The effect of a two-fluid atmosphere on relativistic stars

We model the physical behaviour at the surface of a relativistic radiating star in the strong gravity limit. The spacetime in the interior is taken to be spherically symmetrical and shear-free. The heat conduction in the interior of the star is governed by the geodesic motion of fluid particles and a nonvanishing radially directed heat flux. The local atmosphere in the exterior region is a two-component system consisting of standard pressureless (null) radiation and an additional null fluid with nonzero pressure and constant energy density. We analyse the generalised junction condition for the matter and gravitational variables on the stellar surface and generate an exact solution. We investigate the effect of the exterior energy density on the temporal evolution of the radiating fluid pressure, luminosty, gravitational redshift and mass flow at the boundary of the star. The influence of the density on the rate of gravitational collapse is also probed and the strong, dominant and weak energy conditions are also tested. We show that the presence of the additional null fluid has a significant effect on the dynamical evolution of the star.


Introduction
A compact star is formed when a massive star (M * ≥ 8M ⊙ ) breaks away from a state of hydrostatic equilibrium and collapses under its own gravity. This situation usually 1 email:maharaj@ukzn.ac.za arises when the massive star has reached the end of the first phase of its evolution.
The stellar object that is formed at the end of the gravitational collapse process is very dense and small in size with a typical radius of the order of (7 − 20)km for ultracompact quark-gluon stars, strange stars and pulsars (including unmagnetized neutron stars). As a result of the high mass density in the resulting configuration, the gravitational field throughout the interior is significantly strong and the star has effectively transitioned into the so called strong gravity regime. At this point the object is a far more enhanced self gravitating system and is termed a relativistic star since the theory of general relativity would be required if we are to construct a plausible and realistic model for its global dynamics. For detailed and comprehensive reviews on the theory of nonadiabatic gravitational collapse and relativistic stars the reader is referred to the works of Oppenheimer and Snyder 1 , Penrose 2 , Glendenning 3 , Strauman 4 and Shapiro and Teukolsky 5 , among others. During the contraction of the massive star and the evolution of the compact object, gravitational binding energy is converted into heat energy which is used for ionization and dissociation in the dense interior. The excess heat energy is dissipated as radiation across the stellar surface. This process is described adequately for a relativistic star by constructing the junction conditions at the surface. These equations relate the interior and exterior matter variables as well as the respective geometries.
It was in the pioneering work of Oppenheimer and Snyder 1 that the significant role of general relativity in modeling the nonadiabatic gravitational collapse of stellar bodies was established. The dynamics was studied by considering the contraction of a spherically symmetric dust cloud and it was realised that relativistic gravity could adequately describe the outflow of heat energy in the form of pressureless radiation which is often referred to as null dust. While this seminal investigation provided an initial framework in which to construct models for the internal evolution of the dissipating object, it was not completely clear what the conditions were at the boundary or surface. Moreover it was also not clear as to what the appropriate geometry of the exterior region was, i.e. there was no known exact solution to the Einstein field equations that described the gravitational field of a radiating star. Interestingly, it was some time later that the first solution that describes this scenario was obtained, and this has since then become a crucial and essential ingredient when studying stellar like objects with heat flow and strong gravitational fields in astrophysics. The Vaidya 6 metric is the most well known and widely used solution in general relativity that describes the exterior gravitational field of a dense object exhibiting outward radial heat flow. It defines outgoing null radiation and is written in terms of the mass of the radiating body. This result provided a marked advancement in the modeling process and it was eventually when the so called junction conditions for radiating stars was generated by Santos 7 that general relativity provided a satisfactory framework in which these objects could be studied in greater detail. The Santos junction conditions are physically very meaningful, for localised astrophysical objects because they reveal that the pressure of the radiating stellar fluid is not zero at the boundary but rather that the pressure is proportional to the magnitude of the heat flux. This seemingly simple mathematical relation has to be solved as a highly nonlinear differential equation; the solutions of which provide the evolution of the gravitational field potentials and in turn the behaviour of the matter field. However, it must be noted that this standard framework only describes the emission of pressureless null radiation (photons) into the exterior region of the dissipating relativistic star. It has not been used to probe the outflow of any other type of observable radiation or elementary particles like neutrinos, which are thought to be significant carriers of heat energy in stars and released by particle production processes at the stellar surface. Recently, Maharaj et al 8 have generalised the Santos junction condition by matching the interior geometry of a spacetime containing a shear-free heat conducting stellar fluid to the geometry of an exterior region that is described by the generalised Vaidya metric which contains an additional Type II null fluid. This new result provides a greater array of possibilities with regards to the modeling of relativistic objects in astrophysics as it depicts a more general exterior region for a radiating star, which is comprised of a two-fluid system: a combination of the standard null radiation as in the case of the Santos framework, and an additional fluid distribution which is more general and can be taken to be another form of radiation or better still, a field of particles such as neutrinos, as mentioned above. An interesting feature of this generalised junction condition is that the radiating fluid pressure at the boundary is not only coupled to the internal heat flux but also to the non-vanishing energy density of the Type II null fluid. This conveys a direct relationship between the evolution of the interior and exterior matter and gravitational fields, at the boundary of the star and consequently may yield physical behaviour that is far different and perhaps more realistic from that which arises in the standard scenario.
A substantial amount of work on relativistic radiating stars has been carried out in the standard Santos framework. In the investigations of de Oliveira et al 9 and Maharaj and Govender 10 , the Santos junction conditions were generalised to include the effects of an electromagnetic field and shearing anisotropic stresses during dissipative stellar collapse. More recently, analytical models for shear-free nonadiabatic collapse in the presence of electric charge were obtained by Pinheiro and Chan 11 . The influence of pressure anisotropy, shear and bulk viscosity on the density, mass, luminosity and effective adiabatic index of an 8M ⊙ contracting radiating star was studied by Chan 12,13 .
These results were later extended by Pinheiro and Chan 14 . Misthry et al 15 generated nonlinear exact models in the shear-free regime for relativistic stars with heat flow, using a group theoretic approach that involves the Lie symmetry analysis of the Santos junction condition. Following this, Abebe et al 16 , employing the same technique, investigated the behaviour of radiating stars in conformally flat spacetime manifolds.
Abebe et al 17,18 used Lie analysis to find radiating Euclidean stars with an equation of state. Collapse models with internal pressure isotropy and vanishing Weyl stresses were also probed by Maharaj and Govender 19 . They investigated the dynamical stability of the dissipating stellar fluid and demonstrated that the configuration was more unstable close to the centre than in the outer regions. It is also well understood that the thermal This paper is organised as follows: In section 2 we present the basic theory for relativistic radiating stellar models with spherically symmetric shear-free interiors and two-fluid exteriors. The generalised junction condition is then integrated as a nonlinear second order differential equation at the boundary, and an exact solution for constant null fluid energy density is generated. The luminosities and gravitational redshift are then defined for the case when the null fluid is present on the outside.

