Table of Contents
ISRN Astronomy and Astrophysics
VolumeΒ 2011, Article IDΒ 749396, 5 pages
http://dx.doi.org/10.5402/2011/749396
Research Article

Varieties of Parametric Classes of Exact Solutions in General Relativity Representing Static Fluid Balls

Department of Mathematics, National Defence Academy, Khadakwasla, Pune 411023, India

Received 4 November 2011; Accepted 18 December 2011

Academic Editor: H.Β Zhao

Copyright Β© 2011 Neeraj Pant. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We have presented a method of obtaining parametric classes of spherically symmetric analytic solutions of the general relativistic field equations in canonical coordinates. A number of previously known classes of solutions have been rediscovered which describe perfect fluid balls with infinite central pressure and infinite central density though their ratio is positively finite and less than one. From the solution of one of the newly discovered classes, we have constructed a causal model in which outmarch of pressure and density is positive and monotonically decreasing, and pressure-density ratio is positive and less than one throughout within the balls. Corresponding to this model, we have maximized the Neutron star mass 2.40π‘€Ξ˜ with the linear dimensions of 28.43 kms and surface red shift of 0.4142.

1. Introduction

Due to nonlinearity of the Einstein’s field equations, many attempts have been reported to obtain parametric classes of exact solutions of representing perfect fluid ball in equilibrium such as Tolman [1], Wyman [2], Kuchowich [3], Pant and Sah [4], D.N. Pant and N. Pant [5], Pant [6, 7], Pant [8], Delgaty, Lake [9], Lake [10], Tewari and Pant [11], and Maurya, Gupta [12]. These solutions have four arbitrary constants. The usual boundary conditions determine three arbitrary constants leaving one undetermined; such a solution represents a class of solutions, with undetermined constant being a parameter. The edge of a parametric class of solutions over an ordinary solution lies in the choice of associated parameter which provides us various models of relativistic stars with realistic equation of state. Moreover, by imposing realistic conditions on parameter, one may get physical and significant solutions. Thus in the light of importance of parametric class, in this paper, we present a variety of parametric classes of solutions.

2. Field Equations and Method of Obtaining Parametric Classes of Solutions

We consider the static and spherically symmetric metric in canonical coordinates𝑑𝑠2=βˆ’π‘’πœ†π‘‘π‘Ÿ2βˆ’π‘Ÿ2ξ€·π‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€Έ+π‘’πœπ‘‘π‘‘2,(1) where πœ† and 𝜐 are field variables and functions of π‘Ÿ only. The field equations of gravitation for a nonempty space-time areπ‘…π‘–π‘—βˆ’12𝑅𝑔𝑖𝑗=βˆ’8πœ‹πΊπ‘4𝑇𝑖𝑗,(2) where 𝑅𝑖𝑗 is a Ricci tensor, 𝑇𝑖𝑗 is energy-momentum tensor, and 𝑅 the scalar curvature. The energy-momentum tensor 𝑇𝑖𝑗 is defined as𝑇𝑖𝑗=𝑝+πœŒπ‘2ξ€Έπ‘£π‘–π‘£π‘—βˆ’π‘π‘”π‘–π‘—,(3) where 𝑝 denotes the pressure distribution, 𝜌 the density distribution, and 𝑣𝑖 the velocity vector, satisfying the relation𝑔𝑖𝑗𝑣𝑖𝑣𝑗=1.(4)

Since the field is static, therefore, only a nonzero component of velocity is 𝑣4𝑣4=1βˆšπ‘”44.(5)

Thus, under these conditions, the field equations of general relativity for a perfect fluid ball the physical variables 𝑝(π‘Ÿ) and 𝜌(π‘Ÿ) are (Tolman[1])8πœ‹πΊπ‘4𝑝=π‘’βˆ’πœ†ξ‚΅πœ/π‘Ÿ+1π‘Ÿ2ξ‚Άβˆ’1π‘Ÿ2,(6)8πœ‹πΊπ‘2𝜌=π‘’βˆ’πœ†ξ‚΅πœ†/π‘Ÿβˆ’1π‘Ÿ2ξ‚Ά+1π‘Ÿ2𝑑,(7)ξ‚΅π‘’π‘‘π‘Ÿβˆ’πœ†βˆ’1π‘Ÿ2ξ‚Ά+π‘‘ξ‚΅π‘’π‘‘π‘Ÿβˆ’πœ†πœ/ξ‚Ά2π‘Ÿ+π‘’βˆ’πœ†βˆ’πœπ‘‘ξ‚΅π‘’π‘‘π‘Ÿπœπœ/ξ‚Ά2π‘Ÿ=0,(8) where prime (/) denotes differentiation with respect to π‘Ÿ. The problem consists of solving (8) for πœ† and 𝜐 and obtaining 𝑝(π‘Ÿ) and 𝜌(π‘Ÿ) from (6) and (7).

