DYNAMICAL INSTABILITY OF COLLAPSING RADIATING FLUID
M. Sharif* M. Azama'b**
" Department of Mathematics, University of the Punjab Lahore-54590, Pakistan
b Department of Mathematics, University of Education Lahore-54590, Pakistan
Received December 3, 2012
We take the collapsing radiative fluid to investigate the dynamical instability with cylindrical symmetry. We match the interior and exterior cylindrical geometries. Dynamical instability is explored at radiative and non-radiative perturbations. We conclude that the dynamical instability of the collapsing cylinder depends on the critical value T < 1 for both radiative and nonradiative perturbations.
DOI: 10.7868/S0044451013060050
1. INTRODUCTION
The subject of dynamical instability of self-gravitating objects has attracted many astrophysicists duo to explosions and evolution of these objects. The evolution of different self-gravitating objects during gravitational collapse for different ranges of instability is an important feature of dynamical instability. The static stellar model would be interesting if it remains stable under fluctuations. In this scenario, Chandrasekhar fl] found the instability range T < 4/3 for the spherically symmetric spacetime with isotropic fluid. After that, many people investigated the effects of physical properties of the fluid in the onset of dynamical instability with spherical symmetry.
In [2], it was found that dissipation at Newtonian (N) limit reduces the stability of the sphere and boosts at post-Newtonian (pN) limit. The conclusion in [3] was that the effects of radiation appears similar to dissipation in the N and pN limits. The same authors [4] examined the instability of a spherically symmetric spacetime with shear viscosity and found that it decreases the instability of the fluid. The dynamical instability of a collapsing radiating star would be increased due to the presence of anisotropic pressure and shear viscosity [5].
Cylindrically symmetric spacetimes are idealized models in general relativity. The study of gravitational
E-mail: msharif.math'fflpu.edu.pk
E-mail: azammath'fflgmail.com
collapse of these astrophysical objects is an important problem. Some recent work [6 10] indicate great interest in cylindrical gravitational collapse with different fluids with and without an electromagnetic field. Recently, Sharif and his collaborators [11] investigated the dynamical instability for spherically and cylindrically symmetric spacetimes in general relativity and f(R) gravity in the N and pN regimes, respectively. They have shown that the electromagnetic field, pressure anisotropy, dissipation, and f(R) models have great relevance in the range of instability. The same authors [12] have also explored this problem for the thin-shell wormholes in nonlinear electrodynamics.
In this paper, we explore the dynamical instability of a collapsing radiating cylinder in the N and pN approximations. The paper is organized as follows. In Sec. 2, the field equations and matching conditions are developed. In Sec. 3, we formulate the dynamical instability at nonradiative and radiative perturbations. Finally, we discuss our results in Sec. 4.
2. FIELD EQUATIONS AND MATCHING CONDITIONS
The matter under consideration is assumed to be locally isotropic with pure radiation inside a cylindrical surface E. The energy momentum tensor for such a fluid has the form
TapI = (/'■ + />)'<« tr.~; + PfJaH + ^Jfi, ( 1 )
and the following conditions arc satisfied in conioving coordinates:
= i ',);;. r = .i 'à',1-/.? 'a;1.
wawa = —1, lala = 0.
Hero //., p, wa, e, and Ia arc the energy density, the isotropic pressure, the four-velocity, the radiation density, and the null four-vector of the fluid. We consider the nonstatic cylindrically Symmetrie spacetime inside the hypersurface S with
ds2 = —W2(t,r) dt2 + X2(t,r) dr2 +
+ Y2(t,r)d02 +dz2, (2)
where
^oc <t<oo, 0 ^ r < oc,
—oo < ; < 00, 0 ^ <i> ^ 2TT.
