Specified constant-density star and vacuum boundary model

Within the adopted model Static spherical Einstein-fluid model with a supplied equation of state, choose a constant rest mass density ${\rho }_0>0$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, so the constant rest energy density is ${\epsilon }_0={\rho }_0{c}^2$ in $\mathrm{J}{\mathrm{m}}^{-3}$. Fix an areal boundary radius $R>0$ in metres, define $M_R=4\pi {\rho }_0{R}^3\mathrm{/}3$ in kilograms and $u=2G{M}_R\mathrm{/}(c^{2}R)$, and restrict the model parameters to $0<u<8\mathrm{/}9$. For this specialization take $I_P=(0,\mathrm{\infty })$ and the supplied equation of state $\mathcal{E}(P)={\epsilon }_0$. Use its explicit $C^{\mathrm{\infty }}$ extension $\widehat{\mathcal{E}}:\mathbb{R}\to (0,\mathrm{\infty })$ given by $\widehat{\mathcal{E}}(P)={\epsilon }_0$; it agrees with $\mathcal{E}$ on $I_P$ and defines the fluid EOS on an open pressure interval containing zero and every nonnegative pressure in the branch. On the open interior fluid interval $0<r<R$, the pressure profile $P:(0,R)\to I_P$ is smooth and has the continuous interior boundary value $P(R)=0$, interpreted using $\widehat{\mathcal{E}}$. Prescribe vacuum stress separately for $r\ge R$, with exterior mass $M_R$; the vacuum region is not governed by the fluid EOS.

This is a constitutive and boundary preparation within the stated static Einstein-fluid model. The uniform-density relation is a model input; it is not a finite-sound-speed causal transport law. The range $0<u<8\mathrm{/}9$ defines this selected branch and does not assert a general Buchdahl bound, a realistic stellar equation of state, an observed star, or stability.