Journal of Modern Physics, 2011, 2, 481-487
doi:10.4236/jmp.2011.26058 Published Online June 2011 (http://www.SciRP.org/journal/jmp)
Copyright © 2011 SciRes. JMP
Well Behaved Class of Charge Analogue of Adler’s
Relativistic Exact Solution
Mamta Joshi Pant, Bipin Chandra Tewari
Department of Mathematics, S S J Campus, Kumaun University, Almora, India
E-mail: drbctewari@yahoo.co.in
Received January 11, 2011; revised March 6, 2011; accepted April 1, 2011
Abstract
We present a well behaved class of charge analogue of Alders (1974) [1]. This solution describes charge
fluid balls with positively finite central pressure and positively finite central density; their ratio is less than
one and causality condition is obeyed at the centre. The outmarch of pressure, density, pressure-density ratio
and the adiabatic speed of sound is monotonically decreasing, however, the electric intensity is monotoni-
cally increasing in nature. The solution gives us wide range of parameter K (0.96 K 5.2) for which the
solution is well behaved and appropriate for relativistic theory; therefore, suitable for modeling of super
dense star. For this solution the mass of a star is maximized with all degrees of suitability and by assuming
the surface density ρb = 2 × 1014 g/cm3. Corresponding to K = 0.96 and X = 0.35, the maximum mass of the
star comes out to be 3.43 MΘ with linear dimension 32.66 Km and central redshift and surface redshift
1.09374 and 0.5509 respectively.
Keywords: Charge Fluid, Reissner-Nordstrom, General Relativity, Exact Solution
1. Introduction
It is well known that the Reissner-Nordstrom solution for
the external field of a ball of charged mass has two dis-
tinct singularities at finite radial positions other than at
the centre. Thus the solution describes a bridge (worm
hole) between two asymptotically flat spaces and an
electric flux flowing across the bridge. Graves and Bill
[2] pointed out that the region of minimum radius or the
throat of worm hole pulsates periodically between these
two surfaces due to Maxwell pressure of the electric field.
Consequently, unlike Schwarzschild’s exterior solution
of chargeless matter in Reissner-Nordstrom solution has
no surface which can catastrophically hit the geometric
singularity at r = 0. All these aspects show that the pres-
ence of some charge in a spherical material distribution
provides an additional resistance against the gravitational
contraction by means of electric repulsion, thereby the
catastrophic collapse of the entire mass to a point singu-
larity can be avoided.
The above result has been supported by a physically
reasonable charge spherical model of Bonnor [3], that a
dust distribution of arbitrarily large mass and small ra-
dius can remain in equilibrium against the pull of gravity
by a repulsive force produced by a small amount of
charge. Thus it is desirable to study the implications of
Einstein-Maxwell field equations with reference to the
general relativistic prediction of gravitational collapse.
For this purpose charged fluid ball models are required.
Eventually, the external field of such ball is to be
matched with Reissner-Nordstrom solution.
For obtaining significant charged fluid ball models of
Einstein-Maxwell field equations, the Astrophysicists
have been using exact solutions with finite central pa-
rameters of Einstein field equations, as seed solutions.
There are two type of exact solutions of this category.
Type 1. If the solutions are well behaved (Delgaty-
Lake [4], Pant [5]). These solutions their self completely
describe interior of the Neutron star or analogous super
dense astrophysical objects with chargeless matter. Del-
gaty-Lake [4] studied most of the exact solutions so far
obtained and pointed out that only nine solutions are
regular and well behaved; out of which only six of are
well behaved in curvature coordinates. Pant [5] obtained
two new well behaved solutions in curvature coordinates.
Type 2. If the solutions are not well behaved but with
finite central parameters; such solutions are taken as seed
solutions of super dense star with charge matter since at
centre the charge distribution is zero.
1) Schwarzschild’s interior solution-The solution is
M. J. PANT ET Al.
Copyright © 2011 SciRes. JMP
482
insignificant as it gives us infinite speed of sound
throughout with in the ball. However, charge Analogues
of the solution is well behaved for wide rang of constant
(Gupta-kumar [6], Gupta-Gupta [7], Florides [8] etc).
2) Many of the authors electrified the well known ex-
act solutions which are not well behaved, as seed solu-
tions e.g. Kuchowicz solutions by Nduka [9], Tolman
solution by Cataldo-Mitskievic [10] and Durga-
pal-Fuloria solution [11] by Gupta-Maurya [12] Durga-
pal solution [13] by Pant [14] etc. These coupled solu-
tions are well behaved and completely describe interior
of the Neutron star or pulsar with charge matter.
In the present paper we have charged the Adler [1]
solution, which is not well behaved with chargeless mat-
ter as the speed of sound is monotonically increasing
from centre to boundary Durgapal [13]. The present pa-
per charge analogue of Adler solution is well behaved in
all respects and simple in terms of expression. Though
the charge Analogues of the Adler solution has also been
studied by Singh-Yadav [15], however the solution is
silent about its well behaved nature and not simple in
terms of expression.
For well behaved nature of the solution in curvature
coordinates, the following conditions should be satisfied
(Pant N [5]).
1) The solution should be free from physical and geo-
metrical singularities i.e. finite and positive values of
central pressure, central density and non zero positive
values of e
and e
. i.e. 00p an 00
. For such
solutions the tangent –3 space at the centre is flat and it
is an essential condition. For curvature coordinates,
mathematically it is expressed as