By using the transformationπ‘ˆ=π‘Ÿπ‘šβˆ’1π‘’π‘šπœ/2,𝑉=π‘’βˆ’πœ†.(9) Equation (8) reduces to the following linear differential equation in 𝑉:π‘‘π‘‰ξƒ―π‘‘π‘‘π‘Ÿβˆ’2ξ‚»ξ‚΅π‘Ÿπ‘‘π‘Ÿlog4βˆ’1/π‘šπ‘ˆ1βˆ’1/π‘šπ‘Ÿπ‘ˆ/βˆ’+π‘ˆξ‚Άξ‚Ό2π‘šπ‘ˆπ‘Ÿξ€·π‘Ÿπ‘ˆ/𝑉=+π‘ˆβˆ’2π‘šπ‘ˆπ‘Ÿξ€·π‘Ÿπ‘ˆ/ξ€Έ.+π‘ˆ(10) On solving (10), we get𝑉=π‘’βˆ’πœ†=π‘Ÿ8βˆ’2/π‘šπ‘ˆ2(1βˆ’1/π‘š)ξ€·π‘Ÿπ‘ˆ/ξ€Έ+π‘ˆ2ξƒ¬ξ€œξ€·π΄βˆ’2π‘šπ‘Ÿπ‘ˆ/ξ€Έπ‘ˆ+π‘ˆβˆ’1+2/π‘šπ‘’βˆ«(4π‘šπ‘ˆπ‘‘π‘Ÿ/π‘Ÿ(π‘Ÿπ‘ˆ/+π‘ˆ))π‘Ÿ9βˆ’2/π‘šξƒ­π‘’π‘‘π‘Ÿβˆ’βˆ«(4π‘šπ‘ˆπ‘‘π‘Ÿ/π‘Ÿ(π‘Ÿπ‘ˆ/+π‘ˆ)),(11)where β€œπ΄β€ is an arbitrary constant. Our aim is to explore the possibilities of choosing π‘ˆ such that the right-hand side of (11) becomes integrable. In this paper, we assumeπ‘’βˆ«(4π‘šπ‘ˆπ‘‘π‘Ÿ/π‘Ÿ(π‘Ÿπ‘ˆ/+π‘ˆ))=π‘Ÿπ‘™ξ€·π‘Ÿπ‘ˆ/ξ€Έ+π‘ˆπ‘›;(12)𝑙, π‘š, and 𝑛 are arbitrary constants. Equation (12) results into a second degree homogenous differential equation in Uπ‘›π‘Ÿ2π‘ˆ//+(𝑙+2𝑛)π‘Ÿπ‘ˆ/+(π‘™βˆ’4π‘š)π‘ˆ=0.(13) The solution isπ‘ˆ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’1+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’1,(14) whereπ‘Ž=π‘›βˆ’π‘™12𝑛,𝑏=2𝑛(π‘›βˆ’π‘™)2+16π‘šπ‘›,(15) provided, 𝑛≠0.

𝐢1 and 𝐢2 are arbitrary constants. Also (11) is simplified into𝑉=π‘’βˆ’πœ†=π‘Ÿ8+π‘›βˆ’π‘™βˆ’(π‘Žβˆ’π‘)(𝑛+2/π‘š)[]π΄βˆ’2𝐼(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2𝑛+2𝐢1π‘Ÿ2𝑏+𝐢2ξ€Έ2(βˆ’1+1/π‘š),(16) where ξ€œπ‘ŸπΌβ‰‘π‘šπ‘™βˆ’π‘›βˆ’9+(π‘Žβˆ’π‘)(𝑛+2/π‘š)ξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2𝑛+1×𝐢1π‘Ÿ2𝑏+𝐢2ξ€»βˆ’1+2/π‘šπ‘‘π‘Ÿ.(17) The solution is complete if (17) is integrated. In the foregoing sections, we shall discuss the method of solving (17), which will result into the variety of classes of solutions.

