Dynamical Instability of Gaseous Sphere in the Reissner-Nordstrom Limit
In this paper, we study the dynamical instability of gaseous sphere under radial oscillations approaching the Reissner-Nordström limit. For this purpose, we derive linearized perturbed equation of motion following the Eulerian and Lagrangian approaches. We formulate perturbed pressure in terms of adiabatic index by employing the conservation of baryon numbers. A variational principle is established to evaluate characteristic frequencies of oscillations which lead to the criteria for dynamical stability. The dynamical instability of homogeneous sphere as well as relativistic polytropes with different values of charge in Newtonian and post-Newtonian regimes is explored. We also find their radii of instability in terms of the Reissner-Nordstörm radius. We conclude that dynamical instability occurs if the gaseous sphere contracts to the Reissner-Nordstörm radius for different values of charge.
Keywords: Gravitational collapse; Instability; Electromagnetic
field; Relativistic fluids.
PACS: 04.20.-q; 04.25.Nx; 04.40.Dg; 04.40.Nr.
It is a well-known fact that any relativistic model will be physically interesting if it is stable under fluctuations. The stability/instability of celestial objects has significant importance in general relativity (GR). This study is closely related to the evolution and structure formation of self-gravitating objects. Initially, any stable gaseous mass remains in state of hydrostatic equilibrium for which the gravitational force is counter balanced by the internal pressure of the body acting in the opposite direction. The effect of gravity over the internal pressure causes the matter to collapse and the star contracts to a point under its own gravitational force forming compact stars.
The dynamics of massive stars can be discussed in weak as well as strong-field regimes. The idea of weak-field approximation (Newtonian and post-Newtonian approximations (pN))  has remarkable importance in the context of relativistic hydrodynamics. The analysis of dynamical instability in strong-field regimes becomes complicated due to non-linear terms, so the weak-field approximation schemes are used as an effective tool. Chandrasekhar  was the pioneer who studied the dynamical instability of Newtonian perfect fluid sphere approaching the Schwarzschild limit in terms of adiabatic index. He used Eulerian approach for hydrodynamic equations and developed a variational principle to find characteristic frequencies applicable to the radial oscillations at Newtonian and pN limits. He concluded that the system would be dynamically stable or unstable according to the numerical value of adiabatic index, i.e., or , respectively. The same author  also investigated the stability of gaseous sphere under radial and non-radial oscillations at pN limit.
The dynamical instability of self-gravitating spherical objects has been studied by using various techniques. Herrera et al.  explored dynamical instability of spherical collapsing system for non-adiabatic fluid using perturbation scheme. They showed that heat conduction increases the instability range in Newtonian limit but decreases in pN limit. Later, many researchers  discussed the role of various physical factors on the dynamical instability of spherical systems using perturbation scheme and found interesting results.
The stability of self-gravitating objects in the presence of electromagnetic field has a primordial history starting with Rosseland . There is a general consensus that astrophysical objects do not have charge in large amount  but there are some mechanisms which induce large amount of electric charge in collapsing stars. Stettner  showed that presence of net surface charge enhances the stability of sphere with uniform density. Glazer  investigated the dynamical stability of perfect fluid sphere pulsating radially with electric charge. Ghezzi  studied stability of neutron stars and found that the stars having a charge greater than the extreme value would explode. Sharif and collaborators  discussed the role of electric charge in dynamical instability at Newtonian and pN regimes.
Polytropes are useful self-gravitating objects as they provide simplified models for internal structures of stellar objects. The polytropic equation of state deals with various fundamental astrophysical issues . Tooper  studied the internal structure of gaseous sphere obeying polytropic equation of state and obtained Newtonian polytropes using numerical solution of the Lane-Emden equation. The effect of electromagnetic field on the dynamics of polytropic compact stars has also been studied . Herrera and Barreto  analyzed both Newtonian as well as relativistic polytropes in spherical symmetry. Recently, Breysse et al.  have discussed the dynamical instability of cylindrical polytropic fluid systems under radial and non-radial modes of oscillations.
In this paper, we study the dynamical instability of spherically symmetric gaseous systems following Chandrasekhar’s approach  in the vicinity of electromagnetic field. The paper is organized as follows. The next section deals with matter distribution and the Einstein-Maxwell field equations. In section 3, we discuss motion of the system under radial oscillations following the Eulerian approach. Section 4 provides the formulation of perturbed pressure and adiabatic index in terms of Lagrangian displacement using conservation of baryon number. In section 5, we develop conditions for dynamical instability of homogeneous sphere and relativistic polytropes. Finally, we conclude our results in the last section.
