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Physics & Astronomy International Journal

Research Article Volume 2 Issue 5

Non–vacuum perfect fluid static cylindrically symmetric solutions in f(R) gravity and their energy distribution

Farhat Imtiaz, M Jamil Amir

Department of Mathematics, University of Lahore, Sargodha Campus?40100, Pakistan

Correspondence: M Jamil Amir, Department of Mathematics, University of Lahore, Sargodha Campus–40100, Pakistan

Received: August 28, 2018 | Published: October 25, 2018

Citation: Imtiaz F, Amir MJ. Non–vacuum perfect fluid static cylindrically symmetric solutions in f(R) gravity and their energy distribution. Phys Astron Int J. 2018;2(5):489-496. DOI: 10.15406/paij.2018.02.00131

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Abstract

Much attention has been given towards modified theories of gravity especially towards f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity during the last two decades to understand the reason behind the accelerated expansion of the universe. Recently, Sharif and Sadia explored the non–vacuum static cylindri–cally symmetric solutions for dust case and their energy contents too. In this paper, we are intend to extend their work for perfect fluid case. Regarding this, we obtain the non–vacuum field equations of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity for perfect fluid static solution using cylindrically sym–metric background. The field equations turn out to be complected and hence can’t be solved analytically so we have to put some restric–tions to solve the field equations. For this purpose, we utilize metric approach and constant curvature assumption. Further, we explore the energy distributions of the obtained solutions by using general–ized Landau–Lifshitz energy–momentum prescription in the context of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theory of gravity for a specific choice of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ models. Moreover, we examined the stability conditions for the obtained solutions.

Keywords: f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity, vacuum solutions, perfect fluid, static cylindrically symmetric, generalized landau–lifshits complex

Introduction

Out of many other alternative theories of Einstein’s theory of gravity, the f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theory of gravity has a long history. During the last few years, there has been tremendous change in this area bringing about a lot of study which yields exciting results. On behalf of some evidences, it has been concluded that Weyl1 was the first who worked in the field of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity while many others are in view that it was Eddington2 who was the pioneer to contribute in this field. Later, Buchdahl3 reflected his ideas and efforts in the same field. A comprehensive review has been presented here which consists of relevant literature and important features and characteristics of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theories of gravity.4

Using the relation with the week field limit, Capozziello et al.5 obtained exact spherical symmetric solutions in f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@  gravity for constant Ricci scalar as well as for Ricci scalar as function of radial coordinates. Later on, the same authors6 discussed spherically symmetric solution in f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity via Noether symmetric approach. Kainulanianen et al.7 investigated spherically symmetric spacetimes in f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theories of gravity using analytical and numerical approaches. Multam"a"ki and Vilja8 explored exact spherically symmetric static empty space solution. These results showed that a huge number of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theories have exact solutions as Schwarzschild de sitter metric. The said authors9 also extracted non–vacuum solutions using static spherical symmetric background.

Hollenstein and Lobo10 coupled f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity to non–linear electrody–namics in order to produce static spherically symmetric solutions. In an extended study of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity, Shojai11 pondered on static spherically symmetric interior solutions. Caramês and Mello12 in a higher dimensional spacetime, scrutinizes spherically symmetric vacuum solutions. Sharif and Kousar13 determined non–vacuum static spherically symmet–ric solutions. Much work is available in literature carried out by different authors.14–18

In a detailed study of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity, Sharif and Shamir19 researched static plane symmetric vacuum solutions. Amir and Maqsood20 discussed some non–vacuum plane symmetric solutions using metric approach and non–varying Ricci scalar assumption. They also explored the energy contents of these solutions. Shamir21 has an elaborated contributions in examining plane symmetric vacuum Bianchi type III cosmology in f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity. Momeni and Gholizade22 proved that exact solution of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity constant curvature can be applied to the exterior of a string. Azadi et al.23 using Weyl coordinates in the framework of the metric f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theories of gravity, explored the static cylindrically symmetric vacuum solutions.

Amir and Sattar24 found locally rotationally symmetric vacuum solu–tions in f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ models. Amir and Naheed25 explored vacuum solutions of spatially homogeneous rotating spacetimes. Sharif and Arif26 worked on dust particle for the investigation of static cylindrically symmetric solutions in metric f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity.

In GR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGhbGaae Ouaaaa@3714@ , energy localization is a very serious problem and it is not still sorted out exactly. Much work has been done on problem in the frame–work of GR to resolve this issue. Different scientists gave their own energy momentum complexes and found the energy momentum distribution of many spactimes but could not present a concrete conjuncture. Methodology of energy–momentum pseudo tensor was the first effort to solve this matter and this step was taken by Einstein. Following him many authors, like Landau–Lifshitz, M"o"ller, Bergmann–Thomson and Weinberg gave their own energy–momentum prescriptions. Virbhadra and Parikh27 investi–gated the energy–momentum distribution of several spacetimes, such as Kerr–Newmann, Kerr–Schild classes, Einstein–Rosen, Vaidya and Bonnor–Vaidya spacetimes but could not reach at solid and unique conclusion.

There are many authors who are of the view point that this issue may be tackled correctly in the other frameworks, for example, in telepararllel(TP) theory of gravity, modified theories of gravity etc. In the framework of TP gravity, Sharif and Amir28–34 found out energy–momentum distribution of various spacetimes by using TP version of different prescriptions. They conclude that no general conjecture can be made here because results in some cases are consistent while others are inconsistent. Landau–Lifshitz energy momentum complex was generalized to evaluate energy–momentum distribu–tion for Schwarzschild de Sitter spacetime. The energy density was calculated with the help of generalized Landau–Lifshitz prescription for the plane sym–metric static solution and cosmic string space time by Sharif and Shamir35.

This paper of current study explores the non–vacuum perfect fluid static cylindrically symmetric solutions by using some basics of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity. We utilize generalized Landau–Lifshitz EMC to evaluate energy density for the solution having constant scalar curvature. The following arrangement has been used in this paper. In section 2, the modification of field equations in metric approach of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gravity has been precisely discussed. Section 3 contains the derivation of generalized Landau–Lifshitz EMC. In section 4, the exact non vacuum solution by using the constant curvature condition, of perfect fluid static cylindrically symmetric solutions is described. In section 5, by using Landau–Lifshitz EMC, we have calculated the energy density of the solution that was obtain in the previous section. In the last section results have been summarized.

Field equations in f(R) theory of gravity

We will extract the field equations for this section. For this reason, we utilize the metric approach of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theory of gravity. In this approach, the variation of the action is done with respect to the metric tensor only. The action of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theory is expressed as

S= g ( 1 16πG f(R)+ L m ) d 4 x, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtbGaaG ypaKqbaoaapeaakeqaleqabeqcLbsacqGHRiI8aKqbaoaakaaakeaa jugibiabgkHiTiaadEgaaSqabaqcLbsacaaIOaqcfa4aaSaaaOqaaK qzGeGaaGymaaGcbaqcLbsacaaIXaGaaGOnaiabec8aWjaadEeaaaGa amOzaiaaiIcacaWGsbGaaGykaiabgUcaRiaadYeajuaGdaWgaaqcba saaKqzadGaamyBaaWcbeaajugibiaaiMcacaWGKbWcdaahaaqcbasa beaajugWaiaaisdaaaqcLbsacaWG4bGaaiilaaaa@539D@  (1)

where f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ is a arbitrary function of Ricci scala R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbaaaa@364C@ and L m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGmbGcdaWgaaqcbasaaKqzadGaamyBaaWcbeaaaaa@3DB7@ is the matter Lagrangian. In the standard Einstien–Hilbert action the replacement of R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbaaaa@364C@ by f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ gives us this action. By varying this action with respect to metric tensor g μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGNbWcda WgaaqcbasaaKqzadGaeqiVd0MaeqyVd4gajeaibeaaaaa@3B7D@ , these corresponding field equations can be derived,

F(R) R μν 1 2 f(R) g μν μ ν F(R)+ g μν F(R)=κ T μν , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfacaaIPaGaamOuaSWaaSbaaKqaGeaajugWaiabeY7aTjab e27aUbqcbasabaqcLbsacqGHsisljuaGdaWcaaGcbaqcLbsacaaIXa aakeaajugibiaaikdaaaGaamOzaiaaiIcacaWGsbGaaGykaiaadEga lmaaBaaajeaibaqcLbmacqaH8oqBcqaH9oGBaKqaGeqaaKqzGeGaey OeI0Iaey4bIeDcfa4aaSbaaKqaGeaajugWaiabeY7aTbWcbeaajugi biabgEGirVWaaSbaaKqaGeaajugWaiabe27aUbqcbasabaqcLbsaca WGgbGaaGikaiaadkfacaaIPaGaey4kaSIaam4zaSWaaSbaaKqaGeaa jugWaiabeY7aTjabe27aUbqcbasabaqcLbsacaWGgbGaaGikaiaadk facaaIPaGaaGypaiabeQ7aRjaadsfalmaaBaaajeaibaqcLbmacqaH 8oqBcqaH9oGBaKqaGeqaaSGaaiilaaaa@6FE5@ (2)

Where

F(R)= df(R) dR ,f= μ μ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfacaaIPaGaaGypaKqbaoaalaaakeaajugibiaadsgacaWG MbGaaGikaiaadkfacaaIPaaakeaajugibiaadsgacaWGsbaaaiaaiY cacaaMe8UaamOzaiaai2dacqGHhis0juaGdaahaaWcbeqcbasaaKqz adGaeqiVd0gaaKqzGeGaey4bIeDcfa4aaSbaaKqaGeaajugWaiabeY 7aTbqcbasabaqcLbsacaGGUaaaaa@5164@  (3)

Here, μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhis0ju aGdaWgaaqcbasaaKqzadGaeqiVd0galeqaaaaa@3AC3@ represents the covariant derivative, f= μ μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ypaiabgEGirVWaaWbaaKqaGeqabaqcLbmacqaH8oqBaaqcLbsacqGH his0lmaaBaaajeaibaqcLbmacqaH8oqBaKqaGeqaaaaa@4161@ is called D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGebaaaa@363C@ ’ Alem–bert operator and T μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGubWcda WgaaqcbasaaKqzadGaeqiVd0MaeqyVd4gajeaibeaaaaa@3B6A@ is the standard matter energy–momentum tensor de–rived from L m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmbqcfa 4aaSbaaKqaGeaajugWaiaad2gaaSqabaaaaa@394A@ . In the metric tensor these are the fourth order partial dif–ferential equations. For f(R)=R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaGaaGypaiaadkfaaaa@3A3A@ these equations reduce to the famous Einstein field equations of GR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGhbGaae Ouaaaa@3714@ . After contraction the field equations we get are

F(R)R2f(R)+3F(R)=κT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfacaaIPaGaamOuaiabgkHiTiaaikdacaWGMbGaaGikaiaa dkfacaaIPaGaey4kaSIaaG4maiaadAeacaaIOaGaamOuaiaaiMcaca aI9aGaeqOUdSMaamivaaaa@461B@  (4)

and, in vacuum, i.e., when T=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGubGaaG ypaiaaicdaaaa@37CF@ , the last equation turns out be

F(R)R2f(R)+3F(R)=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfacaaIPaGaamOuaiabgkHiTiaaikdacaWGMbGaaGikaiaa dkfacaaIPaGaey4kaSIaaG4maiaadAeacaaIOaGaamOuaiaaiMcaca aI9aGaaGimaiaac6caaaa@44FC@  (5)

We come to know an important relationship between f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ and F(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfacaaIPaaaaa@387C@ from this which helps in the simplification of field equations and to find out the f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ . Any metric with scalar curvature though, as R= R 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbGaaG ypaiaadkfalmaaBaaajeaibaqcLbmacaaIWaaajeaibeaaaaa@3A52@ , is a solution of Equation (5) if the below equation is as following

F( R 0 ) R 0 2f( R 0 )=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfajuaGdaWgaaqcbasaaKqzadGaaGimaaWcbeaajugibiaa iMcacaWGsbWcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacq GHsislcaaIYaGaamOzaiaaiIcacaWGsbWcdaWgaaqcbasaaKqzadGa aGimaaqcbasabaqcLbsacaaIPaGaaGypaiaaicdacaGGUaaaaa@499F@  (6)

This condition of the constant scalar curvature for the vacuum and non–vacuum case will have following form

F( R 0 ) R 0 2f( R 0 )=κT. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfalmaaBaaajeaibaqcLbmacaaIWaaajeaibeaajugibiaa iMcacaWGsbWcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacq GHsislcaaIYaGaamOzaiaaiIcacaWGsbWcdaWgaaqcbasaaKqzadGa aGimaaqcbasabaqcLbsacaaIPaGaaGypaiabeQ7aRjaadsfacaGGUa aaaa@4B0C@  (7)

These conditions have vital role to find the acceptability of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ models.

