Compact anisotropic stellar distribution with linear barotropic equation of state in energy–momentum trace coupling modification of general relativity

Abstract

There are no known exact isotropic or anisotropic stellar models with an equation of state in energy–momentum trace-coupling (EMTC) modifications of general relativity — the simplest linear case of $f(R,T)$ gravity. The difficulty lies in the intricate entanglement of the density and pressure functions that generate intractable governing equations when an equation of state is introduced. For example there is no interior Schwarzschild incompressible star analogue in the well studied $f(R,T)$ theory. In Einstein’s theory it is straightforward to find anisotropic stellar models with a linear equation of state since the system is under-determined and there remains one more choice to nominate any of the variables. This is also true in EMTC theories however, the master field equation is more formidable. If interpreted as a linear second order equation, no viable solutions emerge by prescribing one of the gravitational potentials or by suppressing some terms in the spirit of Tolman (1939). Rewriting as a nonlinear first order equation and speculating on a power-law form of the temporally directed gravitational potential results in success in finding a physically viable compact star distribution. Specifically the model has monotonic decrease of both pressure and density and a surface of vanishing pressure exists. Several stability tests are imposed and the model performs according to expectations. The singularity at the stellar center may be cured by insertion of a regular core enveloped by our fluid model in a multi-layered star.