2 Field Equations and Matter Configuration
We consider a spherically symmetric system in the interior region given by
where and are the gravitational potentials. The corresponding Einstein field equations can be written as
where dot denotes derivative w.r.t . We assume the energy-momentum tensor corresponding to charged perfect fluid in the form
where is the four-velocity, is the pressure and is the energy density. The electromagnetic field tensor can be defined in terms of four potential, , which satisfies the Maxwell field equations as
where is the four current. The only non-vanishing radial component of electromagnetic field tensor () implies that
whose integration yields
where is the total amount of charge within the sphere.
The energy-momentum tensor follows the conservation identity , which governs hydrodynamics of the fluid and leads to the following relations
where . The non-zero components of energy-momentum tensor are
All the quantities governing the motion remain independent of time during the state of hydrostatic equilibrium. The surface stresses describing equilibrium state are denoted by zero subscript. In this context, Eqs.(2), (3) and (7) take the form
We take the Reissner-Nordström (RN) spacetime in the exterior region as
where corresponds to the total mass of the sphere. The hydrostatic equilibrium describes the state of fluid in which pressure gradient force is balanced by the gravitational force. When one of these forces overcome the other, the stability of the system is disturbed leading to an unstable system. The equation describing hydrostatic equilibrium is obtained by eliminating from Eqs.(9) and (10) as
where the left and right hand sides correspond to pressure gradient and gravitational terms, respectively and
is the Misner-Sharp mass function.
3 Equations Governing Radial Oscillations
Here we discuss the motion of gaseous masses undergoing radial oscillations. The non-zero components of four-velocity are given by
where is the radial velocity component. These components can be calculated with respect to spacetime coordinates by . The stability of any gaseous mass under perturbation ultimately gives rise to the dynamical evolution of gravitating system. We assume that an equilibrium configuration is perturbed such that it does not affect the spherical symmetry. We consider only linear terms so that the respective values in the perturbed state become
Let us introduce a Lagrangian displacement such that . Integration of Eq.(19) leads to
Using Eq.(11), this equation takes the form
Using Eq.(10), it follows that
which in accordance of Eq.(11) leads to
Now we assume time dependent perturbations in the form of Lagrangian displacement, i.e., , where is the characteristic frequency to be evaluated. The Lagrangian displacement connects the fluids elements in equilibrium with corresponding one in the perturbed configuration. Since the equations have natural modes of oscillations, so they will depend on time. Considering and as time dependent amplitudes of the respective quantities, Eq.(20) with (27) can be rewritten as
4 The Conservation of Baryon Number
In order to discuss the perturbed state of pressure in terms of Lagrangian displacement , an additional assumption is required which can relate physical aspects of relativistic theory with the gaseous mass undergoing adiabatic radial oscillations. In this context, the required supplementary condition can be satisfied by conservation of baryon number in the framework of GR as , or
where is the baryon number per unit volume. The conservation of baryon number plays a vital role in collecting different models of the universe. According to this law, the number of particles may vary but their total number will remain conserved during the fluid flow. This change occurs due to loss or gain of net fluxes. Here we consider fluid obeying this identity. Equation (29) through (15) leads to
We assume the perturbation
keeping only the linear terms in , Eq.(30) takes the form
whose integration in terms of Lagrangian displacement leads to
Using Eq.(22), it follows that
We consider an equation of state in the form
and is the adiabatic index (ratio of specific heats) defined by
This relates the pressure and density fluctuations and measures the stiffness of the fluid.
5 Pulsation Equation and Variational Principle
The linear pulsation corresponds to the oscillation frequencies and different modes of small perturbations applied to equilibrium spherical configuration. Inserting the values of and from Eqs.(23) and (36) in (28), it follows that
Substituting from Eq.(10) in the above equation, we have
where . Under the equilibrium condition, Eq.(4) yields
This is the required pulsation equation which satisfies the boundary conditions, i.e., at and at . This constitutes a characteristic value problem for obtained by multiplying the equation with and integrating over values of as
The corresponding orthogonality condition is defined as
where and give proper solutions associated with different characteristic values of . To investigate dynamical instability of spherical star, the right-hand side of Eq.(43) should vanish by choosing a trial function satisfying the given boundary conditions. In the following, we discuss the conditions for dynamical instability by taking two special models.
5.1 The Homogeneous Model of Sphere
First we consider the homogeneous sphere with constant energy density and study the conditions for its dynamical instability. Equations (13) and (14) governing the hydrostatic equilibrium allow the integration  such that we can write
where and . The solutions of the relevant physical quantities can be determined in terms of and as