Positive-scalar conformastatic chart data

Let $D\subseteq {\mathbb{R}}^3$ be open and let $H:D\to (0,\mathrm{\infty })$ be of class $C^3$, meaning that every partial derivative of total order at most three exists and is continuous. Set $\mathcal{M}:={\mathbb{R}}_{x^0}\times D$, an open subset of ${\mathbb{R}}^4$, with global coordinates $(x^0,x^1,x^2,x^3)$; each coordinate is assigned length units, with $x^0=ct$ when a time coordinate is wanted. The function $H$ depends only on $(x^1,x^2,x^3)$ and is dimensionless.

Give $\mathcal{M}$ the coordinate orientation determined by $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$.

Define the symmetric coordinate tensor $g_H$ by $g_H=-H^{-2}(d{x}^0{)}^2+H^2{\sum }_{i=1}^3(d{x}^i{)}^2,$ where $H^{-1}:=1\mathrm{/}H$ and $H^{-2}:=(1\mathrm{/}H{)}^2$. This exact coordinate form is called the positive-scalar conformastatic chart tensor in this item; no field equation or physical solution property is part of the definition.

For this $C^3$ chart tensor, a time orientation means a continuous selection at each $p\in \mathcal{M}$ of one of the two connected components of $\{v\in T_p\mathcal{M}:g_H(v,v)<0\}$, with the topology from the coordinate identification $T\mathcal{M}=\mathcal{M}\times {\mathbb{R}}^4$. Specify the selection globally by $C_p^{+}:=\{v=v^0{\mathrm{\partial }}_0+{\sum }_{i=1}^3{v}^i{\mathrm{\partial }}_i:g_H(v,v)<0,v^0>0\}\mathrm{.}$ Here the selection is continuous in the explicit sense that the fiberwise linear coordinate map $(p,v)\mapsto (p,v^0\mathrm{/}H(p),H(p)v^1,H(p)v^2,H(p)v^3)$ is a continuous bundle isomorphism carrying $C^{+}$ to the constant cone $\{(u^0,\mathbf{u}):u^0>\mathrm{\parallel }\mathbf{u}{\mathrm{\parallel }}_2\}$; this map and its inverse are $C^3$. That cone and its negative are convex and nonempty, and a timelike vector has nonzero $u^0$, so they are exactly the two timelike components. Define a nonzero causal vector by $g_H(v,v)\le 0$ and call it future-directed when $v^0>0$. Set $N_H:=H{\mathrm{\partial }}_0$, a $C^3$ vector field (smooth if $H$ is smooth). The Lorentzian lemma below verifies $g_H(N_H,N_H)=-1$, so $N_H\in C_p^{+}$ everywhere. No smoothness beyond $C^3$ is required for this definition.

Because $H>0$, the functions $H^{-1}$ and $H^{-2}$ are compositions and products of $H$ with positive-base powers of exponents $-1$ and $-2$. The derivative formula for these powers, repeated through any finite order, and closure of $C^k$ functions under composition and finite products show that the displayed coefficients are $C^3$ when $H$ is $C^3$, and smooth when $H$ is smooth.

For later component formulas define $H_i:={\mathrm{\partial }}_{i}H$, $H_{ij}:={\mathrm{\partial }}_i{\mathrm{\partial }}_{j}H$, $q:={\sum }_{i=1}^3{H}_i^2$, and $\mathrm{\Delta }H:={\sum }_{i=1}^3{H}_{ii}$.

The tensor coefficients are $C^3$ by the stated regularity and positivity of $H$. Its nondegeneracy, Lorentzian signature, and the timelike property needed for the chosen time orientation are proved by The positive-scalar conformastatic chart tensor is Lorentzian ↗; this definition alone does not assert them.