Abstract. In this paper, we have obtained plane symmetric cosmological models of domain wall with viscous field coupled with electromagnetic field in general relativity by using equation of state ( ) 1 p = γ − ρ . And their physical and geometrical properties have been discussed

A G Deshmukh

D D Pawar

V R Patil

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Romanian Reports in Physics, Vol. 62, No. 4, P. 722–730, 2010 
ON THICK DOMAIN WALLS WITH VISCOUS FIELD COUPLED 
WITH ELECTROMAGNETIC FIELD IN GENERAL RELATIVITY 
V. R. PATIL1, D. D. PAWAR2, A. G. DESHMUKH3 
1Department of Mathematics, Arts, Science & Commerce College, Chikhaldara Dist Amravati - 
444807 (India). E-mail: vrpatil2007@radiffmail.com 
2Department of Mathematics, Govt. Vidarbha Institute of Science and Humanities,  
Amravati-444 604 (India).  E-mail: dypawar@yahoo.com 
3Department of Mathematics, G. V. I. S. H, Amravati; Higher Education,  
Nagpur Division, Nagpur - 440022 (India). E-mail: dragd2003@yahoo.co.in 
(Received May 7, 2009) 
Abstract. In this paper, we have obtained plane symmetric cosmological models of domain 
wall with viscous field coupled with electromagnetic field in general relativity by using equation of 
state ( )1p = γ − ρ . And their physical and geometrical properties have been discussed. 
Key words: thick domain walls, plane symmetry, viscous field, electromagnetic field. 
1. INTRODUCTION 
The exact solutions to the Einstein Maxwell equation is important for many 
physical applications in particular in astrophysics. The formation of galaxies is due 
to domain wall and cosmic string Vilenkin [1], Hill et al. [10]. The space-time of 
the cosmological domain walls in general relativity has been studied by Sikivie and 
Ispser [15], Bonjour [8], Mukherji [14]. The topological object such as domain 
walls, monopoles and string have an important role in the formation of universe. 
The domain walls are formed when the universe undergoes a series of phase 
transition with discrete symmetry being spontaneously broken Vilenkin [1, 2]. 
After symmetry breaking different regions of the universe can be settling into 
different parts of the vacuum with domain walls forming boundaries between their 
regions. In the recent years many of the researchers interested in the study of large 
scale structure of the universe because of the fact that the origin of the structure in 
the universe is one of the greatest cosmological mysteries even today. The light 
domain walls of large thickness may have produced during the late time phase 
transitions such as those occurring after the decoupling of matter and radiations 
2 Viscous field coupled with electromagnetic field 
 
723 
Hill and Schramm [12], D R K Reddy [6].  Zel’dovich [5] pointed out the stress 
energy of the domain walls composed of the surface energy density and string 
tension into spatial directions with magnitude of tension equal to the surface energy 
density, this is interesting because there are several indication that tension acts 
repulsive source of gravity in general relativity. On other hand, bulk viscosity 
coefficient ξ  plays an important role for understanding the field nature of the 
universe around us. Krori [13] and Mukherji [14] have discussed cosmological 
model with early universe. A lot of work has been done on thick domain walls in 
Lyra geometry by Rahamann [9,10,11]. Recently, Pradhan [3, 4] obtained general 
solution for bulk viscous domain walls in Lyra geometry. Reddy and Rao [7] 
studied axially symmetric domain walls in Lyra geometry. Pawar et al. [16] has 
been obtained bulk viscous fluid plane symmetric dust magnetized string 
cosmological model in general relativity. In present paper, we obtained 
cosmological models of thick domain walls with viscous field coupled with 
electromagnetic field in general relativity and also discussed the nature of the 
existing model.        
2. FIELD EQUATIONS 
Consider the plane symmetric space-time, 
 ( )2 2 2 2 2 2 2d d d d ds t A x y B z= − + − , (1) 
where the metric potential A and B are function of cosmic time t only. 
Thick domain walls are characterized by the energy momentum tensor of a 
viscous field with electromagnetic field which has the form 
 ( ) ,~ ikkikiikik EwwpwwgT +++= ρ with 1 ,j ii iu u w w= − =  (2) 
where .p p= − ξθ   
 Here , , andp pρ ξ  are the energy density, the pressure in the direction 
normal to the plane of the walls, the effective pressure and bulk viscous coefficient 
and  iw   is unit space like vector in the same direction. The energy momentum 
tensor for electromagnetic field ikE  be given by, 
 1 1
4 4ik i k ik
E F F g g F Fαβ αβα β αβ
 = − π  
, (3) 
where ikF  is electromagnetic field tensor which satisfy Maxwell equations, 
 [ ]; 0ij kF =      i.e   ( ) ; 0ij jF g− = .  (4) 
 V.R. Patil, D.D. Pawar, A.G. Deshmukh 3 724 
 In commoving co-ordinate system the incident magnetic field is taken along 
z-axis with the help of Maxwell equations (4), the components of ikF except 12F  
are vanish i.e. 12F  is only non-vanishing component along z-axis, 
 12F = constant = H (say). 
From equation (3) we have, 
 
