String Cloud with Quark Matter in Plane Symmetric Space-Time Admitting Conformal

String Cloud with Quark Matter in Plane Symmetric Space-Time Admitting Conformal

P. D. Shobhane

Rajiv Gandhi College of Engineering & Research, Nagpur (India)

S. D. Deo

1. S. Sc. and Arts College, Bhadrawati, Chandrapur (India)

   Abstract: In this paper, we have examined the string cloud with quark matter in the plane symmetric space- time admitting one-parameter group of conformal motions. Also, we have discussed the properties of the solutions obtained.

   Keywords Plane symmetry; String cloud; quark matter; conformal motion

   General relativity provides a rich arena to use symmetries in order to understand the natural relation between geometry and matter furnished by Einstein equations. Symmetries of geometrical/ Physical relevant quantities of this theory are known as collineations and the most useful collineation is conformal killing vector defined by

   1. INTRODUCTION

      In general relativity, it is a subject of long- standing

      £ gij

      = i;j

      + j;i

      = gij

      , = ( xi ),

      interest to look for the exact solutions of Einsteins field equations. To know the exact physical situation at early stage of the formation of our universe is still challenging subject of study. At the very early stages of evolution of the universe, it is generally assumed that during phase transition (as the universe passes through its critical temperature) the symmetry of the universe is broken spontaneously. It can give rise to topologically stable defects such as strings, domain walls and monopoles.

      Sahoo and Mishra [1] studied plane symmetric space- time with quark matter attached to the string cloud and domain wall in the context of Rosens biometric theory and observed that, in this theory, string cloud and domain walls do not exist and biometric relativity does not help to describe the early era of the universe. Sahoo and Mishra [2] also studied axially symmetric space- time with strange quark matter attached to the string cloud in Rosens biometric theory and shown that there is no contribution from strange quark matter and hence vacuum model is presented. Deo [3] studied spherically symmetric Kantowski- Sachs space- time in the context of Rosens biometric theory with the source matter cosmic strings and domain walls and observed that the space- time does not accommodate the cosmic strings as well as domain walls and it is observed that the resulting space- time represents Robertson-Walker flat space- time which expands according to the signature of the parameter uniformly along the space directions with time. Yilmaz [4] obtained Kaluza- Klien cosmological solutions for quark matter coupled to the string cloud and domain wall in the context of general relativity by using anisotropy feature of the universe. Rao and Neelima [5] studied the anisotropic Bianchi type- VI space- time with strange quark matter attached to string cloud in Barbers second self creation theory and general relativity and noticed that the presence of scalar field does not affect the geometry of the space- time but changes the matter distribution.

      where £ signifies the Lie derivative along i and = (xi) is the conformal factor. In particular, is a special conformal killing vector, if ; ij = 0 and ,i 0 . Here (;) and (,) denote covariant and ordinary derivatives respectively.

      Conformal killing vectors provide a deeper insight into the space- time geometry and facilitate generation of exact solutions to the field equations. Sharif [6] classified the static plane symmetric space- time according to their matter collineations. Aktas and Yilmaz [7] solved Einsteins field equations for spherical symmetric space- time via conformal motions and examined magnetized quark and strange quark matter in spherical symmetric space- time admitting one- parameter group of conformal motions. Kandalkar, Wasnik and Gawande [8] investigated spherically symmetric string cosmological model with magnetic field admitting conformal motion. Shobhane and Deo [9] examined the wet dark fluid matter in the spherical symmetric space- time admitting one parameter group of conformal motion.

      The paper is outlined as follows:

      In Sec.2, we have obtained Einstein field equations for string cloud with quark matter in plane symmetric space- time. In Sec.3, the solutions of the field equations are obtained for string cloud with quark matter in plane symmetric space- time admitting one parameter group of conformal motion. In Sec.4, concluding remarks are given.

   2. EINSTEINS FIELD EQUATIONS

      The metric for static plane symmetric space time is given by

      2 = 2 2 (2 + 2), (1)

      where and are functions of x alone and x1,2,3,4 = x, y, z, t.

      Einsteins field equations can be expressed as

      2

      2

      1 = 8 . (2)

      Here, we shall use geometrized units so that 8G = c =1.

      1 = , (10)

      ,1 2

      The energy momentum tensor Tij for string cloud is given

      by

      = , (3)

      where is the rest energy density for cloud of strings with particles attached to them and is the string tension density and they are related by the equation

      = + = . (4)

      Here p is the particle energy density. The string is free to vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins. So, we will take quarks instead of particle in the string cloud.

      In this case, from (4) , we get

      = + + + = , (5)

      where q is the quark energy density and Bc is the vacuum energy density (called as the bag constant). Further, ui is the

      4 = c (constant), (11)

      1 = , (12)

      1 = , (13)

      and

      2 = 3 = 0. (14)

      Using (12) and (13), we get

      = . (15)

      On integration, we get

      = + c1 or e = c2e , (16) where c1 and c2 ( > 0) are the constants.