3. Varieties of Classes of Solutions

To explore the integrability of (17), we substituteπ‘Ÿ2𝑏2=𝑦,(18)π‘™βˆ’π‘›βˆ’9+(π‘Žβˆ’π‘)𝑛+π‘šξ‚=2π‘π›Όβˆ’1.(19)

So that (17) transforms intoπ‘šπΌβ‰‘ξ€œπ‘¦2π‘π›Όβˆ’1ξ€Ί(π‘Ž+𝑏)𝐢1𝑦+(π‘Žβˆ’π‘)𝐢2𝑛+1×𝐢1𝑦+𝐢2ξ€»βˆ’1+2/π‘šπ‘‘π‘¦.(20) It is easy to see that right-hand side of (20) can be integrated by parts with the restriction on the exponents of integrand to be nonnegative integral values. Thus we arrive at varieties of classes of solutions.

3.1. Class I (new class of solutions)

We assume 𝑛+1=0 and βˆ’1+2/π‘š=0. In view of (18) and (20), the resulting class of solutions is π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’2+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’2ξ€Έ,π‘’βˆ’πœ†=ξ€Ίπ΄π‘Ÿ7βˆ’π‘™πΆβˆ’4/(π‘™βˆ’7)ξ€»ξ€Ί1π‘Ÿ2𝑏+𝐢2ξ€»ξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2ξ€»,(21) whereπ‘Ž=1+𝑙2√,𝑏=(1+𝑙)2βˆ’32.βˆ’2(22) We observe that (π‘’πœ)π‘Ÿ=0 becomes singular for all values except √32βˆ’1≀𝑙≀5. It may be pointed out here that for 𝑙=5, we rediscover Tolman’s IV solution which is the only member of the class giving rise to finite central pressure and central density.

4. Properties of the New Class of Solutions

In view of (21), we obtain from (6) and (7), the pressure and density distribution, respectively,8πœ‹πΊπ‘41𝑝=π‘Ÿ2ξƒ―ξ€Ίπ΄π‘Ÿ7βˆ’π‘™βˆ’4/π‘™βˆ’7ξ€»ξ€Ί(π‘Ž+π‘βˆ’1)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘βˆ’1)𝐢2ξ€»ξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2ξ€»ξƒ°,βˆ’18πœ‹πΊπ‘21𝜌=π‘Ÿ2ξƒ¬ξ€Ίξ€·π΄π‘Ÿ7βˆ’π‘™ξ€Έβˆ’4/π‘™βˆ’74𝑏2𝐢1𝐢2π‘Ÿ2𝑏+ξ€·(π‘™βˆ’8)π΄π‘Ÿ7βˆ’π‘™ξ€ΈΓ—ξ€·πΆ+4/π‘™βˆ’71π‘Ÿ2𝑏+𝐢2ξ€Έξ€·(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2ξ€Έξ€»ξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2ξ€»2ξƒ­.+1(23)

The central values of pressure and density are infinite; however, the limiting value of their ratio is finite and equal to the limiting value of 𝑑𝑝/π‘‘πœŒ:ξ‚΅π‘πœŒπ‘2ξ‚Άπ‘Ÿβ†’0=1𝑐2ξ‚΅π‘‘π‘ξ‚Άπ‘‘πœŒπ‘Ÿβ†’0[]=βˆ’4(π‘Ž+π‘βˆ’1)+(π‘™βˆ’7)(π‘Ž+𝑏)[].4+(π‘™βˆ’7)(π‘Ž+𝑏)(24)

The causality condition at the centre is valid for all those values of 𝑙 satisfying the inequality √32βˆ’1≀𝑙≀5.

In addition to the parameter 𝑙, the solutions (21) contain three arbitrary constants 𝐢1, 𝐢2, and 𝐴. These are to be determined by matching the solutions (21) with Schwarzschild exterior solution for a ball of mass 𝑀 and linear dimension 2π‘Ÿπ‘:π‘ξ€·π‘Ÿπ‘ξ€Έπ‘’=0,βˆ’πœ†(π‘Ÿπ‘)𝑒=1βˆ’2𝑒,𝜐(π‘Ÿπ‘)=1βˆ’2𝑒,(25) where𝑒=𝐺𝑀𝑐2π‘Ÿπ‘.(26) Consequently,𝐢1[]=βˆ’(π‘Žβˆ’π‘βˆ’1)(1βˆ’2𝑒)βˆ’12π‘π‘Ÿπ‘π‘Ž+π‘βˆ’2,𝐢(27)2=[](π‘Ž+π‘βˆ’1)(1βˆ’2𝑒)βˆ’12π‘π‘Ÿπ‘π‘Žβˆ’π‘βˆ’2,(28)𝐴=π‘Ÿπ‘π‘™βˆ’7[]ξ€Ί1βˆ’2𝑒(π‘Ž+𝑏)𝐢1π‘Ÿπ‘2𝑏+(π‘Žβˆ’π‘)𝐢2𝐢1π‘Ÿπ‘2𝑏+𝐢2ξ€»+4ξƒ­π‘™βˆ’7.(29)

