have the forms

𝜇 = −3𝑘𝑔2

+ 3𝛼 (−𝛼

2𝑘+ 𝑔) + 4𝑘(

−𝛼

2𝑘+ 𝑔)

2

,

𝑝 = 3𝑘𝑔2

− (−𝛼

2𝑘+ 𝑔) (𝛼 + 2𝑘𝑔) ,

𝑑𝑝

𝑑𝜇= 1 −

𝛼

2𝑘𝑔,

(60)

respectively, and we have made the redefinitions 𝑔 =𝐶1cosh√𝑘𝑥 + 𝐶

2sinh√𝑘𝑥 and 𝑔 = 𝐶

2cosh√𝑘𝑥 +

𝐶1sinh√𝑘𝑥. In order to satisfy the weak, strong, and domi-

nant energy conditions, we require that each of

𝜇 − 𝑝 = −6𝑘𝑔2

+ 2 (−𝛼

2𝑘+ 𝑔) (𝛼 + 4𝑔) ,

𝜇 + 𝑝 = 2𝑘𝑔 (−𝛼

2𝑘+ 𝑔) ,

𝜇 + 3𝑝 = 6𝑘𝑔2

+(𝛼 + 𝑘𝑔) (𝛼 − 2𝑘𝑔)

𝑘

(61)

be positive.Now, it remains to select suitable values of the four

constants in the problem in order to generate an astrophys-ical model that satisfies elementary conditions for physicalreality. In order to reduce the arbitrariness in selecting theseconstants, we may consider the behaviour of the dynamicalvariables at the central axis corresponding to 𝑥 = 0—withthe idea being that the constants should be compatible with


a regular centre in the first place. Using the subscript 0 todenote central values, we obtain

𝜌0=6𝑘2𝐶2

2− 8𝑘2𝐶2

1+ 2𝛼𝑘𝐶

1+ 𝛼2

𝑘≤ 0, (62)

𝑝0=6𝑘2𝐶2

2− 4𝑘2𝐶2

1− 𝛼2

𝑘≥ 0, (63)

0 < (𝑑𝑝

𝑑𝜌)

0

= 1 −𝛼

2𝑘𝐶1

< 1, (64)

where it must be recalled that 𝛼 and 𝑘 are positive. The right-most inequality of (64) implies that 𝐶

1> 0, and the left

inequality requires that 𝐶1> 𝛼/2𝑘. In addition, (62) and

(63) lead to 𝛼2 + 4𝑘2𝐶21≤ 6𝑘2𝐶2

2≤ 8𝑘2𝐶2

1+ 2𝛼𝑘𝐶

1+ 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 𝐶

2= −2 which produces pleasing physical

properties. Graphical representations of the dynamical andenergy profiles are shown in Figures 1 and 2, respectively.

From Figure 1, we observe that both the energy andpressure are positive definite and monotonically decreasingfunctions of the space variable 𝑥. Importantly, there existsa pressure-free hypersurface at 𝑥 = 0, 11458 which iden-tifies a boundary for the distribution of the perfect fluid.Therefore, this model may succeed in modeling astrophysicalphenomena. In addition, it can be seen that the adiabaticsound-speed index 𝑑𝑝/𝑑𝜇 remains constrained between 0and 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 thegravitational field. Figure 2 reveals that the weak, strong, anddominant 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, pressureand sound-speed index are given by

𝜇 = 3𝑘𝑗2

+ (−𝛼

2𝑘+ 𝑗) (𝛼 + 4𝑘𝑗) ,

𝑝 = −3𝑘𝑗2

− (−𝛼

2𝑘+ 𝑗) (𝛼 + 2𝑘𝑗) ,

𝑑𝑝

𝑑𝜇= 1 −

𝛼

2𝑘𝑗,

(65)

where we have defined 𝑗 = 𝐶1cos√−𝑘𝑥 + 𝐶

2sin√−𝑘𝑥 and

𝑗 = 𝐶2cos√−𝑘𝑥 − 𝐶

1√−𝑘𝑥. The quantities

𝜇 − 𝑝 = 6𝑘𝑗2

+(−𝛼 + 2𝑘𝑗) (𝛼 + 3𝑘𝑗)

𝑘, (66)

𝜇 + 𝑝 =𝑘𝑗 (−𝛼 + 𝑘𝑗)

𝑘, (67)

𝜇 + 3𝑝 = −6𝑘𝑗2

−(−𝛼 + 2𝑘𝑗) (𝛼 + 𝑘𝑗)

𝑘(68)

will assist in studying the energy conditions.