2 2
1 2 3 4
1 2 3 44 4
and
8 8
H HE E E E
A A
= = = = −
π π
.  (5) 
Equation (2) yields, 
 
2 2 2
1 2 3 4
1 2 3 44 4 4
, and .
8 8 8
H H HT T T p T
A A A
= = ρ + = − − = ρ −
π π π
  (6) 
 The Einstein field equations for the space-time (1), choosing gravitational 
units such that c = 1, 8π G = 1 as  
 
2
4
8A AB B H
A AB B A
+ + = πρ + ,  (7) 
 
2 2
2 4
2 8A A Hp
A A A
 
− + = π + 
 
,  (8) 
 
2 2
2 4
2 8A AB H
A AB A
+ = πρ − . (9) 
The dot (·) over letters indicate differentiation with respect to time t. 
3. SOLUTIONS OF THE FIELD EQUATIONS 
 The set of field equations (7–9) are non-linear differential equations in five 
unknowns ( ), , , ,A B pξ ρ , so for solving the field equations completely, we assume 
following conditions: 
(i) The equation of state  ( )1p = γ − ρ ;  
(ii) Metric potential relation nRAB = , where R and n are constant. 
From equation (7) and (9), we get 
 4
2
2
2 2
A
H
A
A
B
B
AB
BA
A
A
=−+− . (10) 
4 Viscous field coupled with electromagnetic field 
 
725 
Using second condition nRAB = , equation (10) yields, 
 ( ) ( ) 3
22
2 2121
A
H
A
AnnAn =−−++ .  (11) 
Substituting A = η  and  2 wη =  in equation (11) we get, 
 
( )
( ) ( )
2 2
3
2 2 1d 4
d 1 1
n nw w H
A n A n A
− −
+ =
+ +
. (12) 
On solving equation (12) finally, we get 
 
( )( )
1
2 2
2 2d 2
d 1 1
A H A cA
t n
− − α
 
= + 
+ α −  
, (13) 
 
( )( )
1
2 2
2 22d d ,
1 1
HA A cA t
n
− − α
 
⇒ = + 
+ α −  
 
where 
( )
( )
2 2 1
1
n n
n
− −
α =
+
 and c is constant of integration. 
Again from equation (8) and (9) by using equation of state,  
 ( )
2 2
2 4
2 2 1 2 8 ( )A A Hn n
A A A
 + γ + − = πξθ + −γ  . (14) 
In order to obtain the solution of equation (13), we consider two different cases. 
Case (I): The constant of integration 0c = , then the equation (13) becomes, 
 
