Einstein's cluster mimicking compact star in the teleparallel equivalent of general relativity

Abstract

We present a physically plausible solution representing Einstein's cluster mimicking the behaviors of compact star in the context of teleparallel equivalent of general relativity. The Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. We focus on two particularly interesting scenarios. First, we develop the Einstein clusters in TEGR field equations using effective energy-momentum tensor for diagonal as well as off-diagonal tetrad. We then study the clusters in modified f (T )−gravity for anisotropic fluid distribution. Based on these two theories, we further study the solution without net electric charge and then for charged solution. For charge parameter k → 0, the charged solution reduces to neutral one. Our calculations show that when charge increases, the stiffness of the EoS also increases. This is due to increase in adiabatic index and sound speed approaching speed of light. When the charge increase beyond a certain limit (0 ≤ k ≤ 1.3 × 10 −5 and 0 ≤ k ≤ 1 × 10 −6 ), the compactness parameter crosses the Buchdahl limit i.e. 2M/R > 8/9 and the solution start violating the causality condition. We test the Tolman-Oppenheimer-Volkoff (TOV) limit for such compact objects. We analyze the static stability criterion of the Einstein clusters for both charged and uncharged case, and the stability of such compact objects is enhanced by the presence some net electric charge. In addition, we present and discuss the energy conditions, causality condition and the adiabatic index close to the stability limit. After analyzing these problems, we conclude that the Einstein clusters do exists only if f (T ) is a linear function of the torsion scalar T , that is in the case of Teleparallel Equivalent of General Relativity. Finally, we compare our solution for pure general relativity. As a result, we concluded that the Einstein cluster solution do exist in pure GR, however, physically unfit to mimic compact stars. We have also extend our findings by assuming the diagonal or off-diagonal tetrad and specific case of f (T ). In such models, Einstein's cluster solutions do exist however can't mimic the properties of a compact star.

Deflection angle and lensing signature of covariant f(T) gravity

“…However, one can alternatively start from the equivalent formulation of gravity in terms of torsion, namely the Teleparallel Equivalent of General Relativity (TEGR) [6][7][8], and extend the corresponding Lagrangian, namely the torsion scalar T , in various ways, resulting in f (T ) gravity [9][10][11], in f (T, T G ) gravity [12,13], f (T, B) gravity [14][15][16], scalartorsion theories [17][18][19], etc. Torsional gravity, and in particular f (T ) theory, has been shown to lead to interesting behavior, both in cosmological applications , as well as in the case of spherically symmetric solutions [46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64].…”

Stability of motion and thermodynamics in charged black holes in $f(T)$ gravity

“…and positive heat capacity implies thermodynamic stability. Substituting ( 50) and ( 53) into (55) we obtain the heat capacity as…”

Neutral compact spherically symmetric stars in teleparallel gravity

“…Therefore, we need extra conditions to be able to solve the above system. The extra conditions are the zero radial pressure, namely, p r = 0 [66,67], and assuming the equation of state (EoS) in terms of the energy density and the tangential pressure. We can represent these conditions as…”

“…Another solution that can be derived from Eq. ( 16) is through the assumption of the unknown function µ to have the form [67]…”

“…2(c), because the two forms are identical up to some constant. The gravitational mass of a spherically symmetric source with the radial dependence r is expressed by [67]…”