Journal of Modern Physics, 2012, 3, 1479-1486
http://dx.doi.org/10.4236/jmp.2012.310182 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
String Cloud and Domain Walls with Quark
Matter in Lyra Geometry
Kamal Lochan Mahanta, Ashwini Kumar Biswal
Department of Mathematics, C. V. Raman College of Engineering, Bhubaneswar, India
Received August 2, 2012; revised September 5, 2012; accepted September 15, 2012
ABSTRACT
We have constructed cosmological models for string cloud and domain walls coupled with quark matter in Lyra geome-
try. For this purpose we have solved the field equations using anisotropy feature of the universe, special law of variation
for Hubble’s parameter proposed by Berman [78] which yields constant deceleration parameter; and time varying dis-
placement field

. Further some properties of the obtained solutions are discussed.
Keywords: Cosmic Strings; Domain Walls; Quark Matter; Lyra Geometry
1. Introduction
In order to geometrize the whole of gravitation and elec-
tromagnetism Weyl [1] proposed a modification of Rie-
mannian manifold. As this theory is physically unsatis-
factory, later on Lyra [2] proposed a further modification
of Riemannian geometry which bears a close resem-
blance to Weyl’s geometry. Sen [3] pointed out that the
static model with finite density in Lyra manifold is simi-
lar to the static model in Einstein theory. Halford [4]
showed that the vector field i in Lyra’s geometry
plays a similar role of cosmological constant Λ in general
theory of relativity. In addition he pointed out that the
energy conservation law does not hold in the cosmologi-
cal theory based on Lyra’s geometry. The scalar-tensor
theory of gravitation in Lyra manifold predicts the same
effects, within observational limits, as in Einstein theory
[5]. Many authors [6-19] constructed different cosmo-
logical models and studied various aspects of Lyra’s
geometry.

In field theories, topological defects are stable field
configurations with spontaneously broken discrete or
continuous symmetries [20,21]. Spontaneous symmetry
breaking is described within the particle physics context
in terms of the Higgs field. The symmetry is said to be
spontaneously broken if the ground state is not invariant
under the full symmetry of the Lagragian density. The
broken symmetries are restored at very high temperatures
in quantum field theories. The topology of the vacuum
manifoldwith 2
M
Z
1 is called domain walls [21,22],
with
M
S2  called strings [23] and one dimensional
textures, with
M
S called monopoles and two di-
mensional textures, and with 3
S is called three
dimensional textures. The topological defects are called
local or global depending on the symmetry is whether
local (gauged) or global (rigid). In the early universe
these defects are expected to be remnants of phase transi-
tions [24].
String theory attracted the attention of many authors as
it possesses the necessary degrees of freedom to describe
other interactions, even a mode to describe the graviton.
The presence of cosmic strings in the early universe is
considered using grand unified theories [25-30]. The
study of various aspects of cosmic strings in different
theories of relativity is available in the literature [31-52].
The topological defects such as strings, domain walls
and monopoles have an important role in the formation
of our universe, Hill et al. [53] pointed out that the for-
mation of galaxies are due to domain walls produced
during a phase transition after the time of recombination
of matter and radiation. Vilenkin [54] and Sikivie and
Ipser [55] discussed thin domain wall in Einstein’s the-
ory. Further Schmidt and Wang [56] discussed the same
in the context of Brans-Dicke theory. Widraw [57] ob-
tained that a thick domain wall with non-zero stress
along a direction perpendicular to the plane of the wall
does not allow static metric to be regular throughout en-
tire space. Goetz [58] constructed thick domain wall
cosmological model where the scalar field responsible
for the symmetry breaking is static while the metric de-
pends on time. Wang [59] obtained a class of exact solu-
tion to the Einstein’s field equations representing the
gravitational collapse of a thick domain wall. Rahaman et
al. [60] found an exact solution of the filed equations for
a thick domain wall in a five dimensional Kaluza-Klein
M
C
opyright © 2012 SciRes. JMP