Generaized landau–lifshitz energy–momentum complex

The generalized Landau–Lifshits EMC is given by,./

τ μν = f ( R 0 ) τ LL μν + 1 6κ { f ( R 0 ) R 0 f( R 0 )} x λ ( g μν x λ g μλ x ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHepaDju aGdaahaaWcbeqcbasaaKqzadGaeqiVd0MaeqyVd4gaaKqzGeGaaGyp aiqadAgagaqbaiaaiIcacaWGsbWcdaWgaaqcbasaaKqzadGaaGimaa qcbasabaqcLbsacaaIPaGaeqiXdq3cdaqhaaqcbasaaKqzadGaamit aiaadYeaaKqaGeaajugWaiabeY7aTjabe27aUbaajugibiabgUcaRK qbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOnaiabeQ7aRbaa caaI7bGabmOzayaafaGaaGikaiaadkfalmaaBaaajeaibaqcLbmaca aIWaaajeaibeaajugibiaaiMcacaWGsbWcdaWgaaqcbasaaKqzadGa aGimaaqcbasabaqcLbsacqGHsislcaWGMbGaaGikaiaadkfalmaaBa aajeaibaqcLbmacaaIWaaajeaibeaajugibiaaiMcacaaI9bqcfa4a aSaaaOqaaKqzGeGaeyOaIylakeaajugibiabgkGi2kaadIhajuaGda ahaaWcbeqcbasaaKqzadGaeq4UdWgaaaaajugibiaaiIcacaWGNbqc fa4aaWbaaSqabKqaGeaajugWaiabeY7aTjabe27aUbaajugibiaadI halmaaCaaajeaibeqaaKqzadGaeq4UdWgaaKqzGeGaeyOeI0Iaam4z aSWaaWbaaKqaGeqabaqcLbmacqaH8oqBcqaH7oaBaaqcLbsacaWG4b WcdaahaaqcbasabeaajugWaiabe27aUbaajugibiaaiMcaaaa@8A9F@ (8)

where τ LL μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHepaDlm aaDaaajeaibaqcLbmacaWGmbGaamitaaqcbasaaKqzadGaeqiVd0Ma eqyVd4gaaaaa@3F27@ is the Landau–Lifshitz EMC in GR and κ=8πG MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH6oWAca aI9aGaaGioaiabec8aWjaadEeaaaa@3B39@ . In the field of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theory of any metric tensor which holds constant scalar curvature late, EMD can be calculated. Energy density is represented by 00–component and as following,

τ 00 = f ( R 0 ) τ LL 00 + 1 6κ ( f ( R 0 ) R 0 f( R 0 ))( x i g 00 X i +3 g 00 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHepaDlm aaCaaajeaibeqaaKqzadGaaGimaiaaicdaaaqcLbsacaaI9aGabmOz ayaafaGaaGikaiaadkfalmaaBaaajeaibaqcLbmacaaIWaaajeaibe aajugibiaaiMcacqaHepaDlmaaDaaajeaibaqcLbmacaWGmbGaamit aaqcbasaaKqzadGaaGimaiaaicdaaaqcLbsacqGHRaWkjuaGdaWcaa GcbaqcLbsacaaIXaaakeaajugibiaaiAdacqaH6oWAaaGaaGikaiqa dAgagaqbaiaaiIcacaWGsbqcfa4aaSbaaKqaGeaajugWaiaaicdaaS qabaqcLbsacaaIPaGaamOuaSWaaSbaaKqaGeaajugWaiaaicdaaKqa GeqaaKqzGeGaeyOeI0IaamOzaiaaiIcacaWGsbqcfa4aaSbaaKqaGe aajugWaiaaicdaaSqabaqcLbsacaaIPaGaaGykaiaaiIcajuaGdaWc aaGcbaqcLbsacqGHciITaOqaaKqzGeGaeyOaIyRaamiEaKqbaoaaCa aaleqajeaibaqcLbmacaWGPbaaaaaajugibiaadEgalmaaCaaajeai beqaaKqzadGaaGimaiaaicdalmaaBaaajeaibaqcLbmacaWGybWcda ahaaqcbasabeaajugWaiaadMgaaaaajeaibeaaaaqcLbsacqGHRaWk caaIZaGaam4zaKqbaoaaCaaaleqajeaibaqcLbmacaaIWaGaaGimaa aajugibiaaiMcacaGGSaaaaa@7E5A@ (9)

where τ LL 00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHepaDlm aaDaaajeaibaqcLbmacaWGmbGaamitaaqcbasaaKqzadGaaGimaiaa icdaaaaaaa@3D2D@ represents the sum of energy–momentum tensor and the energy–momentum pseudo tensor and is given by

τ LL 00 =(g)( T 00 + t LL 00 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHepaDlm aaDaaajeaibaqcLbmacaWGmbGaamitaaqcbasaaKqzadGaaGimaiaa icdaaaqcLbsacaaI9aGaaGikaiabgkHiTiaadEgacaaIPaGaaGikai aadsfalmaaCaaajeaibeqaaKqzadGaaGimaiaaicdaaaqcLbsacqGH RaWkcaWG0bWcdaqhaaqcbasaaKqzadGaamitaiaadYeaaKqaGeaaju gWaiaaicdacaaIWaaaaKqzGeGaaGykaaaa@4FE4@  (10)

and

T 00 = 1 κ ( R 00 1 2 g 00 R), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGubWcda ahaaqcbasabeaajugWaiaaicdacaaIWaaaaKqzGeGaaGypaKqbaoaa laaakeaajugibiaaigdaaOqaaKqzGeGaeqOUdSgaaiaaiIcacaWGsb WcdaahaaqcbasabeaajugWaiaaicdacaaIWaaaaKqzGeGaeyOeI0sc fa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaaaiaadEgalm aaCaaajeaibeqaaKqzadGaaGimaiaaicdaaaqcLbsacaWGsbGaaGyk aiaacYcaaaa@4ECD@  (11)

where R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbaaaa@364C@ is the Ricci scalar and t LL 00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0bWcda qhaaqcbasaaKqzadGaamitaiaadYeaaKqaGeaajugWaiaaicdacaaI Waaaaaaa@3C61@ can be evaluated from the following expression

t LL 00 = 1 2κ [(2 Γ αβ γ Γ γδ δ Γ αδ γ Γ βγ δ Γ αγ γ Γ βδ δ )( g μα g νβ g μν g αβ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0bWcda qhaaqcbasaaKqzadGaamitaiaadYeaaKqaGeaajugWaiaaicdacaaI WaaaaKqzGeGaaGjbVlaai2dacaaMe8Ecfa4aaSaaaOqaaKqzGeGaaG ymaaGcbaqcLbsacaaIYaGaeqOUdSgaaiaaiUfacaaIOaGaaGOmaiab fo5ahTWaa0baaKqaGeaajugWaiabeg7aHjabek7aIbqcbasaaKqzad Gaeq4SdCgaaKqzGeGaeu4KdC0cdaqhaaqcbasaaKqzadGaeq4SdCMa eqiTdqgajeaibaqcLbmacqaH0oazaaqcLbsacqGHsislcqqHtoWrlm aaDaaajeaibaqcLbmacqaHXoqycqaH0oazaKqaGeaajugWaiabeo7a Nbaajugibiabfo5ahTWaa0baaKqaGeaajugWaiabek7aIjabeo7aNb qcbasaaKqzadGaeqiTdqgaaKqzGeGaeyOeI0Iaeu4KdC0cdaqhaaqc basaaKqzadGaeqySdeMaeq4SdCgajeaibaqcLbmacqaHZoWzaaqcLb sacqqHtoWrlmaaDaaajeaibaqcLbmacqaHYoGycqaH0oazaKqaGeaa jugWaiabes7aKbaajugibiaaiMcacaaIOaGaam4zaKqbaoaaCaaale qajeaibaqcLbmacqaH8oqBcqaHXoqyaaqcLbsacaWGNbqcfa4aaWba aSqabKqaGeaajugWaiabe27aUjabek7aIbaajugibiabgkHiTiaadE galmaaCaaajeaibeqaaKqzadGaeqiVd0MaeqyVd4gaaKqzGeGaam4z aSWaaWbaaKqaGeqabaqcLbmacqaHXoqycqaHYoGyaaqcLbsacaGGPa aaaa@A1EE@

+ g μα g βγ ( Γ αδ ν Γ βγ δ + Γ βγ ν Γ αδ δ Γ γδ ν Γ αβ δ Γ αβ ν Γ γδ δ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHRaWkca aMe8Uaam4zaSWaaWbaaKqaGeqabaqcLbmacqaH8oqBcqaHXoqyaaqc LbsacaWGNbWcdaahaaqcbasabeaajugWaiabek7aIjabeo7aNbaaju gibiaaiIcacqqHtoWrlmaaDaaajeaibaqcLbmacqaHXoqycqaH0oaz aKqaGeaajugWaiabe27aUbaajugibiabfo5ahTWaa0baaKqaGeaaju gWaiabek7aIjabeo7aNbqcbasaaKqzadGaeqiTdqgaaKqzGeGaey4k aSIaeu4KdC0cdaqhaaqcbasaaKqzadGaeqOSdiMaeq4SdCgajeaiba qcLbmacqaH9oGBaaqcLbsacqqHtoWrlmaaDaaajeaibaqcLbmacqaH XoqycqaH0oazaKqaGeaajugWaiabes7aKbaajugibiabgkHiTiabfo 5ahTWaa0baaKqaGeaajugWaiabeo7aNjabes7aKbqcbasaaKqzadGa eqyVd4gaaKqzGeGaeu4KdC0cdaqhaaqcbasaaKqzadGaeqySdeMaeq OSdigajeaibaqcLbmacqaH0oazaaqcLbsacqGHsislcqqHtoWrlmaa DaaajeaibaqcLbmacqaHXoqycqaHYoGyaKqaGeaajugWaiabe27aUb aajugibiabfo5ahTWaa0baaKqaGeaajugWaiabeo7aNjabes7aKbqc basaaKqzadGaeqiTdqgaaKqzGeGaaGykaaaa@96E6@

+ g μα g βγ ( Γ αδ ν Γ βγ δ + Γ βγ ν Γ αδ δ Γ γδ ν Γ αβ δ Γ αβ ν Γ γδ δ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHRaWkca aMe8Uaam4zaSWaaWbaaKqaGeqabaqcLbmacqaH8oqBcqaHXoqyaaqc LbsacaWGNbWcdaahaaqcbasabeaajugWaiabek7aIjabeo7aNbaaju gibiaaiIcacqqHtoWrlmaaDaaajeaibaqcLbmacqaHXoqycqaH0oaz aKqaGeaajugWaiabe27aUbaajugibiabfo5ahTWaa0baaKqaGeaaju gWaiabek7aIjabeo7aNbqcbasaaKqzadGaeqiTdqgaaKqzGeGaey4k aSIaeu4KdC0cdaqhaaqcbasaaKqzadGaeqOSdiMaeq4SdCgajeaiba qcLbmacqaH9oGBaaqcLbsacqqHtoWrlmaaDaaajeaibaqcLbmacqaH XoqycqaH0oazaKqaGeaajugWaiabes7aKbaajugibiabgkHiTiabfo 5ahTWaa0baaKqaGeaajugWaiabeo7aNjabes7aKbqcbasaaKqzadGa eqyVd4gaaKqzGeGaeu4KdC0cdaqhaaqcbasaaKqzadGaeqySdeMaeq OSdigajeaibaqcLbmacqaH0oazaaqcLbsacqGHsislcqqHtoWrlmaa DaaajeaibaqcLbmacqaHXoqycqaHYoGyaKqaGeaajugWaiabe27aUb aajugibiabfo5ahTWaa0baaKqaGeaajugWaiabeo7aNjabes7aKbqc basaaKqzadGaeqiTdqgaaKqzGeGaaGykaaaa@96E6@ (12)

Perfect fluid static cylindrically symmetric solutions

In this section, we find the non–vacuum field equations of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ theory for the metric representing static cylindrical symmetric spacetimes by using the con–dition of constant scalar curvature and metric pattern, i.e., (R=constant) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIOaGaam Ouaiaai2daqaaaaaaaaaWdbiaadogacaWGVbGaamOBaiaadohacaWG 0bGaamyyaiaad6gacaWG0bWdaiaacMcaaaa@4033@ . The line element representing static cylindrically symmetric spacetimes is given below

d s 2 =A(r)d t 2 B(r)d ρ 2 C(r)d ϕ 2 Bd z 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKbGaam 4CaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaaGypaiaadgea caaIOaGaamOCaiaaiMcacaWGKbGaamiDaSWaaWbaaKqaGeqabaqcLb macaaIYaaaaKqzGeGaeyOeI0IaamOqaiaaiIcacaWGYbGaaGykaiaa dsgacqaHbpGClmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgk HiTiaadoeacaaIOaGaamOCaiaaiMcacaWGKbGaeqy1dy2cdaahaaqc basabeaajugWaiaaikdaaaqcLbsacqGHsislcaWGcbGaamizaiaadQ halmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaacYcaaaa@5CF8@  (13)

here A(r),B(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbGaaG ikaiaadkhacaaIPaGaaGilaiaadkeacaaIOaGaamOCaiaaiMcaaaa@3C70@ and C(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdbGaaG ikaiaadkhacaaIPaaaaa@3899@ are taken as arbitrary functions of r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYbaaaa@366C@ . For this line element Ricci scalar turn out to be

R= A AB A '2 2 A 2 B + B B 2 B '2 B 3 + C BC C '2 2B C 2 + A C 2ABC . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbGaaG ypaKqbaoaalaaakeaajugibiqadgeagaqbgaqbaaGcbaqcLbsacaWG bbGaamOqaaaacqGHsisljuaGdaWcaaGcbaqcLbsacaWGbbqcfa4aaW baaSqabKqaGeaajugWaiaadEcacaaIYaaaaaGcbaqcLbsacaaIYaGa amyqaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaamOqaa aacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGcbGbauGbauaaaOqaaKqz GeGaamOqaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaaaajugibiabgk HiTKqbaoaalaaakeaajugibiaadkealmaaCaaajeaibeqaaKqzadGa am4jaiaaikdaaaaakeaajugibiaadkeajuaGdaahaaWcbeqcbasaaK qzadGaaG4maaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWG dbGbauGbauaaaOqaaKqzGeGaamOqaiaadoeaaaGaeyOeI0scfa4aaS aaaOqaaKqzGeGaam4qaSWaaWbaaKqaGeqabaqcLbmacaWGNaGaaGOm aaaaaOqaaKqzGeGaaGOmaiaadkeacaWGdbqcfa4aaWbaaSqabKqaGe aajugWaiaaikdaaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGa bmyqayaafaGabm4qayaafaaakeaajugibiaaikdacaWGbbGaamOqai aadoeaaaGaaiOlaaaa@7306@ (14)

where prime shows the derivative with respect to r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYbaaaa@366C@ . For perfect fluid the energy–momentum tensor is given as

T μν =(ρ+p) u μ u ν p g μν , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGubWcda WgaaqcbasaaKqzadGaeqiVd0MaeqyVd4gajeaibeaajugibiaai2da caaIOaGaeqyWdiNaey4kaSIaamiCaiaaiMcacaWG1bqcfa4aaSbaaK qaGeaajugWaiabeY7aTbWcbeaajugibiaadwhalmaaBaaajeaibaqc LbmacqaH9oGBaKqaGeqaaKqzGeGaeyOeI0IaamiCaiaadEgalmaaBa aajeaibaqcLbmacqaH8oqBcqaH9oGBaKqaGeqaaKqzGeGaaiilaaaa @5525@  (15)

where ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbpGCaa a@3735@ is the density of energy and p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWbaaaa@366A@ is the pressure of the fluid and in comoving coordinates the fourth–velocity is given by u μ = g 00 (1,0,0,0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG1bWcda WgaaqcbasaaKqzadGaeqiVd0gajeaibeaajugibiaai2dajuaGdaGc aaGcbaqcLbsacaWGNbWcdaWgaaqcbasaaKqzadGaaGimaiaaicdaaK qaGeqaaaWcbeaajugibiaaiIcacaaIXaGaaGilaiaaicdacaaISaGa aGimaiaaiYcacaaIWaGaaGykaaaa@4778@ . The equation of state given below is satisfied by pressure p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWbaaaa@366A@ and energy density ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbpGCaa a@3735@ .

p=ωρ,0ω1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGWbGaeyypa0JaeqyYdCNaeqyWdiNaaiilaiaaicdacqGH KjYOcqaHjpWDcqGHKjYOcaaIXaaaaa@474A@ (16)

