A Class of Conformally Flat Solutions For Systems Undergoing Radiative Gravitational Collapse
A Class of Conformally Flat Solutions For Systems Undergoing Radiative Gravitational Collapse
A Class of Conformally Flat Solutions For Systems Undergoing Radiative Gravitational Collapse
Sharma et al
systems. Many studies aimed at examining various aspects of gravitationally collapsing stellar systems of different kinds of matter distributions have been carried
out on varying types of space-time background. The space-time in the exterior region of a radiating fluid sphere is known to be described by Vaidya[4] metric. The
formulation of the boundary conditions smoothly joining the interior space-timemetric of the collapsing matter across its boundary with the appropriate form of
Vaidya metric of the exterior space-time, as first proposed by Santos[5] has given
a tremendous impetuous for studies in this direction.
In the present work, we have proposed a relativistic model of a spherically
symmetric matter source, whose collapse is accompanied with dissipation in the
form of radial heat flux following Santos[5]. The background space-time metric is
chosen so as to have vanishing Weyl tensor implying its conformal flatness. Conformally flat space-times, in the context of radiating fluid spheres, were first studied
by Som and Santos[7]. Later, the most general class of conformally flat solutions
for a shear-free radiating star were obtained and examined by Maiti[8], Modak[9],
Banerjee et al[10], Patel and Tikekar[11], Sch
afer and Goenner[12] and Ivanov[13].
Herrera et al[14] have critically examined models of shear-free collapsing fluids
accompanied with dissipation of heat on the space-time background subject to the
constraint that the associated Weyl tensor should vanish. In the present approach,
the geometry of the background space-time of the non-adiabatically collapsing
matter is chosen to be conformal to that of the space-time of Robertson-Walker
metric so that the Weyl tensor is known to vanish since the Robertson-Walker
space-time is already conformally flat.
The paper has been organized as follows. In Sec. 2, the Einstein field equations
for non-adiabatically collapsing spherical matter distributions with radial heat
flow have been formulated on the conformally flat space-time background with
pre-assigned metric form. The pressure isotropy relation in this approach is found
to lead to two classes of solutions for the system. The junction conditions which
smoothly join the interior space-time with the exterior Vaidya[4] metric across
the boundary surface have been obtained in Sec. 3. In Sec. 4, the explicit
expressions for the physical quantities and the constraints on the model-parameters
in view of physical requirements have been obtained. In Sec. 5, specific models
have been examined by considering a couple of solutions of the equation governing
the evolution of the collapse and their physical viability is shown using graphical
methods. Sec. 6 contains a discussion accompanied with some concluding remarks.
1
dr 2
= 2
dt2
r 2 (d2 + sin2 d2 ) .
A (r, t)
1 kr 2
(1)
The evolution with time of the configuration is governed by the function A(r, t)
determined by relativistic field equations. The Weyl tensor for the space-time of
metric (1) will vanish indicating its conformal flatness since the space-time is also
conformal to the Robertson-Walker metric which follows on setting A(r, t) = a(t).
The matter content of the collapsing object in the presence of heat flux is
described by the energy momentum tensor
Tij = ( + p)ui uj pij + qi uj + q j ui ,
(2)
(3)
(4)
(5)
(6)
where an overhead prime ( ) and an overhead dot (.) denote differentiations with
respect to r and t, respectively.
Eqs. (4) and (5) lead to
1
A
A
= 0.
A
rA (1 kr 2 )
(7)
Eq. (7) usually referred to as pressure isotropy condition is easily integrable and
admits two solutions:
Case I: k = 0:
A(r, t) = (t)r 2 + (t).
(8)
Case II: k 6= 0:
A(r, t) = (t)
p
1 kr 2 + (t).
(9)
On setting = 0 one obtains the FRW model of the universe while on setting =
a constant, one obtains a perfect fluid space-time solution without any dissipation
or heat flow. Accordingly, solutions obtained on setting time dependence of the
functions (t) and (t) appropriately will lead to models of collapsing fluid systems
in the presence of heat flux.
