$f(T)$ Corrected Instability of Cylindrical Collapsing Object with Harrison-Wheeler Equation of State

In this paper, we study the dynamical instability of a collapsing object in the framework of generalized teleparallel gravity. We assume a cylindrical object with a specific matter distribution. This distribution contains energy density, isotropic pressure component with heat conduction. We take oscillating states scheme up to first order to check the instable behavior of the object. We construct a general collapse equation for underlying case with non-diagonal tetrad depending on the matter, metric functions, heat conducting term and torsional terms. The Harrison-Wheeler equation of state which contains adiabatic index is used to explore the dynamical instability ranges for Newtonian and post-Newtonian constraints. These ranges depend on perturbed part of metric coefficients, matter parts and torsion.


Introduction
General relativity is one of the most acceptable theory of gravity which describes many natural phenomena of universe. However, this theory faces some issues such as dark matter and dark energy. On the same side, dark matter and dark energy are the important ingredients for the dynamics of the entire universe. That is, 25% part of universe consists of dark matter while dark energy is upto 68%. On the other hand, dark energy is responsible for the expansion of the universe. These problems are the main reason behind the modification of general relativity (GR). The modification of GR is as old as this theory itself. In recent era, f (R) theory of gravity is the simplest modification to GR based on curvature tensor. However, this theory contains fourth order field equations which are very difficult to handle.
An alternative theory to GR is teleparallel gravity which based on torsion using Weitzenböck connection where curvature is zero. The modification of telaparallel gravity results a generalized Teleparallel gravity, that is f (T ) theory of gravity [1]- [11]. One of the advantage of this theory is that it gives field equations of second order which are easier to handle for the discussion of any underlying scenario as compared to f (R) gravity. In this context, Ferraro and Fiorini introduced the f (T ) theory of gravity to solve the partial horizon problem in Born-Infeld strategy which gave singularity free solution [9]. It is also known as non-local Lorentz invariant theory. o figure out this problem, Nashed [2] has used two tetrad matrices for the regularization of f (T ) field equations. He proposed regularized process with general tetrad field to figure out the effect of local Lorentz invariance. Some authors also investigated this problem for reference see [10]- [11].
The dynamical collapse of self gravitating spherically symmetric object has been widely discussed in f (T ) theory of gravity [12]- [17]. The collapse process occurs when the balanced matter of object become imbalanced and the object fails to maintain its equilibrium. In this way, various dynamical states occur, which may be analyzed through dynamical equations. Chandrasehkar [18] was the first who introduced the concept of dynamical instability analysis with the help of adiabatic index, Γ . Instability ranges through adiabatic index have been critically examined for cylindrically and spherically symmetric collapsing matter in f (R) gravity [19]- [22]. Skripkin [23] analyzed the collapsing non-dissipative spherically symmetric fluid with constant energy density and isotropy. He concluded that under expansion-free condition, Minkowskian cavity is located at the center of the fluid. Similarly under the same conditions, dynamical collapse analysis of spherically and cylindrically symmetric anisotropic fluid is explored by Chan et al. [24]- [27]. Some authors [28]- [29] also studied the instability ranges of spherically symmetric collapsing star with the presence of charge and without charge in f (R) gravity and concluded that these instability ranges are based on geometry, matter and curvature.
In this scenario, Kausar worked on the effects of CDTT model having inverse curvature term on the unpredictable behavior for the cylindrical symmetric object. It was shown that dynamical instability of expansion free gravitational collapse with spherical geometry can be investigated without adiabatic index Γ [30]- [31]. It was concluded that instability ranges depend on energy density, electromagnetic field and anisotropic pressure. In Brans-Dicke gravity, Sharif and Manzoor [32] explore the dynamical instability of cylindrically symmetric collapsing star and concluded the range of adiabatic index which is greater than one in special case and remains less than one for unstable behavior. In f (T ) gravity, dynamics of collapsing spherical symmetric object with expansion and expansion-free, shear-free conditions have been explored [16]- [17].
Jawad et al. [16] discussed the dynamical instability ranges of a cylindrical symmetric object in the framework of f (T ) gravity. They have used anisotropic matter distribution and concluded that the instability ranges effected by the presence of matter, metric coefficients and torsional terms. We extend this paper taking matter contribution which contain isotropic pressure with heat conducting term. The scheme of the paper is as follows. In section 2, we construct the field equations for cylindrical symmetric collapsing object for f (T ) gravity taking non-diagonal tetrad. Section 3 provides the perturbation scheme up to first order to develop a general collapse equation. In the next section 4, we obtain instability ranges under Newtonian and post Newtonian constraints. The last section contains the concluding remarks.