Also, from field Equation (4), we get

f(R)= 3F(R)+F(R)RκT 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaGaaGypaKqbaoaalaaakeaajugibiaaiodacaWG gbGaaGikaiaadkfacaaIPaGaey4kaSIaamOraiaaiIcacaWGsbGaaG ykaiaadkfacqGHsislcqaH6oWAcaWGubaakeaajugibiaaikdaaaGa aiOlaaaa@489D@ (17)

Using this value of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMbGaaG ikaiaadkfacaaIPaaaaa@389C@ in non–vacuum field Equation (2), we have

F(R) R αα α α F(R)κ T αα g αα = F(R)F(R)κT 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGaamOraiaaiIcacaWGsbGaaGykaiaadkfajuaGdaWgaaqcbasa aKqzadGaeqySdeMaeqySdegaleqaaKqzGeGaeyOeI0Iaey4bIe9cda WgaaqcbasaaKqzadGaeqySdegajeaibeaajugibiabgEGirVWaaSba aKqaGeaajugWaiabeg7aHbqcbasabaqcLbsacaWGgbGaaGikaiaadk facaaIPaGaeyOeI0IaeqOUdSMaamivaSWaaSbaaKqaGeaajugWaiab eg7aHjabeg7aHbqcbasabaaakeaajugibiaadEgalmaaBaaajeaiba qcLbmacqaHXoqycqaHXoqyaKqaGeqaaaaajugibiaai2dajuaGdaWc aaGcbaqcLbsacaWGgbGaaGikaiaadkfacaaIPaGaeyOeI0seeuuDJX wAKbsr4rNCHbacfaGae8xOLCLaamOraiaaiIcacaWGsbGaaGykaiab gkHiTiabeQ7aRjaadsfaaOqaaKqzGeGaaGinaaaaaaa@71FC@ (18)

In the above equation, the terms on the right hand side are independent of index α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXoqyaa a@3714@ , so we can write the field equation in the following manner,

A αα = F(R) R αα α α F(R)κ T αα g αα . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbWcda WgaaqcbasaaKqzadGaeqySdeMaeqySdegajeaibeaajugibiaai2da juaGdaWcaaGcbaqcLbsacaWGgbGaaGikaiaadkfacaaIPaGaamOuaS WaaSbaaKqaGeaajugWaiabeg7aHjabeg7aHbqcbasabaqcLbsacqGH sislcqGHhis0lmaaBaaajeaibaqcLbmacqaHXoqyaKqaGeqaaKqzGe Gaey4bIe9cdaWgaaqcbasaaKqzadGaeqySdegajeaibeaajugibiaa dAeacaaIOaGaamOuaiaaiMcacqGHsislcqaH6oWAcaWGubWcdaWgaa qcbasaaKqzadGaeqySdeMaeqySdegajeaibeaaaOqaaKqzGeGaam4z aSWaaSbaaKqaGeaajugWaiabeg7aHjabeg7aHbqcbasabaaaaKqzGe GaaiOlaaaa@65DB@  (19)

Here A αα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbWcda WgaaqcbasaaKqzadGaeqySdeMaeqySdegajeaibeaaaaa@3B27@ is used to represent the traced quantity. By subtracting (00) and (11) components, we get

A C F 4ABC B F 2 B 2 + B '2 F 2 B 3 C F 2BC + C '2 F 4B C 2 + A B F 4A B 2 + B C F 4 B 2 C + A F 2AB + B F 2 B 2 F B κ(p+ρ)=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafaGabm4qayaafaGaamOraaGcbaqcLbsacaaI0aGa amyqaiaadkeacaWGdbaaaiabgkHiTKqbaoaalaaakeaajugibiqadk eagaqbgaqbaiaadAeaaOqaaKqzGeGaaGOmaiaadkeajuaGdaahaaWc beqcbasaaKqzadGaaGOmaaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcba qcLbsacaWGcbqcfa4aaWbaaSqabeaajugibiaadEcajugWaiaaikda aaqcLbsacaWGgbaakeaajugibiaaikdacaWGcbWcdaahaaqcbasabe aajugWaiaaiodaaaaaaKqzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGa bm4qayaafyaafaGaamOraaGcbaqcLbsacaaIYaGaamOqaiaadoeaaa Gaey4kaSscfa4aaSaaaOqaaKqzGeGaam4qaKqbaoaaCaaaleqabaqc LbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcbaqcLbsacaaI0a GaamOqaiaadoealmaaCaaajeaibeqaaKqzadGaaGOmaaaaaaqcLbsa cqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGbbGbauaaceWGcbGbauaaca WGgbaakeaajugibiaaisdacaWGbbGaamOqaKqbaoaaCaaaleqajeai baqcLbmacaaIYaaaaaaajugibiabgUcaRKqbaoaalaaakeaajugibi qadkeagaqbaiqadoeagaqbaiaadAeaaOqaaKqzGeGaaGinaiaadkea lmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaadoeaaaGaey4kaS scfa4aaSaaaOqaaKqzGeGabmyqayaafaGabmOrayaafaaakeaajugi biaaikdacaWGbbGaamOqaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsace WGcbGbauaaceWGgbGbauaaaOqaaKqzGeGaaGOmaiaadkealmaaCaaa jeaibeqaaKqzadGaaGOmaaaaaaqcLbsacqGHsisljuaGdaWcaaGcba qcLbsaceWGgbGbauGbauaaaOqaaKqzGeGaamOqaaaacqGHsislcqaH 6oWAcaaIOaGaamiCaiabgUcaRiabeg8aYjaaiMcacaaI9aGaaGimai aac6caaaa@999E@ (20)

Similarly, we get just two independent equations by subtracting (22) and (33) components from (00).

A F 2AB A '2 F 4 A 2 B C F 2BC + C '2 F 4B C 2 + A F 2AB C F 2BC κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafyaafaGaamOraaGcbaqcLbsacaaIYaGaamyqaiaa dkeaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaCaaale qabaqcLbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcbaqcLbsa caaI0aGaamyqaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGe GaamOqaaaacqGHsisljuaGdaWcaaGcbaqcLbsaceWGdbGbauGbauaa caWGgbaakeaajugibiaaikdacaWGcbGaam4qaaaacqGHRaWkjuaGda WcaaGcbaqcLbsacaWGdbqcfa4aaWbaaSqabeaajugibiaadEcajugW aiaaikdaaaqcLbsacaWGgbaakeaajugibiaaisdacaWGcbGaam4qaK qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaaajugibiabgUcaRKqb aoaalaaakeaajugibiqadgeagaqbaiqadAeagaqbaaGcbaqcLbsaca aIYaGaamyqaiaadkeaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGabm4q ayaafaGabmOrayaafaaakeaajugibiaaikdacaWGcbGaam4qaaaacq GHsislcqaH6oWAcaaIOaGaamiCaiabgUcaRiabeg8aYjaaiMcacaaI 9aGaaGimaiaaiYcaaaa@7547@  (21)

A F 2AB A '2 F 4 A 2 B B F 2 B 2 + B '2 F 2 B 3 A B F 4A B 2 B C F 4 B 2 C + A C F 4ABC + A F 2AB B F 2 B 2 κ(p+ρ)=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafyaafaGaamOraaGcbaqcLbsacaaIYaGaamyqaiaa dkeaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaCaaale qabaqcLbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcbaqcLbsa caaI0aGaamyqaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaam OqaaaacqGHsisljuaGdaWcaaGcbaqcLbsaceWGcbGbauGbauaacaWG gbaakeaajugibiaaikdacaWGcbqcfa4aaWbaaSqabKqaGeaajugWai aaikdaaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaamOqaKqb aoaaCaaaleqabaqcLbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraa GcbaqcLbsacaaIYaGaamOqaSWaaWbaaKqaGeqabaqcLbmacaaIZaaa aaaajugibiabgkHiTKqbaoaalaaakeaajugibiqadgeagaqbaiqadk eagaqbaiaadAeaaOqaaKqzGeGaaGinaiaadgeacaWGcbqcfa4aaWba aSqabKqaGeaajugWaiaaikdaaaaaaKqzGeGaeyOeI0scfa4aaSaaaO qaaKqzGeGabmOqayaafaGabm4qayaafaGaamOraaGcbaqcLbsacaaI 0aGaamOqaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaam4qaa aacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGbbGbauaaceWGdbGbauaa caWGgbaakeaajugibiaaisdacaWGbbGaamOqaiaadoeaaaGaey4kaS scfa4aaSaaaOqaaKqzGeGabmyqayaafaGabmOrayaafaaakeaajugi biaaikdacaWGbbGaamOqaaaacqGHsisljuaGdaWcaaGcbaqcLbsace WGcbGbauaaceWGgbGbauaaaOqaaKqzGeGaaGOmaiaadkeajuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaaaaqcLbsacqGHsislcqaH6oWAca aIOaGaamiCaiabgUcaRiabeg8aYjaaiMcacaaI9aGaaGimaiaac6ca aaa@95DF@ (22)

These are the three non–linear ordinary differential equations in which six unknown variables A,B,C,F,ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbGaaG ilaiaadkeacaaISaGaam4qaiaaiYcacaWGgbGaaGilaiabeg8aYbaa @3D2D@ and p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWbaaaa@366A@ are involved. We can not find the solution of these equations directly. Condition of constant curvature has been used to solve these equations.

Constant curvature solution

Let’s say, for constant curvature R= R 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsbGaaG ypaiaadkfajuaGdaWgaaqcbasaaKqzadGaaGimaaWcbeaaaaa@3AB6@ , it is apparent that the first and second derivatives of F(R)= df(R) dR MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgbGaaG ikaiaadkfacaaIPaGaaGypaKqbaoaalaaakeaajugibiaadsgacaWG MbGaaGikaiaadkfacaaIPaaakeaajugibiaadsgacaWGsbaaaaaa@40E3@ will always reduce to:

F ( R 0 )=0= F ( R 0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGgbGbau aacaaIOaGaamOuaSWaaSbaaKqaGeaajugWaiaaicdaaKqaGeqaaKqz GeGaaGykaiaai2dacaaIWaGaaGypaiqadAeagaqbgaqbaiaaiIcaca WGsbWcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacaaIPaGa aiOlaaaa@448E@ (23)

In the Equation (23), the Equation (20)–(22) and Equation (14) reduce to

A C F 4ABC B F 2 B 2 + B '2 F 2 B 3 C F 2BC + C '2 F 4B C 2 + A B F 4A B 2 + B C F 4 B 2 C κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafaGabm4qayaafaGaamOraaGcbaqcLbsacaaI0aGa amyqaiaadkeacaWGdbaaaiabgkHiTKqbaoaalaaakeaajugibiqadk eagaqbgaqbaiaadAeaaOqaaKqzGeGaaGOmaiaadkeajuaGdaahaaWc beqcbasaaKqzadGaaGOmaaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcba qcLbsacaWGcbqcfa4aaWbaaSqabeaajugibiaadEcajugWaiaaikda aaqcLbsacaWGgbaakeaajugibiaaikdacaWGcbWcdaahaaqcbasabe aajugWaiaaiodaaaaaaKqzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGa bm4qayaafyaafaGaamOraaGcbaqcLbsacaaIYaGaamOqaiaadoeaaa Gaey4kaSscfa4aaSaaaOqaaKqzGeGaam4qaKqbaoaaCaaaleqabaqc LbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcbaqcLbsacaaI0a GaamOqaiaadoeajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaaqc LbsacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGbbGbauaaceWGcbGbau aacaWGgbaakeaajugibiaaisdacaWGbbGaamOqaSWaaWbaaKqaGeqa baqcLbmacaaIYaaaaaaajugibiabgUcaRKqbaoaalaaakeaajugibi qadkeagaqbaiqadoeagaqbaiaadAeaaOqaaKqzGeGaaGinaiaadkea lmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaadoeaaaGaeyOeI0 IaeqOUdSMaaGikaiaadchacqGHRaWkcqaHbpGCcaaIPaGaaGypaiaa icdacaGGSaaaaa@85E3@ (24)

A F 2AB A '2 F 4 A 2 B C F 2BC + C '2 F 4B C 2 κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafyaafaGaamOraaGcbaqcLbsacaaIYaGaamyqaiaa dkeaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaCaaale qabaqcLbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcbaqcLbsa caaI0aGaamyqaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaam OqaaaacqGHsisljuaGdaWcaaGcbaqcLbsaceWGdbGbauGbauaacaWG gbaakeaajugibiaaikdacaWGcbGaam4qaaaacqGHRaWkjuaGdaWcaa GcbaqcLbsacaWGdbqcfa4aaWbaaSqabeaajugibiaadEcajugWaiaa ikdaaaqcLbsacaWGgbaakeaajugibiaaisdacaWGcbGaam4qaKqbao aaCaaaleqajeaibaqcLbmacaaIYaaaaaaajugibiabgkHiTiabeQ7a RjaaiIcacaWGWbGaey4kaSIaeqyWdiNaaGykaiaai2dacaaIWaGaai ilaaaa@675C@  (25)

A F 2AB A '2 F 4 A 2 B B F 2 B 2 + B '2 F 2 B 3 A B F 4A B 2 B C F 4 B 2 C + A C F 4ABC κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafyaafaGaamOraaGcbaqcLbsacaaIYaGaamyqaiaa dkeaaaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamyqaKqbaoaaCaaale qabaqcLbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcbaqcLbsa caaI0aGaamyqaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaam OqaaaacqGHsisljuaGdaWcaaGcbaqcLbsaceWGcbGbauGbauaacaWG gbaakeaajugibiaaikdacaWGcbWcdaahaaqcbasabeaajugWaiaaik daaaaaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaamOqaKqbaoaa CaaaleqabaqcLbsacaWGNaqcLbmacaaIYaaaaKqzGeGaamOraaGcba qcLbsacaaIYaGaamOqaKqbaoaaCaaaleqajeaibaqcLbmacaaIZaaa aaaajugibiabgkHiTKqbaoaalaaakeaajugibiqadgeagaqbaiqadk eagaqbaiaadAeaaOqaaKqzGeGaaGinaiaadgeacaWGcbqcfa4aaWba aSqabKqaGeaajugWaiaaikdaaaaaaKqzGeGaeyOeI0scfa4aaSaaaO qaaKqzGeGabmOqayaafaGabm4qayaafaGaamOraaGcbaqcLbsacaaI 0aGaamOqaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaam4qaa aacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGbbGbauaaceWGdbGbauaa caWGgbaakeaajugibiaaisdacaWGbbGaamOqaiaadoeaaaGaeyOeI0 IaeqOUdSMaaGikaiaadchacqGHRaWkcqaHbpGCcaaIPaGaaGypaiaa icdacaGGSaaaaa@85F1@ (26)

A AB A '2 2 A 2 B + B B 2 B '2 B 3 + C BC C '2 2B C 2 + A C 2ABC R 0 =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaOqaaK qzGeGabmyqayaafyaafaaakeaajugibiaadgeacaWGcbaaaiabgkHi TKqbaoaalaaakeaajugibiaadgeajuaGdaahaaWcbeqaaKqzGeGaam 4jaKqzadGaaGOmaaaaaOqaaKqzGeGaaGOmaiaadgeajuaGdaahaaWc beqcbasaaKqzadGaaGOmaaaajugibiaadkeaaaGaey4kaSscfa4aaS aaaOqaaKqzGeGabmOqayaafyaafaaakeaajugibiaadkeajuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaaaaqcLbsacqGHsisljuaGdaWcaa GcbaqcLbsacaWGcbqcfa4aaWbaaSqabeaajugibiaadEcajugWaiaa ikdaaaaakeaajugibiaadkeajuaGdaahaaWcbeqcbasaaKqzadGaaG 4maaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGdbGbauGb auaaaOqaaKqzGeGaamOqaiaadoeaaaGaeyOeI0scfa4aaSaaaOqaaK qzGeGaam4qaKqbaoaaCaaaleqabaqcLbsacaWGNaqcLbmacaaIYaaa aaGcbaqcLbsacaaIYaGaamOqaiaadoealmaaCaaajeaibeqaaKqzad GaaGOmaaaaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsaceWGbbGb auaaceWGdbGbauaaaOqaaKqzGeGaaGOmaiaadgeacaWGcbGaam4qaa aacqGHsislcaWGsbWcdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqc LbsacaaI9aGaaGimaiaac6caaaa@7960@ (27)

We will solve these equations using power law as well as exponential law assumptions.

