Modeling Neutron Star Stability with a Modified Tolman-Oppenheimer-Volkoff Equation

The Tolman-Oppenheimer-Volkoff (TOV) equation represents the solution to the Einstein field equations where the source of curvature is given by the stress-energy tensor of a perfect fluid. In flat space it has the form $T_{\mu\nu} = (\rho + p)U_\mu U_\nu + p\eta_{\mu\nu}$ and the convention for curved space-time is to just replace the Minkowski metric with $g_{\mu\nu} $. For our research we instead use a modified stress-energy tensor of the form $T_{\mu\nu} = (\rho + p)U_\mu U_\nu + pg_{\mu\nu} + \pi_{\mu\nu} $ where the anisotropic $\pi_{\mu\nu} $ is a symmetric, traceless rank two tensor which obeys $U^\mu\pi_{\mu\nu} = 0$. The motivation is that such a term in the stress-energy tensor can account for effects due to the curvature of space-time and would not be present in the tensor describing flat space.The final revised TOV equation is of the form $-r^2p' = GM\rho[1+\frac{p-2q}{\rho}][1+\frac{4\pi r^3(p-2q)}{M}][1-\frac{2GM}{r}]^{-1}-2r^2q' - 6rq $ where the primes indicate differentiation with respect to the radial coordinate and the q terms arise from the components of $\pi_{\mu\nu} $. The equation was then solved numerically with both a polytropic and a MIT bag model equations of state. The result is a changed prediction for the stability range of neutron stars.