Field Equations of f (T ) Gravity for Collapsing Star
In this section, we provide the basic formulation of f (T ) gravity. We discuss the basics of collapsing star in cylindrical symmetry and obtain the f (T ) field equations in this scenario. We have taken cylindrically symmetric line element as interior spacetime while exterior spacetime in retarded time coordinate in the framework of f (T ) gravity. The collapse process happens when stability of matter disturbed and at long last experiences collapse which leads to different structures. We have taken the self-gravitating object as cylindrically symmetric collapsing star. We consider cylindrically symmetric collapsing star as an interior region defined by where A, B and C are functions of t, r and cylindrical coordinates satisfy the In order to keep the cylindrical symmetry, it must satisfy some conditions in the beginning of collapse which is not a trivial process. That is, if the symmetry does not contain any curvature singularity then we know the conditions to apply. Otherwise, on the singular symmetry, it is hard to impose any condition. In the underlying case, the inside region of cylinder is assumed to be flat, that is axis is regular. Thus, we may apply the following conditions [33]- [34].
• The spacetime must hold the following condition in order to preserve the flatness near the symmetry axis. When r → 0 + , the condition is given by 1 4X ∂X ∂x α ∂X ∂x β g αβ → 1.
• There must not be any closed timelike curves (rather this is easy to introduce in cylindrical spacetimes). We impose the following condition in order to avoid these curves throughout the spacetime.
• The cylindrical spacetime cannot be asymptotically flat in the axial direction, but must be along radial direction.
The exterior region in terms of retarted time coordinates υ and gravitational mass M is given by [35], where α is a constant having dimension L. The action of f (T ) gravity [9] is defined as here, h = √ −g = det(h a α ), h a α represents tetrad components, L m is Lagrangian density of matter, f is the function of torsion scalar and κ = 8πG is the coupling constant with G is the gravitational constant. The variation of action (3) with respect to tetrad leads to the following field equations where f T , f T T stand for first and second order derivatives with respect to T and torsion scalar is T ≡ S σ µν T σ µν . The torsion tensor and superpotential tensor are defined as In terms of Einstein tensor, the f (T ) field equations take the following form where Φ (T) µγ represents the torsion part which is defined by The term Φ µγ represents the matter of the universe. Here we take the matter under consideration as where u µ and q µ are the four-velocity and heat conduction satisfy the fol- ρ is the energy density of fluid and p denotes the isotropic pressure.
The selection of vierbein field is crucial for the framework of f (T ) gravity. For cylindrical and spherical geometries, the good tetrad are non-diagonal tetrads that keep the modification of theory with no condition applied on torsion scalar. The diagonal tetrads take the torsion scalar to be constant or vanish in these geometries named as bad choice of tetrad in f (T ) gravity. Therefore, we take the non-diagonal tetrad for the interior spacetime. In order to construct non-diagonal tetrad which are suitable for cylindrical symmetry, different methods are adopted in literature like the local Lorentz rotations form rotated tetrad and local Lorentz boosts introduce boosted tetrad. We choose here another approach in which general coordinate transformation is used [36]. Consider static cylindrically symmetric spacetime (1). The general coordinate transformation law is given as where M T = ∂X ν ′ ∂X µ denotes the transformation matrix from Cartesian to cylindrical polar coordinates while {X ν ′ } and {X µ } are Cartesian and cylindrical polar coordinates, respectively, i.e., X 1 ′ = r cos φ, X 2 ′ = r sin φ, Now we compare the metric (1) with Minkowski metric in cylindrical polar coordinates given as Notice that g 00 and g 11 of the metric (1) are obtained by multiplying the corresponding g 00 and g 11 of Minkowski metric by A 2 and B 2 , respectively and replacing r by C for g 22 . Thus multiplying zeroth and first columns of the above matrix by A and B, respectively, while replacing r by C, we obtain the tetrad components of static cylindrically symmetric spacetime Using the non-zero components of torsion and super-potential tensors, we get torsion scalar as where dot and prime represent derivatives with respect to time and radial coordinates, respectively. The f (T ) field equations of cylindrically symmetric collapsing star using Eqs. (1) and (6) yield Using Eqs. (15)- (16), we get the following relatioṅ In order to match interior and exterior regions of cylindrical symmetric collapsing star, we use junction conditions defined by Darmois. For this purpose, we consider the C-energy, i.e., mass function representing matter inside the cylinder is given by [37] m(t, r) = where E is the gravitational energy per unit specific length L of the cylinder. The areal radiusr is defined asr = µL where circumference radius has the relation µ 2 = χ (1)i χ i (1) while L 2 = χ (2)i χ i (2) . The terms χ (1) = ∂ ∂φ and χ (2) = ∂ ∂z are the Killing vectors corresponding to cylindrical systems. For interior region, the C-energy turns out The continuity of Darmois conditions (junction conditions) establishes the following constraints where Σ (e) indicates the outer region for measurements with r = r Σ (e) =constant. The conservation of total energy of a system is obtained from Bianchi identities through dynamical equations. This relation might be useful in order to simplify the collapse equation. To analyze the conservation of total energy of a collapsing star, contracted Bianchi identities in the framework of f (T ) gravity are used. These are defined as Consequently, we obtain the following two dynamical equations