Using Eqs. (1) and (2), we can write the Einstein field equations
f< 11\.-; = Gaß
as
KA2(/I. ■
kWXS =
X'Y' _ r XY ~ T YW XY'
XY XY
kX2(P -
Kp =
Kp =
W
wx
X
ir:i.Y W"2X .V:iU"
ir.v wx' irr
(3)
(4)
(5)
(6)
WX2 I P.Y ir:i.Y U".Y:i \Y*Y im X'Y' Y" W'Y' XY
Xray
X2Y WX2Y I \ "-' X ) "
(7)
The IIlclSS function proposed by Thome [13] in the form of gravitational C-energy per unit specific length is defined as
1
m(t,r) =
o Of "
and satisfies the relations
P2 = V(0)aV%> l2 = V(
(S)
f = pi,
where /, p, f, i/o, and are the specific length, circumference radius, areal radius, and two Killing vectors for the cylindrical geometry. Hence, the specific energy of the collapsing cylinder becomes [14]
fn(r,t) =
0)
For the radiative fluid, the conservation of the stress energy tensor,
(T^y = 0,
yields
fi-
el x
(10)
P
. X o ^ IV' Y'
£W + ('l+P + 2e) ly + £T
+ + = (11)
We use the Darmois conditions [15] for the continuity of inner and outer manifolds. The cylindrical manifold in the exterior region is [16]
ds2 = — ( dv2
2 dv (IR + + R2((W2
2 7 '
7 at
(12)
where v is known as the retarded time coordinate and 7 has the dimension of 1/r. From the Darmois conditions. as discussed in fill, it follows that
v 1 v
m — M = —, p= 0.
(13)
This shows that the difference of interior and exterior masses is equal to 1/8 at the boundary surface and the isotropic pressure vanishes.
3. THE PERTURBATION SCHEME AND DYNAMICAL INSTABILITY
We perturb the field equations, conservation equations, and physical functions of the fluid (initially in hydrostatic equilibrium) up to the first order in 0 < A -C 1. We use the perturbation scheme [11]
ir(/.r) = IUo(r) + A T(t)tv(r), (14)
X(t, r) = X0(r) + A T(t)x(r), (15)
Y(t,r) = rX(t,r)[l + XT(t)y(r)}, (16)
fi(t, r) = (i.0(r) + \p(t,r), (17)
p(t,r) = Po('i') + Ap(t,r), (18)
e(t,r) = A e(t,r), (19)
mil. r) = m0(r) + \m(t,r). (20)
3 >K9T<£>, libiii. 6
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The static (unperturbed) and perturbed forms of Eqs. (3) (5) are given by
1
An
1 X0
Kp0 =
1 H7, (x'0
r A0
r
Xr
XQ W o V .V„ r
(21) (22)
T
TT-v M#
mi
i
Xq
' T
LQ V
A0
(rx'Y
An V A
A0
2
r
T /3.1:
ry
/i'W qXQS =
W'oA'o
V
^ + (23)
y
Wo An
V
Wo
T, (24)
x
T0 + s
T
W
0
T A
o
w
I \"') I X
y
v W"o / W"o \ A'0 Similarly, Eqs. (10) and (11) turn out to be Ho _ p'o
1 + 1o r Ao
Tv
2k— po. (25) Ao
Wo (fio+Po)
(26)
2W1 Wn
An (/'o -
1
H -
r
" Po)
Mo An
Ar
§ T=0, (27)
/1' + e' + e^ + (//o + po )T (
111) \ H'o
l + ^.)+(p+p+ 2e)p = 0. (28) r A0/ H'o
Inserting Eq. (24) in (27) and integrating, we obtain ' 2.i:
/'■ =
Ar
1
kA p
iJ ) (tio+Po)T
Ao Wo
± + (29)
where $(•/•) =
WqA'O
Wo 1
y
Wo
• (30)
vr Xo,
The unperturbed form of the IIlclSS function becomes
m0 =
1
1
1
r Xc
which can be rewritten as
Xn 1 ,-
-71 =---h \/l - 8m0.