Power Law Assumption

Power law assumption is used to solve these equations i.e., A r m ,B r n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbGaey yhIuRaamOCaSWaaWbaaKqaGeqabaqcLbmacaWGTbaaaKqzGeGaaGil aiaadkeacqGHDisTcaWGYbWcdaahaaqcbasabeaajugWaiaad6gaaa aaaa@4224@ and C r q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdbGaey yhIuRaamOCaSWaaWbaaKazba4=beqaaKqzadGaamyCaaaaaaa@3CF2@ , where m,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTbGaaG ilaiaad6gaaaa@3810@ and q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXbaaaa@366B@ are any real numbers. Therefore, we use A= k 1 r m ,B= k 2 r n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbGaaG ypaiaadUgajuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiaa dkhajuaGdaahaaWcbeqcbasaaKqzadGaamyBaaaajugibiaaiYcaca WGcbGaaGypaiaadUgalmaaBaaajeaibaqcLbmacaaIYaaajeaibeaa jugibiaadkhalmaaCaaajeaibeqaaKqzadGaamOBaaaaaaa@4975@ , and C= k 3 r q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdbGaaG ypaiaadUgalmaaBaaajeaibaqcLbmacaaIZaaajeaibeaajugibiaa dkhalmaaCaaajeaibeqaaKqzadGaamyCaaaaaaa@3E60@ where k 1 , k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRbWcda WgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacaaISaGaam4AaSWa aSbaaKqaGeaajugWaiaaikdaaKqaGeqaaaaa@3D6D@ and k 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRbWcda WgaaqcbasaaKqzadGaaG4maaqcbasabaaaaa@38D0@ are constants of proportionality. By inserting these values of A,B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbbGaaG ilaiaadkeaaaa@37B8@ and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdbaaaa@363D@ in Equations (24)–(26) and subtracting them, we attain

m 2 2m2nmnnqmq=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGTbGcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGH sislcaaIYaGaamyBaiabgkHiTiaaikdacaWGUbGaeyOeI0IaamyBai aad6gacqGHsislcaWGUbGaamyCaiabgkHiTiaad2gacaWGXbGaeyyp a0JaaGimaiaacYcaaaa@4E4B@ (28)

m 2 2m+ q 2 2mn2nq2q=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGTbWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGH sislcaaIYaGaamyBaiabgUcaRiaadghalmaaCaaajeaibeqaaKqzad GaaGOmaaaajugibiabgkHiTiaaikdacaWGTbGaamOBaiabgkHiTiaa ikdacaWGUbGaamyCaiabgkHiTiaaikdacaWGXbGaeyypa0JaaGimai aacYcaaaa@519A@ (29)

q 2 2q+2n+mqmnnq=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpe ea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpe peKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcba qcLbsacaWGXbGcdaahaaWcbeqcbasaaKqzadGaaGOmaaaajugibiab gkHiTiaaikdacaWGXbGaey4kaSIaaGOmaiaad6gacqGHRaWkcaWGTb GaamyCaiabgkHiTiaad2gacaWGUbGaeyOeI0IaamOBaiaadghacqGH 9aqpcaaIWaGaaiOlaaaa@4E4A@ (30)

Also when we put these values in Equations (27) and compare coefficient, we obtain

m 2 2m2n+ q 2 2q+mq=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTbWcda ahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHsislcaaIYaGaamyB aiabgkHiTiaaikdacaWGUbGaey4kaSIaamyCaSWaaWbaaKqaGeqaba qcLbmacaaIYaaaaKqzGeGaeyOeI0IaaGOmaiaadghacqGHRaWkcaWG TbGaamyCaiaai2dacaaIWaGaaiOlaaaa@4AB2@ (31)

To solve these equations, we consider the following cases

I. m=0,n=q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2gacaaI9a GaaGimaiaaiYcacaWGUbGaaGypaiaadghaaaa@3CAF@  II. n=0,m=q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6gacaaI9a GaaGimaiaaiYcacaWGTbGaaGypaiaadghaaaa@3CAF@ , III. q=0,m=n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghacaaI9a GaaGimaiaaiYcacaWGTbGaaGypaiaad6gaaaa@3CAF@

Case I:

When we put m=0,n=q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbGaaGypaiaaic dacaaISaGaamOBaiaai2dacaWGXbaaaa@3C20@ in Equation (31), we obtain the following equation

q 2 4q=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghalmaaCa aajeaibeqaaKqzadGaaGOmaaaajugibiabgkHiTiaaisdacaWGXbGa aGypaiaaicdacaaISaaaaa@3F74@

which gives two cases, i.e., either q=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghacaaI9a GaaGimaaaa@394D@  or q=4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadghacaaI9a GaaGinaaaa@3951@ . In former case we get trivial solution while the later case yields the non–trivial solution, given as

d s 2 = k 1 d t 2 k 2 r 4 d ρ 2 k 3 r 4 d ϕ 2 k 2 r 4 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGRbWaaSbaaSqaaiaaigdaaeqa aOGaamizaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacqaHbpGCdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaiodaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacqaHvpGzdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacaWG6bWaaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa@5797@  (32)

For this solution it is evaluated that:

R=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaaGypaiaaic dacaaISaaaaa@3955@

ρ= 4F κ(1+ω) k 2 r 6 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCcaaI9aWaaS aaaeaacaaI0aGaamOraaqaaiabeQ7aRjaaiIcacaaIXaGaey4kaSIa eqyYdCNaaGykaiaadUgadaWgaaWcbaGaaGOmaaqabaGccaWGYbWaaW baaSqabeaacaaI2aaaaaaakiaaiYcaaaa@456E@

which is a non–vacuum solution.

Case II:

In this case, we insert n=0,m=q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbGaaGypaiaaic dacaaISaGaamyBaiaai2dacaWGXbaaaa@3C20@ in Equation (31) and have

3 m 2 4m=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIZaGaamyBamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaaisdacaWGTbGaaGypaiaaicda caaISaaaaa@3DBD@

which gives two cases, i.e., either m=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbGaaGypaiaaic daaaa@38BA@ or m= 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbGaaGypamaala aabaGaaGinaaqaaiaaiodaaaaaaa@398B@ . In first case we get trivial solution while in second case we have the following non–trivial solution

d s 2 = k 1 r 4 3 d t 2 k 2 d ρ 2 k 3 r 4 3 d ϕ 2 k 2 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGRbWaaSbaaSqaaiaaigdaaeqa aOGaamOCamaaCaaaleqabaWaaSaaaeaacaaI0aaabaGaaG4maaaaaa GccaWGKbGaamiDamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadUga daWgaaWcbaGaaGOmaaqabaGccaWGKbGaeqyWdi3aaWbaaSqabeaaca aIYaaaaOGaeyOeI0Iaam4AamaaBaaaleaacaaIZaaabeaakiaadkha daahaaWcbeqaamaalaaabaGaaGinaaqaaiaaiodaaaaaaOGaamizai abew9aMnaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadUgadaWgaaWc baGaaGOmaaqabaGccaWGKbGaamOEamaaCaaaleqabaGaaGOmaaaaki aac6caaaa@5745@ (33)

For this solution, the Ricci scalar and the energy density have been evaluated as

R=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaaGypaiaaic dacaaISaaaaa@3955@

ρ= 8F 27κ(1+ω) k 2 r 6 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCcaaI9aWaaS aaaeaacaaI4aGaamOraaqaaiaaikdacaaI3aGaeqOUdSMaaGikaiaa igdacqGHRaWkcqaHjpWDcaaIPaGaam4AamaaBaaaleaacaaIYaaabe aakiaadkhadaahaaWcbeqaamaalaaabaGaaGOnaaqaaiaaiodaaaaa aOGaaGilaaaaaaa@47BC@

which is obviously a non–vacuum solution.

Case III:

Here, we put q=0,m=n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbGaaGypaiaaic dacaaISaGaamyBaiaai2dacaWGUbaaaa@3C20@  in Equation (31) and obtain that n 2 4n=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6galmaaCa aajeaibeqaaKqzadGaaGOmaaaajugibiabgkHiTiaaisdacaWGUbGa aGypaiaaicdacaaISaaaaa@3F6E@

which have again two cases either q=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbGaaGypaiaaic daaaa@38BE@ or q=4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGXbGaaGypaiaais daaaa@38C2@ . In previous case, the solution is trivial. But later case yields the following non–trivial, presented as

d s 2 = k 1 r 4 d t 2 k 2 r 4 d ρ 2 k 3 d ϕ 2 k 2 r 4 d z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGRbWaaSbaaSqaaiaaigdaaeqa aOGaamOCamaaCaaaleqabaGaaGinaaaakiaadsgacaWG0bWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0Iaam4AamaaBaaaleaacaaIYaaabeaa kiaadkhadaahaaWcbeqaaiaaisdaaaGccaWGKbGaeqyWdi3aaWbaaS qabeaacaaIYaaaaOGaeyOeI0Iaam4AamaaBaaaleaacaaIZaaabeaa kiaadsgacqaHvpGzdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacaWG6bWaaWbaaSqabeaacaaIYaaaaaaa@56DB@ . (34)

For this solution, we evaluated R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbaaaa@371E@ and ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCaaa@3807@ as

R=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaaGypaiaaic dacaaISaaaaa@3955@

ρ= 8F κ(1+ω) k 2 r 6 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCcqGH9aqpda WcaaqaaiaaiIdacaWGgbaabaGaeqOUdSMaaGikaiaaigdacqGHRaWk cqaHjpWDcaaIPaGaam4AamaaBaaaleaacaaIYaaabeaakiaadkhada ahaaWcbeqaaiaaiAdaaaaaaOGaaGOlaaaa@45B3@

Again it proves that the solution is non–vacuum.

Exponential law assumption

By using exponential law assumption, i.e., inserting A(r)= e 2μ(r) ,B(r)= e 2ν(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaaGikaiaadk hacaaIPaGaaGypaiaadwgadaahaaWcbeqaaiaaikdacqaH8oqBcaaI OaGaamOCaiaaiMcaaaGccaaISaGaamOqaiaaiIcacaWGYbGaaGykai aai2dacaWGLbWaaWbaaSqabeaacaaIYaGaeqyVd4MaaGikaiaadkha caaIPaaaaaaa@4AA6@ and C(r)= e 2λ(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaaGikaiaadk hacaaIPaGaaGypaiaadwgadaahaaWcbeqaaiaaikdacqaH7oaBcaaI OaGaamOCaiaaiMcaaaaaaa@4015@ so that Equations (20)–(22) and (14) be

( μ λ ν λ '2 λ + μ ν + ν λ )F e 2ν F e 2ν + ν F e 2ν κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGafqiVd0Mbau aacuaH7oaBgaqbaiabgkHiTiqbe27aUzaafyaafaGaeyOeI0Iaeq4U dW2aaWbaaSqabeaacaWGNaGaaGOmaaaakiabgkHiTiqbeU7aSzaafy aafaGaey4kaSIafqiVd0MbauaacuaH9oGBgaqbaiabgUcaRiqbe27a UzaafaGafq4UdWMbauaacaaIPaGaamOraiaadwgadaahaaWcbeqaai abgkHiTiaaikdacqaH9oGBaaGccqGHsislceWGgbGbauGbauaacaWG LbWaaWbaaSqabeaacqGHsislcaaIYaGaeqyVd4gaaOGaey4kaSIafq yVd4MbauaaceWGgbGbauaacaWGLbWaaWbaaSqabeaacqGHsislcaaI YaGaeqyVd4gaaOGaeyOeI0IaeqOUdSMaaGikaiaadchacqGHRaWkcq aHbpGCcaaIPaGaaGypaiaaicdacaGGSaaaaa@6B19@ (35)

( μ '2 + μ λ '2 λ )F e 2ν +( μ λ ) F e 2ν κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGaeqiVd02aaW baaSqabeaacaWGNaGaaGOmaaaakiabgUcaRiqbeY7aTzaafyaafaGa eyOeI0Iaeq4UdW2aaWbaaSqabeaacaWGNaGaaGOmaaaakiabgkHiTi qbeU7aSzaafyaafaGaaGykaiaadAeacaWGLbWaaWbaaSqabeaacqGH sislcaaIYaGaeqyVd4gaaOGaey4kaSIaaGikaiqbeY7aTzaafaGaey OeI0Iafq4UdWMbauaacaaIPaGabmOrayaafaGaamyzamaaCaaaleqa baGaeyOeI0IaaGOmaiabe27aUbaakiabgkHiTiabeQ7aRjaaiIcaca WGWbGaey4kaSIaeqyWdiNaaGykaiaai2dacaaIWaGaaiilaaaa@5FD0@ (36)

