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

Figure 1

The mass-radius relation of neutron stars (a) and the relation between the neutron star mass and the central baryon density (b) calculated from the Walecka model.

Figure 2

The gravitational potential $\mathrm{\Phi }$ in the interior of neutron stars computed from the Walecka model. (a) $\mathrm{\Phi }$ at the surface and at the core as functions of the central baryon density $n_{\mathrm{B}}^{\mathrm{c}}$. The colored dashed lines with arrows denote the corresponding baryon chemical potentials ${\mu }_{\mathrm{B}}^{\mathrm{c}}$ at the core. (b) The profile $\mathrm{\Phi }(r)$ (thin dashed lines) in the interior of neutron stars with different central densities. The thick black line denotes the potential at the surface.

Figure 3

The baryon chemical potentials ${\mu }_{\mathrm{B}}^{\mathrm{c}}$ at the core, ${\mu }_{\mathrm{B}}^{\mathrm{s}}$ at the surface, and ${\mu }_{\mathrm{B}}^{\infty }$ at infinity. (a) The three chemical potentials corresponding to the neutron stars on the mass-radius curve. (b) The three chemical potentials as functions of the central baryon density.

Figure 4

The EoS $p({\mu }_{\mathrm{B}})$ and the corresponding baryon density $n_{\mathrm{B}}({\mu }_{\mathrm{B}})$ of cold neutron star matter computed from the Walecka model.