International Journal of Applied Mathematics & Statistical Sciences(IJAMSS) ISSN: 2319-3972 Vol.1, Issue 2 Nov 2012 17-26 © IASET
STATIC SPHERICALLY SYMMETRIC INTERIOR SOLUTIONS WITH NONUNIFORM DENSITY 1
M. A. KAUSER & 2 Q. ISLAM
1
Department of Mathematics, Chittagong University of Engineering & Technology, Chittagong-4349, Bangladesh
2
Research Centre for Mathematical and Physical Sciences, University of Chittagong, Chittagong-4331, Bangladesh
ABSTRACT For the highly nonlinearity conditions, it is so much difficult to obtain exact solution of Einstein’s field equations. Many authors have been working on the investigation of exact solution of Einstein’s equation. One of these solutions, Schwarzschild uniform density solution is unphysical. In this paper, we have applied the Adler method to obtain Einstein’s equations solution in use of different equation of state. We got a new solution although it is Tolman’s solution no IV. But our solution is different and original. We have described the obtain solution in terms of two new variables. Also we have shown that maximum mass of 1/3 the fluid radius (in geometric units) which is less than Adler’s 2/5 or Schwarzschild 4/9.
KEYWORDS: Non-Uniform Density, Exact Solution, Equation of State, Uniform Density INTRODUCTION Uniform density solution [1] is unphysical because velocity of sound becomes infinite if
ρ constant. A small
number of solutions by authors [2, 3, 4, 5] with non-uniform density distribution which have been obtained are valuable and interesting because they provide some insight into the effects of relativity on the interior structures of stars. In this paper, one such solution is studied. In section-2, the solution found by Adler [2] is reviewed. There is a misprint in equation (3.3d) of Adler’s paper which we have corrected in this section. Some interesting properties of Adler’s solution are studied in section 3. In section 4 we have shown our new solution by using the method found by Adler. This is a new solution but no different from Tolman’s [3] solution no IV. Finally some interesting properties of the solution are studied and defined the results in two new variables x and y in section 5. For this solution we have shown that for a finite radius ( r0 ) of the fluid sphere
m0 1 m 4 , where m 0 is the mass of the fluid sphere. The star collapses if 0 exceeds . r0 3 r0 9
Buchdahl [4] has shown that the condition
m0 4 is true in general. r0 9
ADLER’S STATIC SPHERICALLY SYMMETRIC INTERIOR SOLUTION Let us take the static spherically symmetric line element in the following form, ds 2 = -N 2 (r)dt 2 +
dr 2 G 2 (r)
+ r 2 (d θ 2 sinθ 2 d 2 )
(1)
Einstein equation without cosmological constant is
R μν
1 g μν R 8π T μν 2
(2)
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M. A. Kauser & Q. Islam
For a perfect fluid energy momentum tensor T is given by
Tμν (p ρ)uμ u ν pgμν
(3)
For the metric (1), equation (2) gives
8π ρ
1 dG 2 (r G 2 1) 2 dr r
(4)
8π p
1 dN (2rG 2 N NG 2 ) 2 dr Nr
(5)
8π p
1 dN 1 dG 2 d 2 N r dN dG 2 (G 2 N rG 2 2 ) Nr dr 2 dr 2 dr dr dr
(6)
From (5) and (6) we obtain
dG 2 2(N rN - r 2 N )G 2 2N dr r(N r N ) r(N r N ) or,
dT P(r)T Q(r) dr
(7)
where
T(r) G 2 (r) , 2(N rN - r 2 N ) P(r) r(N r N ) and
Q(r)
2N r(N r N )
(8)
(9)
Solution of (7) becomes simple if we put P(r) = Q(r). Now the P(r) and Q(r) are the equal if
rN - r 2 N = 0 or,
N 1 N r
Integrating this we obtain,
log( N ) log r log(2B) or, N 2Br
(10)
Again integrating it we get
N A Br 2
(11)
Static Spherically Symmetric Interior Solutions with Nonuniform Density
19
where A and B are constants. P(r) dr I.F.of (7) e
Therefore, the solution of (7) is P(r) dr P(r)dr Te Q(r) e dr
or, T
Now, P(r)
1 Ce
P(r) dr
dr
(12)
2(A Br 2 ) [putting the value of N from (11)] r(A 3Br 2 )
2 4Br) P(r)dr ( )dr r A 3Br 2
2logr e
P(r)dr
2 log(A 3Br 2 ) 3
r2
2
(A 3Br )
2 3
So from (12) we get
r2
T 1 C
2
(A 3Br 2 ) 3
Cr
i.e, G 2 1
2
(13)
(A 3Br 2 )
2 3
Sometimes it is convenient to replace G(r) by another function m(r) defined by
G 2 (r) 1
2m(r) r
So from (13) we get
1
Cr 2 2 2 3
1
(A 3Br ) or, m(r) 1
Cr
2m(r) r 3
2
2(A 3Br ) If
2 3
r0 is the radius of the fluid sphere and m 0 its mass, then we have
(14)
20
M. A. Kauser & Q. Islam
m0 r0
ε
Cr 0
2
2(A 3Br
(15) 2 0
)
2 3
From (5) we get
8π p
2 2Br 1 G 2 2 (1 G 2 ) 2 r A Br r
On the surface of the sphere p = 0,
r r0 .
