Coordinate cones and the separate stationary generator

Since $g({\mathrm{\partial }}_t,{\mathrm{\partial }}_t)=g_{00}<-C^{-1}$ on $O_1$, the coordinate time vector is uniformly timelike there.

A nonzero tangent vector to $t=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ has the form $v={\sum }_i{Z}^i{\mathrm{\partial }}_{x^i}$ with $Z\mathrm{\ne }0$, so $g(v,v)=g_{ij}{Z}^i{Z}^j\ge C^{-1}\mathrm{\mid }Z{\mathrm{\mid }}^2>0$. The bound on $g(T,T)$ is separately imposed in end clause (3), on ${\mathrm{\Sigma }}_1$.

For the countermodel take $M={\mathbb{R}}^4$ with its standard smooth structure and orientation, $g=-d{t}^2+(d{x}^1{)}^2+(d{x}^2{)}^2+(d{x}^3{)}^2$, time orientation $N={\mathrm{\partial }}_t$, and $T={\mathrm{\partial }}_t$. Set $\mathrm{\Sigma }=\{t=0\}$, ${\mathrm{\Sigma }}_1=\{t=0,\mathrm{\mid }x\mathrm{\mid }>1\}$, ${\mathrm{\Sigma }}^{\mathrm{\prime }}=\{t=0,\mathrm{\mid }x\mathrm{\mid }\le 1\}$ and $O_1=\{(t,x):\mathrm{\mid }x\mathrm{\mid }>1\}$. These are the standard embedded hyperplane, open exterior and closed ball, with their usual smooth structures. For a future causal vector $(a,b)$, the metric and orientation give $a\ge \mathrm{\mid }b\mathrm{\mid }$ and $a>0$. Thus along every nonconstant future causal curve the locally absolutely continuous coordinate $t$ is strictly increasing, by integrating its positive almost-everywhere derivative on any compact parameter subinterval. Consequently $\mathrm{\Sigma }$ is acausal.

Take $R_1=1$, $E=\{z:\mathrm{\mid }z\mathrm{\mid }>1\}$, $\bar{E}=\{z:\mathrm{\mid }z\mathrm{\mid }\ge 1\}$, $e(z)=(0,z)$, $j(z)=(0,z)$ and $h=j{\mathrm{\mid }}_{\mathrm{\partial }\bar{E}}$. The sphere $S=\mathrm{\partial }{\mathrm{\Sigma }}^{\mathrm{\prime }}=\{t=0,\mathrm{\mid }x\mathrm{\mid }=1\}$ is connected; $h$ is a diffeomorphism onto $S$, $j(\bar{E})\cap {\mathrm{\Sigma }}^{\mathrm{\prime }}=S$ and $j(\bar{E})\cup {\mathrm{\Sigma }}^{\mathrm{\prime }}=\mathrm{\Sigma }$. With $C=2$, $k=2$, $\alpha =\lambda =1$, all metric differences, positive-order derivatives and Hölder differences in end clause (1) vanish. Its remaining bounds are $-1<-1\mathrm{/}2$ and $\mathrm{\mid }Z{\mathrm{\mid }}^2\ge \mathrm{\mid }Z{\mathrm{\mid }}^2\mathrm{/}2$, and $r>1$ throughout $O_1$. The complete flow ${\varphi }_u(t,x)=(t+u,x)$ preserves $g$, so $T$ is Killing, $g(T,T)=-1<-1\mathrm{/}2$, and the asymptotics hold with $A=1$ and zero error. This verifies all three end clauses.

Define $K={\mathrm{\partial }}_t+{\mathrm{\partial }}_{x^1}$ and $H=\{t=x^1\}$. Its flow ${\psi }_u(t,x^1,x^2,x^3)=(t+u,x^1+u,x^2,x^3)$ is defined for all real $u$ and preserves the constant metric, so $K$ is a complete Killing field distinct from $T$. The map $(s,y,z)\mapsto (s,s,y,z)$ smoothly embeds ${\mathbb{R}}^3$ onto $H$; the induced form is $d{y}^2+d{z}^2$, with radical spanned by the everywhere nonzero tangent vector $K$. Equivalently, $g(K,\cdot )=-dt+d{x}^1$ annihilates $TH$. Thus $H$ is a Killing horizon for $K$ under [F2]. But $g(K,K)=-1+1=0$ everywhere, including on ${\mathrm{\Sigma }}_1$ and $O_1$. The fully verified end therefore does not force even pointwise timelikeness of a separate complete horizon-normal field, while retaining all coordinate and stationary bounds.