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 a compact star in the context of the 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 the TEGR field equations using an effective energy-momentum tensor for a diagonal as well as an 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 a charged solution. For charge parameter $k\ensuremath{\rightarrow}0$, the charged solution reduces to neutral one. Our calculations show that when the charge increases the stiffness of the equation of state also increases. This is due to an increase in the adiabatic index and sound speed approaching the speed of light. When the charge increases beyond a certain limit ($0\ensuremath{\le}k\ensuremath{\le}1.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ and $0\ensuremath{\le}k\ensuremath{\le}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$), the compactness parameter crosses the Buchdahl limit, i.e., $2M/R>8/9$, and the solution starts violating the causality condition. We test the Tolman-Oppenheimer-Volkof limit for such compact objects. We analyze the static stability criterion of the Einstein clusters for both the charged and uncharged cases, and the stability of such compact objects is enhanced by the presence of some net electric charge. In addition, we present and discuss the energy conditions, causality condition, and 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 $T$, which is in the case of the teleparallel equivalent of general relativity. Finally, we compare our solution for pure general relativity. As a result, we conclude that the Einstein cluster solution does exist in pure general relativity; however, it is physically unfit to mimic compact stars. We 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 but cannot mimic the properties of a compact star.