( μ '2 + μ + μ λ ν μ ν ν λ )F e 2ν +( μ ν ) F e 2ν κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGaeqiVd02aaW baaSqabeaacaWGNaGaaGOmaaaakiabgUcaRiqbeY7aTzaafyaafaGa ey4kaSIafqiVd0MbauaacuaH7oaBgaqbaiabgkHiTiqbe27aUzaafy aafaGaeyOeI0IafqiVd0MbauaacuaH9oGBgaqbaiabgkHiTiqbe27a UzaafaGafq4UdWMbauaacaaIPaGaamOraiaadwgadaahaaWcbeqaai abgkHiTiaaikdacqaH9oGBaaGccqGHRaWkcaaIOaGafqiVd0Mbauaa cqGHsislcuaH9oGBgaqbaiaaiMcaceWGgbGbauaacaWGLbWaaWbaaS qabeaacqGHsislcaaIYaGaeqyVd4gaaOGaeyOeI0IaeqOUdSMaaGik aiaadchacqGHRaWkcqaHbpGCcaaIPaGaaGypaiaaicdacaGGSaaaaa@68E0@ (37)

( μ '2 + μ + ν + λ '2 + λ + μ λ )F e 2ν R 0 2 =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGaeqiVd02aaW baaSqabeaacaWGNaGaaGOmaaaakiabgUcaRiqbeY7aTzaafyaafaGa ey4kaSIafqyVd4MbauGbauaacqGHRaWkcqaH7oaBdaahaaWcbeqaai aadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWkcuaH 8oqBgaqbaiqbeU7aSzaafaGaaGykaiaadAeacaWGLbWaaWbaaSqabe aacqGHsislcaaIYaGaeqyVd4gaaOGaeyOeI0YaaSaaaeaacaWGsbWa aSbaaSqaaiaaicdaaeqaaaGcbaGaaGOmaaaacaaI9aGaaGimaiaac6 caaaa@56A7@ (38)

Now, we get four non linear differential equations and six unknown μ,ν,λ,F,p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8oqBca aISaGaeqyVd4MaaGilaiabeU7aSjaaiYcacaWGgbGaaGilaiaadcha aaa@3F2F@ and ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHbpGCaa a@3735@ . By assumption of constant scalar curvature, we calculate the solution of these equations For constant curvature,these equation are as follows

( μ λ ν λ '2 λ + μ ν + ν λ )F e 2ν κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGafqiVd0Mbau aacuaH7oaBgaqbaiabgkHiTiqbe27aUzaafyaafaGaeyOeI0Iaeq4U dW2aaWbaaSqabeaacaWGNaGaaGOmaaaakiabgkHiTiqbeU7aSzaafy aafaGaey4kaSIafqiVd0MbauaacuaH9oGBgaqbaiabgUcaRiqbe27a UzaafaGafq4UdWMbauaacaaIPaGaamOraiaadwgadaahaaWcbeqaai abgkHiTiaaikdacqaH9oGBaaGccqGHsislcqaH6oWAcaaIOaGaamiC aiabgUcaRiabeg8aYjaaiMcacaaI9aGaaGimaiaacYcaaaa@5CC9@  (39)

( μ '2 + μ λ '2 λ )F e 2 κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGaeqiVd02aaW baaSqabeaacaWGNaGaaGOmaaaakiabgUcaRiqbeY7aTzaafyaafaGa eyOeI0Iaeq4UdW2aaWbaaSqabeaacaWGNaGaaGOmaaaakiabgkHiTi qbeU7aSzaafyaafaGaaGykaiaadAeacaWGLbWaaWbaaSqabeaacqGH sislcaaIYaaaaOGaeyOeI0IaeqOUdSMaaGikaiaadchacqGHRaWkcq aHbpGCcaaIPaGaaGypaiaaicdacaGGSaaaaa@5209@  (40)

( μ '2 + μ + ν + λ '2 + λ + μ λ )F e 2ν R 0 2 =0( μ '2 + μ + μ λ ν μ ν ν λ )F e 2ν κ(p+ρ)=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGaeqiVd02aaW baaSqabeaacaWGNaGaaGOmaaaakiabgUcaRiqbeY7aTzaafyaafaGa ey4kaSIafqyVd4MbauGbauaacqGHRaWkcqaH7oaBdaahaaWcbeqaai aadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWkcuaH 8oqBgaqbaiqbeU7aSzaafaGaaGykaiaadAeacaWGLbWaaWbaaSqabe aacqGHsislcaaIYaGaeqyVd4gaaOGaeyOeI0YaaSaaaeaacaWGsbWa aSbaaSqaaiaaicdaaeqaaaGcbaGaaGOmaaaacaaI9aGaaGimaiaaiI cacqaH8oqBdaahaaWcbeqaaiaadEcacaaIYaaaaOGaey4kaSIafqiV d0MbauGbauaacqGHRaWkcuaH8oqBgaqbaiqbeU7aSzaafaGaeyOeI0 IafqyVd4MbauGbauaacqGHsislcuaH8oqBgaqbaiqbe27aUzaafaGa eyOeI0IafqyVd4MbauaacuaH7oaBgaqbaiaaiMcacaWGgbGaamyzam aaCaaaleqabaGaeyOeI0IaaGOmaiabe27aUbaakiabgkHiTiabeQ7a RjaaiIcacaWGWbGaey4kaSIaeqyWdiNaaGykaiaai2dacaaIWaGaai ilaaaa@7C7B@  (41)

( μ '2 + μ + ν + λ '2 + λ + μ λ )F e 2ν R 0 2 =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGaeqiVd02aaW baaSqabeaacaWGNaGaaGOmaaaakiabgUcaRiqbeY7aTzaafyaafaGa ey4kaSIafqyVd4MbauGbauaacqGHRaWkcqaH7oaBdaahaaWcbeqaai aadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWkcuaH 8oqBgaqbaiqbeU7aSzaafaGaaGykaiaadAeacaWGLbWaaWbaaSqabe aacqGHsislcaaIYaGaeqyVd4gaaOGaeyOeI0YaaSaaaeaacaWGsbWa aSbaaSqaaiaaicdaaeqaaaGcbaGaaGOmaaaacaaI9aGaaGimaiaac6 caaaa@56A7@ (42)

The subtraction of Equations (40), (41) from (39) and similarly, the subtraction of Equations (41) from (40). Also comparing coefficient of Equations (42). we have,

μ '2 + μ + ν μ ν ν λ μ λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHRaWk cuaH9oGBgaqbgaqbaiabgkHiTiqbeY7aTzaafaGafqyVd4Mbauaacq GHsislcuaH9oGBgaqbaiqbeU7aSzaafaGaeyOeI0IafqiVd0Mbauaa cuaH7oaBgaqbaiaai2dacaaIWaGaaiilaaaa@4E80@ (43)

μ '2 + μ + λ '2 + λ 2 μ ν 2 ν λ )=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHRaWk cqaH7oaBdaahaaWcbeqaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdW MbauGbauaacqGHsislcaaIYaGafqiVd0MbauaacuaH9oGBgaqbaiab gkHiTiaaikdacuaH9oGBgaqbaiqbeU7aSzaafaGaaGykaiaai2daca aIWaGaaiilaaaa@506D@ (44)

λ '2 + λ + μ λ ν μ ν ν λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWk cuaH8oqBgaqbaiqbeU7aSzaafaGaeyOeI0IafqyVd4MbauGbauaacq GHsislcuaH8oqBgaqbaiqbe27aUzaafaGaeyOeI0IafqyVd4Mbauaa cuaH7oaBgaqbaiabg2da9KqzGeGaaGimaiaacYcaaaa@4F4A@  (45)

μ '2 + μ + ν + λ '2 + λ + μ λ =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHRaWk cuaH9oGBgaqbgaqbaiabgUcaRiabeU7aSnaaCaaaleqabaGaam4jai aaikdaaaGccqGHRaWkcuaH7oaBgaqbgaqbaiabgUcaRiqbeY7aTzaa faGafq4UdWMbauaacaaI9aGaaGimaiaac6caaaa@4C75@ (46)

We take into account the following three cases, in order to solve these equations

 I. λ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeU7aSzaafa GaaGypaiaaicdaaaa@3A17@ , II. μ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeY7aTzaafa GaaGypaiaaicdaaaa@3A19@ , III. ν =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbe27aUzaafa GaaGypaiaaicdacaaIUaaaaa@3AD3@

Case I:

We consider the value of λ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqbaiaai2 dacaaIWaaaaa@3988@ , It implies that

λ= c 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBcaaI9aGaam 4yamaaBaaaleaacaaIXaaabeaakiaacYcaaaa@3B4B@  (47)

where c 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJbWcda WgaaqcbasaaKqzadGaaGymaaqcbasabaaaaa@38C6@ is an integration constant. Using this value in Equation (43)–(46), we get

μ '2 + μ + ν μ ν =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHRaWk cuaH9oGBgaqbgaqbaiabgkHiTiqbeY7aTzaafaGafqyVd4Mbauaaca aI9aGaaGimaiaacYcaaaa@45A0@  (48)

μ '2 + μ 2 μ ν =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHsisl caaIYaGafqiVd0MbauaacuaH9oGBgaqbaiaai2dacaaIWaGaaiilaa aa@43AB@  (49)

ν μ ν =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislcuaH9oGBga qbgaqbaiabgkHiTiqbeY7aTzaafaGafqyVd4MbauaacaaI9aGaaGim aiaacYcaaaa@3FA7@  (50)

μ '2 + μ + ν =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHRaWk cuaH9oGBgaqbgaqbaiaai2dacaaIWaGaaiOlaaaa@412F@  (51)

By using the Equation (51) into Equation (48), we obtain

μ ν =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBgaqbaiqbe2 7aUzaafaGaaGypaiaaicdaaaa@3B4E@  (52)

Now, we use Equation (52) into Equation (49) and (50), we get

μ + μ '2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeY7aTzaafy aafaGaey4kaSIaeqiVd02cdaahaaqcbasabeaajugWaiaadEcacaaI YaaaaKqzGeGaaGypaiaaicdaaaa@4038@ (53)

ν =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbe27aUzaafy aafaGaaGypaiaaicdacaGGUaaaaa@3AD8@  (54)

We can easily get the solution for the above equations as

μ=ln( a 1 r+ a 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeY7aTjaai2 daciGGSbGaaiOBaiaaiIcacaWGHbWcdaWgaaqcbasaaKqzadGaaGym aaqcbasabaqcLbsacaWGYbGaey4kaSIaamyyaSWaaSbaaKqaGeaaju gWaiaaikdaaKqaGeqaaKqzGeGaaGykaiaacYcaaaa@46E2@ (55)

ν= b 1 r+ b 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe27aUjaai2 dacaWGIbWcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbsacaWG YbGaey4kaSIaamOyaSWaaSbaaKqaGeaajugWaiaaikdaaKqaGeqaaK qzGeGaaiOlaaaa@439F@  (56)

Thus, we have found the following non–vacuum solution:

d s 2 =( a 1 r+ a 2 ) 2 d t 2 e 2 b 1 r+2 b 2 d ρ 2 c 2 d ϕ 2 e 2 b 1 r+2 b 2 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadsgacaWGZb WcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacaaI9aGaaGikaiaa dggalmaaBaaajeaibaqcLbmacaaIXaaajeaibeaajugibiaadkhacq GHRaWkcaWGHbGcdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiaa iMcalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaadsgacaWG0b WcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGHsislcaWGLbWc daahaaqabeaajugWaiaaikdacaWGIbWcdaWgaaqaaKqzadGaaGymaa WcbeaajugWaiaadkhacqGHRaWkcaaIYaGaamOyaSWaaSbaaeaajugW aiaaikdaaSqabaaaaKqzGeGaamizaiabeg8aYPWaaWbaaSqabKqaGe aajugWaiaaikdaaaqcLbsacqGHsislcaWGJbGcdaWgaaqcbasaaKqz adGaaGOmaaqcbasabaqcLbsacaWGKbGaeqy1dyMcdaahaaqcbasabe aajugWaiaaikdaaaqcLbsacqGHsislcaWGLbGcdaahaaqcbasabeaa jugWaiaaikdacaWGIbGcdaWgaaqcbasaaKqzadGaaGymaaqcbasaba qcLbmacaWGYbGaey4kaSIaaGOmaiaadkgakmaaBaaajeaibaqcLbma caaIYaaajeaibeaaaaqcLbsacaWGKbGaamOEaOWaaWbaaSqabKqaGe aajugWaiaaikdaaaqcLbsacaGGUaaaaa@8146@  (57)

For this solution, we obtain

R=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadkfacaaI9a GaaGimaiaaiYcaaaa@39E4@

ρ= 2 a 1 b 1 F κ(1+ω)( a 1 r+ a 2 ) e 2 b 1 r+2 b 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCcaaI9aWaaS aaaeaacaaIYaGaamyyamaaBaaaleaacaaIXaaabeaakiaadkgadaWg aaWcbaGaaGymaaqabaGccaWGgbaabaGaeqOUdSMaaGikaiaaigdacq GHRaWkcqaHjpWDcaaIPaGaaGikaiaadggadaWgaaWcbaGaaGymaaqa baGccaWGYbGaey4kaSIaamyyamaaBaaaleaacaaIYaaabeaakiaaiM cacaWGLbWaaWbaaSqabeaacaaIYaGaamOyamaaBaaabaGaaGymaaqa baGaamOCaiabgUcaRiaaikdacaWGIbWaaSbaaeaacaaIYaaabeaaaa aaaOGaaGOlaaaa@5433@

Case II:

We have assumed the value of μ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBgaqbaiaai2 dacaaIWaaaaa@398A@ , It implies that

μ= a 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcaaI9aGaam yyamaaBaaaleaacaaIZaaabeaakiaacYcaaaa@3B4D@ (58)

where a 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadggakmaaBa aajeaibaqcLbmacaaIZaaaleqaaaaa@3A07@ is an integration constant. Using this value in Equations (43)–(46), we get

ν ν λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbe27aUzaafy aafaGaeyOeI0IafqyVd4MbauaacuaH7oaBgaqbaiaai2dacaaIWaGa aiilaaaa@3F47@  (59)

λ '2 + λ 2 ν λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeU7aSPWaaW baaSqabeaajugibiaadEcajugWaiaaikdaaaqcLbsacqGHRaWkcuaH 7oaBgaqbgaqbaiabgkHiTiaaikdacuaH9oGBgaqbaiqbeU7aSzaafa GaaGypaiaaicdacaGGSaaaaa@4680@  (60)

λ '2 + λ ν ν λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHsisl cuaH9oGBgaqbgaqbaiabgkHiTiqbe27aUzaafaGafq4UdWMbauaaca aI9aGaaGimaiaacYcaaaa@45A5@  (61)

λ '2 + λ ν ν λ =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHsisl cuaH9oGBgaqbgaqbaiabgkHiTiqbe27aUzaafaGafq4UdWMbauaaca aI9aGaaGimaiaac6caaaa@45A7@  (62)

From the Equations (59) we have

ν = ν λ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH9oGBgaqbgaqbai aai2dacuaH9oGBgaqbaiqbeU7aSzaafaGaaiOlaaaa@3D13@  (63)

