Static isotropic Einstein-fluid and supplied EOS model

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.

Assume the adopted model Static spherical Einstein-fluid model with a supplied equation of state with its supplied $C^{\mathrm{\infty }}$ positive EOS $\mathcal{E}:I_P\to (0,\mathrm{\infty })$ on an open pressure interval $I_P\subset (0,\mathrm{\infty })$. Let $P_c\subset I_P$ be a nonempty compact interval, let $\delta >0$ be such that its closed $\delta$-neighbourhood lies in $I_P$, and choose a central pressure $P_0\in P_c$ and a dimensionless central lapse $e^{{\varphi }_c}>0$. Under the exact mathematical convention $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ of Regular-center TOV local existence and dependence, put $K=4\pi G\mathrm{/}{c}^4$. There is a common $R_0>0$ and, for each $P_0\in P_c$, smooth even profiles $P,v$ on $[-R_0,R_0]$ with $P(r)\in I_P$ for every $\mathrm{\mid }r\mathrm{\mid }\le R_0$ and satisfying $v(r)=K{\int }_0^1{s}^2\mathcal{E}(P(rs))ds,P(r)=P_0-{\int }_0^{r}t\frac{(\mathcal{E}(P(t))+P(t))(v(t)+KP(t))}{1-2t^{2}v(t)}dt,$ with $P(0)=P_0$, $v(0)=K\mathcal{E}(P_0)\mathrm{/}3$, $b(r):=1-2r^{2}v(r)\ge 1\mathrm{/}2$, and $P^{\mathrm{\prime }}(0)=v^{\mathrm{\prime }}(0)=0$. Defining $M(r)=\frac{c^2}{G}{r}^{3}v(r),\varphi (r)={\varphi }_c+{\int }_0^{r}t\frac{v(t)+KP(t)}{1-2t^{2}v(t)}dt,$ gives a positive-lapse $C^2$ metric center and a local solution of the adopted Einstein-fluid equations. On $r>0$ its equations are $M^{\mathrm{\prime }}=\frac{4\pi r^2\mathcal{E}(P)}{c^2},{\varphi }^{\mathrm{\prime }}=\frac{GM\mathrm{/}{c}^2+4\pi G{r}^{3}P\mathrm{/}{c}^4}{r^{2}b},P^{\mathrm{\prime }}=-\frac{G(\mathcal{E}(P)+P)(M+4\pi r^{3}P\mathrm{/}{c}^2)}{c^2{r}^{2}b}\mathrm{.}$ No conclusion is made that the branch reaches a finite zero-pressure surface, remains horizon-free globally, or is dynamically stable.

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.

Assume $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ exactly as in Proper spherical entropy and particle constrained functionals. Adopt a separate static, spherically symmetric, isotropic $\mathrm{\Lambda }=0$ Einstein-fluid preparation using the geometry and $(-,+,+,+)$, $x^0=ct$ conventions of Smooth static spherical geometry and state domain, without a separate barotropic relation $\epsilon =\mathcal{E}(P)$. The adopted field equation is $G^a{}_b=(8\pi G\mathrm{/}{c}^4)T^a{}_b$ with $T^a{}_b=\mathrm{diag}(-\epsilon (r),P(r),P(r),P(r))$ and the signed curvature convention in Smooth static spherical geometry and state domain. Fix a radius $R>0$, and supply a $C^2$ open state chart $\mathcal{U}\subset (0,\mathrm{\infty }{)}^2$ with coordinates energy density $\epsilon$ in $\mathrm{J}{\mathrm{m}}^{-3}$ and fixed-composition marker number density $n$ in ${\mathrm{m}}^{-3}$. Supply $s\in C^2(\mathcal{U},\mathbb{R})$ in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$ with $s_{\epsilon }>0$, and define $\mathrm{\Theta }:=1\mathrm{/}{s}_{\epsilon },\mu :=-\mathrm{\Theta }{s}_n,P:=\mathrm{\Theta }s+\mu n-\epsilon \mathrm{.}$ Assume this chart is compatible with the adopted material energy-per-marker function $e(v,{\sigma }_p)$ of Restricted local equilibrium parcel with adopted material state function: $v=1\mathrm{/}n$, ${\sigma }_p=s\mathrm{/}n$, $\epsilon =ne(1\mathrm{/}n,s\mathrm{/}n)$, and $\mathrm{\Theta }=e_{{\sigma }_p}>0$ on the supplied chart. The symbols $\mathrm{\Theta },\mu ,P$ respectively have units kelvins, joules, and $\mathrm{J}{\mathrm{m}}^{-3}$. The local identities $d\epsilon =\mathrm{\Theta }ds+\mu dn$ and $dP=sd\mathrm{\Theta }+nd\mu$ are those of the exact mathematical chart definition Proper spherical entropy and particle constrained functionals.