This implies
Br 0
2
A Br 0 or,
2
(1 2ε ) 2ε 2
ε A r0 (5ε 2)B 0
(16)
Also we have by matching with the Schwarzschild exterior solution 2
A B r0 1 2ε or,
(17)
2
ε A ε B r0 ε 1 2ε
(18)
Subtracting (16) from (18) we get
ε
B
2r 0
2
(19)
(1 - 2ε )
Putting this value of B in (17) we get A
2 - 5ε
(20)
2 (1 - 2ε )
From (15) we get 2ε C - 2 r0
(1 - ε )
2 3
(1 - 2 ε )
(21) 1 3
Now from (11) N
2 - 5ε 2 (1 - 2 ε )
ε r ( )2 2 (1 - 2ε ) r 0 1
or, N (1 - 5 ε 1 ε y 2 )(1 - 2ε ) 2 2 2
Where
y
r . r0
From (13) we get
(22)
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Static Spherically Symmetric Interior Solutions with Nonuniform Density
2
2ε (1 - ε ) 3 y 2 G2 12 5 1 (1 - ε ε y 2 ) 3 2 2
(23)
Also from (14) we get 2
m 0 (1 - ε ) 3 y 3
m(r)
(1 -
5 1 ε ε y2) 2 2
(24) 2 3
From (5) pressure p(r) is given by 2
p(r)
2
ε 5 1 5 3 [G 2 (1 - ε ε y 2 ) -1 (1 - ε ) 3 (1 - ε ε y 2 ) 3 ] 2 2 2 2 2 4π r0
(25)
There is a misprint in equation (3.3d) of Adler’s paper which has been correct in equation (25). Similarly from (4), density distribution is given by 2
ε ρ(r) { 4π r 02
(1 - ε ) 3 2
5 3 (1 - ε ε y 2 ) 3 2 2
}{3 -
2ε y 2 } 5 3 1- ε ε y2 2 2
(26)
PROPERTIES OF THE SOLUTION Central density
ρc is obtained by putting y = 0 in (26). We get 2
ρc
3ε 4 π r 02
1- ε 1- 5 ε 2
3
We see from (27) that the central density
function of
(27)
ρc is a function of ε
ρ0 . m0 is maximum if
dm0 0 dρ c m0 ) r0 0 [Since by assumption dρ c
d(
dε 0 dρ c
r0 = constant]
m0 . Clearly for a fixed r0 , the mass m0 is a r0
22
M. A. Kauser & Q. Islam
dρ c dε
(28)
From (28) we get
ε
2 5
m0
Thus for a fixed
m0
r0
2 r0 5 maximum is given by m 0
2 r0 . This may be compared with the maximum mass 5
1 r0 obtained in the Schwarzschild interior solution by demanding that the Schwarzschild radius not be exterior to 2
be fluid sphere. It should be noted that
2 1 2 r0 < r0 . In the solution found by Adler we see that m 0 cannot exceed r0 5 2 5
whereas in the Schwarzschild interior solution
m0
is bounded from above by m 0
1 r0 . 2
p c is obtained by putting y = 0 in (25). We get
Central pressure
2
pc
Now
pc
ε 4 π r 02
1- ε 1 [ 5 5 1 - ε 1 - ε 2 2
becomes infinite if ε
Hence m 0
3 ]
(29)
2 2 m 0 r0 5 5
2 r0 5
Schwarzschild interior solution has the similar property that
pc
becomes infinite when ε
4 4 which implies m 0 < r0 . 9 9
A NEW SOLUTION [TOLMAN’S SOLUTION NO. IV] In this section we have shown that a new solution derived by using the Adler’s method. It is nothing but Tolman’s solution no IV.