Using Equation (63) in Equations (60)–(62). we obtain

λ '2 + λ 2 ν =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHsisl caaIYaGafqyVd4MbauGbauaacaaI9aGaaGimaiaacYcaaaa@41F0@  (64)

λ '2 + λ +2 ν =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWk caaIYaGafqyVd4MbauGbauaacaaI9aGaaGimaiaac6caaaa@41E7@  (65)

These equations can be simplified we get

λ=ln( c 3 r+ c 4 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBcaaI9aGaci iBaiaac6gacaaIOaGaam4yamaaBaaaleaacaaIZaaabeaakiaadkha cqGHRaWkcaWGJbWaaSbaaSqaaiaaisdaaeqaaOGaaGykaiaacYcaaa a@424B@ (66)

ν= b 3 r+ b 4 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcaaI9aGaam OyamaaBaaaleaacaaIZaaabeaakiaadkhacqGHRaWkcaWGIbWaaSba aSqaaiaaisdaaeqaaOGaaiOlaaaa@3F06@ (67)

Thus, we get the following non–vacuum solution

d s 2 = a 4 d t 2 e 2 b 3 r+2 b 4 d ρ 2 ( c 3 r+ c 4 ) 2 d ϕ 2 e 2 b 3 r+2 b 4 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGHbWaaSbaaSqaaiaaisdaaeqa aOGaamizaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGLb WaaWbaaSqabeaacaaIYaGaamOyamaaBaaabaGaaG4maaqabaGaamOC aiabgUcaRiaaikdacaWGIbWaaSbaaeaacaaI0aaabeaaaaGccaWGKb GaeqyWdi3aaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGikaiaadoga daWgaaWcbaGaaG4maaqabaGccaWGYbGaey4kaSIaam4yamaaBaaale aacaaI0aaabeaakiaaiMcadaahaaWcbeqaaiaaikdaaaGccaWGKbGa eqy1dy2aaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyzamaaCaaale qabaGaaGOmaiaadkgadaWgaaqaaiaaiodaaeqaaiaadkhacqGHRaWk caaIYaGaamOyamaaBaaabaGaaGinaaqabaaaaOGaamizaiaadQhada ahaaWcbeqaaiaaikdaaaGccaGGUaaaaa@6407@ (68)

Corresponding this solution has been evaluated as

R=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadkfacaaI9a GaaGimaiaaiYcaaaa@39E4@

ρ= 2 b 3 c 3 F κ(1+ω)( c 3 r+ a 4 ) e 2 b 3 r+2 b 4 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCcaaI9aWaaS aaaeaacaaIYaGaamOyamaaBaaaleaacaaIZaaabeaakiaadogadaWg aaWcbaGaaG4maaqabaGccaWGgbaabaGaeqOUdSMaaGikaiaaigdacq GHRaWkcqaHjpWDcaaIPaGaaGikaiaadogadaWgaaWcbaGaaG4maaqa baGccaWGYbGaey4kaSIaamyyamaaBaaaleaacaaI0aaabeaakiaaiM cacaWGLbWaaWbaaSqabeaacaaIYaGaamOyamaaBaaabaGaaG4maaqa baGaamOCaiabgUcaRiaaikdacaWGIbWaaSbaaeaacaaI0aaabeaaaa aaaOGaaGOlaaaa@5443@

Case III:

When ν =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH9oGBgaqbaiaai2 dacaaIWaaaaa@398C@  we get on integrating that

ν= b 5 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH9oGBcaaI9aGaam OyamaaBaaaleaacaaI1aaabeaakiaacYcaaaa@3B52@ (69)

where b 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadkgakmaaBa aajeaibaqcLbmacaaI1aaaleqaaaaa@3A0A@ is an integration constant. Substituting this value of< ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabe27aUbaa@388E@ in Equations (40)–(46), we get

μ '2 + μ μ λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHsisl cuaH8oqBgaqbaiqbeU7aSzaafaGaaGypaiaaicdacaGGSaaaaa@42EB@ (70)

μ '2 + μ + λ '2 + λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0MbauGbauaacqGHRaWk cqaH7oaBdaahaaWcbeqaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdW MbauGbauaacaaI9aGaaGimaiaacYcaaaa@455E@ (71)

λ '2 + λ + μ λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWk cuaH8oqBgaqbaiqbeU7aSzaafaGaaGypaiaaicdacaGGSaaaaa@42DC@ (72)

λ '2 + λ + μ '2 + μ + μ λ =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacqGHRaWk cqaH8oqBdaahaaWcbeqaaiaadEcacaaIYaaaaOGaey4kaSIafqiVd0 MbauGbauaacqGHRaWkcuaH8oqBgaqbaiqbeU7aSzaafaGaaGypaiaa icdacaGGUaaaaa@49C4@ (73)

After making use of Equation (71) in (73), we get

μ λ =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBgaqbaiqbeU 7aSzaafaGaaGypaiaaicdacaGGUaaaaa@3BFC@  (74)

By using last equation in Equation (70) and Equation (72), we obtain

μ + μ '2 =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBgaqbgaqbai abgUcaRiabeY7aTnaaCaaaleqabaGaam4jaiaaikdaaaGccaaI9aGa aGimaiaacYcaaaa@3E7C@ (75)

λ '2 + λ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBdaahaaWcbe qaaiaadEcacaaIYaaaaOGaey4kaSIafq4UdWMbauGbauaacaaI9aGa aGimaiaacYcaaaa@3E78@ (76)

whose solutions can be easily obtained as

μ=ln( a 5 r+ a 6 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBcaaI9aGaam iBaiaad6gacaaIOaGaamyyamaaBaaaleaacaaI1aaabeaakiaadkha cqGHRaWkcaWGHbWaaSbaaSqaaiaaiAdaaeqaaOGaaGykaiaacYcaaa a@424D@ (77)

λ=ln( c 5 r+ c 6 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBcaaI9aGaam iBaiaad6gacaaIOaGaam4yamaaBaaaleaacaaI1aaabeaakiaadkha cqGHRaWkcaWGJbWaaSbaaSqaaiaaiAdaaeqaaOGaaGykaiaac6caaa a@4251@ (78)

Thus, the corresponding values of A,B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGbbGaaGilaiaabk eaaaa@3886@  and C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGdbaaaa@370D@ are

A=( a 5 r+ a 6 ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaaGypaiaaiI cacaWGHbWaaSbaaSqaaiaaiwdaaeqaaOGaamOCaiabgUcaRiaadgga daWgaaWcbaGaaGOnaaqabaGccaaIPaWaaWbaaSqabeaacaaIYaaaaO Gaaiilaaaa@406C@  (79)

B= b 6 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbGaaGypaiaadk gadaWgaaWcbaGaaGOnaaqabaGccaGGSaaaaa@3A62@  (80)

C=( c 5 r+ c 6 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaaGypaiaaiI cacaWGJbWaaSbaaSqaaiaaiwdaaeqaaOGaamOCaiabgUcaRiaadoga daWgaaWcbaGaaGOnaaqabaGccaaIPaWaaWbaaSqabeaacaaIYaaaaO GaaiOlaaaa@4074@  (81)

When we use these values in Equation (14), we evaluate the Ricci scalar as

R= 2 a 5 c 5 b 6 ( a 5 r+ a 6 )( c 5 r+ c 6 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaaGypamaala aabaGaaGOmaiaadggadaWgaaWcbaGaaGynaaqabaGccaWGJbWaaSba aSqaaiaaiwdaaeqaaaGcbaGaamOyamaaBaaaleaacaaI2aaabeaaki aaiIcacaWGHbWaaSbaaSqaaiaaiwdaaeqaaOGaamOCaiabgUcaRiaa dggadaWgaaWcbaGaaGOnaaqabaGccaaIPaGaaGikaiaadogadaWgaa WcbaGaaGynaaqabaGccaWGYbGaey4kaSIaam4yamaaBaaaleaacaaI 2aaabeaakiaaiMcaaaGaaiilaaaa@4CE4@ (82)

which is not constant. For the sake of constant Ricci scalar, we must take either a 5 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaSbaaSqaai aaiwdaaeqaaOGaaGypaiaaicdaaaa@39A3@ or c 5 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaSbaaSqaai aaiwdaaeqaaOGaaGypaiaaicdaaaa@39A5@ . In first case, we have

A= a 6 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaaGypaiaadg gadaqhaaWcbaGaaGOnaaqaaiaaikdaaaGccaGGSaaaaa@3B1D@  (83)

B= b 6 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbGaaGypaiaadk gadaWgaaWcbaGaaGOnaaqabaGccaGGSaaaaa@3A62@  (84)

C=( c 5 r+ c 6 ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaaGypaiaaiI cacaWGJbWaaSbaaSqaaiaaiwdaaeqaaOGaamOCaiabgUcaRiaadoga daWgaaWcbaGaaGOnaaqabaGccaaIPaWaaWbaaSqabeaacaaIYaaaaO GaaiOlaaaa@4074@  (85)

Now for the second case, these turn out to be

A=( a 5 r+ a 6 ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbGaaGypaiaaiI cacaWGHbWaaSbaaSqaaiaaiwdaaeqaaOGaamOCaiabgUcaRiaadgga daWgaaWcbaGaaGOnaaqabaGccaaIPaWaaWbaaSqabeaacaaIYaaaaO Gaaiilaaaa@406C@  (86)

B= b 6 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGcbGaaGypaiaadk gadaWgaaWcbaGaaGOnaaqabaGccaGGSaaaaa@3A62@ (87)

C= c 6 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbGaaGypaiaado gadaqhaaWcbaGaaGOnaaqaaiaaikdaaaGccaGGUaaaaa@3B23@  (88)

Finally, the corresponding solutions take the forms:

d s 2 = a 6 2 d t 2 b 6 d ρ 2 ( c 5 r+ c 6 ) 2 d ϕ 2 b 6 d z 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGHbWaa0baaSqaaiaaiAdaaeaa caaIYaaaaOGaamizaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsi slcaWGIbWaaSbaaSqaaiaaiAdaaeqaaOGaamizaiabeg8aYnaaCaaa leqabaGaaGOmaaaakiabgkHiTiaaiIcacaWGJbWaaSbaaSqaaiaaiw daaeqaaOGaamOCaiabgUcaRiaadogadaWgaaWcbaGaaGOnaaqabaGc caaIPaWaaWbaaSqabeaacaaIYaaaaOGaamizaiabew9aMnaaCaaale qabaGaaGOmaaaakiabgkHiTiaadkgadaWgaaWcbaGaaGOnaaqabaGc caWGKbGaamOEamaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@5888@  (89)

d s 2 =( a 5 r+ a 6 ) 2 d t 2 b 6 d ρ 2 c 6 2 d ϕ 2 b 6 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaaIOaGaamyyamaaBaaaleaacaaI 1aaabeaakiaadkhacqGHRaWkcaWGHbWaaSbaaSqaaiaaiAdaaeqaaO GaaGykamaaCaaaleqabaGaaGOmaaaakiaadsgacaWG0bWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaamOyamaaBaaaleaacaaI2aaabeaaki aadsgacqaHbpGCdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGJbWa a0baaSqaaiaaiAdaaeaacaaIYaaaaOGaamizaiabew9aMnaaCaaale qabaGaaGOmaaaakiabgkHiTiaadkgadaWgaaWcbaGaaGOnaaqabaGc caWGKbGaamOEamaaCaaaleqabaGaaGOmaaaakiaac6caaaa@5888@  (90)

It is mentioned here that the energy density of these solutions vanish and hence these are the vacuum solutions. Energy Density of the Non–Vacuum Per–fect Fluid Static Cylindrically Symmetric Solutions. In this portion, we calculate energy density of the non–vacuum perfect fluid static cylindrically symmetric solutions (32), (33) (34), (57) and (68), which is obtained in the context of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ theory of gravity in the last portion. We use generalized Landau–Lifshitz EMC in the framework of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ gravity for this purpose. By substituting the value of g 00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGNbWaaWbaaSqabe aacaaIWaGaaGimaaaaaaa@38D4@ , the Equation (9), it will be

τ 00 = f ( R 0 ) τ LL 00 + 1 6κ {( f ( R 0 ) R 0 f( R 0 ))( r A A 2 + 3 A )}. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiqadAgagaqbaiaaiIcacaWGsbWa aSbaaSqaaiaaicdaaeqaaOGaaGykaiabes8a0naaDaaaleaacaWGmb GaamitaaqaaiaaicdacaaIWaaaaOGaey4kaSYaaSaaaeaacaaIXaaa baGaaGOnaiabeQ7aRbaacaaI7bGaaGikaiqadAgagaqbaiaaiIcaca WGsbWaaSbaaSqaaiaaicdaaeqaaOGaaGykaiaadkfadaWgaaWcbaGa aGimaaqabaGccqGHsislcaWGMbGaaGikaiaadkfadaWgaaWcbaGaaG imaaqabaGccaaIPaGaaGykaiaaiIcadaWcaaqaaiaadkhaceWGbbGb auaaaeaacaWGbbWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRmaala aabaGaaG4maaqaaiaadgeaaaGaaGykaiaai2hacaGGUaaaaa@5E4F@  (91)

Now, by calculating the values of T 00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaWbaaSqabe aacaaIWaGaaGimaaaaaaa@38C1@  and t LL 00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0bWaa0baaSqaai aadYeacaWGmbaabaGaaGimaiaaicdaaaaaaa@3A83@  from Equations (11) and (12) respectively and then using in Equation (10), the final expressions of τ LL 00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaqhaaWcba GaamitaiaadYeaaeaacaaIWaGaaGimaaaaaaa@3B4F@ , for the solutions (32), (33), (34), (57) and (68), take the form

τ LL 00 = k 2 2 k 3 r 12 (ρ 28 k 2 κ r 6 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaqhaaWcba GaamitaiaadYeaaeaacaaIWaGaaGimaaaakiaai2dacaWGRbWaa0ba aSqaaiaaikdaaeaacaaIYaaaaOGaam4AamaaBaaaleaacaaIZaaabe aakiaadkhadaahaaWcbeqaaiaaigdacaaIYaaaaOGaaGikaiabeg8a YjabgkHiTmaalaaabaGaaGOmaiaaiIdaaeaacaWGRbWaaSbaaSqaai aaikdaaeqaaOGaeqOUdSMaamOCamaaCaaaleqabaGaaGOnaaaaaaGc caaIPaGaaiilaaaa@4F19@  (92)

