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Academic Editor:B. G. Konopelchenko

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

Received 10 November 2013; Accepted 6 December 2013; 9 January 2014

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.

1. Introduction

Spherically symmetric gravitating models with heat flow, in the absence of shear, are important in the study of various cosmological processes and the evolution of relativistic astrophysical bodies. For a variety of applications in the presence of inhomogeneity, see Krasinski [1]. Heat flow models are also important in analysing gravitational collapse and relativistic stellar processes. Astrophysical studies in which heat flow is important include the shear-free models of Wagh et al. [2], Maharaj and Govender [3], Misthry et al. [4], and Herrera et al. [5]. By studying shear-free models, we avail ourselves with a rather simpler avenue where we only need to provide solutions to the generalised condition of pressure isotropy containing two metric functions. A complete study of shear-free heat conducting fluids with charge was completed by Nyonyi et al. [6] using Lie's group theoretic approach applied to differential equations. Shearing models where heat flow is significant have been recently studied by Thirukkanesh et al. [7] for radiating spherically symmetric spheres. It turns out that the resulting nonlinear equations with shear are much more difficult to analyse.

A generic method of obtaining new solutions to the Einstein field equations with heat flow was provided by Deng [8]. Using this general method, we can regain existing results and obtain new classes of solutions. Nyonyi et al. [6], Ivanov [9], and Msomi et al. [10] have obtained new solutions using the Lie group theoretic approach and other methods, by solving the underlying pressure isotropy condition. These investigations are applicable to four dimensions. Extensions to higher dimensions have also been considered by many authors because of physical requirements; for example, Bhui et al. [11] showed the absence of horizons in nonadiabatic gravitational collapse. Studies of this type motivated the Lie symmetry analysis of heat conducting fluids by Msomi et al. [12] in dimensions greater than four. In the present treatment, we extend the Deng [8] algorithm to higher dimensions and show that new results are possible.

2. The Model

We consider the line element of a shear-free, spherically symmetric (n+2) -dimensional manifold in the form [figure omitted; refer to PDF] where n...5;2 . The gravitational potential components D and V are functions of r and t with [figure omitted; refer to PDF] For a heat conducting fluid, the energy momentum tensor is given by [figure omitted; refer to PDF] where ρ is the energy density, p is the kinetic pressure, q is the heat flux tensor, and U is a timelike (n+2) -velocity vector. For a comoving observer, we have ^Ua =(1/D,0,0,...,0) and ^qa =(0,q,0,...,0) .

Utilizing (1)-(3), we obtain the Einstein field equations [figure omitted; refer to PDF] which are consistent with the derivation of Bhui et al. [11]. Equations (4b) and (4c), together with the transformation u=^r2 , give the pressure isotropy condition [figure omitted; refer to PDF] which is the master equation for the gravitating fluid in (n+2) -dimensions.

Deng [8] provided a general recipe for generating a series of solutions of the isotropy condition (5) for n=2 . This technique may be extended to the master equation (5). Note that the isotropy condition is an ordinary differential equation in u (since no time derivatives appear) which may be reduced to a simpler differential equation in D if V is known and vice versa. In this technique, the elementary forms of either V or D are chosen that are in turn substituted into the equation to be solved so as to obtain the form of the remaining term. Below we give a brief outline of the method.

(1) Take a simple form of V , say V=_V1 , and substitute it into the master equation to find the most general solution of D , say D=_D1 . The pair V=_V1 and D=_D1 provides the first class of solutions to (5).

(2) Take D=_D1 and substitute it into the master equation. This gives an equation in V with V=_V1 already a solution. We are now in a position to obtain a second solution V=_V2 linearly independent of _V1 . The linear combination _V3 =a_V2 +b_V1 gives the general solution that satisfies the master equation. The pair V=_V3 and D=_D1 is the second class of solutions to (5).

(3) Take V=_V3 and substitute it into the master equation. We obtain an equation in D with D=_D1 already a solution. We are then in a position to obtain D=_D2 in the same way we obtained _V2 . The pair V=_V3 and D=c_D1 +d_D2 is the third class of solutions to (5).

(4) Repeat the above process to obtain an infinite sequence of solutions.

It is important to note that, in principle, this is a nonterminating process for obtaining solutions, and an infinite number of solutions can be listed. The difficulty arises in obtaining subsequent solutions in the process because the integration may become more complicated. However, the algorithm proves to be a powerful mechanism for generating new solutions.

