Static hydrostatic background and equation-of-state data

Let $\mathrm{\Sigma }$ be a nonempty compact smooth three-manifold without boundary, let $h$ be a smooth Riemannian metric on $\mathrm{\Sigma }$, and give $\mathbb{R}$ its usual topology and the standard smooth structure generated by the global identity chart ${\mathrm{id}}_{\mathbb{R}}:\mathbb{R}\to \mathbb{R}$. Give $M=\mathbb{R}\times \mathrm{\Sigma }$ the canonical product smooth structure. Adopt a constant $c>0$ in metres per second as the length/time conversion, write $x^0=c{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ for the standard length coordinate on $\mathbb{R}$, where $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is time in seconds, and use spatial coordinates measured in metres. For a smooth function $N:\mathrm{\Sigma }\to (0,\mathrm{\infty })$, extend $N$ constantly along $\mathbb{R}$ and define the metric on $M$ by $g=-N^{2}d{x}^0\otimes d{x}^0+h$, where $h$ is pulled back from $\mathrm{\Sigma }$. The metric components and lapse $N$ are dimensionless in these length coordinates. This is a static spacetime model: the coefficients are independent of $x^0$, ${\mathrm{\partial }}_{x^0}$ is the selected future timelike Killing field, and its lapse is defined here by $N_{\xi }:=\sqrt{-g(\xi ,\xi )}$ for $\xi ={\mathrm{\partial }}_{x^0}$, so $g({\mathrm{\partial }}_{x^0},{\mathrm{\partial }}_{x^0})=-N^2$ and $N_{\xi }=N$.

The Riemannian volume density of $h$ is denoted by $d{V}_h$; in length coordinates it has units of cubic metres and finite total volume, and no orientation of $\mathrm{\Sigma }$ is required. In a fixed SI gauge, let $\mathrm{\Phi }:\mathrm{\Sigma }\to \mathbb{R}$ be a smooth electrostatic scalar potential in volts, let $q\in \mathbb{R}$ be a fixed charge in coulombs, let $T_0>0$ be a reference temperature in kelvins, and let ${\mu }_0\in \mathbb{R}$ be a reference chemical-potential parameter in joules. The sign convention is that the local electrochemical parameter is ${\mu }_0-q\mathrm{\Phi }$. For the static Killing field $\xi ={\mathrm{\partial }}_{x^0}$, its lapse is $N_{\xi }=N$; fix the Killing-normalized temperature $T_K:=N_{\xi }T$ to be the constant $T_0$, so the local temperature is $T(x)=T_0\mathrm{/}N(x)$. Define the local chemical-potential field by $\mu (x)=({\mu }_0-q\mathrm{\Phi }(x))\mathrm{/}N(x)$ for $x\in \mathrm{\Sigma }$; then $q\mathrm{\Phi }$ and $\mu$ have units of joules.

Let $D\subset (0,\mathrm{\infty })\times \mathbb{R}$ be an open domain for the pair $(T,\mu )$, measured in kelvins and joules, and let $p:D\to \mathbb{R}$ be a supplied $C^2$ pressure function with values in pascals. Require $(T(x),\mu (x))\in D$ at every $x\in \mathrm{\Sigma }$. The function $p$ is model input; this definition does not derive an equation of state or assert that the prescribed data have a material realization.

Smooth manifolds and their smooth charts

The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Killing fields of a Lorentzian metric

Products of smooth manifolds have a canonical product smooth structure

Riemannian metric and riemannian manifold

Riemannian volume density

The riemannian volume density is coordinate independent

Riemannian volume is the radon measure of the riemannian density

Measurable integration extends smooth density integration

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)