τ LL 00 = k 2 2 k 3 r 4 3 (ρ 4 9 k 2 κ r 6 3 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaqhaaWcba GaamitaiaadYeaaeaacaaIWaGaaGimaaaakiaai2dacaWGRbWaa0ba aSqaaiaaikdaaeaacaaIYaaaaOGaam4AamaaBaaaleaacaaIZaaabe aakiaadkhadaahaaWcbeqaamaalaaabaGaaGinaaqaaiaaiodaaaaa aOGaaGikaiabeg8aYjabgkHiTmaalaaabaGaaGinaaqaaiaaiMdaca WGRbWaaSbaaSqaaiaaikdaaeqaaOGaeqOUdSMaamOCamaaCaaaleqa baWaaSaaaeaacaaI2aaabaGaaG4maaaaaaaaaOGaaGykaiaacYcaaa a@4FFD@ (93)

τ LL 00 = k 2 2 k 3 r 8 (ρ 8 k 2 κ r 6 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaqhaaWcba GaamitaiaadYeaaeaacaaIWaGaaGimaaaakiaai2dacaWGRbWaa0ba aSqaaiaaikdaaeaacaaIYaaaaOGaam4AamaaBaaaleaacaaIZaaabe aakiaadkhadaahaaWcbeqaaiaaiIdaaaGccaaIOaGaeqyWdiNaeyOe I0YaaSaaaeaacaaI4aaabaGaam4AamaaBaaaleaacaaIYaaabeaaki abeQ7aRjaadkhadaahaaWcbeqaaiaaiAdaaaaaaOGaaGykaiaacYca aaa@4DA8@  (94)

τ LL 00 = c 2 e 4 b 1 r+4 b 2 (ρ 2 b 1 2 κ e 2 b 1 r+2 b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaqhaaWcba GaamitaiaadYeaaeaacaaIWaGaaGimaaaakiaai2dacaWGJbWaaSba aSqaaiaaikdaaeqaaOGaamyzamaaCaaaleqabaGaaGinaiaadkgada WgaaqaaiaaigdaaeqaaiaadkhacqGHRaWkcaaI0aGaamOyamaaBaaa baGaaGOmaaqabaaaaOGaaGikaiabeg8aYjabgkHiTmaalaaabaGaaG OmaiaadkgadaqhaaWcbaGaaGymaaqaaiaaikdaaaaakeaacqaH6oWA caWGLbWaaWbaaSqabeaacaaIYaGaamOyamaaBaaabaGaaGymaaqaba GaamOCaiabgUcaRiaaikdacaWGIbWaaSbaaeaacaaIYaaabeaaaaaa aOGaaGykaaaa@5715@  (95)

and

τ LL 00 = e 4 b 3 r+4 b 4 ( c 3 r+ c 4 ) 2 (ρ c 3 2 +(2 b 3 2 ( c 3 r+ c 4 )+4 b 3 c 3 )( c 3 r+ c 4 ) κ e 2 b 3 r+2 b 4 ( c 3 r+ c 4 ) 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaqhaaWcba GaamitaiaadYeaaeaacaaIWaGaaGimaaaakiaai2dacaWGLbWaaWba aSqabeaacaaI0aGaamOyamaaBaaabaGaaG4maaqabaGaamOCaiabgU caRiaaisdacaWGIbWaaSbaaeaacaaI0aaabeaaaaGccaaIOaGaam4y amaaBaaaleaacaaIZaaabeaakiaadkhacqGHRaWkcaWGJbWaaSbaaS qaaiaaisdaaeqaaOGaaGykamaaCaaaleqabaGaaGOmaaaakiaaiIca cqaHbpGCcqGHsisldaWcaaqaaiaadogadaqhaaWcbaGaaG4maaqaai aaikdaaaGccqGHRaWkcaaIOaGaaGOmaiaadkgadaqhaaWcbaGaaG4m aaqaaiaaikdaaaGccaaIOaGaam4yamaaBaaaleaacaaIZaaabeaaki aadkhacqGHRaWkcaWGJbWaaSbaaSqaaiaaisdaaeqaaOGaaGykaiab gUcaRiaaisdacaWGIbWaaSbaaSqaaiaaiodaaeqaaOGaam4yamaaBa aaleaacaaIZaaabeaakiaaiMcacaaIOaGaam4yamaaBaaaleaacaaI ZaaabeaakiaadkhacqGHRaWkcaWGJbWaaSbaaSqaaiaaisdaaeqaaO GaaGykaaqaaiabeQ7aRjaadwgadaahaaWcbeqaaiaaikdacaWGIbWa aSbaaeaacaaIZaaabeaacaWGYbGaey4kaSIaaGOmaiaadkgadaWgaa qaaiaaisdaaeqaaaaakiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqa aOGaamOCaiabgUcaRiaadogadaWgaaWcbaGaaGinaaqabaGccaaIPa WaaWbaaSqabeaacaaIYaaaaaaakiaaiMcacaGGUaaaaa@7DE5@ (96)

When we use Equations (92)–(96) in Equation (91), the 00–components of the general–ized Landau–Lifshitz EMC turn out respectively to be

τ 00 =S f ( R 0 ) k 2 2 k 3 r 12 (ρ 28 k 2 κ r 6 )+ 1 2 k 1 κ ( f ( R 0 ) R 0 f( R 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadofaceWGMbGbauaacaaIOaGa amOuamaaBaaaleaacaaIWaaabeaakiaaiMcacaWGRbWaa0baaSqaai aaikdaaeaacaaIYaaaaOGaam4AamaaBaaaleaacaaIZaaabeaakiaa dkhadaahaaWcbeqaaiaaigdacaaIYaaaaOGaaGikaiabeg8aYjabgk HiTmaalaaabaGaaGOmaiaaiIdaaeaacaWGRbWaaSbaaSqaaiaaikda aeqaaOGaeqOUdSMaamOCamaaCaaaleqabaGaaGOnaaaaaaGccaaIPa Gaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaiaadUgadaWgaaWcbaGa aGymaaqabaGccqaH6oWAaaGaaGikaiqadAgagaqbaiaaiIcacaWGsb WaaSbaaSqaaiaaicdaaeqaaOGaaGykaiaadkfadaWgaaWcbaGaaGim aaqabaGccqGHsislcaWGMbGaaGikaiaadkfadaWgaaWcbaGaaGimaa qabaGccaaIPaGaaGykaaaa@6411@  (97)

τ 00 = f ( R 0 ) k 2 2 k 3 r 4 3 (ρ 4 9 k 2 κ r 6 3 )+ 13 18 k 1 r 4 3 κ ( f ( R 0 ) R 0 f( R 0 )), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiqadAgagaqbaiaaiIcacaWGsbWa aSbaaSqaaiaaicdaaeqaaOGaaGykaiaadUgadaqhaaWcbaGaaGOmaa qaaiaaikdaaaGccaWGRbWaaSbaaSqaaiaaiodaaeqaaOGaamOCamaa CaaaleqabaWaaSaaaeaacaaI0aaabaGaaG4maaaaaaGccaaIOaGaeq yWdiNaeyOeI0YaaSaaaeaacaaI0aaabaGaaGyoaiaadUgadaWgaaWc baGaaGOmaaqabaGccqaH6oWAcaWGYbWaaWbaaSqabeaadaWcaaqaai aaiAdaaeaacaaIZaaaaaaaaaGccaaIPaGaey4kaSYaaSaaaeaacaaI XaGaaG4maaqaaiaaigdacaaI4aGaam4AamaaBaaaleaacaaIXaaabe aakiaadkhadaahaaWcbeqaamaalaaabaGaaGinaaqaaiaaiodaaaaa aOGaeqOUdSgaaiaaiIcaceWGMbGbauaacaaIOaGaamOuamaaBaaale aacaaIWaaabeaakiaaiMcacaWGsbWaaSbaaSqaaiaaicdaaeqaaOGa eyOeI0IaamOzaiaaiIcacaWGsbWaaSbaaSqaaiaaicdaaeqaaOGaaG ykaiaaiMcacaaISaaaaa@690A@ (98)

τ 00 = f ( R 0 ) k 2 2 k 3 r 8 ( ρ 8 k 2 κ r 6 )+ 7 6 k 1 r 4 κ ( f ( R 0 ) R 0 f( R 0 ) ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabes 8a09aadaahaaWcbeqaa8qacaaIWaGaaGimaaaakiabg2da9iqadAga paGbauaapeWaaeWaa8aabaWdbiaadkfapaWaaSbaaSqaa8qacaaIWa aapaqabaaak8qacaGLOaGaayzkaaGaam4Aa8aadaqhaaWcbaWdbiaa ikdaa8aabaWdbiaaikdaaaGccaWGRbWdamaaBaaaleaapeGaaG4maa WdaeqaaOWdbiaadkhapaWaaWbaaSqabeaapeGaaGioaaaakmaabmaa paqaa8qacqaHbpGCcqGHsisldaWcaaWdaeaapeGaaGioaaWdaeaape Gaam4Aa8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqaH6oWAcaWG YbWdamaaCaaaleqabaWdbiaaiAdaaaaaaaGccaGLOaGaayzkaaGaey 4kaSYaaSaaa8aabaWdbiaaiEdaa8aabaWdbiaaiAdacaWGRbWdamaa BaaaleaapeGaaGymaaWdaeqaaOWdbiaadkhapaWaaWbaaSqabeaape GaaGinaaaakiabeQ7aRbaadaqadaWdaeaapeGabmOza8aagaqba8qa daqadaWdaeaapeGaamOua8aadaWgaaWcbaWdbiaaicdaa8aabeaaaO WdbiaawIcacaGLPaaacaWGsbWdamaaBaaaleaapeGaaGimaaWdaeqa aOWdbiabgkHiTiaadAgadaqadaWdaeaapeGaamOua8aadaWgaaWcba Wdbiaaicdaa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIcacaGLPaaa caGGSaaaaa@6943@  (99)

τ 00 = f ( R 0 ) c 2 2 e 4 b 1 r+4 b 2 (ρ 2 b 1 2 κ e 2 b 1 r+2 b 2 )+ 1 6κ ( f ( R 0 ) R 0 f( R 0 )) 5 a 1 r+3 a 2 ( a 1 r+ a 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiqadAgagaqbaiaaiIcacaWGsbWa aSbaaSqaaiaaicdaaeqaaOGaaGykaiaadogadaqhaaWcbaGaaGOmaa qaaiaaikdaaaGccaWGLbWaaWbaaSqabeaacaaI0aGaamOyamaaBaaa baGaaGymaaqabaGaamOCaiabgUcaRiaaisdacaWGIbWaaSbaaeaaca aIYaaabeaaaaGccaaIOaGaeqyWdiNaeyOeI0YaaSaaaeaacaaIYaGa amOyamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiabeQ7aRjaadw gadaahaaWcbeqaaiaaikdacaWGIbWaaSbaaeaacaaIXaaabeaacaWG YbGaey4kaSIaaGOmaiaadkgadaWgaaqaaiaaikdaaeqaaaaaaaGcca aIPaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOnaiabeQ7aRbaacaaI OaGabmOzayaafaGaaGikaiaadkfadaWgaaWcbaGaaGimaaqabaGcca aIPaGaamOuamaaBaaaleaacaaIWaaabeaakiabgkHiTiaadAgacaaI OaGaamOuamaaBaaaleaacaaIWaaabeaakiaaiMcacaaIPaWaaSaaae aacaaI1aGaamyyamaaBaaaleaacaaIXaaabeaakiaadkhacqGHRaWk caaIZaGaamyyamaaBaaaleaacaaIYaaabeaaaOqaaiaaiIcacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaamOCaiabgUcaRiaadggadaWgaaWc baGaaGOmaaqabaGccaaIPaWaaWbaaSqabeaacaaIYaaaaaaaaaa@79AF@  (100)

and

τ 00 = f ( R 0 ) e 4 b 3 r+4 b 4 ( c 3 r+ c 4 ) 2 (ρ c 3 2 +(2 b 3 2 ( c 3 r+ c 4 )+4 b 3 c 3 )( c 3 r+ c 4 ) κ e 2 b 3 r+2 b 4 ( c 3 r+ c 4 ) 2 )+ 1 2 a 4 κ ( f ( R 0 ) R 0 f( R 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiqadAgagaqbaiaaiIcacaWGsbWa aSbaaSqaaiaaicdaaeqaaOGaaGykaiaadwgadaahaaWcbeqaaiaais dacaWGIbWaaSbaaeaacaaIZaaabeaacaWGYbGaey4kaSIaaGinaiaa dkgadaWgaaqaaiaaisdaaeqaaaaakiaaiIcacaWGJbWaaSbaaSqaai aaiodaaeqaaOGaamOCaiabgUcaRiaadogadaWgaaWcbaGaaGinaaqa baGccaaIPaWaaWbaaSqabeaacaaIYaaaaOGaaGikaiabeg8aYjabgk HiTmaalaaabaGaam4yamaaDaaaleaacaaIZaaabaGaaGOmaaaakiab gUcaRiaaiIcacaaIYaGaamOyamaaDaaaleaacaaIZaaabaGaaGOmaa aakiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqaaOGaamOCaiabgUca RiaadogadaWgaaWcbaGaaGinaaqabaGccaaIPaGaey4kaSIaaGinai aadkgadaWgaaWcbaGaaG4maaqabaGccaWGJbWaaSbaaSqaaiaaioda aeqaaOGaaGykaiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqaaOGaam OCaiabgUcaRiaadogadaWgaaWcbaGaaGinaaqabaGccaaIPaaabaGa eqOUdSMaamyzamaaCaaaleqabaGaaGOmaiaadkgadaWgaaqaaiaaio daaeqaaiaadkhacqGHRaWkcaaIYaGaamOyamaaBaaabaGaaGinaaqa baaaaOGaaGikaiaadogadaWgaaWcbaGaaG4maaqabaGccaWGYbGaey 4kaSIaam4yamaaBaaaleaacaaI0aaabeaakiaaiMcadaahaaWcbeqa aiaaikdaaaaaaOGaaGykaiabgUcaRmaalaaabaGaaGymaaqaaiaaik dacaWGHbWaaSbaaSqaaiaaisdaaeqaaOGaeqOUdSgaaiaaiIcaceWG MbGbauaacaaIOaGaamOuamaaBaaaleaacaaIWaaabeaakiaaiMcaca WGsbWaaSbaaSqaaiaaicdaaeqaaOGaeyOeI0IaamOzaiaaiIcacaWG sbWaaSbaaSqaaiaaicdaaeqaaOGaaGykaiaaiMcaaaa@91FC@  (101)