For positive even $C^2$ profiles $(\epsilon (r),n(r))$ with compact range in $\mathcal{U}$, define $M,b,S,N_b,E$ exactly by Proper spherical entropy and particle constrained functionals and require $b(r)\ge b_{\ast }>0$ on $[0,R]$. The adopted proper entropy and particle number are $S=4\pi {\int }_0^{R}s{r}^2{b}^{-1\mathrm{/}2}dr$ and $N_b=4\pi {\int }_0^{R}n{r}^2{b}^{-1\mathrm{/}2}dr$, while the constrained mass-energy is $E=c^{2}M(R)=4\pi {\int }_0^R\epsilon r^{2}dr$. Restrict variations to the supplied positive-$b$ chart at fixed $R$ and fixed $E,N_b$, and choose the boundary time normalization $e^{\varphi (R)}=\sqrt{b(R)}$.

This model supplies local material thermodynamics and a gravitational mass constraint as assumptions. Its variation problem is not an independent derivation of all Einstein equations. A stationary profile is only a first-variation critical point; no entropy maximum, existence of a maximizing profile, boundary vacuum matching, or physical observation is assumed.

Assume the constant-density preparation Specified constant-density star and vacuum boundary model and the exact $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ mathematical convention used by Exact regular constant-density solution with verified matching. Put $a=\sqrt{1-u}$ and $B(r)=\sqrt{1-u{r}^2\mathrm{/}{R}^2}$. For $0\le r\le R$, the adopted model has the explicit profiles $M(r)=M_R(r\mathrm{/}R{)}^3,b(r)=B(r{)}^2,P(r)={\epsilon }_0\frac{B(r)-a}{3a-B(r)},e^{\varphi (r)}=\frac{3a-B(r)}{2}\mathrm{.}$ For $r\ge R$, set $M(r)=M_R$, $b(r)=1-2G{M}_R\mathrm{/}(c^{2}r)$, $P(r)=0$, $\epsilon (r)=0$, and $e^{\varphi (r)}=\sqrt{b(r)}$. Then the interior is a regular static Einstein-fluid solution through the center, the exterior is vacuum, and the join at $r=R$ is $C^1$ and piecewise $C^2$ with no delta-supported surface stress. The central pressure is finite and equals $P(0)={\epsilon }_0(1-a)\mathrm{/}(3a-1)$. This conclusion is specific to the constant-density branch $0<u<8\mathrm{/}9$.

Assume the model preparation Supplied thermodynamic chart and gravitational entropy variation model, including a positive even $C^2$ profile pair with compact state-chart range, fixed $R>0$, $b\ge b_{\ast }>0$, fixed constraints $E,N_b$, and boundary normalization $\varphi (R)=\frac{1}{2}\log b(R)$. Use the exact $\mathrm{Z}\mathrm{F}+{\mathrm{A}\mathrm{C}}_{\omega }$ convention of Constrained entropy criticality, Tolman–Klein equivalence and TOV consequence. An admissible profile curve is $C^1$ in the sup-norm space $C^0([0,R]{)}^2$ on an open interval about zero, takes values in the even $C^2$ positive-b profile domain with range in a compact subset of the open state chart, and has a smooth derivative at zero compactly supported in $(0,R)$ and extended evenly at the center. Define constrained stationarity to mean that $S$ has zero derivative at the base profile along every such curve preserving both $E$ and $N_b$. Then constrained stationarity is equivalent to the existence of constants $\alpha ,{\mathrm{\Theta }}_K,{\mu }_K$ such that on $(0,R)$ $\frac{\mu }{\mathrm{\Theta }}=\alpha ,\mathrm{\Theta }{e}^{\varphi }={\mathrm{\Theta }}_K,\mu e^{\varphi }={\mu }_K,$ and the pressure obeys $P^{\mathrm{\prime }}=-(\epsilon +P){\varphi }^{\mathrm{\prime }}=-\frac{G(\epsilon +P)(M+4\pi r^{3}P\mathrm{/}{c}^2)}{c^2{r}^{2}b}\mathrm{.}$ Here ${\mathrm{\Theta }}_K=\mathrm{\Theta }(R)\sqrt{b(R)}$ and ${\mu }_K=\mu (R)\sqrt{b(R)}$. This is a conditional first-variation result in the adopted model, not a claim that nature selects such a profile or that it is an entropy maximum.