01
r
e
and
0r
e
= positive constant ( Leibovitz [16] Pant [5]).
2) The solution should have positive and monotoni-
cally decreasing expressions for fluid parameters (p and
ρ) with the increase of r i.e.
a)

0
d0
dx
pve
x


 , (b)
0
d0
dx
x



3) The solution should have positive and monotoni-
cally decreasing expression for fluid parameter 2
p
c
with the increase of r i.e .
2
0
1d 0
dx
p
x
c






.
4) The solution should have positive and monotoni-
cally decreasing expressions for fluid parameter d
d
p



with the increase of r
dd 0
dd xo
p
x






.
5) The solution should have causality condition at
centre of the ball i.e.
2
0
1d
01
dr
p
c


 .
6) The solution should have positive value of ratio of
pressure-density and less than 1 at the centre of the ball
i.e.
0
2
0
01
p
c
.
7) The central red shift Z0 and surface red shift Zb
should be positive and finite i.e.
/2
00
10
r
Ze
 and 210
b
b
Ze



; both
should be bounded and red shift should be monotonically
decreasing with the increase of r i.e.
0
d0
dx
z
x


 .
8) Electric intensity is positive and monotonically in-
creasing from centre to boundary and at the centre the
Electric intensity is zero i.e.
2
10
d0
dx
E
xc







.
2. Einstein-Maxwell Equat i o n f o r C h ar g e d
Fluid Distribution
Let us consider a spherical symmetric metric in curvature
coordinates
2222222
ded dsinded
s
rr t


 (1)
where the functions
r
and

vr satisfy the Ein-
stein-Maxwell equations

2
44
8π18π
2
11
4π4
iiii i
j
jj jj
imi mn
jmj mn
GG
TR Rc pvvp
cc
FF FF

 

 


(2)
where ,, ,
i
ij
pv F
denote energy density, fluid pressure,
velocity vector and skew-symmetric electromagnetic
field tensor respectively.
In view of the metric (1), the field equation (2) gives
(Dionysiou, [17])

2
24 4
1e 8π
eqr
vG
p
rrc r

(3)
M. J. PANT ET Al.
Copyright © 2011 SciRes. JMP
483

2
2
44
8π
e
24 42
qr
vvvv G
p
rcr


 
 


(4)


2
22 4
1e 8π
eqr
G
rrcr

(5)
where, prime (/) denotes the differentiation with respect
to r and q(r) represents the total charge contained within
the sphere of radius r.
Now let us set

