Prepared for submission to JCAP

Exact Radiation Model For PerfectFluid Under Maximum EntropyPrinciple

Abdul Aziz,a Sourav Roy Chowdhury,a Debabrata Deb,a FarookRahaman,b Saibal Ray,c B.K. Guha.a

aDepartment of Physics, Indian Institute of Engineering Science and Technology, Shibpur,Howrah 711103, West Bengal, IndiabDepartment of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, IndiacDepartment of Physics, Government College of Engineering and Ceramic Technology, Kolkata700010, West Bengal, India

E-mail: azizmail2012@gmail.com, sourav.rs2016@physics.iiests.ac.in,ddeb.rs2016@physics.iiests.ac.in, rahaman@associates.iucaa.in,saibal@associates.iucaa.in, bkguhaphys@gmail.com

Abstract. We find the Euler-Lagrangian equation by maximising the total entropy. Hencewe obtain an expression for mass of the spherically symmetric system by solving the Euler-Lagrangian equation where the Homotopy Perturbation Method has been employed. Withthe help of this expression and the Einstein field equations we obtain an interior solution set.Thereafter, we explain different aspects of the solution describing the system in connectionto the mass, density, pressures, energy, stability, mass-radius ratio, compactness factor andsurface redshift. This analysis shows that all the physical properties, in connection to browndwarf stars, are valid with the observed features.

Keywords: general relativity, homotopy perturbation method, maximum entropy principle,compact stars

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Contents

1 Introduction 1

2 The methodology for mass of the spherical symmetric system of radiatingstar 22.1 The Maximum Entropy Principle (MEP) 22.2 The Homotopy Perturbation Method (HPM) 4

3 The Einstein field equations and their solutions 5

4 Physical features of the radiation model 64.1 Mass, density and pressure 64.2 Stability of stellar model 74.3 Energy conditions 74.4 Compactness and redshift 8

5 Conclusion 10

1 Introduction

There has always been search for the interior solution of the spherically symmetric system.Scientists have tried to find out more than several hundreds different type of interior solutions.But very few solutions made its physical acceptance in all aspects describing the system sofar.

However, in the present paper we have studied a spherically symmetric system of ra-diating star under the homotopy perturbation method (HPM) which was introduced anddeveloped by [1–7] and others [8–12]. This is a series expansion method used in the solutionof non-linear partial differential equations, in the present case the Einstein field equationsof general relativity. The method in principle employs a homotopy transform to generate aconvergent series solution of differential equations. He [3] advocated in favour of homotopy aswell as perturbation technique to solve non-linear problems. Subsequently other researchersalso applied the HPM in various field of pure and applied mathematics (as an exhaustiveappliance), and in physics and astrophysics (as a new field of application) to solve relatednon-linear differential equations in an extraordinary simplified way [13–16].

In connection to self-gravitating radiation model Sorkin et al. [17] have examined theentropy of self-gravitating radiation confined to a spherical box with finite radius in thecontext of general relativity. Their results are expected to supports the validity of self-gravitating systems of the Bekenstein upper limit on the entropy to energy ratio of materialbodies.

Rahaman et al. [18] proposed and analysed a model for the existence of strange stars.They predicted a mass function for the ultra dense strange stars. The interpolation techniquehas been used to estimate the cubic polynomial that yield the following expression for themass as a function of the radial coordinate m(r) = ar3 − br2 + cr − d with a, b, c and dall being numerical constants. Their analysis is based on the MIT bag model and yieldsphysically valid energy density, radial and transverse pressures.

– 1 –

However, in the above mentioned work of Rahaman et al. [18] the target object wasa strange star. In our present investigation, we start with the intention to develop a basicinterior solution of the Einstein equations valid for any radiating model under a similarexpression for the mass as a function of the radial coordinate, i.e., m(r). Then we match ourtheoretically obtained solution set with the observational results for practical validity of themodel and find that our model is the best fit for the brown dwarf star of E0 type.

