Static spherical Einstein-fluid model with a supplied equation of state

Assume the countable-choice convention ZF+${\mathrm{A}\mathrm{C}}_{\omega }$ used by the exact geometry supplier Smooth static spherical geometry and state domain. Use its product spacetime, areal radius $r$, time coordinate $x^0=ct$, signature $(-,+,+,+)$, and static metric $g=-e^{2\varphi (r)}(d{x}^0{)}^2+b(r{)}^{-1}d{r}^2+r^2\gamma ,$ where $\gamma$ is the unit round metric, $b(r)>0$, and the metric coefficients are independent of $x^0$. Here $x^0$ and $r$ are in metres, $\varphi$ and $b$ are dimensionless, $G>0$ is in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and $c>0$ is in $\mathrm{m}{\mathrm{s}}^{-1}$, as in Thermodynamics-GR fields, currents, reference normalization and units and Smooth static spherical geometry and state domain. The static unit fluid velocity is $u=e^{-\varphi }{\mathrm{\partial }}_{x^0}$.

For an open pressure interval $I_P\subset (0,\mathrm{\infty })$, supply a $C^{\mathrm{\infty }}$ constitutive function $\mathcal{E}:I_P\to (0,\mathrm{\infty })$, with pressure and energy-density values both in $\mathrm{J}{\mathrm{m}}^{-3}$. On any open radial interval $J\subset (0,\mathrm{\infty })$ occupied by the fluid, let $P\in C^{\mathrm{\infty }}(J,I_P)$; that is, $P$ is a smooth real-valued function and $P(r)\in I_P$ for every $r\in J$. Identify $P$ with the pressure component $p{\mathrm{\mid }}_J$ of Smooth static spherical geometry and state domain and set $\epsilon {\mathrm{\mid }}_J=\mathcal{E}\circ P\in C^{\mathrm{\infty }}(J,(0,\mathrm{\infty }))$. Adopt the $\mathrm{\Lambda }=0$ Einstein field equation on $J$ $G^a{}_b=\frac{8\pi G}{c^4}{T}^a{}_b,T^a{}_b{\mathrm{\mid }}_J=\mathrm{diag}(-\mathcal{E}(P(r)),P(r),P(r),P(r)),$ with the signed curvature and Einstein-tensor conventions of Smooth static spherical geometry and state domain. Thus “perfect fluid” here means exactly this static isotropic rest-frame stress tensor on the stated fluid interval; the EOS is a supplied model input, not derived from gravity. This formulation asserts no smoothness across an interface with vacuum or another matter region; any piecewise matching is a separate condition.

This is an adopted classical model and source of conditional predictions. The item asserts no observed star, measured EOS, global zero-pressure surface, causal finite-sound-speed law, or radial/dynamical stability. The cited treatments motivate the static-fluid and EOS conventions; the later local branch and constant-density results are established only by their displayed conditional arguments and mathematical suppliers.