Temporal chart with lapse, shift and positive slice metric

Assume countable choice ACω, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), for the smooth cotangent-bundle carrier convention used by this page; no full Axiom of Choice is assumed.

Let $(M,g,T)$ be a smooth, time-oriented Lorentzian four-manifold of signature $(-+++)$, with the geometric meaning fixed by Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Here $T$ denotes the future-cone field called $N$ in that definition. Let $I\subset \mathbb{R}$ be a nonempty open interval, let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be open, and let $U\subset M$ be open. A temporal product chart is a smooth diffeomorphism $\chi :U\to I\times \mathrm{\Omega }$. Its coordinate functions $x^0,x^1,x^2,x^3$ are the four component functions of $\chi$. Spatial indices $i,j$ range over $1,2,3$, spacetime indices $a,b$ range over $0,1,2,3$, and repeated upper/lower indices are summed. We require $x^0$ and each $x^i$ to have length units in the physical interpretation.

The chart has lapse-shift data $(N,\beta ,h)$ when $N:I\times \mathrm{\Omega }\to (0,\mathrm{\infty })$, ${\beta }^i:I\times \mathrm{\Omega }\to \mathbb{R}$, and $h_{ij}:I\times \mathrm{\Omega }\to \mathbb{R}$ are smooth, $h_{ij}=h_{ji}$, and the matrix $(h_{ij})$ is positive definite at each point, with

$g=-N^2(d{x}^0{)}^2+h_{ij}(d{x}^i+{\beta }^{i}d{x}^0)(d{x}^j+{\beta }^{j}d{x}^0)\mathrm{.}$

Write $h^{ij}$ for the inverse matrix. The normal field is

$n=N^{-1}({\mathrm{\partial }}_0-{\beta }^i{\mathrm{\partial }}_i)\mathrm{.}$

Direct contraction gives $g(n,n)=-1$; require that $n$ is future-directed in the supplied time orientation. This field of observers is distinct from the coordinate vector ${\mathrm{\partial }}_0$, which may be spacelike. The coefficients $N,{\beta }^i,h_{ij}$ are dimensionless when all four coordinates are in metres. This definition is local to the specified product chart; it asserts no global temporal foliation.