As a consequence, in the present study our sole motivation is to find the interior solutionof the spherically symmetric system considering radiation effects in a more general way. Forthis we consider a metric with unknown time-time component. By using the maximumentropy principle (MEP) we obtain an Euler-Lagrangian equation which gives second-orderdifferential equation for mass of the spherical distribution. Then we find out solution for massby the homotopy perturbation method (HPM). The Einstein field equations are solved fordensity, time-time component of metric and pressures of the system. Thus, the main purposeof the present investigation is to take data set arbitrarily and see the physical nature andfrom there to predict whether this type of radiation model can effectively be used.

The work plan of the study is as follows: in section 2 we develop the methodology forthe mass of the spherical symmetric system of radiating star under (i) the maximum entropyprinciple, and (ii) the homotopy perturbation method. Next in the section 3 we providethe Einstein field equations for self-gravitating radiation system and discuss several physicalaspects of the solution describing the system in connection to mass, density, pressures, energy,stability, mass-radius ratio, compactness factor and surface redshift. In section 4 it has beenshown in details that all the physical features in connection to brown dwarf stars of type E0are valid with the observed features. We have made some specific remarks on the radiationmodel in the section 5.

2 The methodology for mass of the spherical symmetric system of radiat-ing star

2.1 The Maximum Entropy Principle (MEP)

We consider the metric of the spherical system

ds2 = −gtt(r)dt2 +

(1− 2m(r)

r

)−1dr2 + r2(dθ2 + sin2θdφ2), (2.1)

where m(r) is the mass distribution of the spherically symmetric body and gtt(r) is the time-time component of metric. These are functions of the radial coordinate r only.

The equation of state of the system with thermal radiation is given by

p =1

3ρ, (2.2)

where p and ρ are the radial pressure and density of the matter distribution. Compatiblewith spherically symmetry, we assume the general energy-momentum tensor in the interiorof the isotropic star as

Tµν = diag(ρ,−p,−p,−p). (2.3)

For the locally measured temperature T of the black body radiation, the rest frameenergy density and entropy density, respectively, assume the following forms [17]

ρ = bT 4, (2.4)

– 2 –

s =4

3bT 3, (2.5)

where b is a constant. Here we have assumed that in Planck units G = c = h = k = 1 [19].By substituting eq. (2.4) into eq. (2.5), one can easily obtain the following relation

s = α(ρ)3/4, (2.6)

where α = 43b

1/4.For the matter distribution up to radius r ≤ R, the total entropy is given by

S = 4π

∫ R

0s(r)

[1− 2m(r)

r

]−1/2r2dr, (2.7)

which can be simplified as

S = (4π)14α

∫ R

0Ldr, (2.8)

where the Lagrangian L is given by

L = (m′)34

[1− 2m(r)

r

]− 12

r12 , (2.9)

with constraint equation which we get from Einstein’s field equation

m′(r) = 4πr2ρ. (2.10)

Also, we have the Euler-Lagrangian equation in the form

d

dr

(∂L

∂m′

)− ∂L

∂m= 0, (2.11)

which reduces tom′ +m′′m− 1

2m′′r − 2

3(m′)2 − 4mm′

r= 0. (2.12)

Here the above eq. (2.12) is the master equation for a self-gravitating radiation system.If it is possible to solve this highly non-linear differential equation, then one would get thecomplete structure of the self-gravitating radiation system and therefore could be used todescribe a model of a fluid sphere or stellar configuration. In the following section, we shallsolve this equation by using a newly developed technique known as homotopy perturbationmethod (HPM) [3].

Substituting eq. (2.10) in eq. (2.12), we obtain relativistic Tolman-Oppenheimer-Volkoff(TOV) equation as follows:

d

dr

(ρ3

)= −4ρ

9r

(3m+ rm′

r − 2m

). (2.13)

The relativistic Tolman-Oppenheimer-Volkoff (TOV) equation for the isotropic stablesystem given in eq. (2.13) can be derived from the Tolman-Whittaker formula and the Einsteinfield equations in the following explicit structure

− MG (ρ+ p)

r2e

λ−ν2 − dp

dr= 0. (2.14)

– 3 –

Here MG = MG(r) is the effective gravitational mass within the sphere up to radius rwhich takes form

MG(r) =1

2r2e

ν−λ2 ν ′. (2.15)

The above eq. (2.14), can be expressed as

Fg + Fh = 0, (2.16)

whereFg = −2

3ρ

(g′ttgtt

), (2.17)

Fh = −ρ′

3. (2.18)

The above eq. (2.14) or eq. (2.16), therefore indicates that subject to the gravitationaland hydrostatic forces the equilibrium is achieved for the fluid sphere.

