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Neutron stars are regarded as natural laboratories for the study of densestrong interaction matter. The equation of state (EoS) of dense matter computedin flat spacetime is used to predict the structure of neutron stars by solvingthe Tolman-Oppenheimer-Volkoff (TOV) equation. Recently, it has been reportedthat the curved spacetime effect or specifically gravitational time dilationeffect on the EoS of dense matter leads to a significant increase of themaximum mass limit of neutron stars [Phys. Rev. D \textbf{104}, 123005 (2021)and J. Cosmol. Astropart. Phys. 02 (2021) 026]. However, in this work, we showthat to study the hydrostatic equilibrium of dense matter within the frameworkof general relativity and relativistic fluid dynamics, the grand canonical EoSof dense matter, $p(T,\mu)$, should be the same as that computed in flatspacetime, otherwise it is not consistent with local thermodynamic relationsand energy-momentum conservation of the fluid. The gravitation influences thepressure $p$ only through enhancing the temperature $T$ and the chemicalpotential $\mu$, known as Tolman's law and Klein's law. We rewrite the TOVequation as an alternative version so that the grand canonical EoS computed byusing field theoretical methods can be used as a direct input. This may providea tool to study the grand canonical EoS of dense matter via deep learning.

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