Oscillating States and Collapse Equation
The selection of f (T ) model is very important to establish the collapse equation representing the instability dynamics of cylindrically symmetric object in the framework of f (T ) gravity. Here we assume the f (T ) model in powerlaw form up to quadratic order which is defined by f (T ) = T + χT 2 , where χ is an arbitrary constant. This model is widely used in the literature as it indicates the accelerated expansion of the universe in the phantom phase. The possibility of realistic wormhole solutions under this model are found. Also, the instability conditions for a collapsing star under spherically symmetric collapsing object are discussed. In order to construct the dynamical equations and explore instability ranges, we assume the linear perturbation strategy to find the instable behavior. For this we choose the perturbation scheme in such a way that in the static configuration (non-perturbed part), metric and matter parts are only radial dependent while perturbed part contains both radial and time dependency upto first order where 0 < λ ≤ 1 [?]- [28]. The metric functions under perturbation strategy become The static and perturbed parts of matter components are ρ(t, r) = ρ 0 (r) + λρ(t, r), p(t, r) = p 0 (r) + λp(t, r), q(t, r) = λq(t, r), while mass function, torsion scalar and f (T ) model are perturbed as follows m(t, r) = m 0 (r) + λm(t, r), T (t, r) = T 0 (r) + λω(r)∆(t), f (T ) = T 0 (1 + χT 0 ) + ωλ∆(1 + 2χT 0 ), f T (T ) = 1 + 2χT 0 + 2χλ∆ω.
The quantities having zero subscript show the zero order perturbation. Applying the above set of equations, the perturbed part of the first dynamical equation is Similarly, the second dynamical equation yields The second dynamical equation can be rewritten as where The Eq.(29) is called as general collapse equation for a cylindrical symmetric object in f (T ) gravity. To discuss the instability ranges, we have to find the values of ρ, p, q and ∆. For this purpose, we do some mathematics in the following. After applying perturbation on Eq.(15), we obtain the following equation Putting the above value ofq in Eq.(28), we get We can write this equation as where Integrating the above equation w.r.t "t", we get The Harrison-Wheeler equation of state defined by where Γ is named as adiabatic index. The adiabatic index defines the instability ranges of a self-gravitating collapsed objects. Here we discuss these ranges for cylindrically symmetric self gravitational fluid in the frame-work of f (T ) gravity both in Newtonian and post Newtonian Regimes through adiabatic index under collapse equation. Equation (34) can be written as To find the value of p, we insert the value ofρ in Eq. (35), which gives In order to determine the value of ∆(t), we perturbed Eq.(17) which is given by The solution of this equation can be found through matching of interior and exterior regions which leads r = r Σ = constant r, on boundary surface. Under this condition and using value of p, Eq.(37) yields where In the above equation, all the terms with subscript Σ are as follows.
The solution of Eq.(38) is This solution exhibits the stability as well as instability configurations and a 1 and a 2 are constants. Here to discuss the instability analysis, we take only static solution of cylindrical collapsing star which implies a 2 = 0, while Putting all the correspondent values in general collapse equation, we attain the collapse equation of cylindrically symmetric object in f (T ) gravity as where ∆ is given in Eq.(39).