Use the finite relative Kepler subsystem and Gibbs-volume entropy convention of Confined relative Kepler shell and volume entropy preparation, with $R>16r_0$. In that model, choose $E_G=-e_{\ast }$ with $A\mathrm{/}R<e_{\ast }<A\mathrm{/}(16r_0)$, strictly inside the energy branch, and restrict to an open interval ${\mathcal{I}}_G$ about $x_{\ast }=E_G$ whose energies stay in that interior. The supplied formula $\mathrm{\Omega }(-e)=C{e}^{-3\mathrm{/}2}J(e{r}_0\mathrm{/}A)$ has $J(\delta )={\int }_{\delta }^1{s}^{1\mathrm{/}2}(1-s{)}^{3\mathrm{/}2}ds$ and $J^{\mathrm{\prime }}(\delta )=-{\delta }^{1\mathrm{/}2}(1-\delta {)}^{3\mathrm{/}2}$; because $\delta$ stays in $(0,1\mathrm{/}16)$, this derivative is smooth there, so $J$, $\mathrm{\Omega }$, and $S_G=k_B\log \mathrm{\Omega }$ are $C^2$ on ${\mathcal{I}}_G$. The mathematical result Negative Gibbs-volume capacity in a finite regulated shell gives $T_G(x_{\ast })=T_{\ast }>0$ and finite $C_G(x_{\ast })<0$ on this branch; restrict ${\mathcal{I}}_G$ if necessary so these strict signs persist. For a physical Newtonian slow-motion interpretation, also retain the upstream condition $2A\mathrm{/}(\mu r_0{c}^2)\le {\delta }_{\mathrm{N}\mathrm{R}}\ll 1$ for a stated tolerance ${\delta }_{\mathrm{N}\mathrm{R}}$; the scalar exchange calculation itself uses only the explicitly adopted model.

As a separate finite bath, choose $C_b>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$, reference energy $E_0$ in joules, reference temperature $T_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ in kelvins, and entropy constant $S_0$ in $\mathrm{J}{\mathrm{K}}^{-1}$. For bath energy $E_b>E_0$, adopt $S_b(E_b)=C_b\log \text{\hspace{0.17em}⁣}\left(\frac{E_b-E_0}{C_b{T}_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)+S_0\mathrm{.}$ The logarithm argument is dimensionless, and this stipulated bath has $T_b=(d{S}_b\mathrm{/}d{E}_b{)}^{-1}=(E_b-E_0)\mathrm{/}{C}_b$ and constant heat capacity $C_b$. Fix total energy $E_{\mathrm{t}\mathrm{o}\mathrm{t}}=x+E_b$, choose $E_b=E_0+C_b{T}_{\ast }$ to match the selected subsystem temperature, and restrict $0<C_b<\mathrm{\mid }{C}_G(x_{\ast })\mathrm{\mid }$.

Finally adopt the effective scalar energy-exchange law $\dot{x}=L\frac{d}{dx}[S_G(x)+S_b(E_{\mathrm{t}\mathrm{o}\mathrm{t}}-x)],L>0\mathrm{J}\mathrm{K}{\mathrm{s}}^{-1}\mathrm{.}$ The law is a selected one-dimensional model with time measured in seconds; it does not assert a relativistic transport equation or a mechanism for thermalization. The literature citations give gravitational-thermodynamic context only; the explicit bath and ODE are adopted here, and their mathematical consequences are proved conditionally by the linked local supplier.

Assume the finite-bath preparation Selected finite bath and effective energy-exchange preparation, which supplies $C^2$ subsystem and bath entropies on open energy intervals, with subsystem temperature $T_{\ast }>0$, finite capacity $C_G<0$, bath capacity $C_b$ satisfying $0<C_b<\mathrm{\mid }{C}_G\mathrm{\mid }$, conserved total energy, and exchange law $\dot{x}=L{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{\prime }}(x)$ for $L>0$. At the matched-temperature point $x_{\ast }$, $S_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}(x_{\ast })=-\frac{1}{T_{\ast }^2}(\frac{1}{C_G}+\frac{1}{C_b})<0.$ Consequently $x_{\ast }$ is a strict local entropy maximum in this energy-exchange direction, the adopted ODE has a forward-invariant local basin with exponential attraction to $x_{\ast }$, and along each solution $d{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{/}dt=L(S_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{\prime }}{)}^2\ge 0$. These are properties of the selected scalar exchange model only.