0.02 0.04 0.06 0.08 0.10 0.12 0.14

2

4

6

8

dp

d𝜇p

𝜇

Figure 1: Graph of energy density (𝜇), pressure (𝑝) and sound-speedindex (𝑑𝑝/𝑑𝜇).

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

2

4

6

8

10

12𝜇 + 3p

𝜇 + p

𝜇 − p

Figure 2: Graph of 𝜇 − 𝑝, 𝜇 + 𝑝, and 𝜇 + 3𝑝.

As above we now need to find suitable constants in orderto create a viable fluid model. Again we examine the abovequantities at the central axis 𝑥 = 0. We obtain

𝜌0=6𝑘2𝐶2

2+ (−𝛼 + 2𝑘𝐶

1) (𝛼 + 4𝑘𝐶

1)

2𝑘> 0, (69)

𝑝0=−6𝑘2𝐶2

2− (−𝛼 + 2𝑘𝐶

1) (𝛼 + 2𝑘𝐶

1)

2𝑘≥ 0, (70)

0 ≤𝑑𝑝

𝑑𝜇= 1 −

𝛼

2𝑘𝐶1

< 1 (71)

as constraints on the constants. Now, from (71), we infer that𝑟 < 0 and also that 𝛼 < 2𝑘𝐶

1recalling that 𝛼 > 0 and 𝑘 < 0 in

this case. Furthermore, reconciling (69) and (70), we obtainthe relations 𝛼2 − 4𝑘2𝐶2

1< 6𝑘2𝐶2

2< 𝛼2+ 2𝛼𝑘𝐶

1− 8𝑘2𝐶2

1.

Disregarding the middle term in this inequality, we derivethat 𝛼 > 2𝑘𝐶

1which contradicts our earlier conclusion.

Therefore, it is impossible to satisfy the three most importantphysical requirements 𝜌 > 0, 𝑝 ≥ 0, and 0 < 𝑑𝑝/𝑑𝜇 < 1with any combination of the constants in the problem. Inother words, the case 𝑘 < 0 cannot represent a realistic fluid.Therefore, we do not pursue any further consideration of thiscase.


6. Nonstatic Solutions

Finding nonstatic exact solutions of the Einstein equations fora particular configuration ofmatter is nontrivial.Themethodof conformal transformations gives us the ability to constructnon-static solutions from known static metrics. In the caseunder investigation, we suppose that the conformal factor hasthe simple form 𝑈 = 𝑈(𝑡). The field equations then have thefamiliar form

𝜇 = 3𝑓2

𝑡+ 𝑓 (4𝑘𝑓 + 3𝛼) ,

𝑝 = −3𝑓2

𝑡+ 𝑓 (2𝑓

𝑡𝑡− 𝛼) ,

tanh V =− (2𝑘𝑓 + 𝛼)

2𝑓𝑡𝑡+ 2𝑘𝑓 + 𝛼

,

0 = (2𝑘𝑓 + 𝛼) (2𝑓𝑡𝑡+ 2𝑘𝑓 + 𝛼) ,

(72)

where now 𝑓 = 𝑓(𝑡) only. Once again, the conclusion𝑓(𝑡) = 𝛼/2𝑘 leads to a trivial constant conformal factor, sowe disregard it. We examine the other possibility

2𝑓𝑡𝑡+ 2𝑘𝑓 + 𝛼 = 0 (73)

more closely. Equation (73) is a simple second-order differen-tial equation and is solved by

𝑓 (𝑥) =

{{{{{{

{{{{{{

{

−𝛼

4𝑡2+ 𝐶1𝑡 + 𝐶2, 𝑘 = 0,

−𝛼

2𝑘+ 𝐶1cos√𝑘𝑡 + 𝐶

2sin√𝑘𝑡, 𝑘 > 0,

𝛼

2𝑘+ 𝐶1cosh√−𝑘𝑡 + 𝐶

2sinh√−𝑘𝑡, 𝑘 < 0,

(74)

where 𝐶1and 𝐶

2are integration constants. Let us examine

each case in turn.

6.1. The Case 𝑘 = 0. This case degenerates into a conformalflat metric and so it is not worthy of further attention as suchmetrics have already been discovered.