Note that, in the context of shear-free radiating collapse, the conformally flat
space-time metric obtained earlier by Herrera et al[14] has the form (in coordinates
(t, r, , ))
ds2 =
2
1
[(
t)
r 2 + 1]2 2
d
r + r2 (d2 + sin2 d2 .
dt
2
2
2
2
t)
t)
[(
r + (t)]
[(
r + (t)]
(10)
Obviously, the solution (8) is a sub-class of the Herrera et al[14] model which can
be obtained by setting (
t) = 0 in Eq. (10). Further, if we set
= k and make the
following coordinate transformations
r =
(1 +
r
,
1 kr 2 )
t
t = ,
2
Sharma et al
the solution (9) can be obtained from (10) by identifying the parameters = +/k
and = /k.
Hence, the solution (9) also turns out to be a sub-case
= a
constant of Eq. (10). Since all conformally flat space-times are conformally related,
this is expected.
It should be stressed here that even though the classes of solutions obtained in
this paper are special cases of the conformally flat solution of [14], the solutions
have been obtained by adopting a different approach. The conformally flat solution
of [14] has been obtained by equating the Weyl tensor of the associated space-time
to zero while the solutions reported in this paper have been obtained by imposing the pressure isotropy condition in a space-time with geometry conformal to
a conformally flat Robertson-Walker space-time. The space-time geometry of the
non-adiabatically collapsing fluid sphere continues to be conformal to the homogeneous geometry of the RW space-time as the collapse proceeds. In addition, the
t) = (k/4)
two classes of solutions have the following distinctive features. If (
(t),
(
t) = (k/4), on choosing a new time coordinate dT = dt/
(t), one finds that the
space-time of (10) corresponds to that of a FRW fluid sphere without heat flux.
In our model, the space-time degenerates to that of a FRW fluid sphere without
heat flux when (t) = 0. It is noteworthy that our choice of curvature coordinates
brings out to attention the role of the curvature parameter k of RW space-time
explicitly in the evolution of the collapse.
3 Junction conditions
The space-time exterior to the collapsing matter source will be filled with radiation
and is appropriately described by the Vaidya[4] metric for a radiating star
ds2+ =
2m(v)
1
dv 2 + 2dvdr r2 [d2 + sin2 d2 ].
r
(11)
The interior space-time (1) should smoothly be joined with that of the Vaidya
metric (11) across the boundary surface separating the interior (r r ) and the
exterior (r r ) space times. Following Santos[5], we shall stipulate the boundary
conditions which ensure continuity of (i) the metrics of the interior and exterior
space times with that on , and (ii) the extrinsic and intrinsic curvatures of .
Let gij be the intrinsic metric to such that
ds2 = gij d i d j ,
i = 1, 2, 3
(12)
and g
be the metric corresponding to the exterior (+)/interior () regions so
that
= 0, 1, 2, 3.
(13)
d
ds2 = g
d ,
(14)
+
) ,
) = (Kij
(Kij
(15)
where,
Kij
n
i j
i j
(16)
In (16), n
denote the components of the normal vector to in coordinates.
We express the intrinsic metric on as
(17)
Using the condition (14) for the intrinsic metric (17) of the interior space-time (1),
we obtain
dt
= A(r , t),
d
r = R( )A(r , t).
(18)
(19)
(20)
is obtained as
n
= {0,
1
, 0, 0}.
A 1 kr 2
(21)
The intrinsic curvatures corresponding to the interior region have the explicit
expressions
K = A (1 kr 2 )1/2
K
=
r2 A
r
A
A2
(22)
(1 kr 2 )1/2
(23)
Using the condition (14) for the metric (17) of the exterior space-time (11), we
obtain
r (v) = R( ),
dv
d
2
dr
2m(v)
= 2
.
+1
dv
r
(24)
(25)
(26)
dr
, 1, 0, 0 .
dv
(27)
dr dv
,
, 0, 0 .
d d
(28)
f
=
+
The unit normal to is
n+
=
Sharma et al
The expressions for the extrinsic curvatures corresponding to the exterior region
are found to be
K+
+
K
"
d2 v
d 2
dv
d
1
m dv
2
r d
dr
2m
dv
= r
1
+r
d
d
r
(29)
(30)
where Eq. (25) and its derivative were used to derive Eq. (29). Using Eqs. (18),
(19), (24) and (25) in Eqs. (23) and (30) and imposing the condition
+
(K
) = (K
) ,
(31)
we obtain
m(v) =
"
"
r
A
kr 2 + 2r(1 kr 2 )
+ r2
2A
A
2
2 ##
A
A
2
2
r (1 kr )
. (32)
A
A
It follows from Eq. (32), that the mass within the sphere of radius r at any
instant t can be expressed as
"
A
r
kr 2 + 2r(1 kr 2 )
m(r, t) =
+ r2
2A
A
2
2 #
A
A
2
2
r (1 kr )
.
