Stationary section time and translated maximal sections

PDF p. 23: §4, before Lemma 4.1

Fix $q\in (0,1]$. Use the stationary maximal-graph data: the exterior is identified with $\mathcal{E}={\mathbb{R}}_t\times {\mathrm{\Sigma }}_1^{\circ }$, the stationary field is $T={\mathrm{\partial }}_t$, and the supplied section is the graph ${\mathrm{\Sigma }}_q^{\circ }=\{(u_q(x),x):x\in {\mathrm{\Sigma }}_1^{\circ }\}$. Assume that $g_q$ is a smooth time-oriented Lorentzian metric invariant under the $T$-flow, that this graph is spacelike and maximal for $g_q$, and that its lapse relative to the future unit normal is positive. A maximal spacelike section means a spacelike hypersurface whose mean curvature vanishes. A section ${\mathrm{\Sigma }}_q$ with interior ${\mathrm{\Sigma }}_q^{\circ }$ and boundary $S$ is supplied separately.

Define the stationary section time by $v_q(t,x)=t-u_q(x)$. For any local spatial coordinates $(x^i)$ on ${\mathrm{\Sigma }}_1^{\circ }$, use the same coordinate functions on the product, so that they are constant along $T$-orbits. For $a\in \mathbb{R}$, set ${\mathrm{\Sigma }}_{q,a}^{\circ }=\{p\in \mathcal{E}:v_q(p)=a\}$ and ${\mathrm{\Sigma }}_{q,a}={\mathrm{\Sigma }}_{q,a}^{\circ }\cup S$; in particular ${\mathrm{\Sigma }}_{q,0}={\mathrm{\Sigma }}_q$. The boundary notation records the supplied common set $S$ and does not by itself assert smooth boundary regularity for every translate.

On ${\mathrm{\Sigma }}_q^{\circ }$, let $n_q$ be the future-directed unit normal, $h_q=g_q{\mathrm{\mid }}_{T{\mathrm{\Sigma }}_q^{\circ }}$ the induced metric, $N_q=-g_q(T,n_q)$ the lapse, and $X_q=T-N_q{n}_q$ the tangent shift. Thus $T=N_q{n}_q+X_q$ and $N_q>0$ by hypothesis. All chronological and causal relations used with these conventions are relative to the exterior $\mathcal{E}$ with metric $g_q$: their curves are required to remain in $\mathcal{E}$.

Stationary maximal graphs and normalized graph heights

Stationary Lorentzian data and tensor conventions

Spacelike sections, maximality and lapse–shift data

Lapse, shift, induced metric and second fundamental form

Causal exterior, chronological boundaries and Cauchy data

**Given:**The product, graph, metric, flow, positive lapse and boundary data in the Definition.

Goal: Justify the section time, transported coordinates, translated sets and lapse–shift notation.

1.1 Smoothness of $u_q$ makes $v_q$ smooth. Because $T={\mathrm{\partial }}_t$, one has $T{v}_q=1$, while $v_q(u_q(x),x)=0$. Conversely, each point $(t,x)$ has the unique representation as the flow by $t-u_q(x)$ of the graph point $(u_q(x),x)$. Hence $v_q$ is the unique function with these two properties, the transported $x^i$ are constant on each orbit, and ${\mathrm{\Sigma }}_{q,a}^{\circ }$ is the flow translate of ${\mathrm{\Sigma }}_q^{\circ }$ by parameter $a$. [given, F1, F2]

2.1 Write ${\mathrm{\Phi }}_a(t,x)=(t+a,x)$ for the supplied flow. The field $T={\mathrm{\partial }}_t$ is nowhere zero on the product exterior and is Killing for $g_q$ by the assumed invariance, so [F5] gives preservation of its Levi–Civita connection ${\mathrm{\nabla }}^q$ on the flow domains. The maps ${\mathrm{\Phi }}_a$ are globally defined here by the product, and this connection identity is local at every point. Metric invariance makes their translated interiors spacelike. Put $n_{q,a}={\mathrm{\Phi }}_{a\ast }{n}_q$ along ${\mathrm{\Sigma }}_{q,a}^{\circ }$. It is unit timelike and orthogonal to that interior. It is future-directed because, for each graph point $p$, the continuous path $s\mapsto {\mathrm{\Phi }}_{s\ast }{n}_q(p)$ consists of timelike vectors based at ${\mathrm{\Phi }}_s(p)$ and starts future-directed at $s=0$, so it cannot change time cone. Let $h_{q,a}$ and $k_{q,a}$ be the induced metric and second fundamental form with this normal, using the convention in [F3]. For graph-tangent vectors $U,V$, metric and connection preservation give $h_{q,a}({\mathrm{\Phi }}_{a\ast }U,{\mathrm{\Phi }}_{a\ast }V)=h_q(U,V),k_{q,a}({\mathrm{\Phi }}_{a\ast }U,{\mathrm{\Phi }}_{a\ast }V)=-g_q({\mathrm{\nabla }}_{{\mathrm{\Phi }}_{a\ast }U}^q({\mathrm{\Phi }}_{a\ast }{n}_q),{\mathrm{\Phi }}_{a\ast }V)=-g_q({\mathrm{\Phi }}_{a\ast }({\mathrm{\nabla }}_U^q{n}_q),{\mathrm{\Phi }}_{a\ast }V)=k_q(U,V)\mathrm{.}$ Taking the trace with the transported induced metric yields $H_{q,a}\circ {\mathrm{\Phi }}_a=-{\mathrm{tr}}_{h_q}{k}_q=H_q=0$ on ${\mathrm{\Sigma }}_q^{\circ }$. Thus every translated interior is maximal. [given, F2, F3, F5, step 1.1]

3.1 The supplied boundary set defines the displayed unions without an additional boundary-regularity assertion. The hypersurface interface supplies the unique future unit normal and the induced metric, and the lapse–shift formulas give the tangent decomposition; positivity is the stated hypothesis. The causal convention specifies the domain in which the relevant curves are considered. [given, F3, F4, step 1.1, step 2.1] ∎