The interior and exterior geometry
The dynamics of the gravitational field in the stellar interior, in the absence of shearing stresses, is governed by the spacetime line element where A(r, t) and B(r, t) are the relativistic gravitational potentials. The matter inside the star is defined by a relativistic fluid with heat conduction in the form of the energy momentum tensor Here µ and p are the fluid energy density and isotropic pressure respectively. The components g ab represent the metric tensor field and u and q, are the fluid four-velocity and the radial heat flux, respectively. The Einstein field equations G − ab = T − ab for the interior may be written as where dots and primes denote differentiation with respect to coordinate time t and radial distance r respectively.
In the local region outside the star the dynamics of the gravitational field may be described by the generalised Vaidya outgoing radiation metric which has the following form where m(v, r) represents the mass flow function at the surface, and is related to the gravitational energy within a given radius r. The characteristic feature about the metric (4) is that the mass function also contains a spatial dependence in the radial direction during dissipation; this is significantly different form the standard scenario in which the mass at the boundary only has a time dependance. Husain 29 and Wang and Wu 30 have shown that an energy momentum tensor that is consistent with (4) is which is a superposition of null radiation and an arbitrary null fluid. Here ε is the energy density of the photon radiation, and ρ and P are the energy density and pressure of the additional null fluid, respectively. In general, T + ab represents a Type It is interesting to note that in the standard framework, the Santos 7 junction condition tells us that on the boundary of the star the pressure of the stellar fluid is proportional to the magnitude of the heat flux This condition has recently been extended to incorporate the generalised Vaidya radiating solution. Maharaj et al 8 have shown from first principles that the junction condition (describing the dissipation) for the metric (4) can be written in the form Note here, that this result is more general and contains the Santos equation (7) as a special case when ρ = 0, i.e. when the additional null fluid is absent and the outside contains only pure radiation. It is also important to observe that the generalised boundary equation (8) is new for spherically symmetric nonadiabatic stellar models and consequently have no reported exact solutions. Furthermore, the new result is quite extensive and may be applicable to more physically realistic astrophysical scenarios as it includes the effect of the null fluid energy density. As mentioned earlier, the null fluid and its nonvanishing energy density have been explored in the context of cosmological and more localised string fluid distributions in four dimensions. It may also be appropriate for the description of neutrino outflow from compact relativistic stellar objects in which nonadiabatic and particle production processes prevail. In view of this it is important to emphasize that the generalised null fluid models should allow for significant improvement on the results obtained by Glass 40 , following the seminal treatment by Misner 41 . that an explicit relationship between the energy density ρ and the pressure P can be obtained. By simply differentiating through (6b) with respect the exterior radial coordinate r and using (6c) we see that

