#### Dynamics of charged bulk viscous collapsing cylindrical source with heat flux

Eur. Phys. J. C
Dynamics of charged bulk viscous collapsing cylindrical source with heat flux
S. M. Shah 0
G. Abbas 0
0 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur 63100 , Pakistan
In this paper, we have explored the effects of dissipation on the dynamics of charged bulk viscous collapsing cylindrical source which allows the out-flow of heat flux in the form of radiations. The Misner-Sharp formalism has been implemented to drive the dynamical equation in terms of proper time and radial derivatives. We have investigated the effects of charge and bulk viscosity on the dynamics of collapsing cylinder. To determine the effects of radial heat flux, we have formulated the heat transport equations in the context of Müller-Israel-Stewart theory by assuming that thermodynamics viscous/heat coupling coefficients can be neglected within some approximations. In our discussion, we have introduced the viscosity by the standard (non-causal) thermodynamics approach. The dynamical equations have been coupled with the heat transport equation; the consequences of the resulting coupled heat equation have been analyzed in detail.
1 Introduction
The stars composed of some nuclear matter which is
continuously gravitating and is attracted toward its center due to the
gravitational interaction of its particles. This phenomenon in
the theory of general reactivity is known as gravitational
collapse. The description of this phenomenon is the main
objective of the relativistic theories of gravity (including general
relativity) [
1–3
]. Oppenheimer and Snyder [
4
] theoretically
illustrated the process of collapse in 1939, they addressed the
contraction of a highly idealized spherically symmetric dust
cloud. They used the exterior and interior spacetimes of the
Schwarzschild metric and a Friedman like solution,
respectively. An enormous amount of contributions in the research
of gravitational collapse have been added by Vaidya [
5
], who
provided the exterior gravitational field of a stellar body
sending out radiations. Misner and Sharp [
6,7
] studied a perfect
fluid spherically symmetric collapse and also some authors
[
8–20
] considered it in different situations.
Rossland [
21
] proved that the atoms are converted into
ions with great strength and the law of central force should
be observed by the forces between the free particles. Its order
of magnitude should be greater than that of remaining forces
acting between neutral atoms. The effect of electrical forces
is fairly large if the star is built of heavy elements with 1.5
times the solar mass and a mean molecular weight 2.8 unit.
Eddington [
22
] explored the fact that, in the internal
electrical field of star, the electric potential φ directly relates
the gravitational potential ψ , the mass m and charge e of a
proton, a scalar parameter α affected by the density ni of
the ions, the atomic weight Ai of the ions and the effective
charge e Zi . Mitra [
23
] introduced the fact that the
formation and evolution of stars would happen due to gravitational
collapse, which is a high energy dissipating process and can
be characterized by two respective cases: the free streaming
approximation and the diffusion approximation. In the free
streaming approximation case, Tewari added some models
[
24–26
] by the solution of the Einstein field equation with
a different approach. A number of distinguished researchers
such as Bonner et al. [
27
], Bowers and Liang [
28
], de Oliveria
[
29
], Mahraj and Govender [
30
], Ivanov [
31
] and Phinheiro
and Chan [32] discussed many realistic models in the
diffusion approximation with anisotropy, inhomogeneity,
viscosity, and an electromagnetic field, and one also addressed the
different dissipative processes analytically.
Since then, a huge amount of literature [
8–20
] on
gravitational collapse considered the spherical symmetry of the
star, which is the simplest geometry. In order to determine
a realistic model of gravitational collapse, it would be
interesting to study the dynamics of a collapsing star with a
nonspherical background. It would be implied by the existence of
gravitational waves that cylindrical and plane symmetries are
more important for a non-spherical background. The
cylindrical sources may serve as a test bed for numerical
relativity, quantum gravity, and for probing the cosmic censorship
and hoop conjecture, among other important issues, and they
represent a natural tool to seek the physics that lies behind
the two independent parameters in the Levi-Civita metric
[34]. Herrera et al. [33] have discussed gravitational collapse
and junction/interface criteria for a gravitating source which
has cylindrical geometry. Sharif and Ahmad [35] have
predicted that gravitational radiations can be emitted during a
gravitational collapse of two perfect fluids. Di Prisco et al.