2.2 The Homotopy Perturbation Method (HPM)

Primarily our task is to find the solution of eq. (2.12) by using the HPM. Therefore to havea basic idea about HPM, let us consider the following nonlinear differential equation

A(u)− f(r) = 0, (2.19)

under the boundary condition

B

(u,∂u

∂n

)= 0. (2.20)

Here we identify A as a general differential operator, B as boundary operator, f(r) asa known analytical function and ∂

∂n as directional derivative. The operator A is divided intolinear and nonlinear part, namely L and N . So eq. (2.19) takes the form

L(u) +N(u)− f(r) = 0. (2.21)

Now a homotopy is constructed as

H(v, p) = L(v)− L(u0) + hL(u0) + h[N(v)− f(r)], (2.22)

where h is an embedding parameter and u0 is an initial approximation. As h changes from0 to 1 monotonically, H(v, 0) is continuously transformed into H(v, 1). In topology this iscalled continuous deformation and hence H(v, 0) and H(v, 1) are homotopic functions.

Under the above formalism, now to solve eq. (2.12) we consider linear part of thedifferential equation as m′ = dm/dr and the rest of the terms as non-linear. So the homotopyequation can be written as [3]

m′ −m′0 + h

[m′0 +mm′′ − 1

2m′′r − 2

3(m′)2 − 4mm′

r

]= 0, (2.23)

where m0 is the initial guess of the solution. For h = 0, we get initial approximation solutionand for h = 1, one obtains the desired solution for mass.

– 4 –

We assume the initial solution as m0 = ar3. According to HPM, therefore, we considerthe general solution as

m = m0 + hm1 + h2m2 + h3m3 + .... (2.24)

Substituting this in eq. (2.23) and equating coefficients of the different orders of h, weget the solution for mass up to second order correction as follows

m(r) = ar3 +36

5a2r5 +

312

35a3r7 + 3Aar2 +B, (2.25)

where A and B are constants of integration.The boundary conditions in the present system are

m(0) = 0, m′(R) = 0. (2.26)

Using eqs. (2.25) and (2.26) we obtain the constants of integration, A and B in thefollowing form

A = −(R

2+ 6aR3 +

52

5a2R5

), (2.27)

B = 0, (2.28)

where R is the radius of the spherical distribution.Therefore, substituting A in eq. (2.25), we obtain

m(r) = ar3 +36

5a2r5 +

312

35a3r7 − 3ar2

(R

2+ 6aR3 +

52

5a2R5

). (2.29)

3 The Einstein field equations and their solutions

The Einstein field equations for our system are given by

2m′

r2= 8πρ, (3.1)

2m

r3−(

1− 2m

r

)g′ttgtt

1

r= −8πp, (3.2)

−(

1− 2m

r

)[1

2

g′′ttgtt− 1

4

(g′ttgtt

)2

+1

2r

g′ttgtt

]

−(m

r2− m′

r

)(1

r+

1

2

g′ttgtt

)= −8πp. (3.3)

Now, substituting eq. (2.29) into eq. (3.1) we obtain an expression for density as follows

ρ =1

4π

(3a+ 36a2r2 +

312

5a3r4

)− 3a

2πr

(R

2+ 6aR3 +

52

5a2R5

), (3.4)

whereas, using eq. (3.4) in eq. (2.2), we get an expression for the radial pressure as

p =1

12π

[3a+ 36a2r2 +

312

5a3r4 − 6a

r

(R

2+ 6aR3 +

52

5a2R5

)]. (3.5)

– 5 –

Again by using eq. (2.2) and eq. (3.1), we solve eq. (3.2) for gtt as follows:

gtt =K exp

∫4

3(r−2m)dr

r(r − 2m)1/3, (3.6)

where K is a constant.

