Do we need dense matter equation of state in curved spacetime for neutron stars?

Neutron stars are regarded as natural laboratories for the study of dense strong interaction matter. The equation state (EoS) matter computed in flat spacetime is used to predict structure neutron by solving Tolman-Oppenheimer-Volkoff (TOV) equation. Recently, it has been reported that curved effect or specifically gravitational time dilation on EoS leads a significant increase maximum mass limit [Phys. Rev. D 104, 123005 (2021) and J. Cosmol. Astropart. Phys. 02 026]. However, this work, we show hydrostatic equilibrium within framework general relativity relativistic fluid dynamics, matter, $p(T,\ensuremath{\mu})$, should be same spacetime, otherwise not consistent with local thermodynamic relations energy-momentum conservation fluid. gravitation influences pressure $p$ only through enhancing temperature $T$ chemical potential $\ensuremath{\mu}$, known Tolman's law Klein's law. We rewrite TOV an alternative version so using field theoretical methods can direct input. This may provide tool via deep learning.