K. L. MAHANTA, A. K. BISWAL
1480
space time within the frame work of Lyra geometry.
Pradhan et al. [61] studiedplane symmetric domain wall
in Lyra geometry. Rahaman and Mukherji [62] con-
structed two models of domain walls in Lyra geometry. In
one of their model the space time is non-singular both in
its spatial and temporal behaviour and the gravitational
field experienced by a test particle is attractive. In the
other model they presented a spherical domain wall with
non vanishing stress components in the direction perpen-
dicular to the plane of the wall. Rahaman et al. [63] stu-
died two types of thin domain wall models in Lyra ge-
ometry. In their first model the pressure of the domain
wall is negligible along perpendicular and transverse
direction to the wall. In their second model pressure of the
domain wall in the perpendicular direction is negligible
but transverse pressures are existed. Further they showed
that the thin domain walls have no particle horizon and the
gravitational force due to them is attractive. Pradhan et al.
[64] obtained general solutions of the field equations for
bulk viscous domain walls in Lyra geometry.
It is believed that one of the transitions during the phase
transitions of the universe could be Quark Gluon Plasma
(QGP)  hadron gas called quark-hadron phase transition
when cosmic temperature was T ~200 MeV. The quark-
hadron phase transition in the early universe and conver-
sion of neutron stars into strange ones at ultrahigh densi-
ties are two ways of formation of strange quark matter
[65-67]. In quark bag models based on strong interaction
theories it is considered that breaking of physical vacuum
takes place inside hadrons. As a result vacuum energy
densities inside and outside a hadron become essentially
different and the vacuum pressure on the bag wall equi-
librates the pressure of quarks and stabilizes the system. It
is pointed out that if the hypothesis of the quark matter is
true, then some of neutron stars could actually be strange
stars built entirely of strange matter [68,69]. The quark
matter is modeled with an equation of state (EOS) based
on phenomenological bag model of quark matter in which
quark confinement is described by an energy term pro-
portional to the volume. In the frame work of this model
the quark matter is composed of mass less u, d quarks,
massive s quarks and electrons. In the simplified version
of the bag model, assuming the quarks are mass
less and non-interacting we have 3
q
q
p

, where q

is
the quark energy density. The total energy density is given
by
mqc
B


mqc
PPB
(1)
and the total pressure by
(2)
Therefore the equation of state for strange quark matter
[70,71] is given by

14
3
mmC
PB  
(3)
Cosmic string is free to vibrate and different vibration
modes of the string represent the different particle types
since different modes are seen as different masses or spins.
Therefore it is plausible to attach quark matter to the string
cloud and domain walls. Yilmaz [72] studied rotating
cosmological models for domain walls with strange quark
matter and normal matter in the non-static and stationary
Gödel universes with cosmological constant. Further
Yilmaz [73] obtained Kaluza-Klein cosmological solu-
tions of the Einstein’s field equations for quark matter
coupled with the string cloud and domain wall. Adhav et
al. [74] constructed n-dimensional Kaluza-Klein cosmo-
logical models for quark matter coupled with string cloud
and domain walls in general relativity. Khadekar et al. [75]
studied Kaluza-Klein type Robertson Walker cosmo-
logical model by considering variable cosmological term
Λ in the presence of strange quark matter with domain
wall. Recently Mahanta et al. [76] constructed Bianchi
type-III cosmological model with strange quark matter
attached to the string cloud in Barber’s second self-crea-
tion theory of gravitation.
Motivated by the aforesaid discussion in this paper we
consider quark matter coupled to the string cloud and
domain walls in the context of Lyra geometry.
2. Field Equations and Their Solutions for
the String Cloud with Quark Matter
In this section we consider the metric of the form


22222 22
ddddd
s
tAxy Bz 
ijijsij
Tuu xx
(4)
where A and B are functions of cosmic time t only. The
energy momentum tensor for string cloud given by Le-
telier [32] and Stachel [31] is


 (5)
Here

is the rest energy density for the cloud of
strings with particles attached to them,
s

is the string
tension density, i

is the four velocity for the cloud of
particles,


1
0,0,0,
i
x
B  is the four vector which
represents the strings direction which is the direction of
anisotropy and
p
s

 (6)


where
p

is the particle energy density. We know that
string is free to vibrate. The different vibrational modes of
the string represent different types of particles because
these different modes are seen as different masses or spins.
Therefore, in this section we take quarks instead of parti-
cles in the string cloud. Hence we consider strange quark

K. L. MAHANTA, A. K. BISWAL
C JMP
1481
qsc
B
matter energy density instead of particle energy density of
the string cloud. Thus from (6) we get
we consider
opyright © 2012 SciRes.
 

ijs ij
uuxx  
i
ui
(7)
From (5) and (7) we get the energy momentum tensor
for strange quark matter attached to the string cloud as

ijqs c
TB   (8)
Moreover and
x
satisfy the standard relations
1
ii
i
xx
i
 0
ii
uxuu and (9)
The Einstein’s field equations based on Lyra’s geome-
try proposed by Sen [3] and Sen and Dunn [16] in normal
gauge are
13 3
22 4
ij iji jij
RgRkk ij g T  
 
,0,0,0t 
(10)
where i is the displacement vector and
other symbols have their usual meaning as in Riemannian
geometry.