[36] studied the shear-free conditions and cylindrical
gravitational waves by taking the Einstein–Rosen spacetime in the
exterior of a general non-static cylindrical spacetime. Nakao
and Morisawa [37] have explored the gravitational radiations
from the collapse of a hollow cylinder.
Since Einstein and Rosen [38] initially predicted
cylindrical gravitational waves theoretically, the observational
evidence of gravitational waves through advance detectors such
as LIGO [39] and GEO [40] has motivated researchers to
study the cylindrically symmetric gravitating source. The
formation of a naked singularity during the generic gravitational
collapse would be expected during a cylindrical gravitational
collapse. Several numerical approximations [41] depict the
emission of gravitational cylindrical waves from a cylindrical
gravitating source. These results have been verified
analytically by Nakao and Morisawa [42]. During the recent years,
many attempts [43–45] have been made to study the
dynamics of collapsing cylindrical sources, but all these involve
cylindrical spacetimes which are very similar to spherical
spacetimes. In the current study, we have taken the
nontrivial cylindrical spacetime.
In the present study, we have considered the two types
of dissipation processes, heat dissipation associated to the
radial heat flux and bulk viscosity. These both dissipative
terms have been included in the stress energy tensor of the
gravitating source. In order to see the effects of these terms
on the dynamics of the collapse, we used the heat
transport equations in the context of Müller–Israel–Stewart
theory [
46–49
]. Such equations provide the physically
reasonable heat transportation process as compared to Landau–
Eckart approach [
50,51
] (by neglecting the
thermodynamics viscous/heat coupling coefficients). The bulk viscosity
has been described according to standard (non-causal)
irreversible thermodynamics approach in the stress energy tensor
of the gravitating source. The inclusion of the bulk
viscosity in the fluid implies that we are assuming the
relativistic Stokes equations, which corresponds to the irreversible
thermodynamics. This equation does not satisfy causality,
because implicitly it is assumed that the corresponding bulk
viscosity relaxation time vanishes, and this assumption is
valid within some approximations.
The plan of the paper is as follows: in Sect. 2 we present a cylindrical source and the field equations. The dynamical
equations with Misner–Sharp approach have been presented
in Sect. 3. The derivation of heat transport equation and its
coupling is given Sect. 4. The last section is devoted to a
summary of the results of this paper.
2 Gravitating source and field equations
In this section, we shall briefly introduce matter source,
geometry of star for both interior and exterior regions and
the field equations for the charged radiating bulk viscous
source. The cylindrically symmetric spacetime [
52
] is
ds−2 = −X 2(r, t )dt 2 + Y 2(r, t )dr 2 + R2(r, t )dθ 2 + dz2,
where −∞ ≤ t ≤ ∞, 0 ≤ r , −∞ ≤ z ≤ ∞, 0 ≤ θ ≤ 2π .
Inside the cylindrical star, we take a charged, anisotropic, bulk viscous fluid with radial heat flux, which has the following form of energy momentum tensor:
Tαβ = (μ + Pr )Vα Vβ − ( Pr − Pz )Sα Sβ + ( Pr − Pθ )χαχβ
−(gαβ + Vα Vβ )ξ
1
+ 4π
1
Fαγ Fβγ − 4 F γ δ Fγ δ gαβ ,
+ qα Vβ + Vαqβ + Pr gαβ
where μ is the energy density, Pr is the pressure
perpendicular to z direction, Pθ is the pressure in the θ direction, Pz
is the pressure in z direction, Vα is the four-velocity, ξ is the
coefficient of the bulk viscosity, is the expansion scalar and
qα is the radial heat flux. Also, Fαβ = −φα,β + φβ,α is the
Maxwell field tensor with four-potential φα. Moreover, Sα
and χα are the unit four-vectors, which satisfy the following
relations:
χ αχα = Sα Sα = 1, V α Vα = −1,
V α Sα = Sαχα = V αχα = 0.
The four-vector velocity Vα and four-vectors χα and Sα can
be defined as follows:
χα = Rδα2 , vα = −X δα0 , Sα = δα.
3
The Maxwell field equations are
F α;ββ = 4π J α, F[αβ;γ ] = 0,
(1)
(2)
(3)
where Jα is the four-current. It is assumed that there exists
only a non-vanishing electric scalar potential and the
magnetic vector potential will be zero; then the four-potential
takes the following form:
φα = φδα0 , J α = ζ V α,
where ζ (r, t ) is the charge density and φ (r, t ) is the electric
scalar potential.
