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 $k \rightarrow 0$, 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 ($0\le k \le 1.3 \times 10^{-5}$ and $0\le k \le 1 \times 10^{-6}$), 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. 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.