2
2
1
e1
vBcr (6)
which is the same as that of the metric obtained by Adler
[1].
Putting (6) into (3)-(5), we have

2
1
24
1
1
418π
1
Zcq
Z
Gp
xx c
xc

(7)

2
1
22
1
1d18π
2d
Zcq
Z
G
xx c
xc
 (8)
and Z satisfying the equation


2
1
12
d1 1
d13 13
xcq
Zx Z
xxxxxx

 



(9)
where 2
1
x
cr, e
Z
.
Our task is to explore the solutions of equation (9) and
obtain the fluid parameters p and
from Equation (7)
and Equation (8).
3. New Class of Solutions
In order to solve the differential Equation (9), we con-
sider the electric intensity E of the following form

2
21
13
2
1
13
2
cq
EK
x
x
cx
  (10)
where K is a positive constant. The electric intensity is so
assumed that the model is physically significant and well
behaved i.e. E remains regular and positive throughout
the sphere. In addition, E vanishes at the centre of the
star.
In view of Equation (10) differential Equation (9)
yields the following solution



2
22
33
1
e1
213 13
xx
K
Ax
Z
x
x
 

(11a)
where A is an arbitrary constant of integration.

2
e1
vBx (11b)
Using (11a), (11b) into Equations (7) and (8), we get
the following expressions for pressure and energy den-
sity


 

2
422
133
87115
18π4
21
13113
xx Ax
GK
p
cx
cxxx

 

(12)



32
255
133
263516 335
18π
213 13
xxx
A
x
GK
cc
x
x

 

(13)
4. Properties of the New Class of Solutions
The central values of pressure and density are given by
0
4
1
18π4
2
GK
pA
cc
 (14)
0
2
1
18π33
2
G
K
A
cc
 (15)
For 0
p and 0
must be positive and 0
0
1
p
, we
have
2
22
K
K
A
 , and 0, 0KA (16)
Differentiating (12) and (13) w.r.t. x., we get;


 

32
45
13
2
52
23
325528 5
18πd
d2 13 1
15 4
2
1
113
xxx
Gp K
xcx
cxx
x
A
x
xx





(17)



32
28
13
8
3
1048338 1
18πd
.d2 13
1
10
13
xxx
GK
cx
cx
x
A
x

 
(18)
4
10
18πd5
24
d2
x
GpK A
cx
c



 (19)

4
10
18πd0
dx
Gp ve
cx
c




The expression of right hand side of (19) is negative,
thus by virtue of theorem, the pressure p is maximum at
the centre and monotonically decreasing.
M. J. PANT ET Al.
Copyright © 2011 SciRes. JMP
484
2
10
18πd10
d2
x
GK
A
cx
c



 (20)
2
10
18πd0
dx
G
cx
c




(21)
The expression of right hand side of (20) is negative,
thus the density ρ is maximum at the centre and mono-
tonically decreasing.
and hence the velocity of sound v is given by the follow-
ing expression
2d
d
p
v







5
322 3
22
32
1 33255285 1415813
1d
d1048338120 11
xK xxxxAxx
p
cKx xxAxx







(22)
2
0
1d548 1
d20
r
pKA
KA
c
 


 , for all values of K and A
satisfied by (16)
Using Equations (12) and (13)
2
1p
c
(23)
where



 

2
22
33
871154
21
131 13
xx Ax
K
x
xxx

 





32
55
33
263516335
21313
xxx
A
x
K
x
x

 

Differentiating (22) w.r.t. x
22
dd
1d dd
d
p
x
x
x
c



 (24)

22
22
0
146032112 64
1d
d36
x
KAKKAA
p
x
cKA
 






(25)
The expression of right hand side of (25) is negative,
thus the pressure–density ratio 2
p
c
is maximum at the
centre and monotonically decreasing for all values of K
and A satisfied by (16).
Differentiating Equation (22) w.r.t. x,we get,
22
dd
1dd ddd
ddd
p
x
x
xx
c