+ π2 E2 Y 2
X R
= X R +
Y 2
X
R¨ X˙ R˙
− R + X R
,
1 ∂
Dt = X ∂t .
(4)
(5)
(6)
(7)
,
(8)
(9)
(10)
(11)
(12)
κ Pr − ξ
κ Pθ − ξ
κ Pz − ξ
− π2 E2
− π2 E2
=
Here l is the constant specific length of the cylinder.
Let be a boundary surface, which separates the interior
region (defined in Eq. (1)) from the exterior region, the
exterior region is described for a cylindrically symmetric
manifold in the retarded time coordinate by [
52
]
ds+2 = −
−
2M (ν)
q˜ 2(ν)
where M (ν) and q˜ (ν) are the mass and the charge,
respectively, and γ 2 = − 3 , is the cosmological constant. Using
the continuity of line elements and extrinsic curvature of line
elements given by Eqs. (1) and (11) and the field equations,
we get [
53
]
Pr − ξ
= (qY ),
m − M =
Qˆ 2l2 =
q˜ 2,
l =
4R˜ .
l
8
,
The expansion scalar is 1
= X
2Y˙ R˙
Y + R
,
where a dot and a prime denote differentiation with respect
to t and r , respectively.
The set of Einstein–Maxwell field equations is
κ μ − π2 E2 X 2 = YY˙ RR˙ +
κq X Y 2 = RR˙ − YY˙RR − RR˙XX ,
X 2
Y
X R R
X R − R
,
These are the necessary conditions for the smooth matching
of internal and external geometries of cylindrical stars over
the hypersurface . For the assumed cylindrical source the
difference of M and m (specific energy) is non-zero in general
and the constraint l = 4R˜ must be satisfied over .
3 Dynamical equations
According to Misner and Sharp [6,7], we introduce the proper
time derivative Dt as follows:
The velocity U of the fluid collapse may be stated in terms of Eq. (13) as
U = Dt R < 0
Hence Eq. (10) yields
R
´
Y =
8m
1 + U 2 − l +
1
4Qˆ 2l 2
R
= Eˆ ,
(in the case of collapse).
where Eˆ is the energy of an element of the fluid that
undergoes collapse. The proper time derivative of the mass function
described in Eq. (10) takes the following form:
Dt m = l
R˙ R¨ R˙2 X˙ R R˙ R 2Y˙
4X 3 − Y X 4 − 4Y 2 X + 4X Y 3
R˙ Qˆ 2l2
− 2X R2 .
Using Eqs. (5) and (7) and E = 2πQˆR , we obtain
Dt m = −2πl Eˆ q B + U ( Pr − ξ ) R.
These equations provide the rate of change of the total energy
available inside the cylinder of radius R. Here, we briefly
explain the effect of each term on the change of total internal
energy, on the right hand side of the above equation with the
term Eˆ q B being the multiple of negative sign. This stands
for the amount of heat energy leaving the surface of the
cylindrical star. In other words, the out-flow of heat from the
collapsing system reduces the total energy of the system. In the
second term U ( Pr − ξ ) < 0 (as U < 0, and < 0 due
to collapse and ξ > 0), hence this having a pre-factor −2π ,
increases the energy inside the collapsing source. The proper
radial derivative DR is used to address the dynamics of the
collapsing system, which is defined as follows:
(13)
(14)
(15)
(16)
(17)
1 ∂
DR = R ∂r .
(18)
Using Eqs. (10) and (18), we have
Now Eqs. (5), (6) and (19) yield
DRm = 2π Rl 4μ +
qY
+
Qˆ Qˆ l2
R R
−
This expression yields the change in total energy contained
inside the various cylindrical surfaces of different radii. The
U qY increases the energy as for the physically
term 4μ + E
ˆ
realistic fluid μ > 0, although it is affected by the heat flux
and U < 0 reduces μ. The second term implies the
presence of an electromagnetic field inside the gravitating source.
After the integration of Eq. (20), we obtain
0
Here, we have assumed that m(0) = 0.
Now we obtain Dt U , which is the acceleration of the col
lapsing matter inside the . From Eq. (13), we get the
following relation:
1 ∂
Dt U = X ∂t
R
˙
X
R¨ R˙ X˙
⇒ Dt U = X 2 − X 3 .