A
A
(33)
Using Eqs. (5), (6), (18), (19), (22), (23), (24), (29), (30) and (32) and imposing
the condition
(34)
(K+ ) = (K ) ,
we finally obtain
p=
q
A 1 kr 2
(35)
(36)
q = 4r(r 2 + )2 .
2 2
(37)
(38)
The total mass of fluid contained within the spherical region of radius r at
any instant t has the explicit expression
r 3 [4 + (r 2 + )
2]
.
2
3
2(r + )
(39)
= u
; = 3A = 3(r + ).
(40)
m(r, t) =
The parameter of expansion is
2 2
2
= 0.
+ ) + 12r
8(r
(41)
This relation governs evolution of the physical quantities. Eq. (41) is a highly nonlinear equation relating two unknown arbitrary functions ((t), (t)). A suitable
choice of one of the two functions is essential to determine the other. The functions
are further constrained by the requirements of (weak) energy conditions (, p, q >
0) and other physical requirements. For a collapsing sphere should be negative
> 0. Positive heat flux is ensured if < 0. From (36), we
implying (r 2 + )
2 + ).
conclude that d/dr = 12r (r
Since, (r 2 + )
> 0 and < 0, it follows
that d/dr < 0. The implications of other physical requirements can be examined
only by adopting numerical procedures.
i
p
1 kr 2 + 2 ,
p
p
p = [3k2 r 2 2 k( 1 kr 2 + )2 + 4k((1 kr 2 ) + 1 kr 2 )
p
p
p
2 + 2( 1 kr 2 + )( 1 kr 2 + )],
3( 1 kr 2 + )
p
p
q = 2kr 1 kr 2 ( 1 kr 2 + )2 .
= 3 k 2 + k 2 + (1 kr 2 )2 + 2
(42)
(43)
(44)
The presence of dissipation in this case also is observed to give rise to inhomogeneities in matter density distribution of the collapsing fluid.
The total mass of the fluid enclosed within the spherical region of radius r of
the configuration of this class at any instant t is
p
r3
k( 2 + 2 ) + (1 kr 2 )2 + 2 1 kr 2 + 2 .
2( 1 kr 2 + )3
(45)
The expansion parameter is
m(r, t) =
= 3(
1 kr 2 + ).
(46)
Sharma et al
q
q
q
2 +
2 + )
2 + )(
) 2kr ( 1 kr
1 kr
1 kr
q
q
2 2 2
2 + )
2 + )2
k 3( 1 kr
+3r
2 k( 1 kr
q
2
2 ] = 0,
) + 1 kr
+4k[(1 kr
(47)
which is the relation governing evolution of the various physical quantities. Again,
one needs to make a suitable choice of one of the two arbitrary functions to determine the other. The functions are further required to comply with the requirements
of energy conditions (, p,
q > 0) and physical plausibility requirements. In this
case, will be negative if 1 kr 2 + > 0. Positivity of heat flux is ensured
if > 0 for k > 0 and < 0 for k < 0. The implications of other physical
requirements can be examined using numerical procedures.
5 Particular models
In view of the complexity of the relation governing the evolution with time of the
collapse, we have used two approaches for examining the physical plausibility of
the evolution.
5.1 Approach: I
The evolution of the collapsing systems is governed by Eqs. (41) and (47). In order
to analyze physical viability of the model, it is essential to solve these equations.
Note that the metric (1) is conformal to the Roberson-Walker metric for which
we have k = 0, 1. For k = +1, we have 0 < r 1. However, there is no such
restriction on r (0, ) for k = 0 and k = 1.
We consider a particular case k = +1 for which the maximum value of the
coordinate parameter is r = 1. Substituting these values in Eq. (47), we obtain
2 2 3 2 + 3 2 2 = 0.