To get the final expression for energy density, we have to consider a suitable f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ model. It is important to mention here that we must be careful in choosing the f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ model specially when R=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaaGypaiaaic daaaa@389F@ . It is because if the model contains the logarithmic function of Ricci scalar R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbaaaa@371E@ or a linear superposition of R n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaWbaaSqabe aacqGHsislcaWGUbaaaaaa@392B@ , where n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGUbaaaa@3666@ is positive integer, then we can not find this EMC. Hence, we consider the following f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ model

f(R)=R+ε R 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaGaaGypaiaadkfacqGHRaWkcqaH1oqzcaWGsbWaaWbaaSqa beaacaaIYaaaaOGaaiilaaaa@400F@  (102)

where ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=x b9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH1oqzaa a@371C@ is a positive real number. Consequently, the Equations (32), (33), (34), (57) and (68), results the 00–component of generalized Landau–Lifshitz EMC as

τ 00 = k 2 2 k 3 r 12 (ρ 28 k 2 κ r 6 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadUgadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccaWGRbWaaSbaaSqaaiaaiodaaeqaaOGaamOCamaaCa aaleqabaGaaGymaiaaikdaaaGccaaIOaGaeqyWdiNaeyOeI0YaaSaa aeaacaaIYaGaaGioaaqaaiaadUgadaWgaaWcbaGaaGOmaaqabaGccq aH6oWAcaWGYbWaaWbaaSqabeaacaaI2aaaaaaakiaaiMcacaGGUaaa aa@4D79@ (103)

τ 00 = k 2 2 k 3 r 4 3 (ρ 4 9 k 2 κ r 6 3 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadUgadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccaWGRbWaaSbaaSqaaiaaiodaaeqaaOGaamOCamaaCa aaleqabaWaaSaaaeaacaaI0aaabaGaaG4maaaaaaGccaaIOaGaeqyW diNaeyOeI0YaaSaaaeaacaaI0aaabaGaaGyoaiaadUgadaWgaaWcba GaaGOmaaqabaGccqaH6oWAcaWGYbWaaWbaaSqabeaadaWcaaqaaiaa iAdaaeaacaaIZaaaaaaaaaGccaaIPaGaaiilaaaa@4E5B@ (104)

τ 00 = k 2 2 k 3 r 8 (ρ 8 k 2 κ r 6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadUgadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccaWGRbWaaSbaaSqaaiaaiodaaeqaaOGaamOCamaaCa aaleqabaGaaGioaaaakiaaiIcacqaHbpGCcqGHsisldaWcaaqaaiaa iIdaaeaacaWGRbWaaSbaaSqaaiaaikdaaeqaaOGaeqOUdSMaamOCam aaCaaaleqabaGaaGOnaaaaaaGccaaIPaaaaa@4B56@

τ 00 = e 4 b 3 r+4 b 4 ( c 3 r+ c 4 ) 2 (ρ c 3 2 +(2 b 3 2 ( c 3 r+ c 4 )+4 b 3 c 3 )( c 3 r+ c 4 ) κ e 2 b 3 r+2 b 4 ( c 3 r+ c 4 ) 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadwgadaahaaWcbeqaaiaaisda caWGIbWaaSbaaeaacaaIZaaabeaacaWGYbGaey4kaSIaaGinaiaadk gadaWgaaqaaiaaisdaaeqaaaaakiaaiIcacaWGJbWaaSbaaSqaaiaa iodaaeqaaOGaamOCaiabgUcaRiaadogadaWgaaWcbaGaaGinaaqaba GccaaIPaWaaWbaaSqabeaacaaIYaaaaOGaaGikaiabeg8aYjabgkHi TmaalaaabaGaam4yamaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgU caRiaaiIcacaaIYaGaamOyamaaDaaaleaacaaIZaaabaGaaGOmaaaa kiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqaaOGaamOCaiabgUcaRi aadogadaWgaaWcbaGaaGinaaqabaGccaaIPaGaey4kaSIaaGinaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaWGJbWaaSbaaSqaaiaaiodaae qaaOGaaGykaiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqaaOGaamOC aiabgUcaRiaadogadaWgaaWcbaGaaGinaaqabaGccaaIPaaabaGaeq OUdSMaamyzamaaCaaaleqabaGaaGOmaiaadkgadaWgaaqaaiaaioda aeqaaiaadkhacqGHRaWkcaaIYaGaamOyamaaBaaabaGaaGinaaqaba aaaOGaaGikaiaadogadaWgaaWcbaGaaG4maaqabaGccaWGYbGaey4k aSIaam4yamaaBaaaleaacaaI0aaabeaakiaaiMcadaahaaWcbeqaai aaikdaaaaaaOGaaGykaiaacYcaaaa@7C41@  (105)

τ 00 = c 2 2 e 4 b 1 r+4 b 2 (ρ 2 b 1 2 κ e 2 b 1 r+2 b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadogadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccaWGLbWaaWbaaSqabeaacaaI0aGaamOyamaaBaaaba GaaGymaaqabaGaamOCaiabgUcaRiaaisdacaWGIbWaaSbaaeaacaaI YaaabeaaaaGccaaIOaGaeqyWdiNaeyOeI0YaaSaaaeaacaaIYaGaam OyamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiabeQ7aRjaadwga daahaaWcbeqaaiaaikdacaWGIbWaaSbaaeaacaaIXaaabeaacaWGYb Gaey4kaSIaaGOmaiaadkgadaWgaaqaaiaaikdaaeqaaaaaaaGccaaI Paaaaa@5630@  (106)

and

τ 00 = e 4 b 3 r+4 b 4 ( c 3 r+ c 4 ) 2 (ρ c 3 2 +(2 b 3 2 ( c 3 r+ c 4 )+4 b 3 c 3 )( c 3 r+ c 4 ) κ e 2 b 3 r+2 b 4 ( c 3 r+ c 4 ) 2 ) τ 00 = k 2 2 k 3 r 8 (ρ 8 k 2 κ r 6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaahaaWcbe qaaiaaicdacaaIWaaaaOGaaGypaiaadwgadaahaaWcbeqaaiaaisda caWGIbWaaSbaaeaacaaIZaaabeaacaWGYbGaey4kaSIaaGinaiaadk gadaWgaaqaaiaaisdaaeqaaaaakiaaiIcacaWGJbWaaSbaaSqaaiaa iodaaeqaaOGaamOCaiabgUcaRiaadogadaWgaaWcbaGaaGinaaqaba GccaaIPaWaaWbaaSqabeaacaaIYaaaaOGaaGikaiabeg8aYjabgkHi TmaalaaabaGaam4yamaaDaaaleaacaaIZaaabaGaaGOmaaaakiabgU caRiaaiIcacaaIYaGaamOyamaaDaaaleaacaaIZaaabaGaaGOmaaaa kiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqaaOGaamOCaiabgUcaRi aadogadaWgaaWcbaGaaGinaaqabaGccaaIPaGaey4kaSIaaGinaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaWGJbWaaSbaaSqaaiaaiodaae qaaOGaaGykaiaaiIcacaWGJbWaaSbaaSqaaiaaiodaaeqaaOGaamOC aiabgUcaRiaadogadaWgaaWcbaGaaGinaaqabaGccaaIPaaabaGaeq OUdSMaamyzamaaCaaaleqabaGaaGOmaiaadkgadaWgaaqaaiaaioda aeqaaiaadkhacqGHRaWkcaaIYaGaamOyamaaBaaabaGaaGinaaqaba aaaOGaaGikaiaadogadaWgaaWcbaGaaG4maaqabaGccaWGYbGaey4k aSIaam4yamaaBaaaleaacaaI0aaabeaakiaaiMcadaahaaWcbeqaai aaikdaaaaaaOGaaGykaiabes8a0naaCaaaleqabaGaaGimaiaaicda aaGccaaI9aGaam4AamaaDaaaleaacaaIYaaabaGaaGOmaaaakiaadU gadaWgaaWcbaGaaG4maaqabaGccaWGYbWaaWbaaSqabeaacaaI4aaa aOGaaGikaiabeg8aYjabgkHiTmaalaaabaGaaGioaaqaaiaadUgada WgaaWcbaGaaGOmaaqabaGccqaH6oWAcaWGYbWaaWbaaSqabeaacaaI 2aaaaaaakiaaiMcaaaa@90A0@ (107).

Furthermore, the stability condition for this f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@  model is also satisfied by all these solutions (as R=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGsbGaaGypaiaaic daaaa@389D@  for every solution) as

1 ε(1+2ε R 0 ) = 1 ε >0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaaigdaae aacqaH1oqzcaaIOaGaaGymaiabgUcaRiaaikdacqaH1oqzcaWGsbWa aSbaaSqaaiaaicdaaeqaaOGaaGykaaaacaaI9aWaaSaaaeaacaaIXa aabaGaeqyTdugaaiaai6dacaaIWaGaaiOlaaaa@4552@  (108)

Summary and conclusion

The objectives of this work are two folded: Firstly we explore the non–vacuum static cylindrically symmetric solutions in f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ gravity and the energy distri–bution of the obtain solution for the perfect fluid case, i.e, ρ,p0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHbpGCcaaISaGaam iCaiabgcMi5kaaicdaaaa@3C33@ . For this purpose, we obtain the field equations of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ gravity and solve these equa–tions for static cylindrically symmetric spacetimes considering non–vacuum case while using assumption R= R 0 =constant MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadkfacaaI9a GaamOuaSWaaSbaaKqaGeaajugWaiaaicdaaKqaGeqaaKqzGeGaaGyp aabaaaaaaaaapeGaam4yaiaad+gacaWGUbGaam4CaiaadshacaWGHb GaamOBaiaadshaaaa@44BB@ . We use power law assumption, to solve these equation and obtain three non vacuum solutions as,

d s 2 = k 1 d t 2 k 2 r 4 d ρ 2 k 3 r 4 d ϕ 2 k 2 r 4 d z 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGRbWaaSbaaSqaaiaaigdaaeqa aOGaamizaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacqaHbpGCdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaiodaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacqaHvpGzdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacaWG6bWaaWbaaSqabeaacaaIYaaaaOGaaiilaaaa@5795@  (109)

d s 2 = k 1 r 4 3 d t 2 k 2 d ρ 2 k 3 r 4 3 d ϕ 2 k 2 d z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGRbWaaSbaaSqaaiaaigdaaeqa aOGaamOCamaaCaaaleqabaWaaSaaaeaacaaI0aaabaGaaG4maaaaaa GccaWGKbGaamiDamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadUga daWgaaWcbaGaaGOmaaqabaGccaWGKbGaeqyWdi3aaWbaaSqabeaaca aIYaaaaOGaeyOeI0Iaam4AamaaBaaaleaacaaIZaaabeaakiaadkha daahaaWcbeqaamaalaaabaGaaGinaaqaaiaaiodaaaaaaOGaamizai abew9aMnaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadUgadaWgaaWc baGaaGOmaaqabaGccaWGKbGaamOEamaaCaaaleqabaGaaGOmaaaaaa a@5689@  (110)

and

d s 2 = k 1 r 4 d t 2 k 2 r 4 d ρ 2 k 3 d ϕ 2 k 2 r 4 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGRbWaaSbaaSqaaiaaigdaaeqa aOGaamOCamaaCaaaleqabaGaaGinaaaakiaadsgacaWG0bWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0Iaam4AamaaBaaaleaacaaIYaaabeaa kiaadkhadaahaaWcbeqaaiaaisdaaaGccaWGKbGaeqyWdi3aaWbaaS qabeaacaaIYaaaaOGaeyOeI0Iaam4AamaaBaaaleaacaaIZaaabeaa kiaadsgacqaHvpGzdaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaamOCamaaCaaaleqabaGaaGinaaaa kiaadsgacaWG6bWaaWbaaSqabeaacaaIYaaaaOGaaiOlaaaa@5797@  (111)

We solve these equations again for exponential law assumption and examine three cases ( μ =0, ν =0, λ =0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIOaGafqiVd0Mbau aacaaI9aGaaGimaiaaiYcacuaH9oGBgaqbaiaai2dacaaIWaGaaGil aiqbeU7aSzaafaGaaGypaiaaicdacaaIPaaaaa@42E1@ and get three solutions from which one is vacuum while the other two are non vacuum solutions, which are given respectively as,

d s 2 =( a 1 r+ a 2 ) 2 d t 2 e 2 b 1 r+2 b 2 d ρ 2 c 2 d ϕ 2 e 2 b 1 r+2 b 2 d z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaaIOaGaamyyamaaBaaaleaacaaI XaaabeaakiaadkhacqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaaeqaaO GaaGykamaaCaaaleqabaGaaGOmaaaakiaadsgacaWG0bWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaamyzamaaCaaaleqabaGaaGOmaiaadk gadaWgaaqaaiaaigdaaeqaaiaadkhacqGHRaWkcaaIYaGaamOyamaa BaaabaGaaGOmaaqabaaaaOGaamizaiabeg8aYnaaCaaaleqabaGaaG OmaaaakiabgkHiTiaadogadaWgaaWcbaGaaGOmaaqabaGccaWGKbGa eqy1dy2aaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyzamaaCaaale qabaGaaGOmaiaadkgadaWgaaqaaiaaigdaaeqaaiaadkhacqGHRaWk caaIYaGaamOyamaaBaaabaGaaGOmaaqabaaaaOGaamizaiaadQhada ahaaWcbeqaaiaaikdaaaaaaa@633B@  (112)

and

d s 2 = a 4 d t 2 e 2 b 3 r+2 b 4 d ρ 2 ( c 3 r+ c 4 ) 2 d ϕ 2 e 2 b 3 r+2 b 4 d z 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbGaam4CamaaCa aaleqabaGaaGOmaaaakiaai2dacaWGHbWaaSbaaSqaaiaaisdaaeqa aOGaamizaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGLb WaaWbaaSqabeaacaaIYaGaamOyamaaBaaabaGaaG4maaqabaGaamOC aiabgUcaRiaaikdacaWGIbWaaSbaaeaacaaI0aaabeaaaaGccaWGKb GaeqyWdi3aaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGikaiaadoga daWgaaWcbaGaaG4maaqabaGccaWGYbGaey4kaSIaam4yamaaBaaale aacaaI0aaabeaakiaaiMcadaahaaWcbeqaaiaaikdaaaGccaWGKbGa eqy1dy2aaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyzamaaCaaale qabaGaaGOmaiaadkgadaWgaaqaaiaaiodaaeqaaiaadkhacqGHRaWk caaIYaGaamOyamaaBaaabaGaaGinaaqabaaaaOGaamizaiaadQhada ahaaWcbeqaaiaaikdaaaGccaGGUaaaaa@6407@  (113)

Secondly, we evaluate the energy density of these obtained solutions by using generalized Landau–Lifshtiz EMC of f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ gravity for a suitable f(R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqkY=MjY=wqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0d meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaaGikaiaadk facaaIPaaaaa@396E@ model and we obtain a well–defined expression for components of energy densities of these solution in this case. Moreover, it is also checked that the stability conditions are fulfilled by these solutions.36–44

Acknowledgements

None.

Conflict of interest

The author declares no conflict of interest.

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