Qualitative features of the exterior null fluid
This equation indicates that there exists, in general, a linear connection between the pressure and the energy density. Moreover, (9) also suggests that given a form for the density, we can determine both the magnitude and the sign of the pressure. The sign of the pressure is crucial as it reveals more about the nature of the null fluid and the interaction of the fluid with the gravitational field. If an arbitrary functional form is prescribed for the density then it is more difficult to determine the sign of the pressure, as this would depend on the actual expressions for the terms in brackets. However, if as in the case of the model presented in this work, the density at the surface is taken to be constant then it is straightforward to obtain the signature of P since for physically meaningful applications the energy density has to be strictly positive (ρ > 0).
With this in mind it turns out that for a constant null fluid energy density at the surface, the pressure P is always negative. This is consistent with the ideas exploited in cosmological scenarios where a fluid characterised by the cosmological constant or as dark energy or a scalar field, has to have a negative pressure in order to counteract the attractive force of gravity induced by matter and large scale structure in the universe.
In a more localised astrophysical setting as in this study, the interpretation may be somewhat different. We aim now to generalise the definitions for the fluid luminosity and gravitational redshift in the context of our new model. The null radiation emitted by a relativistic object with heat flow, experiences a gravitational redshift (change in energy) due to its motion in the gravitational field. This redshift is generally written as

Luminosity and gravitational redshift
where L Σ is the luminosity of the radiating fluid at the surface and L ∞ is the luminosity of the radiating fluid as seen by a stationary observer at an infinite distance away from the object. These are given respectively, by and In the above m is the mass flow across the boundary Σ, τ is the timelike coordinate on the boundary, and v is the timelike coordinate in the exterior region described by the generalised Vaidya spacetime (4). (For details see Maharaj et al 8 ). The term dv dτ is given in terms of the interior gravitational potentials by The derivative dt dτ is written as The mass flow is given by Differentiating (15) results in the expression Then using the generalised junction condition (8) as well as (3b) in (16) gives which is the generalised result for the mass flow rate when the null fluid is present in the exterior with the generalised Vaidya metric with m = m(v, r). When the null fluid is absent (ρ = 0) we then get Using the derivatives (17), (14), and (13) in the definitions (11) and (12) we generate the following expressions for the surface luminosity L Σ and the asymptotic luminosity . We observe, once again, that when the null fluid is absent in the exterior (ρ = 0), (19a) and (19b) reduce to the following forms, Utilising the system (19) and the definition (10) we may write down the form for the gravitational redshift of the radiating fluid at the surface as Although it appears that the structural form of the redshift has not changed, the magnitude of z Σ will be different since the potentials A and B are obtained by solving equation (8)