For π‘’πœ to be definitely positive in the region 0β‰€π‘Ÿβ‰€π‘Ÿπ‘, we must have 𝐢1,𝐢2β‰₯0. Thus in view of (27) and (28), we haveπ‘Ž+π‘βˆ’2π‘Ž+π‘βˆ’1≀2π‘’β‰€π‘Žβˆ’π‘βˆ’2.π‘Žβˆ’π‘βˆ’1(30)

5. Particular Member of Class I for 𝑙=4.75

In this section, we shall present a detail study of the particular solution corresponding to 𝑙=4.75.

The solution is π‘’πœ=𝐢1π‘Ÿ0.35962+𝐢2π‘Ÿ1.39ξ€Έ,π‘’βˆ’πœ†=ξ€Ίπ΄π‘Ÿ2.25𝐢+1.78ξ€»ξ€Ί1+𝐢2π‘Ÿ1.03ξ€»ξ€Ί2.359𝐢1+3.39𝐢2π‘Ÿ1.03ξ€»,8πœ‹πΊπ‘41𝑝=π‘Ÿ2ξƒ¬ξ€Ίπ΄π‘Ÿ2.25+1.78ξ€»ξ€Ί1.359𝐢1+2.39𝐢2π‘Ÿ1.03ξ€»ξ€Ί2.359𝐢1+3.39𝐢2π‘Ÿ1.03ξ€»ξƒ­,βˆ’18πœ‹πΊπ‘21𝜌=π‘Ÿ2ξƒ¬ξ€Ίξ€·π΄π‘Ÿ2.25ξ€Έ+1.781.06𝐢1𝐢2π‘Ÿ1.03+ξ€·βˆ’3.25π΄π‘Ÿ2.25ξ€ΈΓ—ξ€·πΆβˆ’1.781+𝐢2π‘Ÿ1.03ξ€Έξ€·2.359𝐢1+3.39𝐢2π‘Ÿ1.03ξ€Έξ€»ξ€Ί2.359𝐢1+3.39𝐢2π‘Ÿ1.03ξ€»2ξƒ­.+1(31) In view of (30) π‘’πœ to be definitely positive in the region 0β‰€π‘Ÿβ‰€π‘Ÿπ‘, then0.13225≀𝑒≀0.29079.(32) Corresponding to 𝑒=0.25 and in view of (27), (28), and (29), the constants are𝐢1π‘Ÿπ‘0.35962=0.1891,𝐢2π‘Ÿπ‘1.39=0.3106,π΄π‘Ÿπ‘2.25=βˆ’0.2808.(33)

In Table 1 the march of pressure, density, pressure-density ratio, and square of adiabatic sound speed 𝑑𝑝/π‘‘πœŒ is given for 𝑒=0.25. We observe that pressure and density decrease monotonically with the increase of radial coordinate, pressure-density ratio, and square of adiabatic sound speed, which is positive and less than 1 throughout within the ball.

tab1
Table 1: The march of pressure, density, pressure-density ratio, and square of adiabatic sound speed within the ball corresponding to 𝑙=4.75 with 𝑒=0.25.

We now present here a model of Neutron star based on the particular solution discussed above. The Neutron star is supposed to have a surface density: πœŒπ‘=2Γ—1014gm-cmβˆ’3 suggested by Brecher and Caporaso [13]. The resulting causal model has the mass 𝑀=2.40π‘€Ξ˜ and the linear dimension 2π‘Ÿπ‘β‰ˆ28.43km. The surface red shift 𝑍𝑏=[(1βˆ’2𝑒)βˆ’0.5βˆ’1]β‰ˆ0.4142.

6. Some More Parametric Classes of Solutions

By assigning the different nonnegative integral values to the exponents of (20), we obtain the following different parametric classes of solutions.