6.2. The Case 𝑘 > 0. If we make the change of variables 𝑙 =𝐶1cos√𝑘𝑡+𝐶

2sin√𝑘𝑡, and �̃� = 𝐶

2cos√𝑘𝑡−𝐶

1sin√𝑘𝑡 then

the relevant dynamical quantities have the following forms:

𝜇 = −3𝑘�̃�2

−1

2𝑘(𝛼 − 2𝑘𝑙) (𝛼 + 4𝑘𝑙) ,

𝑝 = 3𝑘�̃�2

+1

2𝑘(𝛼 − 2𝑘𝑙)

2

,

𝑑𝑝

𝑑𝜇=

𝛼 − 14𝑘𝑙

2 (𝛼 + 𝑘𝑙).

(75)

It is pleasing to note that the positive definiteness of pressurecriterion is valid everywhere for this case 𝑘 > 0. We alsoexpect all physical constraints to be satisfied at the initial

condition 𝑡 = 0. Denoting initial values with the subscript0, we have that the following inequalities must be satisfied:

𝜇0=−6𝑘2𝐶2

2− 𝛼2− 2𝛼𝑘𝐶

1+ 8𝑘2𝐶2

1

2𝑘> 0, (76)

𝑝0=6𝑘2𝐶2

2+ 𝛼2− 4𝛼𝑘𝐶

1+ 4𝑘2𝐶2

1

2𝑘≥ 0, (77)

0 < (𝑑𝑝

𝑑𝜇)

0

=𝛼 − 14𝑘𝐶

1

2 (𝛼 + 𝑘𝐶1)< 1. (78)

The first two relationships constrain 𝑘 as

− 𝛼2

+ 4𝛼𝑘𝐶1− 4𝑘2

𝐶2

1

≤ 6𝑘2

𝐶2

2< −𝛼2

− 2𝛼𝑘𝐶1+ 8𝑘2

𝐶2

1,

(79)

and the outer two expressions must satisfy 𝐶1< 0 or 𝐶

1>

𝛼/2𝑘. The adiabatic sound-speed criterion (78) forces thevalue of 𝐶

1as −𝛼/16𝑘 < 𝐶

1< 𝛼/14𝑘 since 𝛼 > 0 and

𝑘 > 0. An additional constraint on 𝐶1is implied by requiring

the positivity of the right hand side of (79). This is satisfiedprovided that 𝐶

1< −𝛼/4𝑘 or 𝐶

1> 𝛼/2𝑘. Now, it can

immediately be seen that these inequalities are never satisfiedsimultaneously for any value of 𝐶

1. Accordingly, we deduce

that no physical reasonable model will result from this case.That is, positivity of pressure and energy density as well as asubluminal sound speed cannot all be achieved for any valueof the constant𝐶

1.There is little value in considering the value

of 𝐶2in 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

𝜇 =6𝑘 ∗ Ṽ2 + 5𝛼2 + 14𝛼𝑘V + 8𝑘2V2

2𝑘,

𝑝 =−6𝑘Ṽ2 − 𝛼2 + 4𝑘2V2

2𝑘,

𝑑𝑝

𝑑𝜇= −7 (

𝛼

2𝑘V+ 1) ,

(80)

where we have again redefined V = 𝐶1cosh√−𝑘𝑡 +

𝐶2sinh√−𝑘𝑡 and Ṽ𝐶

2cosh√−𝑘𝑡 + 𝐶

1sinh√−𝑘𝑡. Reasoning

as before togetherwith empirical testing corroborates that it isnot possible to achieve simultaneously a positive pressure andenergy density. Additionally, the sound-speed index is alwaysfound to be negative. Therefore, this model cannot model arealistic distribution of stellar fluid.

7. A Spacetime Foliation 𝑈=𝑈(𝑡, 𝑥)

The Lie group analysis method has proved successful in gen-erating broader classes of solutions than the ones discussedpreviously. The functional forms for 𝑓(𝑡, 𝑥) for ℎ = 0 werepreviously obtained byHansraj et al. [27] using group analysistechniques. We now take these solutions and substituteinto expressions (47) and (48) to establish the dynamical


quantities, energy density (𝜇) and pressure (𝑝).Thereafter, weverify that the pressure and density profiles satisfy conditionsfor physical acceptability. A rigorous analytical treatmentis prohibitive for general 𝑓(𝑡, 𝑥) and ℎ(𝑦, 𝑧), in view ofthe complexity of the resultant expressions. Therefore, forgraphical purposes, it is necessary to consider a simplifiedsituation such as foliations of the distribution in terms of thetemporal and one space variable. It is not possible to visuallyrepresent the complete 4-dimensional solution spacetime.We accordingly select the particular solution