 


 
 
 (26)





5
322 3
1332552851415813xK xxxxAxx




(26a)


2
32
1048338120 11
K
xxxAxx



(26b)

2
22
1dd152808256800
dd 20
xo
pK AKKA
x
cKA

 

 


(27)
The expression of right hand side of (27) is to be neg-
ative, thus the square of adiabatic speed of sound
d
d
p
is maximum at the centre and monotonically de-
creasing.
The expression for gravitational red-shift (z) is given
by

1
11
x
zB

(28)
The central value of gravitational red shift to be non
zero positive finite, we have
10B (28a)
Differentiating Equation (28) w.r.t. x, we get,
0
d1
0
dx
z
xB
 

 (28b)
The expression of right hand side of (28b) is negative,
thus the gravitational red-shift is maximum at the centre
and monotonically decreasing.
Differentiating equation (10) w.r.t. x, we get,


2
23
1
14
d
d2
13
x
EK
xc x

 
 


(29)
M. J. PANT ET Al.
Copyright © 2011 SciRes. JMP
485

2
10
d
d2
x
EK
ve
xc

 




 (29a)
The expression of right hand side of (29a) is positive,
thus the electric intensity is minimum at the centre and
monotonically increasing for all values of K > 0. Also at
the centre it is zero.
5. Boundary Conditions
The solutions so obtained are to be matched over the
boundary with Reissner-Nordstrom metric.;

1
22
2222222
22
2e 2e
d1d dsind1d
GM GM
s
rr t
rr
rr

 
 
 
 
(30)
which requires the continuity of e
, e
and q across
the boundary b
rr

2
22
2e
e1
b
r
bb
GM
cr r
 (31)

2
22
2e
e1
b
r
bb
GM
cr r
 (32)

b
qr e
(33)

0
b
pr
(34)
The condition (34) can be utilized to compute the val-
ues of arbitrary constants A as follows:
On setting 2
1,
b
rr b
x
Xcr
 (b
r being the radius of
the charged sphere)
Pressure at ()
0
b
rr
p
gives




2
23
187113
4
215 15
XX XX
K
A
X
X

 

(35)
The expression for mass can be written as


4
3
2
2
14 4
21515
bX
r
GM X
KX
X
X
c


(36)
In view of (31) and (32) we get,


4
23
213
115
K
XX
B
X
X

 (37)
The surface density is given by




32
2
255
33
263516 335
8π
213 13
bb
XXX
A
X
GK
rX
cXX



 




(38)
Table 1. The variation of various physic al parameters at the centre, surface density, electric field intensity on the boundary,
mass and linear dimension of stars with different values of K and X.
K 2
1b
Xcr 0
4
1
18πG
p
cc 0
2
1
18πG
cc
0
2
0
1
p
c
2
1d
d
x
o
p
c


 0
z
2
1b
r
E
c



2
2
8π
bb
Gr
c
Μ
M
2rb in Km
0.96 0.1 0.6607 9.9687 0.0679 0.2390 0.2889 0.0523 0.705 1.28 27.52
0.96 0.23 1.00720 8.9777 0.1122 0.2427 0.6694 0.1315 0.974 2.60 32.35
0.96 0.35 1.08886 8.73342 0.12467 0.24373 1.09374 0.213 0.993 3.43 32.66
2 0.1 0.5183 10.44 0.0496 0.1738 0.2937 0.1091
0.691 1.31 27.24
2 0.25 0.61276 10.16 0.0603 0.1732 0.7998 0.3012 0.890 2.94 30.92
2 0.3
0.559141 10.32258 0.05416 0.17357 1.03058 0.985 0.857 3.34 30.34
5.2 0.05 0.0954 11.71 0.0081 0.05926 0.1501 0.1362 0.436 0.63 21.64
5.2 0.1 0.029 11.91 0.0024 0.06062 0.3088 0.2837
0.649 1.40 26.40
7 0.01
0.009635 11.97 0.00080 0.01888 0.02999 0.035 0.112 0.07 9.04
M. J. PANT ET Al.
Copyright © 2011 SciRes. JMP
486
Table 2. The Derivatives of various physical parameters at the centre with different values of K and X.
K 2
1b
Xcr 4
10
18πd
dx
Gp
cc x



