DYNAMICS OF CHARGED PLANAR GEOMETRY IN TILTED

AND NONTILTED FRAMES

M. Sharif* M. Zaeern Ul Haq Bhatti**

Department of Mathematics, University of the Punjab Lahore-54590, Pakistan

Received October 29, 2014

We investigate the dynamics of charged planar symmetry with an anisotropic matter field subject to a radially moving observer called a tilted observer. The Einstein-Maxwell field equations are used to obtain a relation between nontilted and tilted frames and between kinematical and dynamical quantities. Using the Taub mass formalism and conservation laws, two evolution equations are developed to analyze the inhomogeneities in the tilted congruence. It is found that the radial velocity (due to the tilted observer) and the electric charge have a crucial effect on the inhomogeneity factor. Finally, we discuss the stability in the nontilted frame in the pure diffusion case and examine the effects of the electromagnetic field.

DOI: 10.7868/S0044451015050080

1. INTRODUCTION

I11 any physical phenomenon, the significance of observers cannot be ignored. During the last few years, there has been a renewed interest in the study based on relative motion of observers. One of the reasons for this interest is to study any realistic picture of the evolution of the early universe. Physical quantities like the Hubble parameter depend on the choice of congruence and are consequently referred to as congruence-dependent quantities. Cosmological models have two timelike vector fields (congruences): the unit vector field orthogonal to the surface (geometric congruence) and the four-velocity of the matter distribution (fluid congruence). If the four-velocity is not aligned with the unit vector field, then it is called a tilted, and otherwise nontilted congruence. There has been an extensive study of homogeneous and anisotropic cosmological models describing evolution of the early universe.

Many theoretical and observational reasons have motivated the researchers to study anisotropic and in-homogcncous models including the Tolnian, Szekeres, Gowdy, and some plane symmetric solutions. At small scales, the observed galaxy distribution is found to be inhoniogcncous, while it is expected to become spatially

E-mail: msharif.math'&pu.edu.pk E-mail: mzaeem.math'&pu.edu.pk

homogeneous 011 theoretical grounds. A11 inhoniogcncous matter distribution may lead to the formation of a naked singularity, compared to the homogeneous fluid configuration, where a black hole is more likely to be formed [1]. Some inhoniogcncous solutions of a plane symmetric spacetime for a viscous fluid distribution were found in [2]. I11 [3], the Lemaitre Tolnian Bondi metric was studied in spherical coordinates and the effect of anisotropy and inhomogeneity 011 the collapse of a dust cloud was analyzed. The impact of inhomogeneity 011 different parameters of a spherically symmetric collapsing star radiating away its energy in the form of radial heat flux was explored in [4]. A cosmological model for isotropic expansion of an inhoniogcncous universe was proposed in [5], where some exact inhoniogcncous solutions for spherical and axial symmetries were also obtained.

In most of the cases, the fluid distribution is considered to be isotropic in pressure. However, pressure anisotropy and heat dissipation are also expected to play a crucial role in an expanding and collapsing universe. Many researchers [6] have taken keen interest in investigating tilted cosmological models in the presence of a heat flux. Tilted models having a disordered radiating isotropic fluid with heat flux were explored in [7] for a Bianchi type I model. Hydrodynamical and thcr-modynamical properties of a tilted Lemaitre Tolnian Bondi spacetime with an anisotropic matter configuration were studied in [8]. Some dynamical properties of tilted planar geometry with a radiating anisotropic

matter distribution wore explored in [9]. In [10], homogeneous tilted models with a radiating source in plane symmetry were found and the behavior of some physical parameters was examined. Recently, we have discussed the dynamics of a charged spherical star with tilted and nontiltod frames [11].

The occurrence of a magnetic field in the present galactic as well as inter galactic spaces is a well-established fact and its significance is acknowledged in many astrophysical phenomena. According to [12], tilted Bianchi types I, II, and III are possible in the presence of an electromagnetic field. In [13], electromagnetic effects on cylindrically symmetric inhomogo-noous cosmological models with a perfect fluid were explored and different physical and geometrical properties were discussed. The consequences of charge and dissipation (heat flux, shear viscosity, and radiation density) in the dynamics of spherically symmetric collapse were worked out in [14]. Sharif and his collaborators [15, 16] have studied the effects of electric charge on self-gravitating collapsing models with different physical backgrounds.

This paper is devoted to exploring the dynamics of plane symmetric spacetime with the congruence of a tilted observer consisting of radiating anisotropic matter in the presence of the electromagnetic field. The paper has the following format. In Sec. 2, we present the Einstein Maxwell field equations for both tilted and nontiltod observers and find some relations between them. In the latter case, the matter content is no longer charged dissipative but charged dust cloud. Section 3 deals with some kinematical and dynamical quantities that are used to investigate the Ellis evolution equation for the tilted congruence as well as the inhomo-geneity factor. We also discuss stability analysis in the nontiltod frame with the effects of the electromagnetic field. Section 4 concludes our results.