𝑓 (𝑥, 𝑡) = 𝐴 sinh√𝑘 (𝑡2 − 𝑥2) + 𝐵 cosh√𝑘 (𝑡2 − 𝑥2) (81)

presented by Hansraj et al. [27] together with that obtainedfrom 𝑍

1. The conformally related metric (33) may now be

given by

𝑑𝑠2

= (−𝑑𝑡2

+ 𝑑𝑥2

+ 𝑐1𝑒𝑧2/8𝑦2

(𝑧

𝑦)

3/4

×(𝑧2

𝑦2− 1)

1/2

(𝑑𝑦2

+ 𝑑𝑧2

))

× ((𝑒𝑐1+𝑧2/8𝑦2

𝑦3/4

𝑧1/4

+ 𝐵 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐴 sinh [√𝑘 (𝑡2 − 𝑥2)])2

)

−1

,

(82)

and the energy density takes the form

𝜇 = − 3(𝑐1𝑒𝑧2/8𝑦2

(𝑧

𝑦)

3/4

(𝑧2

𝑦2− 1)

1/2

)

−1

× ((𝑒𝑐1+𝑧2/8𝑦2

𝑦3/4

4𝑧3/4+𝑒𝑐1+𝑧2/8𝑦2

𝑧5/4

4𝑦5/4)

2

+(3𝑒𝑐1+𝑧2/8𝑦2

𝑧1/4

4𝑦1/4−𝑒𝑐1+𝑧2/8𝑦2

𝑧9/4

4𝑦9/4)

2

)

+ (𝑒𝑐1+𝑧2/8𝑦2

𝑦3/4

𝑧1/4

+ 𝐵 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐴 sinh [√𝑘 (𝑡2 − 𝑥2)])

× (−2𝑒𝑐1+𝑧2/8𝑦2

𝑘𝑦3/4

𝑧1/4

− 3𝛼

+ 4𝑘 (𝐵 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐴 sinh [√𝑘 (𝑡2 − 𝑥2)]))

+ 3((

𝐴𝑘𝑡 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

+

𝐵𝑘𝑡 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

)

2

−(−

𝐴𝑘𝑥 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

−

𝐵𝑘𝑥 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

)

2

),

(83)

while the pressure is given by

𝑝 = 3(𝑐1𝑒𝑧2/8𝑦2

(𝑧

𝑦)

3/4

(𝑧2

𝑦2− 1)

1/2

)

−1

× ((𝑒𝑐1+𝑧2/8𝑦2

𝑦3/4

4𝑧3/4+𝑒𝑐1+𝑧2/8𝑦2

𝑧5/4

4𝑦5/4)

2

+(3𝑒𝑐1+𝑧2/8𝑦2

𝑧1/4

4𝑦1/4−𝑒𝑐1+𝑧2/8𝑦2

𝑧9/4

4𝑦9/4)

2

)

−𝑘 (3𝑡2+ 𝑥2)

(𝑡2 − 𝑥2)(𝐴 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐵 sinh [√𝑘 (𝑡2 − 𝑥2)])2

+ (𝑒𝑐1+𝑧2/8𝑦2

𝑦3/4

𝑧1/4

+ 𝐵 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐴 sinh [√𝑘 (𝑡2 − 𝑥2)])

×(−𝛼 + 2

×(𝑒𝑐1+𝑧2/8𝑦2

𝑘𝑦3/4

𝑧1/4

+ 𝐴(−

𝑘2𝑡2 cosh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2


+

𝑘 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

+

𝑘𝑡2 sinh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2)

− 𝐴(−

𝑘2𝑥2 cosh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

−

𝑘 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

+

𝑘𝑥2 sinh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2)

− 𝐵(

𝑘𝑥2 cosh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2

−

𝑘2𝑥2 sinh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

−

𝑘 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

)

+ 𝐵(−

𝑘𝑡2 cosh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2

−

𝑘2𝑡2 sinh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

+

𝑘 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

))).