A0 r
(31)
(32)
Now we can write the differential equation by considering Eqs. (24) and (25) and using the condition
Po = 0
as
where
Wo Ao
4<T - 2<t>T = T,
Ar
(33)
1 + 1o
r Ao
w
Wn
Wo VA
(34)
<P(r) =
1 U
o / •(•
2 An \X
^o
y
Wo
W'oA'n
*o An
y
Wn
The solution of the above equation yields T(t) = — oxp
-<t>i
0S + t
(35)
(36)
where we choose
4's >0, <i>s < 0
for the solution to be real. According to the above equation, the cylinder proceeds to collapse at t = —oc and continues with the increase in t.
To find dynamical instability, we introduce the adi-abatic index T defined in [21:
p = T
Po
fio + Po
(37)
where quantities with a bar represent the perturbed energy density and isotropic pressure. Substituting Eq. (29) in the above equation, we obtain
P = -T/)0 1
2x_
To
y T
kXo
1
T
■ (38)
.Wo Wo rj J //„ - />„ Equation (25) with the matching condition leads to
3.1. Nonradiative perturbation
For a nonradiative perturbation, we assume that = 0. Then the integration of Eq. (24) leads to
Po
VI - Sm0 ) • (41)
.(;=W"oA'o, y = ~y
J'o+Po
Using this fact, we see from Eqs. (30) and (35) that
$(»•) = 0, d>(r) = 0.
Substituting these results in Eq. (40), we obtain the instability equation as
w
w>
x
xl ir,?
I + ^o
r A0
X
T W'^
xr!!) T-wAxr!!
W n A'o 1 111
77^ + 77^ + - x Wo Wo r
x T
Po
/to + Po
Ao , T
Wo T
'I'- + Kpo*XQ \ . (39)
2-^- + y ] Tpo
J1 Xo
+ U<o+Po) - + 1T1 \r A0
Xn ( X \ T
T
kpqxXQ + + y ) ^
k. 2
/ ] tpoX'c
■ yo 1 y J
YL
Wo VA
W o I x
o
+ (//o + Po + Tpo)|| = o. (42)
Next, we develop the main equation used for the instability range at nonradiative and radiative perturbations. It follows by inserting Eqs. (24), (29), (38), and (39) in (28) that
3.1.1. Newtonian approximation
To find the instability range in the N approximation, we use that
Wo = 1, Ao = 1,
-p0T + y
X ( +
Po
WqA'O
«A2 V
Wo 1 Wo
<î> 'f ™ +
An ___ T Wo T ^ k X
i a;
l
T
X ~ + 77^r^+U>o+Po)
r XoJ WoAo T
j Xq ( x \ f W'^
x wf UHr^wi
1
K
1 A
x
o \A'o
r A0 y
-I/-9
— + y I KT/jqAq
W0 W0 r
Po
/to + Po
Ao T
—'I>—-K/io-'-Ao
WÔ Wo
1
h-Xr
TW-
2.C
— +•'7 ) (t'o+Po+fpo) X Ao
Ei + Zil + I
Wo Wo r
Po
Wr
0 \ /10 + Po
X
1 ) = 0. (40)
and ignore terms like po//io that are of the order of nio/r in Eq. (42), which gives
-2p'a T + 2 p'0 + V-S//0 = 0,
(43)
assuming p'0 < 0 for the collapsing fluid. Consequently, the instability condition turns out to be
T < 1.
(44)
This equation shows that the instability of the collapsing fluid depends on the critical value 1.
3.1.2. Post-Newtonian approximation
In the pN approximation, we assume that mo ,, , , m0
Wo = 1
Ao = 1
and take the terms of the order oî m0/r. Consequently, Eq. (42) becomes
(2 + y)p'0 T + 2 p'0 + TpQ + 0S(1 + y)
l<o
k( 2 + y)Y/i0p0 + K/i0p0 = 0, (45)
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3*
and the instability range in this limit turns out to be
T < 1 + ^-
bol
^KVoPo - ^i1 + VH'Z
(46)
In the above inequality, the third term, which conies from the static background of energy density, enhances the instability and is decreased by the last term.
3.2. Radiative perturbation
In the radiative case, w
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