6.1. Class II

If we assume π›Όβˆ’1=0 and βˆ’1+2/π‘š=0. In view of (17), (18), and (20), the resulting class of solutions is (see Neeraj Pant [6])π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’2+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’2ξ€Έ,π‘’βˆ’πœ†=ξƒ¬π΄π‘Ÿβˆ’2𝑏(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2𝑛+2βˆ’π‘Ÿβˆ’2𝑏2𝑏(𝑛+2)(π‘Ž+𝑏)𝐢1𝐢1π‘Ÿ2𝑏+𝐢2ξ€».(34)

6.2. Class III

If we assume π›Όβˆ’1=0 and 𝑛+1=0. In view of (17), (18), and (20), the resulting class of solutions is (see Pant [8])π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’π‘š+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’π‘šξ€Έ2/π‘š,π‘’βˆ’πœ†=ξƒ¬π΄π‘Ÿβˆ’2π‘βˆ’π‘š2π‘Ÿβˆ’2𝑏/2𝑏𝐢1ξ€·C1π‘Ÿ2𝑏+C2ξ€Έ2/π‘šξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2Cξ€»ξ€·1π‘Ÿ2𝑏+C2ξ€Έ2(βˆ’1+1/π‘š)ξƒ­.(35)

6.3. Class IV

If we assume 𝑛+1=0 and βˆ’1+2/π‘š=1. In view of (17), (18), and (20), the resulting class of solutions is (see Pant [8])π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’1+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’1ξ€Έ2,π‘’βˆ’πœ†=π‘Ÿβˆ’2π‘π›Όξ€Ίπ΄βˆ’π‘Ÿ2𝑏𝛼𝛼𝐢1π‘Ÿ2𝑏+(𝛼+1)𝐢2ξ€Ύξ€»/𝑏𝛼(𝛼+1)ξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2ξ€»,(36) where𝛼=π‘™βˆ’7+π‘Žβˆ’π‘.2𝑏(37)

6.4. Class V

If we assume π›Όβˆ’1=1 and βˆ’1+2/π‘š=0. In view of (17), (18), and (20), the resulting class of solutions is (New Class of Solutions)π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’2+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’2ξ€Έ,π‘’βˆ’πœ†=π΄π‘Ÿβˆ’4π‘βˆ’4π΅π‘Ÿβˆ’4𝑏𝔄𝔅(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2𝑛+2𝐢1π‘Ÿ2𝑏+𝐢2ξ€Έβˆ’1,(38) where 𝔄 denotes [(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2]𝑛+2 and 𝔅 denotes [(π‘Ž+𝑏)(𝑛+2)𝐢1π‘Ÿ2π‘βˆ’(π‘Žβˆ’π‘)𝐢2], 1𝐡=𝑏(π‘Ž+𝑏)2(𝑛+2)(𝑛+3)𝐢12.(39)

6.5. Class VI

If we assume π›Όβˆ’1=1 and 𝑛+1=0. In view of (17), (18), and (20), the resulting class of solutions is (New Class of Solutions)π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’π‘š+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’π‘šξ€Έ2/π‘š,π‘’βˆ’πœ†=𝔇2𝐢1π‘Ÿ2π‘βˆ’π‘šπΆ2ξ€Έξ€Ί(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2Cξ€»ξ€·1π‘Ÿ2𝑏+C2ξ€Έ2(βˆ’1+1/π‘š)ξƒ­,(40) where 𝔇 denotes π΄π‘Ÿβˆ’4π‘βˆ’2π‘š2π‘Ÿβˆ’4𝑏/4𝑏𝐢12(C1π‘Ÿ2𝑏+C2)2/π‘š.

6.6. Class VII

If we assume π›Όβˆ’1=0 and βˆ’1+2/π‘š=1. In view of (17), (18), and (20), the resulting class of solutions is (see Tewari and Pant [11])π‘’πœ=𝐢1π‘Ÿπ‘Ž+π‘βˆ’1+𝐢2π‘Ÿπ‘Žβˆ’π‘βˆ’1ξ€Έ2,π‘’βˆ’πœ†=π΄π‘Ÿβˆ’2𝑏(π‘Ž+𝑏)𝐢1π‘Ÿ2𝑏+(π‘Žβˆ’π‘)𝐢2𝑛+2ξ€·πΆβˆ’π΅1/+𝐢2/π‘Ÿβˆ’2𝑏.(41)

7. Conclusion

By assigning different nonnegative integral values to the any two pairs of exponents of integrand of (17), that is, π›Όβˆ’1,𝑛+1,βˆ’1+2/π‘š, we arrive at different parametric classes of solutions. All these classes have of the form π‘’πœ=(𝐢1π‘Ÿπ›½+𝐢2π‘Ÿπ›½)𝛾.For meaningful solutions 𝛾, 𝛽 to be nonnegative, thus restriction on parameter ranges.

Acknowledgments

The author expresses his gratitude to Professer O. P. Shukla, Principal NDA, for his encouragement. He is also grateful to the anonymous referees for pointing out the errors in the original paper and making constructive and relevant suggestions.

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