Generating an exact solution
We now turn our attention to the process of generating an exact solution to the new boundary condition (8). Considering the high degree of nonlinearity and other mathematical complexities that surround equation (8)  For geodesic motion of fluid particles we have the gravitational potential A = 1.
The condition of pressure isotropy in the absence of shearing fluid stresses admits the following analytical form for the gravitational potential B. It is also important to recall that the above form corresponds to conformally flat tidal gravitational effects and was first obtained by from (3b) and (3d) respectively. With the above system (23) and the form (22) the generalised junction condition given by (8) can be fully expanded as where we have taken r = r Σ = b(= constant), on the stellar surface. The value r Σ = b is the actual radius of the stellar distribution and for compact and ultracompact relativistic bodies like neutron stars and pulsars, and strange or quark stars, is of the order of 7 − 30kms. It is important to note the appearance of the additional term that arises due to the presence of the null fluid energy density ρ = 0. This parameter is taken to be constant on the stellar boundary and as mentioned earlier, is an appropriate and reasonable assumption. In the limit when the density ρ goes to zero, (for which the stellar exterior is composed only of pure radiation), we regain the corresponding We realise that in order to integrate (24), the following transformation can be utilised Then equation (24) may be rewritten as Equation (26) is a Riccati equation in c 1 but is still difficult to solve in general. If we let u = α (constant) then (26) becomeṡ We now make use of the transformation where w(t) is an arbitrary function. Then equation (27) becomes which is a second order linear ordinary differential equation with constant coefficients.
The general solution to equation (29) is given by In the above solution g 1 (t) and g 2 (t) are functions of integration and Then the functions c 1 (t) and c 2 (t) become Consequently the gravitational potential B has the form and the exact solution in metric form is The new exact solution (34)

Physical analysis
In order to establish the role and influence of the null fluid energy density ρ at the surface Σ of the radiating star we probe the temporal behaviour of the fluid and gravitational variables in the generalised scenario with m = m(v, r) described in this instance by the exact solution (34). We compare this with the standard case with m = m(v) when ρ = 0, using the solution which can be found in Thirukkanesh and Maharaj 38 . We point out that these models are cast in natural units and consequently all physical parameters and mathematical constants are dimensionless.
For the purpose of this investigation we plot the quantities B, p, z, L Σ , L ∞ , m and dm dt in Fig. 1-12. We show that the null density ρ drastically affects the physical behaviour of the model in the graphical plots. For the purpose of the graphs we make the following choice for the constants that appear in the exact solution: For these values, (31) places a constraint on the allowed values for the energy density ρ and it becomes clear that 0 < ρ < 1. This restriction on ρ is unique for this model and our particular choice of values; it is therefore model dependant and will differ from another exact solution.