( )( )
1
2 2
22d d
1 1
HA A t
n
−
 
=  
+ α −  
. (15) 
On solving, we have 
 ( ) ( ) 22 and 2
n
A Lt B R Lt = + β = + β   ,  (16)  
where  
2
2
3 2
HL
n n
=
− −
 and  β  is constant of integration. 
Then the space-time (1) reduces to, 
 V.R. Patil, D.D. Pawar, A.G. Deshmukh 5 726 
 ( )( ) ( )2 2 2 2 2 2d d 2 d d 2 dnns t Lt x y R Lt z= − +β + − +β . (17) 
4. THE PHYSICAL AND GEOMETRICAL PROPERTIES OF MODEL 
Substituting the values of A and B in equation (9), we have 
 ( ) ( )
2 2 18 .
4 2
n nL Lt −
+
πρ = + β  (18) 
Also, the scalar expansion  ( )θ  is given by 
 ; ,u u uαα α β αα βαθ = = + Γ ,  
( ) ( ) 12
2
n
L Lt −
+
⇒ θ = +β . (19) 
Using equation (19), equation (14) gives, 
 
( )
( )
( ) ( )
1 1
8 4
2 2 2
L Lt n
n
n
−+ β +
 π ξ = γ − +
.  (20) 
And equation (8) yields 
 
2 2
2 4
28 8A A Hp
A A A
π = − − + πξθ− .  (21) 
Using equation (16), (19) and (20) in equation (21) gives 
 ( ) ( ) ( )
2 2 18 1
4 2
n nLp Lt −
+
π = +β γ − . (22) 
Again,  2 1
2
ij
ijσ = σ σ  , 
 ( ) ( )2 22 27 2
72
n L Lt −σ = + +β . (23) 
From equation (19) and (23) we have, 
 7lim 0
18t→∞
σ  = ≠ θ 
. (24) 
 Spatial volume ( ) ( )
2 2
2 22
n n
V g R Lt
+  +       = − = + β ,  (25) 
6 Viscous field coupled with electromagnetic field 
 
727 
In the presence of magnetic field, the reality conditions 0, 0p ≥ ρ ≥  are 
satisfied for all n and the coefficient of bulk viscosity  ξ  is positive for 0≥n . The 
scalar expansion ( )θ , pressure ( )p , density ( )ρ are becomes zero, when t →∞  
and they are infinite when 0t → . It means that, the model has initial singularity. 
The spatial volume ( )V is infinite when t →∞ and which is zero for 0t → . The 
quantities pressure ( )p , density ( )ρ , coefficient of bulk viscosity ( )ξ is finite 
provided 1n ≠ − . This shows that the universe is expanding, shearing and start with 
big-bang. 
Since lim 0
t→∞
σ  ≠ θ 
, the model does not approach the isotropy for large values 
of t. Also from equations (18), (20), (22), we have: 
– If 1γ = , the dust model of the universe. 
– If 2γ = , the stiff fluid model of the universe. And for 4
3
γ = , we obtain the 
radiating model of the universe. 
In absence of magnetic field (H = 0) the space-time is 
 ( )2 2 2 2 2d d d d ds t x y R z= −β + − β . 
This is the flat space-time of special relativity. 
In the absence of magnetic field, the quantities pressure ( )p , density ( )ρ , 
coefficient of bulk viscosity ( )ξ is zero i.e. we get the vacuum model. 
It is interesting to note that the role of magnetic field is vital in this model. 
Case (II): The constant of integration 0c ≠ , then the equation (13) becomes, 
 
( )( )
1
2 2
2 22d d
1 1
Ht A cA A
n
−
− − α
 
= + 
+ α −  
. (26) 
For this solution, the geometry of the universe is described by the metric, 
   
( )( ) ( )
21
2 2
2 2 2 2 2 2 2 22d d d d d
1 1
nHs A cA A A x y RA z
n
−
− − α
 
  = + − + −  
+ α −   
 
.   (27) 
Under suitable transformation of coordinates of above metric reduces to, 
 V.R. Patil, D.D. Pawar, A.G. Deshmukh 7 728 
 
( )( ) ( )
1
2
2 2 2 2 2 2 2 22d d d d d
1 1
nHs T cT T T x y RT z
n
−
− − α
 
= + − + − 
+ α −  
. (28) 
5. THE PHYSICAL AND GEOMETRICAL PROPERTIES OF MODEL 
The pressure, density, coefficient of bulk viscosity for the cosmological 
model (28) are given by, 
 
( )
( )( ) ( )
( )
2
2 12 42 2 18 2 1
1 1
n H
H T n cT
n
− α+−
 +
πρ = + + + 
+ α −  
,   (29) 
 