Figure 1. Variation of the mass as a function of the fractional radial distance (r/R) for a = −2.55×10−15 km−2 and R = 0.065 R�

Figure 2. Variation of the density as a function of the fractional radial distance (r/R) for a =−2.55× 10−15 km−2 and R = 0.065 R�

4 Physical features of the radiation model

4.1 Mass, density and pressure

To start with the basic physical studies preferably we choose parameter a = −2.55×10−15km−2

and R = 0.065 solar radius and m(R) = 0.080 solar mass of E0 class of brown dwarf stars(see the Link Star [20]). The reason to consider these data set lies on the physical backgroundthat our target is to investigate a radiating model which is compatible with brown dwarfswhich are cool star-like objects that have insufficient mass to maintain stable nuclear fusion

– 6 –

Figure 3. Variation of the radial pressure as a function of the fractional radial distance (r/R) fora = −2.55× 10−15 km−2 and R = 0.065 R�

in their core regions [21]. It has been argued Rebolo et al. [21] that although brown dwarfsare not stars, they are expected to form in the same way, and they should radiate a largefraction of their gravitational energy at near-infra red wavelengths.

The variation of mass, density and pressure with respect to the radial coordinate isshown in figures 1-3.

4.2 Stability of stellar model

Now we perform some physical test regarding the stability of stellar model. First we checkthe sound speed in the interior of stellar structure. The sound speed is defined as

v2r =dp

dρ. (4.1)

For a physical perfect fluid model square of sound speed must be less than the speed of lightin the interior of star [22]. This leads to the condition v2r < 1. Now from figure 4 we see thatin our model the square of sound speed has constant value which is 1/3.

The adiabatic index for a perfect fluid structure is defined by

Γ =

(ρ+ p

p

)[dp

dρ

]. (4.2)

The value of adiabatic index will be greater than 4/3 for a stable configuration at equilibrium.From figure 5 we find that adiabatic index is equal to 4/3 which is the critical value of adiabaticindex [23–26].

4.3 Energy conditions

Let us now check whether all the energy conditions regarding present radiating model aresatisfied or not. For this purpose, we are considering the following inequalities and plottinggraphs for each case:(i) NEC: for any null vector Kα, TαβK

αKβ ≥ 0, or effectively ρ+ p ≥ 0.(ii) WEC: for any timelike vector V α, TαβV

αV β ≥ 0, or effectively ρ ≥ 0 and ρ+ p ≥ 0.(iii) SEC: for any timelike vector V α, (Tαβ − 1

2Tgαβ)V αV β ≥ 0, or effectively ρ + p ≥0 and ρ+ 3 p ≥ 0.

– 7 –

Figure 4. Variation of the forces as a function of the fractional radial distance (r/R) for a =−2.55× 10−15 km−2 and R = 0.065 R�

Figure 5. Variation of the forces as a function of the fractional radial distance (r/R) for a =−2.55× 10−15 km−2 and R = 0.065 R�

(iv) DEC: for any timelike vector V α,(a) TαβV αV β ≥ 0 and Tαn V n is causal,(b) for any two co-oriented timelike vectors V α and Wα,TαβV

αW β ≥ 0 or effectively ρ ≥ 0 and ρ >| p | .We note from figure 6, that in our model all the energy conditions are satisfied throughout

the interior region.

4.4 Compactness and redshift

By definition compactness of a star can be given by u = m(r)/r which in our case takes thefollowing form:

u = ar2 +36

5a2r4 +

312

35a3r6 − 3ar

(R

2+ 6aR3 +

52

5a2R5

). (4.3)

It can easily be observed that figure 7 exhibits the condition m(R)R < 4

9 .

– 8 –

Figure 6. The energy conditions applicable in the interior of the spherical distribution are plottedagainst r/R for a = −2.55× 10−15 km−2 and R = 0.065 R�

Figure 7. Variation of the compactification factor as a function of the fractional radial distance fora = −2.55× 10−15 km−2 and R = 0.065 R�

Here the value of u is of the order of 10−6 in relativistic unit throughout the star. It canbe noted that the exponential term in eq. 3.6 becomes r

43 under the approximation 2u ≤ 1

and hence we obtain

gtt ≈Kr1/3

(r − 2m)1/3. (4.4)

Using boundary conditions for gtt, one can find out K, which ultimately gives the time-time component of metric (2.1) as

gtt =

[1− 2m(R)

