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The present works deals with gravitational collapse of cylindrical viscous heat conducting anisotropic fluid following the work of Misner and Sharp. Using Darmois matching conditions, the dynamical equations are derived and the effects of charge and dissipative quantities over the cylindrical collapse are analyzed. Finally, using the Miller-Israel-Steward causal thermodynamic theory, the transport equation for heat flux is derived and its influence on collapsing system has been studied.

A challenging but curious issue in gravitational physics as well as in relativistic astrophysics is to know the final fate of a continual gravitational collapse. The stable configuration of a massive star persists as long as the inward pull of gravity is neutralized by the outward pressure of the nuclear fuel at the core of the star. Subsequently, when the star has exhausted its nuclear fuel there is no longer any thermonuclear burning and there will be endless gravitational collapse. However, depending on the mass of the collapsing star, the compact objects such as white dwarfs, neutron stars, and black holes are formed. White dwarf and neutron star gravity are counterbalanced by electron and neutron degeneracy pressure, respectively, while black hole is an example of the end state of collapse.

The study of gravitational collapse was initiated long back in 1939 by Oppenheimer and Snyder [

Although most of the works on collapse dynamics are related to spherical objects, still there is interesting information about self-gravitating fluids for collapsing object with different symmetries. The natural choice for nonspherical symmetry is axis-symmetric objects. The vacuum solution for Einstein field equations in cylindrically symmetric space-time was obtained first by Levi-Civita [

Further, the junction conditions due to Darmois [

Moreover, from realistic point of view, it is desirable to consider dissipative matter in the context of collapse dynamics [

On the other hand, in the context of gravitational waves, the sources must have nonspherical symmetry. Further, cylindrical collapse of nondissipative fluid with exterior containing gravitational waves shows nonvanishing pressure on the boundary surface by using Darmois matching conditions. Recently, it has been verified [

In the present work, following Misner and Sharp, collapse dynamics of viscous, heat conducting charged anisotropic fluid in cylindrically symmetric background will be studied. The paper is organized as follows. Section

Mathematically, the whole four-dimensional space-time manifold having a cylindrical collapsing process can be written as

In

For compact notation, we write

The anisotropic fluid having dissipation in the form of shear viscosity and heat flow has the energy-momentum tensor of the following form [

For the above metric, one may choose the unit time-like vector, space-like vector, and heat flux vector in a simple form as

Also the acceleration vector and the expansion scalar have the following explicit expressions:

If, in addition, we assume the above fluid distribution to be charged then the energy-momentum tensor for the electromagnetic field has the following form:

As the charge per unit length of the cylinder is at rest with respect to comoving coordinates so the magnetic field will be zero in this local coordinate system [

From the law of conservation of charge,

Further, in the interior space-time

The gravitational energy per specific length in cylindrically symmetric space-time is defined as follows [

For the present model with the contribution of electromagnetic field in the interior region, the

It should be noted that the above energy is also very similar to Tabu’s mass function in the plane symmetric space-time [

The exterior space-time manifold

defines the time coordinate on

Similarly, from the perspective of the exterior manifold, the boundary three-surface

so that the exterior metric on

In order to have a smooth matching of the interior and exterior manifolds over the bounding three surfaces (not a surface layer), the following conditions due to Darmois [

(i) The continuity of the first fundamental form is

(ii) The continuity of the second fundamental form is

In the above expression for extrinsic curvature,

Also, in the above, the Christoffel symbols are evaluated for the metric in

The continuity of the 1st fundamental form gives

Now the nonvanishing components of extrinsic curvature

Hence, continuity of the extrinsic curvature together with (

Thus (

From the conservation of energy-momentum, that is,

Using (

Now following the formulation of Misner and Sharp [

so that the fluid velocity in the collapsing situation can be defined as follows [

Using (

Hence, using the field equations for the interior manifold, we obtain the time rate of change of

Further, using the Einstein field equations (

This radial derivative can be interpreted as the energy variation between the adjacent cylindrical surfaces within the matter distribution. The first term on the r.h.s. is the usual energy density of the fluid element, while the second term and third term are the conditions due to the electromagnetic field. The fourth term represents contribution due to the dissipative heat flux and the last term will increase or decrease the energy of the system during the collapse of the cylinder provided

Finally, the collapse dynamics is completely characterized by the equation of motion in (

In causal thermodynamics, due to Miller-Israel-Stewart, the transport equation for heat flow is given by [

Note that although heat dissipation is described above through a physically reasonable transport equation (

Now considering proper derivatives in (

with

The l.h.s. of (

The authors declare that they have no conflicts of interest.

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