In (7) let us put
3 Q P(r) r Q
. Then we get
2N 2rN - 2r 2 N 3 N N 3rN r 2 N r(N r N ) r N r(N r N ) 2
2
2
or, r N r NN - rNN 0 2
Put α N in (30), then
α 2NN , α 2N 2 2NN
(30)
Static Spherically Symmetric Interior Solutions with Nonuniform Density
23
Thus (30) becomes
r 2 α - rα 0 or,
α 1 α r
α ar 2 b , where a and b are integrating constants. N 2 ar 2 b
(31)
N ar 2 b Now integrating factor of (7) is given by I.F =
P dr e 3
=
e ln(r Q)
=
1 rQ 3
Therefore, the solution of (7) is
T
1 1 Q 3 dr rQ rQ 3
-3
= r dr
T
=
=
1 d , where d is a constant of integration. 2r 2
=
2dr 2 1 2r 2
2dr 2 1 3 r Q 2r 2
Qr(2dr 2 1) 2
From (9) we get
Q(r)
2 ar 2 b 2ar 2 r( ar 2 b ) 2 ar 2 b
or, rQ(r) 2(ar (2ar
2 2
b) b)
24
M. A. Kauser & Q. Islam
T (
a 2 r 1 )(1 - cr 2 ) , where –c = 2d b a 2 2 r 1 b (
or,
2
G (r)
1 2 G (r)
For this solution
a 8π ρ b
a 2 r 1)(1 - cr 2 ) b a 2 r2 1 b
2 (
(32)
a 2 r 1 b
a 2 r 1)(1 - cr 2 ) b
ρ and p are given by
3bc 3cr a a 1 2 r2 b
1
a 8π p b
2(ar2 b) 2dr2 1 2 2ar2 b
2
2a b
1 - cr 2 a (1 2 r 2 ) 2 b
bc 3cr 2 a a 1 2 r2 b
(33)
1
(34)
CONCLUDING PROPERTIES OF THE SOLUTION Central density
ρc and central pressure p c ρc
can be obtained by putting r = 0 in (33) and (34) respectively. We obtain
1 a 3bc 2a { (1 ) } 8π b a b
or, ρ c
3 a ( c) 8π b
(35)
Similarly
pc Let
1 a ( c) 8π b
(36)
r0 be the radius of the fluid sphere. Then p = 0 when r = r0 . This implies 2
r0
1 bc (1 ) 3c a
From (35) and (36) we get
(37)
25
Static Spherically Symmetric Interior Solutions with Nonuniform Density
a 4π (ρ c 3p c ) b 3
(38)
4π (ρ c 3p c ) 3
(39)
c
Putting the values of
a and c in (37) we get b
3
2
r0
r0 depends only on
From (40) we see that
Let
m0
r0
0
ρc ρ . It is also clear that for a finite radius we must have c > 3. pc pc
be the mass of the fluid sphere. Matching with the Schwarzschild solution, then gives
2m 1r0
or, m
(40)
ρ ρ 2 π( c 3)( c 3) pc pc
1 [1 2
( 0
(
a 2 2 r0 1)(1 - cr 0 ) b a 2 2 r0 1 b
a 2 2 r0 1)(1 - cr 0 ) b ] a 2 2 r0 1 b
Putting the values of
(41)
a 2 , c and r0 from (38), (39) and (40) in (41) we get b (42)
mo 2 ρc ro 3 pc
From (42) we see that
mo ρc mo 1 mo 1 if ρc pc , if depends only on . Clearly ro pc ro 2 ro 2
m 1 mo 2 if ρc pc . Let us put y o and x 1 2 ro 3 ro ρc pc y = 2x The graph of (43) is a straight line passing through the origin.
ρc pc
and
in (42). Then we get the equation
3 (43)
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M. A. Kauser & Q. Islam
If we assume that
This implies x
0 pc
ρc ρ 1 is positive then the range of x is given by 0 x . But from (40) we have c < 3. pc 3 pc
1 1 1 . Hence the range of x is further limited to 0 x . In 0 x for a fixed ρc we have 6 6 6
ρc . 3
Thus we have the following results:
pc
m ρc 1 and o . 3 ro 3
ACKNOWLEDGEMENTS We would like to thank Prof. Emeritus Jamal Nazrul Islam, Centre for Mathematical and Physical Sciences, University of Chittagong, for his encouragement, support and innumerable valuable interactions throughout this work.
REFERENCES 1.
Schwarzschild, M., et al, Astrophys. J. Vol. 129, Page-637(1959)
2.
Adler, R. J., J. Math. Phys., Vol.15, No.6, Page-727(1974)
3.
Tolman, R. C., Phys. Rev., Vol.55, Page-364(1939)
4.
Buchdahl, H. A., Phys. Rev., Vol. 116, Page-1027(1949)
5.
Islam, Q., The Chittagong Univ. J. Sc. Vol. 22(2) Page-91(1998)
6.
Kauser, M. A., M. Phil. Thesis, University of Chittagong, Bangladesh (2010).