2
10
18πd
dx
Gp
cc x



2
0
1d
dx
p
cx



2
1dd
dd
x
o
p
cx






0
d
dx
z
x



0.96 0.1 –9.2058 –38.5093 –0.6691 –0.0022 –1.2889
0.96 0.23 –8.5456 –35.2058 –0.51178 –0.00428 –1.6694
0.96 0.35 –8.38228 –34.3914 –0.46883 –0.0049 –2.09375
2 0.1 –7.963 –45.817 –0.5447 –0.4241 –1.29
2 0.25 –7.774 –44.872 –0.4988 –0.4325 –1.79
2 0.3
7.88172 –45.4086 0.52526 –0.42772 –2.03059
5.2 0.05 –4.009 –67.64 –0.2952 –1.057 –1.15
5.2 0.1 –4.14
68.30 –0.3333 –1.048 1.30
7 0.01
-1.48073 –78.4036 –0.11842 –1.24925 –1.02999
Table 3. The march of pressure, density, pressure-density ratio and square of adiabatic sound speed within the ball corre-
sponding to K = 0.96 with X = 0.35.
b
rr 2
4
8π
b
G
p
r
c 2
2
8π
b
Gr
c
2
p
c



2
1d
d
p
c



0 1.08886 3.056697 0.356221 0.243732
0.1 1.059853 3.01504 0.351522 0.243685
0.2 0.976679 2.895476 0.337312 0.243187
0.3 0.849993 2.712458 0.313366 0.24119
0.4 0.694975 2.485169 0.279649 0.236025
0.5 0.528536 2.232796 0.236715 0.225541
0.6 0.366874 1.971201 0.186117 0.207331
0.7 0.223863 1.711536 0.130796 0.179042
0.8 0.110333 1.460396 0.07555 0.138766
0.9 0.034033 1.220761 0.027878 0.085434
1.0 0 0.99311 0 0.019145
Centre red shift is given by
1/2
01zB
 (39)
In view of and Table 3 We observe that pressure, den-
sity, pressure-density ratio and square of adiabatic sound
speed decrease monotonically with the increase of radial
coordinate.
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; ρb = 2 × 1014 g/cm3.
The resulting well behaved model has the mass M = 3.43
MΘ and the linear dimension, 2 rb 32.66 km .The sur-
face red shift Zb 0.5509.
6. Discussion
In view of and Table 1, it has been observed that all the
physical parameters (p,
, 2
p
c
, d
d
p
and z) are posi-
tive at the centre and within the limit of realistic equation
of state. From Table 2, the first derivative of all the pa-
rameters are negative at the centre. Thus by virtue of the
M. J. PANT ET Al.
Copyright © 2011 SciRes. JMP
487
theorem all the parameters are monotonically decreasing
in nature for (0.96 5.2K ). In view of (29a), the
electric intensity is minimum at the centre and mono-
tonically increasing. for all values of K > 0. Therefore,
the solution is well behaved for (0.96 5.2K), how-
ever, corresponding to any value of K satisfying the ine-
qualities 00.96K the nature of adiabatic sound
speed is non decreasing in nature.
Corresponding to any value of K > 7 there exist no
value of X for which centre pressure is positive. How-
ever, for (7 > K > 5.2), the length of interval for X con-
verges to zero and the value of X is also approaching to
zero as K approaches to 7.
It has been observed that for higher values of K ap-
proaching to 7, though the solution is well behaved but
the mass of the star is very less than the Chandrasekhar
limit.
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