Using (8) and (9) the field Equations (10) for the metric
(4) leads to the following system of equations
2
2
AA
AA
2
3
4
B
B








 (11)
2
30
4
BAAB
A
AB
 

B

 
(12)
2
2
3
4
2
s
AA
AA



 



 (13)
Here afterwards the dash over the field variable repre-
sents ordinary differentiation with respect to t. Now we
find that Equations (11)-(13) are there independent equa-
tions involving five unknowns A, B,

,

and
s

.
Therefore to obtain exact solution of the field equations
we require two more relations. In view of the anisotropy
of the space time, we assume that expansion



is
proportional to the components of shear tensor


2

which also represents anisotropy of the universe [77]. This
leads to a polynomial relation between the metric coeffi-
cients
n
A
B

(14)
where n is a non-zero constant.
In addition, with the help of special law of variation of
Hubble’s parameter proposed by Berman [78] that yields
constant deceleration parameter models of the universe,
2Constant
RR
qR


  




(15)
213
RAB

where is the overall scale factor. The con-
stant is taken as negative to obtain an accelerating model
of the universe. From (15) we obtain
1
1q
Ratb

 (16)


0a


0b

and where

are constants of integration.
Equation (16) yields the condition of expansion i.e.
10q

. Now we get


11
231q
ABatb  

 
. (17)
Using (14) in (17) we find
1321qn
Batb 

 
(18)
Thus
31 21nqn
Aatb   (19)
Substituting (18) and (19) in (12) we get


2
22
22
41
31
akknnn
kat b




 (20)




121.kqn

where

Now using (18)-(20) in (11) and (13) we obtain string
energy density
2222 22
2
1827933ananaankak
at b




(21)
and string tension density
22 22 22
2
2
1899 33
sanan aankak
katb

 


 
(22)
Using (18) and (19) the line element (4) is expressed as


 
121
22 22
12162
6
dd dd
d
qn
qn
n
n
s
tatb xy
at bz


 

 
(23)
The model (23) represents a string cosmological model
with quark matter in Lyra geometry with negative con-
stant deceleration parameter. For the model (23) we have
string particle density







222222222
2
2
118319 39 13
ps
kannak ka akank
katb


  (24)

K. L. MAHANTA, A. K. BISWAL
1482
Quark energy density n
A
B

2222 2
2
18279 3
qc
anan aan
at b
 

 
2
3
c
k ak
BB


(25)
Quark pressure

2222 2
2
18279 3
33
q
qan anaan
Pat b



 
2
3
3
c
B k ak

 
(26)
Scalar expansion
9
1
a
qatb
  (27)
Shear scalar
 
2
22
27
21
a
qatb

i jij
uuPg  
c
B
22
1
6
  (28)
3. Field Equations and Their Solutions for
Domain Walls with Quark Matter
The energy momentum tensor of a domain wall [79] in the
conventional form is given by
D
ij
TP (29)
This perfect fluid form of the domain wall includes
quark matter [72] (described bymq


mqc
PPB w
and
) as well as domain wall tension

i.e.
mw

  and mw PP

. Further m and m are
related by the bag model equation of state i.e. Equation (3)
and equation of state
P P

1
mm
P

 
12
(30)
where


i
u1
i
uu
is a constant. Here the four velocity
vector is such that i

. We use commoving
coordinate system 0
ii
u

. Using the line element (4), the
filed equations (10) yield
2
2
AA
AA
2
3
4
B
B







 (31)
2
3
4P
AABB
A
AB B



 
 (32)
2
2
3
4P2AA AA

 
 



 (33)
where dash over the field variables denote differentiation
with respect to t. Here Equations (31)-(33) are three in-
dependent equations involving five unknowns A, B,
(34)

where n is a non-zero constant.
Further we consider the power law relation between
time co-ordinate and displacement field [61,64]
0t

(35)

 

is a constant. where
Adding Equation (31) with (32) and (33) we get

, P,
and

. Therefore, in order to obtain exact solution of the
field equations two more relations connecting these va-
riables are required. Due to anisotropy of the space-time
we assume that scalar of expansion



is proportional
to the components of shear tensor

2

which gives

2
3
A
AB BA
P
A
AB BA








(36)
and
2
22 2
A
AAB
P
A
AAB


 



 (37)
From (34), (36), and (37) we obtain
2
20,1
BB
nn
BB
 






(38)
Integrating Equation (38), we find
1
21 n
Bctd  (39)