(48)
We need to specify one of the two unknowns to determine the other. Accordingly,
we assume
n2
= ,
(49)
t
where, n is a constant. Solution of Eq. (48) is then obtained as
n2 e
=
( 3 + 3t) 3Q( 3t 1)
,
3t2 ( 3e 3t + 3Q)
3t
(50)
5.2 Approach: II
In this approach, we adopt an appropriate approximation procedure to examine
the collapsing configurations with k 6= 0. We set r = 1 in equation Eq. (47)
rewriting it as
2(
p
p
p
(1 k) + )( (1 k) + ) 3( (1 k) + )
2
p
p
2k( (1 k) + ) + 4k((1 k) + (1 k))
p
k( (1 k) + )2 + 3k2 2 = 0.
(51)
Since a closed form solution of Eq. (51) is not available, we generate an approximate solution by assuming
(t) = P ent ,
(t) = h(t),
(52)
p
(1 k)(k + n2 )P
p
(t) =
2[k2 + n2 + k(1 + n( (1 k) n)]
"
#
(k+3n
+ Ce
where
(k+3n
(1k)L)t
(1k)
+ De
(1k)+L)t
(1k)
(53)
q
p
L = k(4 + 5k) + 6 (1 k)kn 5(1 + k)n2 ,
6 Discussions
In this work, we have described relativistic conformally flat solutions for spherically
symmetric fluid configurations, undergoing dissipative collapse. It is not difficult
to find a similarity between these solutions and the solutions obtained earlier by
Herrera et al[14] which follow by adopting a different procedure. In the Herrera
10
Sharma et al
References
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2. P. S. Joshi, Global Aspects in Gravitation and Cosmology, Clarendon Press, Oxford (1993).
3. K. P. Thorne, Magic Without Magic: John Archibald Wheeler, edited by J. Klauder,
Frimann, San Francisco (1972).
4. P. C. Vaidya, Proc. Indian Acad. Sci. A33, 264 (1951).
5. N. O. Santos, Mon. Not. R. Astron. Soc. 216, 403 (1985).
6. P. S. Joshi and D. Malafarina, Int. J. Mod. Phys. D 20, 2641 (2011).
7. M. M. Som and N. O. Santos,Phys. Lett. A 87, 89 (1981).
8. S. R. Maiti, Phys. Rev. D 25, 2518 (1982).
9. B. Modak, J. Astrophys. Astron. 5, 317 (1984). (1984).
10. A. Banerjee, S. Choudhury and B. Bhui, Phy. Rev. D 40, 670 (1989).
11. L. K. Patel and R. Tikekar, Mathematics Today IX, 19 (1991).
12. D. Sch
afer and H. F. Goenner, Gen. Relativ. Grav. 42, 2119 (2000).
13. B. V. Ivanov, Gen. Relativ. Grav. 44, 1835 (2012).
14. L. Herrera, G. Le Denmat, N. O. Santos and A. Wang, Int. J. Mod. Phys. D 13, 583
(2004).
15. A. Banerjee, S. Chatterjee and N. Dadhich, Mod. Phys. Lett. A 17, 2335 (2002).
11
1.0
rS
0.5
0.0
150
100
HrS ,t
50
0
0
-50
t
-100
rS
0.5
0.0
20
15
q
10
0
0
-50
t
-100
0.0
0.08
0.06
0.04
mHrS ,t
0.02
0.00
-20
-40
-60
-80
-100
12
Sharma et al
1.0
rS
0.5
0.0
0.0
-0.5
-1.0
-1.5
-2.0
-20
-40
-60
-80
-100
8
r rS = 1
P = -0.332575
C = 100
D = 100
= 0.001
n = 0.824118
k = .2
0
0.0
0.2
0.4
0.6
t
0.8
1.0
13
0.35
0.30
r rS = 1
P = -0.332575
C = 100
D = 100
= 0.001
n = 0.824118
k = .2
0.25
0.20
0.15
0.10
0.05
0.00
0.0
0.2
0.4
0.6
0.8
1.0
0.8
1.0
6
mHrS ,tL
4
0
0.0
0.2
0.4
0.6
t
14
Sharma et al
14
r rS = 1
P = -0.332575
C = 100
D = 100
= 0.001
n = 0.824118
k = .2
12
rS AHrS ,tL
10
8
6
4
2
0
0.0
0.2
0.4
0.6
t
0.8
1.0