( )
( )( ) ( ) ( )( )
( )
2
2 12 42 2 18 1 2 1 1
1 1
n H
p H T n cT
n
− α+−
 +
π = + γ − + + γ − 
+ α −  
,   (30) 
 
( ) ( )( )
( )( )
( ) ( )( )
1
2 2
2 2
2
2
2 2 2 2
1
2 2
2 2
1 28
2 1 1
4 2
1 1
22 1 2 .
1 1
H T cT
n T n
H T cT H T
n
Hn n T cT
n
−
− − α
− − α −
− − α
 
πξ = + ⋅ 
+ + α −  
 −
⋅ − α + γ + 
+ α −  
 
 + γ + − +   + α −  
 (31) 
Spatial volume ( ) 2nV g RT += − = .  (32) 
In the presence of magnetic field, the energy conditions , 0.p ρ ≥ , are 
satisfied provided 1n ≠ −  and 1α ≠ . The scalar expansion ( )θ , pressure ( )p , 
density ( )ρ  is become zero when T →∞ , ( )1 0α + > , and , ,p ρ θ→∞  
as 0T → . 
The spatial volume tends to infinite as T →∞  and it tends to zero as 0T → . 
In this case also lim 0
T→∞
σ  ≠ θ 
, the model does not approach the isotropy for 
large values of T. Also from expression of , ,p ρ ξ  we obtain, for 1γ = , the dust 
model of the universe, for 2γ = , the stiff fluid model of the universe and for 
4
3
γ =   the radiating model of the universe. 
8 Viscous field coupled with electromagnetic field 
 
729 
In absence of magnetic field (H = 0) the space-time is 
 ( )2 2 2 2 2 2 2d d d d dns cT T T x y RT zα= − + − .  (32) 
The pressure, density, coefficient of bulk viscosity for the cosmological 
model (28) are given by, 
 ( ) ( )2 18 2 1n cT − α+πρ = + ,   (33) 
 ( )( ) ( )2 18 2 1 1p n cT − α+π = + γ − , (34) 
 
( ) ( )
1 1
2 2 22 2
2
18 2 2 1 2
2
cT cT n n cT
n T
−− α − α − α−       πξ = α + γ + −      +
. (35) 
In the absence of magnetic field, the pressure ( )p , density ( )ρ , coefficient 
of bulk viscosity ( )ξ is positive, then the model reduces the model with thick 
domain walls and bulk viscous field in general relativity. The energy conditions 
, 0p ρ ≥  are satisfied provided 1
2
n ≠ − . 
The pressure ( )p , density ( )ρ  both tend to zero as T →∞ , ( )1 0α + > and  
,p ρ  tends to infinite as 0T → , ( )1 0α + > . And as the spatial volume V is infinite 
when T →∞  and spatial volume V tends to zero as 0T → , the universe is blow up 
and shrunk with infinite past and future. 
A network of domain walls accelerates the expansion of the universe. An 
interesting result emerged in this work is that pressure perpendicular to the domain 
walls is non-zero i.e. at any instant the domain wall density ρ , pressure p is 
perpendicular direction decreases both side of walls away from the symmetry plane 
and both vanish as T →∞  in both cases. 
6. CONCLUSION 
The plane symmetric cosmological models for thick domain walls with 
viscous fluid coupled with electromagnetic field are obtained. Also we discussed 
the models in absence of electromagnetic field. Generally the models are 
expanding, shearing with big bang. The study of the domain walls in this paper 
successfully describes various features of the universe. 
Acknowledgements. The author V R Patil especially thanks to University Grant Commission, 
New Delhi to award the FIP under the Xth plan of UGC scheme. 
 V.R. Patil, D.D. Pawar, A.G. Deshmukh 9 730 
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ON THICK DOMAIN WALLS WITH VISCOUS FIELD COUPLED WITH ELECTROMAGNETIC FIELD IN GENERAL RELATIVITY

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ON THICK DOMAIN WALLS WITH VISCOUS FIELD COUPLED WITH ELECTROMAGNETIC FIELD IN GENERAL RELATIVITY

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