(84)

In order to obtain an indication of the model’s feasibilityto represent a realistic distribution, we elect to make agraphical study of this solution. The energy density andpressure are given by

𝜇 = 4𝑘 (𝐵 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐴 sinh [√𝑘 (𝑡2 − 𝑥2)])2

− 3((

𝐴𝑘𝑡 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

+

𝐵𝑘𝑡 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

)

2

−(−

𝐴𝑘𝑥 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

−

𝐵𝑘𝑥 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

)

2

),

𝑝 = 2 (𝐵 cosh [√𝑘 (𝑡2 − 𝑥2)] + 𝐴 sinh [√𝑘 (𝑡2 − 𝑥2)])

×(𝐴(−

𝑘2𝑡2 cosh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

+

𝑘 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

+

𝑘𝑡2 sinh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2)

− 𝐴(

−𝑘2𝑥2 cosh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

−

𝑘 cosh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

+

𝑘𝑥2 sinh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2)

− 𝐵(

𝑘𝑥2 cosh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2

−

𝑘2𝑥2 sinh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

−

𝑘 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

)


+ 𝐵(

𝑘𝑡2 cosh [√𝑘 (𝑡2 − 𝑥2)]

𝑡2 − 𝑥2

−

𝑘2𝑡2 sinh [√𝑘 (𝑡2 − 𝑥2)]

(𝑘 (𝑡2 − 𝑥2))3/2

+

𝑘 sinh [√𝑘 (𝑡2 − 𝑥2)]

√𝑘 (𝑡2 − 𝑥2)

))

−𝑘 (3𝑡2+ 𝑥2)

(𝑡2 − 𝑥2)

× (𝐴 cosh [√𝑘 (𝑡2 − 𝑥2)]

+𝐵 sinh [√𝑘 (𝑡2 − 𝑥2)])2

,

(85)

We need to select appropriate values for the parameters𝐴, 𝐵, and 𝑘 to finalise the model. It should be observed atthis stage that, in many situations, it is possible to obtainbounds for the constants by examining the central conditions𝑝(𝑡, 0, 0, 0) and 𝜇(𝑡, 0, 0, 0). We then require that all thephysical conditions are satisfied at the centre. This is notfruitful in the present context. Furthermore, the approachin fixing integration constants in most problems in generalrelativity involves matching of the perfect fluid solutionwith its corresponding vacuum solution. For example, ifwe were devising a model of a static neutral sphere, thenintegration constants can be fixed by demanding continuityof 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 ofradiating spheres, this pressure-free interface does not exist,and this condition is modified to include the effects ofradiation, and the junction conditions must be establishedvia other means. The difficulty in our analysis is that we haveno known vacuum solution. Consequently, our selection ofthe constants is completely random. We are fortunate thatthe available computing technology is able to quickly checkthe efficacy of our choices for the integration constants (seeFigures 3 and 4).

These plots, generated viaMathematica [35], demonstratemany pleasing features. For example, it is evident that, inthe region chosen, the pressure and energy density are bothpositive. These are the most basic requirements for modelsto serve as candidates for realistic celestial phenomena.Therefore, it is possible for this solution to model a realisticconfiguration of perfect fluid.

8. Conclusion

Wehave demonstrated that new static and nonstatic solutionsof the Einstein field equations may be constructed from

y

5

4

3

2

1

x

1 2 3 4 5

f

47464544

Figure 3: Plot of pressure versus 𝑡 and 𝑥 with 𝐴 = 1, 𝐵 = 10, and𝑘 = 0.1.

y

5

4

3

2

1

x

12

34

5

f

500400300200100

Figure 4: Plot of energy density versus 𝑡 and 𝑥 with 𝐴 = 1, 𝐵 =10, and 𝑘 = 0.1.

known solutions through the use of conformal mappings.Even solutions that are considered as defective may serveas seeds for new solutions which are physically reasonable.The reason for the success of this approach is that theseed solution is already a solution of the possibly vacuumEinstein field equations. The Einstein tensor splits up neatlyinto a part containing the original Einstein tensor and anexpression involving the conformal factor. It is this latter partthat must be coupled to an appropriate perfect fluid. Themathematical reason for this behaviour is that the conformalKilling vectors of the original seed metric become Killingvectors of the new metric. This generates symmetries whichhave the effect of simplifying the resultant field equations.While 4-dimensional solutions to the field equations havealready been reported, solutions had not been analysedfor physical plausibility. We have endeavoured to constructexplicit models by considering cases where the conformalfactor is a function of a space variable 𝑈 = 𝑈(𝑥), a functionof the temporal coordinate 𝑈 = 𝑈(𝑡) only and finally thecase of the conformal factor as a function of space and time𝑈 = 𝑈(𝑡, 𝑥). Analyses of each case show that it is possible to


generate physically reasonable models through this methodof conformal transformations.

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