3. Results

We start with a simple case [figure omitted; refer to PDF] Equation (5) reduces to [figure omitted; refer to PDF] This equation can be solved directly to obtain [figure omitted; refer to PDF] where a and b are arbitrary functions of t . The pair of equations (6) and (8) gives the first class of solutions [figure omitted; refer to PDF] Observe that the dimension n is absent in metric (9). In this class of solutions the potentials are independent of the dimensionality; however, the spatial volume depends on n and the matter variables are affected through the field equations. When n=2 , we regain the results of Bergmann [13].

On substituting (8) back into (5) we obtain [figure omitted; refer to PDF] The general solution to (10) is [figure omitted; refer to PDF] with c and d arbitrary functions of t . The pair of equations (8) and (11) gives the second class of solutions [figure omitted; refer to PDF] Again, we observe that the dimension n does not appear explicitly in (12). This means that the potentials in (12) are independent of the dimension. When n=2 , we regain the solutions obtained by Maiti [14] and later generalised by Modak [15] and Sanyal and Ray [16]. We make the general point that, for a linear form of V , the parameter n does not appear in (5). Thus all solutions with a linear form of V do not contain the dimension n , thereby leading to the metrics (9) and (12).

Now, substituting (11) into (5), we obtain [figure omitted; refer to PDF] We require two independent solutions _V1 and _V2 of (13). Note that _V1 in (8) is a solution to (13). We propose the second solution to (13) to be given by [figure omitted; refer to PDF] where the function α(u,t) has to be found explicitly. On substituting (14) into (13), we obtain [figure omitted; refer to PDF] On integrating (15), we obtain α expressed as [figure omitted; refer to PDF] Consequently, the second solution _V2 will depend on the dimension n . To evaluate integral (16), we need to consider two cases: ad=bc and ad...0;bc .

3.1. Case I

When ad=bc we have [figure omitted; refer to PDF] where g is an arbitrary function of t . Therefore _V2 becomes [figure omitted; refer to PDF] This implies that _V2 is proportional to _V1 and is therefore not a second linearly independent solution. The case ad=bc is degenerate.

3.2. Case II

When ad...0;bc , we have [figure omitted; refer to PDF] where a , b , c , d , e , and g are arbitrary functions of t . Therefore the second solution _V2 becomes [figure omitted; refer to PDF] And since the general solution to (5) is a linear combination of _V1 and _V2 , we obtain [figure omitted; refer to PDF] where we have introduced for convenience h(t) and j(t) . The third class of solutions is therefore given by (11) and (21) with metric [figure omitted; refer to PDF] This is a new class of solutions and it is evident that it certainly depends on the dimension n...5;2 . Therefore we can conclude that the dimensionality of the problem does indeed affect the dynamics of the gravitational field with heat flow. The next class of solutions can be obtained by substituting _V3 into (5) and then solving the resulting equation for _D3 . This may be continued to obtain further new solutions. The integration process gets more complicated for further iterations.

We now consider the special case of four dimensions. When e=1 and n=2 , the line element (22) becomes [figure omitted; refer to PDF] We can rewrite (23) in the equivalent form [figure omitted; refer to PDF] where the function κ is given by [figure omitted; refer to PDF] When we set κ=0 in (24), we regain the result of Deng [8]. We interpret (22) as the higher dimensional generalisation of the Deng model with heat flow.

4. Example

We illustrate the validity of our solutions by considering a simple example with physically viable conditions. For the line element (22), we make the simple choice: a=d=0 , b=c=1 , h+jg=1 , and e=j=1 . This gives the simplified forms of the potentials [figure omitted; refer to PDF] Even with this simple example, a qualitative analysis of the matter variables and energy conditions for the interior matter distribution is arduous. We therefore generate graphical plots on a constant timelike hypersurface to illustrate the validity of our solutions using this example. Figures 1, 2, and 3 are the plots for the energy density ρ , the pressure p , and the heat flow q for three different dimensions: n=2 (dashed line), n=3 (solid line), and n=4 (dotted line). It is clearly evident that the matter variables are positive and they decrease with the increase in dimension. This is due to the fact that an increase in dimension translates to an increase in the number of degrees of freedom leading to a decrease in the mass per unit volume of the gravitating fluid. In Figure 4, we have plotted the speed of sound. From Figure 4, we observe that causality is not violated for the dimensions n=2 , 3 , and 4 . In Figures 5, 6, and 7, we have plotted the quantities A=ρ-p+Δ , B=ρ-3p+Δ , and C=2p+Δ , where Δ=(p+q⁾² -4^q2 for n=3 . We observe that A , B , and C are positive; hence, the weak, dominant, and strong energy conditions are satisfied. Therefore the matter distribution for this example is physically reasonable.

Figure 1: Energy density ρ .