Instability Ranges
Now we consider the collapse equation (40) with Newtonian and Post Newtonian regime constraints to figure out the instability ranges of self gravitating object in f (T ) gravity.

Newtonian Order
The constraints for Newtonian regime are given as follows Using these conditions in Eq.(40), the collapse equation turns out as which indicates the hydrostatic state of a cylindrically symmetric self gravitating fluid where J N 2eq represents those terms in J 2eq which comes through Newtonian approximation. This equation contains the matter, metric and torsion scalar components which take part to develop the instability ranges. It is noted that we take adiabatic index with positive sign all over the scenario to preserve the variation between gravitational forces, gradient of heat and pressure components. This system will be unstable if The left hand side of inequality remains positive while the system remains in instable state till this inequality holds. Now we discuss the following three cases.
• Case 1: Eq.(41), then we obtain Γ = 1. This leads to the hydrostatic equilibrium state for this particular case for cylindrically symmetric self gravitating object.
• Case 3: ψ Σ , we get Γ > 1 which establishes instability ranges for collapse of the self-gravitating cylindrically symmetric object in the framework of f (T ) gravity.
It is noted that, we may recover the general relativity in Newtonian limit for instability range as Γ < 4 3 .

Post-Newtonian Order
In the post-Newtonian order, we deal the dynamics of cylindrical symmetric star as A 0 = 1 − mo r , B 0 = 1 + mo r . We have taken m r 0 terms upto first order and neglect all the higher terms. Using these constraints, we obtain some terms of collapse equation (40) as Using these values in collapse equation, we obtain an equation representing hydrodynamical state for post-Newtonian order like Newtonian order. The cylindrical system becomes unstable in post-Newtonian orders if the adiabatic index satisfies the following inequality where J 1pN and J 2pN are the expressions from Eq.(39) under post-Newtonian constraints. Following the Newtonian order cases, we can construct the possibilities of hydrodynamical equilibrium as well as instable states.

Conclusion
The collapse process happens when due to some internal or external disturbance, the matter of object become unbalanced and to maintain its equilibrium, it collapse down and leads to different structure i.e., white dwarfs, black holes and stellar groups. Dynamical instability ranges are used for spherically symmetric object such as galactic halos, globular clusters etc, while cylindrical symmetry and plates are associated with the post-shocked clouds at stellar scale. To study the unstable behavior of spherically symmetric collapsing object due to its own gravity, the adiabatic index is used whose numerical range is less then 4 3 in GR. Chandrasekhar [18] gave the direction to study and explore the dynamical instability in demonstrating the development and shaping of stellar objects that must be stable against fluctuations. During these processes, the self-gravitating fluid happens to under go many phases of dynamical activities that remain in hydrostatic equilibrium for a short span.
After this equilibrium state, the system is changed from its initial static phase to perturbed and oscillating phase. It is necessary to study the evolution of the system immediately after its departure from the equilibrium state. In this work, we have analyzed the instability ranges of self-gravitating cylindrically symmetric collapsing object in f (T ) gravity. We have taken isotropic matter distribution for the interior metric and have found the important results for interior and exterior regimes. For this purpose, we have calculated physical quantities like torsion tensor and super-potential tensor. The selection of tetrad field is crucial for the framework of f (T ) gravity. We have taken non-diagonal tetrad for the interior spacetime. Torsion scalar has been formulated with the help of tetrad field, torsion and super-potential tensor in f (T ) gravity.