The above equation with Eq. (7) gives
Dt U = −
m
R2 + 8π(Pr − ξ )R
X Eˆ Qˆ 2
+ X Y + R
l2
2R2 − 1
l
+ 8R2 (1 + U 2 − Eˆ 2).
By the conservation law (T αββ =0), we deduce the following
dynamical equations: ;
qY 2
˙
X
Q
ˆ
R2
Pr +
− ( P
X
+ ( Pr + μ) X − ξ
X
+ X ξ
+
E R − R E
= 0.
R
− Pr ) R +
qY 2
X
R˙ 3Y˙
R + Y
Using the value of XX from Eq. (24) in Eq. (23) and
considering the field equations, after some algebra we obtain
(Pr + μ − ξ )Dt U
= − (μ + Pr − ξ )
m l2 Qˆ 2 Qˆ 2 lU 2
R2 + 8π R(Pr − ξ ) + 2R3 R + 8R2
−Eˆ2
−Eˆ
The factor ( Pr +μ−ξ ) being a multiple of the acceleration
DT U plays the role of an effective inertial mass density,
while the same factor on the right hand side before the square
bracket is the passive gravitational mass density. This factor
is affected by the radial pressure and bulk viscosity, but it is
independent of the electric charge. The first square bracket
on the right hand side shows the effects of dissipation and
charge on the dynamical process. The second square bracket
gives the effects of the local anisotropy, electric charge and
gravitational mass density. In the last square bracket Pr is the
pressure gradient and the terms involving q, ξ and Qˆ explain
the collective effects of dissipation and the electromagnetic
field on the hydrodynamics of the collapsing source. The
consequences of Dt q will be dealt with in the next section
by deriving the heat transport equation and then performing
the possible coupling of the dynamical equation with the
resulting heat transport equation.
4 Heat transport equation
As already mentioned in the introduction, we shall use
a transport equation that comes from the Müller–Israel–
Stewart [
46–49
] second order phenomenological theory for
dissipative fluids (by neglecting the thermodynamics
viscous/heat coupling coefficients). Since we have introduced
the bulk viscosity in a fluid source, we have to take
accordingly the full causal approach as discussed in [
54–59
], but, for
the sake of simplicity, we neglect the thermodynamics
viscous/heat coupling coefficients and only take into account
the only transportation of heat flux governed bythe Cattaneo
type equation [60] (leading to a hyperbolic equation for the
propagation of a thermal perturbation). Thus according to
[
12,14
], the transport equation for the heat flux is
τ hαβ V γ qβ;γ + qα = −K hαβ (T,β + T aβ )
1 K T 2 τ V β
− 2 K T 2
;β
qα.
(26)
In the above equation hαβ denotes the projection onto the
space orthogonal to V β , K is the thermal conductivity and
T and τ are temperature and relaxation time, respectively.
With the symmetry of the given interior spacetime, the heat transport equation has the following form as regards the independent component:
Y Dt q = −
K Y T
τ
After substituting the value of Y Dt q in Eq. (25), we have
Q2
ˆ
+ R +
l(1 + U 2)
8R2
,
K T Y 2
α = τ ( Pr + μ − ξ )
.
Fhyd = −Eˆ2
1 − τ (Pr + μ − ξ )
(32)
From Eq. (30), it is noted how dissipation affects the final
stage of the charged collapsing cylinder. This fact was
investigated for the first time in [
61
], when the authors discussed
the thermal conduction in systems out of hydrostatic
equilibrium. They analyzed the fact that the evolution of the
gravitating source depends on the parameter α (which is defined in
terms of the thermodynamic variables), further for the
validity of causality, the constraints on α have been determined in
that work.
It is clear that the left hand side of Eq. (30) will be zero as
α → 1, which confirms that the effective inertial mass
density of the fluid element tends to zero. Further, we observe
that the inertial mass will be decreased as α exceeds than 1.
Moreover, Fgrav, being a multiple of (1 − α), is affected by
this factor. Also, it is evident that both inertial mass and
gravitational attraction are affected by the same factor (1 − α).