Temporal evolution of the potential B
The behaviour of the gravitational potentials B ρ=0 and B ρ =0 is given in Fig. 1. It is evident that the potential B is always significantly larger in the case when ρ = 0 than in the limiting scenario when ρ = 0. In the former, B is regular and decreases steadily with time, and increases in the latter. This trend suggests that the exterior null fluid has a suppressing effect on the gravitational field at the surface. We observe that this may in part be due to the nature and structure of our exact solution, and can possibly be somewhat different in another model. Here we examine the temporal behavior of the radiating fluid pressure at the boundary of the object. The pressure, for the two cases, is defined by the junction condition (8). Fig. 2 shows that for the case of pressureless radiation only, the fluid pressure p s , at the boundary is very small (p ≈ 4 × 10 −4 ) and decreases monotonically with time until it is negligible. This is consistent with what has been theoretically established for stellar models in both the Newtonian and relativistic limits. On the other hand, in Figure 3 it can be seen that when ρ = 0, p is slightly suppressed and becomes negative at the boundary. This is not surprising considering the general form of (8). Another reason that supports this result is that when ρ = 0, the pressure is already close to zero, and if the density which is always strictly positive (however small in magnitude it may be) is taken into account it is clear that this will induce a further reduction.
We have also investigated the effect of varying the magnitude of ρ and the family of profiles in Fig. 3 reveal that as the density is increased within the the allowed range, the pressure becomes increasingly more negative. This trend suggests that p scales inversely with ρ at the surface of the radiating star. It should be emphasized that despite this behaviour, the pressure is still only just slightly negative and does not in any way make the model physically unreasonable.  In general, the luminosity is suppressed and changes through negative values as the magnitude of ρ is increased; L Σ at early times is less negative, and at late times tends to zero. The overall behaviour is similar to that found in Fig. 4 and again may be applicable to stars in the strong gravity regime. In Fig. 7, when there is only radiation in the exterior, the luminosity at infinity is singular closer to the centre of the fluid distribution and evolves through positive but small values with r. It is also clear that for large r (r → ∞), L ∞ tends to zero. This suggests that a horizon will form at the end of the collapse of the radiating object. Fig. 6 on the other hand indicates that L ∞ , when ρ = 0, is finite at r = 0 and evolves through negative values until eventually going to zero. Again, this behaviour demonstrates the formation of a horizon at late times. is consistent with what is believed to be the case for most compact objects (see the earlier treatments of Misner 41 and Glass 40 ). In Fig. 11 we observe that mass flow rate across the surface, decreases fairly rapidly and ultimately goes to zero when the flow ceases. The profile in Fig. 12 indicates that for ρ = 0 (dm/dt) is always negative and would seem to contradict the profile in Fig. 10. This is probably due to the form of the gravitational potential in the solution (34) and the choice of parameter values that have been considered.
where ∆ is given by In the limit when ρ = 0 the classical definitions for the energy conditions are regained.
The corresponding numerical profiles for the parameters W , D, and S are given in Fig. 13-15. Fig. 13 demonstrates that the weak energy condition is satisfied since energy condition is also obeyed as seen from the profile for D in Fig. 14. However, the strong energy condition is clearly violated since S is always non-positive, in Fig. 15.
We must point out, though, that this result is not entirely detrimental for the model since it has been established by Kolassis et al 50 , that the strong energy condition can be allowed to fail provided that the overall pressure of the matter distribution is negative. This works perfectly well with the generalised model with m = m(v, r) since the pressure P of the null fluid is negative for a constant density (as highlighted earlier in section 2.2) and in addition, the radiating fluid pressure is also negative (from The rate Θ of gravitational collapse for a spherically symmetrical shear-free fluid with geodesic particles is given by for a comoving 4-velocity u a = δ a 0 . With the solution (34), for the situation where the additional null fluid is present on the outside, the analytical form for the rate of collapse is In the standard Vaidya model considered by Thirukkanesh and Maharaj 38 , with ρ = 0, we have that It is quite clear from equations (38) and (39) that in the general case Θ ρ =0 has a strong dependence on the density ρ and is far more involved than in the standard case Θ ρ=0 .
The numerical profiles corresponding to (38) and (39) are given respectively by Fig.   16 and 17.
From Fig. 16 it is evident that the rate of collapse is negative when the null fluid is present in the exterior of the radiating body. In other words the non-vanishing energy density ρ actually slows down the collapse process until such time when it reaches a complete halt and the relativistic star becomes stable again. It can also be seen that as the magnitude of the density is increased, Θ ρ =0 becomes less negative at earlier times and eventually goes to zero at late times. The plot in Fig. 17 illustrates that for vanishing density, the rate of collapse is always positive and indicates that the collapse itself is occurring reasonably faster.

Discussion
In this paper we modeled a spherically symmetric relativistic radiating star that has vanishing shearing stresses and radial heat flow in the interior while its local exterior region consists of a two We established that the constant null fluid energy density has a marked impact and significantly reduces the pressure p Σ and luminosity L Σ when compared to the limiting case when ρ = 0, where the pressure and luminosity are positive, finite and consistent with well established results for extremely dense stellar objects in the strong gravity regime. The flow of mass across the radiating surface was also probed and it has been demonstrated that the density ρ actually enhances the outgoing flow; this is suggestive that the null fluid on the outside could more likely be interpreted as a distribution of particles that are sourced from nonadiabatic and particle production processes originating within and at the surface of the star. This is in contrast to other treatments where the null fluid has been considered to be an ambient string system in four dimensions. In light of this our model may offer a more realistic description for dense stars in a relativistic setting. By utilising the interior matter variables and the exact solution (34) we also tested the strong, dominant and weak energy conditions when the constant density null fluid is prevalent. Numerical profiles indicate that the weak and dominant conditions are satisfied while the strong condition is violated. However this does not severely affect the validity of the model in any way as the overall pressure is negative.
Since radiating models are crucial for studying the nonadiabatic gravitational collapse of stellar objects, the rate of collapse, for both situations has also been examined. It has been found that ρ slows down the collapse process while in the case of null radiation only it occurs faster. Our model can be further improved on by including the effects of shear and bulk viscosity and analysing the dynamical stability of the stellar matter configuration. Furthermore, if the null fluid is to be taken as a system of particles like neutrinos for example then the model can be appropriately adapted by incorporating non-vanishing neutrino fluxes in the interior. However, these considerations would consequently render the model more complicated and in particular the junction condition increasingly difficult to solve, even with the robust mathematical techniques that are currently available. These endeavors will be the subject of an ongoing investigation.