0c
where

and d are constants of integration. From
(34) and (39) we have

21
n
n
Actd  (40)
Substituting (35), (39) and (40) in (31)-(33) we obtain

22 2
22
0
22
123
4
21
mw nc nct
nctd

 




 (41)







and

22 2
22
0
22
123
4
21
mw nc nc
PP t
nctd






 (42)







Further we find the scalar expansion 3c
ct d
 

and
shear scalar

2
2
2
3
2
c
ct d
 .
From (41) and (42) we observe that the solutions rep-
resent stiff domain walls. To determine the tension of the
domain walls


w

and density and pressure of the quark
matter, we will use the equations of state given by (3) and
(30) separately.
3.1. Case 1
By using equation of state for quark matter i.e. Equation
(3) we get

K. L. MAHANTA, A. K. BISWAL 1483

22 2
22
32
21
nc nc
nctd









22
3
24
moc
tB







(43)
and

22 2
22
12
24
21
nc nc
nctd





22
0
3
mc
Pt
B







(44)
Now from equations (1) and (2) we have

22 2
32
21
qnc nc
nct
22
0
22
3
24
t
d















(45)
and

22 2
12
21
qnc nc22
0
22
3
24
P
t
d







nc
t





 (46)
Again with the help of (41) and (43) we obtain

22 2
12
21
qnc nc
P
nct






22
0
22
3
24
t
d







(47)
In this case domain walls behave like invisible matter
due to their negative tension. Further we find 3
q
q
P

 as
proposed by Bodmer [66] and Witten [67].
3.2. Case 2
If we use (30) in (41) and (42) we get

22 2
12 4
21
mnc nc
nct






22
0
22
3
2t
d







(48)

22 2
12 4
21
mnc nc22 0
22
3
2
P
t
d







nc
t








(49)

22 2
22
0
3
2c
tB







22
12 4
21
q
nc nc
nctd







(50)

22 2
22
0
12 43
2c
nc nctB









22
21
q
P
nc td



 (51)
and

222 222 22 22
00
224
63
4
nc ncnc nctt
22
21
wnc
td






1
 

(52)
In this case, when


with ne tension and dust quark matter. When
we get domain walls solutions
egativ ,
n anwe have domain walls solutions with negative tensiod
quark matter solution like radiation. When 2


, w
clusions
tained exact solutions of the field eq-
loud and domain walls with quark
e
have stiff quark matter solution and domain walls disap-
pear.
4. Con
In this paper we ob
uations for string c
matter in Lyra geometry. In our solutions we observe the
following properties:
1) In the case of string cloud with quark matter for
1n

, we get dust quark matter solution i.e. 0
s


. In
this model the universe starts at an initial epoch tb
a

.
2
At initial epoch the physical parameters

and

di-
verge. As cosmic time t gradually increases

2 and

decrease and finally they vanish when t. Tis is
consistent with the results of Brook-Have nationl
laboratory [80,81]. Here we find
h
n a
0.408


. The present
upper limit of


is 10−15 obtained frndirect argu-
ments concernihe anisotropy of the primordial black
body radiation [82]. The greater value of
om i
ng t


for our
model than the aforesaid limit indicates that e model
represents the early stages of the evolution of thuniverse.
th
e
At initial epoch tb
a

, the gauge function

diverges.
With the increase in cosmic time t, gauge fnction u

decreases and drs as t . Hencehis theory isappea t
te
ed stiff domain wall solution. Since in the more
re
leads to Einstein general theory of relativity at infini
time.
2) In the case of domain walls with quark matter we
obtain
alistic case in which the domain walls interact with the
primordial plasma, the equation of state for domain walls
is expected to be stiffer than that of a radiation, our solu-
tions correspond to the early stages of evaluation of the
universe. Here we note that the universe starts at an initial
epoch td
c

. At the initial epoch the physical parame-
ters

2
and

diverge. With the increase in cosmic
time t scalar expansion

and shear scalar 2

decrease
and fally thvanish as t.
In case 1 of strange quark matter coupleto domain
walls we get
iney
d
4
3


3
qq
P

 as edpropos by Bodmer [66] and
Witten [67]. Inse domain walls behave like invisi-
ble matter due negative tension.
this ca
to their
In case 2 of aforesaid model when 2,

 we get do-
main walls with negative tension and dust quark matter.

K. L. MAHANTA, A. K. BISWAL
1484
When 4
3
 , we have domain walls with negative ten-
sion anrk matter like radiation. When 2, d qua


do-
main walsappear and we have stiff quark matter. ls di
ies o
ical
m
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