In other words, we can say that this equation satisfies the
equivalence principle and we would like to point out that the
factor (1 − α) has no effects on Fhyd. One may observe that a
collapsing cylinder would evolve in such a way that the value
of α keeps on increasing and attains a critical value of 1. With
the passage of time during collapse the rapid decrease in the
force of gravity may gradually result in altering the physical
effects of the right hand side of Eq. (30). As is clear from the
definition of α, it is inversely related to the effective inertial
mass density, so as long as α increases from 1, then there
would be a decrease in the inertial mass density. Physically,
it is only possible when the gravitating source depicts the
bouncing behavior. The factor (1 − α) does not depend on
the charge parameter but it heavily depends on the bulk
viscosity, which is explicitly clear from Eq. (32). Further, one
can see the dependence of the factor (1−α) on the dissipative
variables when a full causal approach [
62
] is used to discuss
the dynamics of dissipative collapse (see Eq. (54) of [
62
]).
5 Conclusion
The cylindrically symmetric systems which combine transla
tions along the axis are exactly known to general relativists.
The study of such systems was started by Weyl [63] and
Levi-Civita [34] in the early 20th century immediately after
the birth of the Einstein theory of relativity. In the beginning
physicists were interested in finding the gravitating objects
that are exactly axially symmetric. The realistic fluids are
very important in the modeling of astronomical objects. So,
one cannot ignore the effects of dissipation during the
gravitational collapse.
Here, we discuss the gravitational collapse of charged
radiating cylindrically symmetric stars. To this end, we
formulated the Einstein field equations and conservation
equation for a non-static charged bulk viscous heat conducting
anisotropic cylindrically symmetric source. Using the Misner
and Sharp formalism, the dynamical equations are derived.
Further, we have considered the two types of dissipation pro
cesses, heat dissipation associated to the radial heat flux and
bulk viscosity. In order to see the effects of these terms on the
dynamics of the collapse, we have excluded thermodynamics
viscous/heat coupling coefficients in the heat transport
equations in the context of Müller–Israel–Stewart theory [
46–49
].
The inclusion of the bulk viscosity in the fluid implies that
we are assuming the relativistic Stokes equations, which
corresponds to irreversible thermodynamics. This equation does
not satisfy causality, because implicitly it is assumed that the
corresponding bulk viscosity relaxation times vanishes. But
this assumption is sensible, because within some
approximations, such relation times could be neglected. A full
causal approach to the dynamics of dissipative collapse has
been analyzed with significant consequences in [
62
], without
excluding the thermodynamics viscous/heat coupling
coefficients for the heat flux and bulk viscosity. As an
implication of their analysis to astrophysical scenario, they pointed
out that in a pre-supernova event, the dissipative parameters
(particularly the thermal conductivity) would be so large as to
produce a significant decreasing in the force of gravity which
leads to the reversal of the collapse. We would like to mention
that we have introduced the bulk viscosity by the standard
(non-causal) irreversible thermodynamics approach, so we
have used the partially causal approach (the
thermodynamics viscous/heat coupling coefficients have been excluded)
to discuss the dynamics of the dissipative source considered.
Further, the form of Eqs. (17), (23), (24), (25), (29), (30), and
(32) depends on the standard (non-causal) irreversible
thermodynamics approach, which we have used in the present
analysis. If one considers the full causal approach to the
discussion of the dynamics of dissipative gravitational collapse
as in [
62
], then in the present case the term −ξ in Eqs. (17),
(23), (24), (25), (29), (30), and (32) will be replaced by a
dissipative variable .
Finally, in our analysis, it has been investigated that during
the evolution of the cylindrical star, charge, bulk viscosity and
anisotropic stresses reduce the energy of the system and we
conclude the following.
• The out flow of heat from the collapsing cylindrical star
reduces the total energy of the system.
• The bulk viscosity reduces the radial pressure of the
collapsing fluid.
• The bulk viscosity and charge of the fluid would affect
the rate of collapse prominently.
• Active/passive gravitational mass density is affected by
the bulk viscosity and it is independent of the
electromagnetic field.
• In Eq. (30), the factor (1 − α) would explain the possible
evolutionary stages of the charged dissipative cylinder.
• The term α is inversely related to the gravitational mass
density, which is affected by the bulk viscosity, while it
is linearly related to temperature of the fluid.
• The inclusion of the bulk viscosity would increase the
value of α.
• For α < 1, α > 1 and α = 1, we have an expanding,
collapsing and bouncing behavior of the fluid distribution,
respectively.
Acknowledgements The constructive comments and suggestions of
an anonymous referee are highly appreciated.
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