      Using (10) and (12), we get

      d1

      four velocity of the particles and xi is the unit spacelike

      vector representing the string direction in cloud i.e. the direction of anisotropy.

      We have

      = /2 and = .

      1

      = 2 dx. (17)

      4 1 The solution of (17) is given by

      uiui = xixi = 1 and uixi = 0.

      Then using (3) we get

      T11 = , T22 = T33 = 0 and T44 =e .

      Using (1) and (2), we get

      1 (2 + 2) = , (6)

      4

      2 + 2 + 2 + 2 + = 0 (7) and

      1 (4 + 32)

      1 = c3 e/2 , c3 >

      0. (18)

      Using (7) and (15), we get

      4 + 32 = 0 . (19)

      Setting = 1 , we get

      n

      4

      = , (8)

      dn dx =

      3 3

      4 i. e. n = 4 x

      where primes denote differentiation w.r.t. x.

   3. SOLUTIONS OF FIELD EQUATIONS

      + k1, (20)

      where k1 is the constant of integration.

      Now, we shall assume that spacetime admits a one- parameter group of conformal motions [7] i.e.

      £gij = i;j + J;i = gij , (9)

      d = dx

      4

      3x + 4k1.

      where £ signifies the Lie derivative along i and is an arbitrary function of x. In particular, is a special conformal killing vector, if ;ij = 0 and ,i 0 . Here (;) and (,) denote covariant and ordinary derivatives respectively.

      Using (1) and (9), by virtue of plane symmetry, we get following expressions:

      On integration, we get

      e = [k2(3x + 4k1)]4/3 = c2e. (21)

      Using (12), (18) and (21), we get

      k

      = (3x + 4k1)1/3 , (22)

      For c2=1, above line element reduce to the anti-De Sitter metric in unusual form [12]:

      2 = ()(2 2 2) 2, (30)

      where k = 4c3 (k2)2/3 > 0 is the constant.

      Using (6), (8), (15) and (21), we get

      = 12

      (3x + 4 k1 )2

      and

      (23)

      where () = () = [2(3 + 41)]4/3.

   4. CONCLUSION

From (23), we observed that the string tension density is negative i.e. < 0. From (23) and (25), we get . The energy condition 0 and 0 are satisfied. Further when 4 13, , and when x ,

0, 0 and .

= 0. (24)

As pointed by Letelier [10], [11], the string tension density

may be positive or negative.

Using (5), we get

p = q + Bc =

ACKNOWLEDGEMENT

We would like to thank Prof. Dr. K. D. Patil, BDCOE, Wardha for his valuable suggestions.

REFERENCES

1. Pradyumn Kumar Sahoo and Bivdutta Mishra (2013), Journal of Theoretical and Applied Physics, Vol. 7 (12). http://www.jtaphys.com/content/7/1/12.

2. P. K. Sahoo and B. Mishra (2013), International Journal of Pure and Applied Mathematics, Vol. 82, No. 1, pp. 87-94. url:http://www.ijpam.eu

   = 12

   = 12

   (3x + 4 k1 )2

   (25)

3. S.D. Deo (2012), International Journal of Applied Computational Science and Mathematics, Vol. 2 , No.1, pp. 23-28. http://www.ripublication.com/ijacsm.htm.

[4] I. Yilmaz (2006), Gen.Rel.Grav., Vol. 38, pp. 1397-1406.

[5] V.U.M. Rao and D. Neelima (2014), The African Review of Physics,

q (3x + 4 k1 )2

Bc (26)

Assuming that the quarks are massless and non interacting

, then we have the quark pressure [4]

Vol. 9:0004, pp. 21-26.

[6] M. Sharif (2003, Oct.),arXiv:gr-qc/0310019v1 .

1. C. Aktas and I. Yilmaz (2007), Gen. Rel. Grav. , Vol. 39, pp. 849- 862.

2. S.P. Kandalkar, A.P. Wasnik & S.P. Gawande (2013, Jan.), Prespacetime Journal, Vol. 4, Issue 1, pp. 48-58. www.prespacetime.com.

3. P.D. Shobhane and S.D. Deo (2016), Advances in Applied Science

q

pq = 3

= 4

(3x + 4 k1 )2

Research, Vol. 7(1), pp. 8-12. www.pelagiaresearchlibrary.com.

[10] P.S. Letelier (1979), Phy. Rev. D. , Vol. 20, pp. 1294.

Bc 3

. (27)

[11] P.S. Letelier (1983), Phy. Rev. D. , Vol. 28, pp. 2414.

[12] Asghar Qadir and M. Ziad (!991, Jun.) International Atomic Energy Agency and UNESCO, International Centre for Theoretical Physics , Internal Report IC/91/158.

Then Total particle pressure is given by

pp = pq

Bc

= 4 [ 1

(3x + 4 k1 )2

Bc ]. (28)

3

Using (1), space-time geometry of string cloud with quark matter is given by

2

()

= 2

2

2

()

(2

+

2), (29)

where () = [2(3 + 41)]4/3.