2. FLUID CONFIGURATION AND BASIC FORMALISM

To investigate inhomogeneities in the present accelerated expansion phase of the universe, we take non-static plane symmetric geometry in the form [17]

ds2 = -A2(t,z)dt2 + B2(t,

(dx2 + dtj2 + C2(f,

I dz

(1)

The energy momentum tensor for a charged dust cloud in a nontiltod frame is

Taß = I"*1\ ".-. + ( F2Fßl

(2)

where ua, p, and Fap are the four-velocity, the energy density, and the Maxwell field tensor. In comoving coordinates, it takes the form

"n = (-4,0,0,0), Faß = -4>a,n + <t>ß,a,

where <f>a is the four-potential. The Maxwell field equations are

F% = I'oJ"

— 0,

(3)

where .Ja and //o = 4?r is the four-current and the magnetic permeability. In comoving coordinates, the four-current and four-potential become

where <f> and £ respectively denote the scalar potential and charge density; both are functions of t and With these used in Eq. (3), the Maxwell-field equations yield the independent components

d2<t> (A' c 2 B'\d<t>_f 2

a^lT + c7 ^ ôF "v'oAC •

d2d) _(A C _ 2B_ \ 30 Qtdz 1.4 + C B j dz

= 0.

(4)

(5)

Here, the prime and the dot respectively stand for i and t differentiation. Integrating Eq. (4) with respect to i leads to

<!>' = S{z)f*AC, whore «(;)= [ SCB2dz, (6) B" I

which equivalently satisfies Eq. (5).

The corresponding Einstein Maxwell field equations in the nontiltod frame yield

&irp ■

2 2 S f>-6

B4

A =

B \ B f A

B)B^{C ~b ~ ~c~) ~ß

0 = ^2

DA' CD'

■s2ti2C2 D4

DA CD

2D ~B

(7)

(8)

2AD

~ ~AD

2A'D' AD

0)

C' B'\ A"

c ~B) 1 ,4

(ô p\ CB

C b) 1 CB

B'C'

BC

(10)

Wo porforni a Lorontz transformation on the nontiltod congruonco of the observer to obtain a tilted frame in which the matter configuration has a radial velocity ui. The unit four-vectors then take the form 1

Ua =

sa =

Ia =

AyT

■ UT

ui

AyT

ul 4

- ur 1

.0.0.

,0,0,

,0,0,

ui

1

• ui-

cVT^ZJ' 1

(11)

ui ■

A/T^ZT1.......C</T— u^

The energy momentum tensor coincides with that of an imperfect matter distribution when we deal with the tilted congruence. We assume the matter content in our systematic analysis to be locally anisotropic in pressure dissipating in both streaming out and diffusion approximation in the presence of an electromagnetic field. Such a fluid distribution is represented by the second-rank stress energy tensor

Ta,i = (P± + p)UaUj - (P± - P, )SaSj + elJt + + (laUfj + P±f)aH + (liiUa +

(12)

where p, P., P±, qa, and e are respectively the energy density, pressure anisotropy, heat flux, and radiation density. The heat flux qa = qSa satisfies the equation Ua'Ia = 0, while

Vn — S Irv — 1 — S S (y — U In

SaUn = 0.

In comoving coordinates, the four-current in the tilted frame takes the form .Ja = £Ua. With this used in Eq. (3), the Maxwell field equations lead to the independent components

_ (£L j. £ _ ^ - ti<oC2A

dz2

C A

B j a

d2<t>

s/T^ZP' Ç(ioCA2uj

(13)

dtdz \c a b I dz s/T^

Integrating Eq. (13) with respect to î leads to ¿(z)fioCA

(14)

<t>' =

where S(z) =

B2

JCB2

(15)

• ui-

which identically satisfies Eq. (14). For the tilted congruence, the Einstein Maxwell field equations yield

8tt.42

1 ^u/2

p + ul2P■

S W ¡JjQ A w

X

B4

2C Ê\ D /A

~c+B)B^{C

"IT ~ ~ ~B ) ~B

x

(16)

4nCA

1 -u;2

|u;(p+ P-_) + (1 + u/2

/)' ¿.4' CB'

1 — ui2 V

B BA CB

S2p2C2

(17)

(18)

SttB2P± '

B

x

B2 \C

£1 _ — [<— _ —

~~B ~ T v"c ~ "B

C C

C + B .4 1 B + C

x

(19)

where p = e + p, P- = e + P-, and q = e + q. By comparing Eqs. (7) (10) and (16) (19), we have some relations between the physical variables of nontiltod and tilted frames:

p + uTPt

2 f*Q

8ttB4

(,s2 — ,S2)(1 — Ui2) = = p(l — Ui2

L0(p+P-_)+(l+uJ2)q = 0, 8ttB2P± + ^(S2^,s2) = 0,

B2

l^LO2

uJ2 p + P-

t'o

8 ttB4

/ -">

3. STRUCTURE SCALARS AND DYNAMICAL EQUATIONS

In this section, we investigate some scalars associated with the kinematical quantities, like acceleration

and expansion scalars in a tilted frame. We also explore the conservation of the energy momentum tensor and express it in terms of these kinomatical quantities. The scalar associated with the Woyl tensor is known as the Woyl scalar, while the scalars associated with the Riemann tensor are known as

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