R

]4/3[1− 2m(r)

r

]1/3 , (4.5)

where m(R) is a constant and can be given by

m(R) = −1

2aR3 − 54

5a2R5 − 156

7a3R7. (4.6)

Again, the surface redshift is defined by the relation

z = (1− 2u)−12 − 1, (4.7)

– 9 –

Figure 8. Variation of the redshift as a function of the fractional radial distance for a = −2.55 ×10−15 km−2 and R = 0.065 R�

which in the present study turns out to be

1 + z =1√

1− 2{ar2 + 365 a

2r4 + 31235 a

3r6 − 3ar(R2 + 6aR3 + 52

5 a2R5

)}. (4.8)

The variation of surface redshift z is shown in figure 8 which is physically within anacceptable profile.

5 Conclusion

Basically the goal of the present paper is to study a radiating star on the basis of electro-magnetic radiation by exploiting the equation of state p = 1

3ρ. For this purpose our first stepis involved in finding out an expression of mass for a spherically symmetric system from theEuler-Lagrangian equation by using Homotopy Perturbation Method. Hence by employingthis expression for the mass and the Einstein field equations we have obtained a set of inte-rior solution. We have explained different physical properties of the solutions describing thesystem. In the present model for radiating brown dwarf stars all the features are seen to bephysically viable.

One may find a few articles where HPM have been used in the field of astrophysics andthe trend is getting it’s way successfully [27–31]. We have applied HPM using particularhomotopy structure which gives a particular solution describing brown dwarf stars.

In connection to all the above aspects with brown dwarf stars we have done a preliminarycalculation taking the data set of Bhar et al. [32] related to highly compact stars such asneutron stars and strange stars. This data set provides negative radial pressure and alsosuffers from instability. Therefore, we suspect that radiation model may not be compatiblewith highly compact stars rather it is compatible with radiating brown dwarf stars. Actually,to relate radiation model with highly compact stars is also not justified due to their exhaustedfuel system and hence burning as well as radiation stage which seems have been stopped muchearlier.

Regarding singularity we note that there are several famous solutions available in theliterature [33–35] which suffer from this unwanted situation by exhibiting infinite curvatureat the centre. In this connection we would also like to point out that according to an analysisby Delgaty and Lake [36] out of 127 published solutions only 16 solutions satisfy all the

– 10 –

physical conditions of general relativity. However, from eq. 4.5 of the present model we havegtt(0) = 0.999993 as finite and positive whereas grr(0) = 1. But the density and pressureare not finite at the centre though we have finite mass. So the solution is geometricallynon-singular with infinite density and pressure at centre of the star. This type of solutionphysically possible as discussed in their work by Fuloria and Durgapal [37]. They arguedthat there is a possibility of having information from the central region of infinite densityand pressure if the central redshift is finite. In our model for the brown dwarf the redshiftin the central region of the star is finite. Inspired by this fact we can think of a core at thecentral region of the star. The central density for a brown dwarf varies from 10 gm/cm3

to 103 gm/cm3 [38]. From figure 2 we see that the density becomes 103 gm/cm3 at certaincritical distance (rc = 369 km) near the centre. Since the star reaches its maximum densityat rc, we consider a small core of constant density 103 gm/cm3 of core radius rc. This is agood approximation as the core is very small compared to the star (the volume of the coreis just 0.000054311 percent of the total volume of the star). Therefore we find the constantdensity core as suitable alternative to the centre of infinite density.

In future work it may be possible to get a non-singular solution with finite centraldensity and pressure using different homotopy structure. As a final remark we would like tostate that though in the present radiation model the brown dwarf star has been successfullydescribed yet some more investigations are needed to perform with different methodologybefore coming to a definite decision regarding applicability of HPM and MEP to radiatingand highly compact stars.

Acknowledgments

FR and SR are thankful to the Inter-University Centre for Astronomy and Astrophysics(IUCAA), Pune, India for providing the Visiting Associateship under which a part of thiswork was carried out. SR also thanks the Institute of Mathematical Sciences (IMSc), Chennai,India for providing the working facilities and hospitality under the Associateship scheme.

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