Causal And Initial Value Boundaries — Examples

(The level sets.) By [L4] each ${\mathrm{\Sigma }}_c$ is an embedded three-dimensional submanifold with tangent space $\mathrm{span}({\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3)$ on which $\eta$ restricts to $\mathrm{diag}(1,1,1)>0$; hence ${\mathrm{\Sigma }}_c$ is a spacelike embedded hypersurface. Its normal line is spanned by ${\mathrm{\partial }}_0$ with $\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1$, and $g({\mathrm{\partial }}_0,N)=g({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1<0$ with $N={\mathrm{\partial }}_0$, so ${\mathrm{\partial }}_0$ is the future unit normal. This proves (i).

(The coordinate is temporal.) The function $t=x^0$ has differential $dt$ with $dt({\mathrm{\partial }}_0)=1$ and $dt({\mathrm{\partial }}_a)=0$ for $a=1,2,3$; the vector field $w:=-{\mathrm{\partial }}_0$ satisfies $\eta (w,X)=dt(X)$ for every $X$ because $\eta (-{\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)(-1)=1$ and $\eta (-{\mathrm{\partial }}_0,{\mathrm{\partial }}_a)=0$; hence $\mathrm{\nabla }t=w$ with $\eta (\mathrm{\nabla }t,\mathrm{\nabla }t)=-1<0$ and $dt({\mathrm{\partial }}_0)=1>0$. This is (ii).

(The double cone is causally convex.) Let $p\in A$, $\epsilon >0$, let $\gamma :[a,b]\to A$ be future causal with $\gamma (a),\gamma (b)\in V_{\epsilon }(p)$ and let $a<s<b$. Since $\gamma (a)\in I^{+}(p-\epsilon e_0)$, [L1] gives $\gamma (a)-(p-\epsilon e_0)\in C^{+}$; since $\gamma {\mathrm{\mid }}_{[a,s]}$ is future causal, [L1] gives $\gamma (s)-\gamma (a)\in \overline{C^{+}}$; by [L3], $\gamma (s)-(p-\epsilon e_0)\in C^{+}$, so $p-\epsilon e_0\ll \gamma (s)$. Similarly $\gamma (b)\in I^{-}(p+\epsilon e_0)$ gives $(p+\epsilon e_0)-\gamma (b)\in C^{+}$; since $\gamma {\mathrm{\mid }}_{[s,b]}$ is future causal, $\gamma (b)-\gamma (s)\in \overline{C^{+}}$ and [L3] gives $(p+\epsilon e_0)-\gamma (s)\in C^{+}$, so $\gamma (s)\ll p+\epsilon e_0$. Hence $\gamma (s)\in V_{\epsilon }(p)$, which is causally convex. The displayed coordinate description follows from [L1]: $q\in V_{\epsilon }(p)$ iff $(x^0(q)-x^0(p)+\epsilon ,z(q)-z(p))\in C^{+}$ and $(x^0(p)+\epsilon -x^0(q),z(p)-z(q))\in C^{+}$, i.e. $\mathrm{\mid }z(q)-z(p){\mathrm{\mid }}_{\mathrm{e}}<\epsilon +x^0(q)-x^0(p)$ and $\mathrm{\mid }z(q)-z(p){\mathrm{\mid }}_{\mathrm{e}}<\epsilon -x^0(q)+x^0(p)$, which is the stated inequality. This proves (iii).

(The diamond is compact.) Let $r\in A$ with coordinates $(t_r,z_r)$. By [L1], $r\in J^{+}(o)$ means $t_r\ge \mathrm{\mid }{z}_r{\mathrm{\mid }}_{\mathrm{e}}$ and $r\in J^{-}(T{e}_0)$ means $T-t_r\ge \mathrm{\mid }{z}_r{\mathrm{\mid }}_{\mathrm{e}}$; this is the displayed description. The set described is closed (non-strict inequalities of continuous functions) and bounded: $t_r\in [0,T]$ and $\mathrm{\mid }{z}_r{\mathrm{\mid }}_{\mathrm{e}}\le \min (t_r,T-t_r)\le T\mathrm{/}2$. By [L5] it is compact, and its $t$-section is the closed ball of radius $\min (t,T-t)$ in ${\mathbb{R}}^3$. This proves (iv).

(The domain of dependence is everything.) Let $p\in A$ with $t(p)\ge c$, and let $\gamma :I\to A$ be any past-inextendible causal curve through $p$. I claim $\inf (t\circ \gamma )(I)=-\mathrm{\infty }$. Otherwise put $m:=\inf (t\circ \gamma )(I)>-\mathrm{\infty }$. Since $t\circ \gamma$ is increasing, it tends to $m$ as the parameter tends to $\inf I$. For every $\epsilon >0$ choose a parameter $a\in I$ with $m\le t(\gamma (a))<m+\epsilon$. For any $u,v\in I$ with $u,v\le a$, the Lipschitz bound [L2] gives $\mathrm{\mid }z(\gamma (u))-z(\gamma (v)){\mathrm{\mid }}_{\mathrm{e}}\le \mathrm{\mid }t(\gamma (u))-t(\gamma (v))\mathrm{\mid }<\epsilon$. Thus the full spatial tail is Cauchy as the parameter tends to $\inf I$; by completeness [L8] it converges to some $z_{\mathrm{\infty }}\in {\mathbb{R}}^3$. Since $t(\gamma (\cdot ))\to m$, the full curve tail converges to $x^{-1}(m,z_{\mathrm{\infty }})\in A$, contradicting past-inextendibility by clause (d) of Causal curves, chronological and causal reachability, and inextendibility. Hence its time range is unbounded below. Since $t\circ \gamma$ is continuous and strictly increasing on the interval $I$ and attains $t(p)\ge c$ at the parameter of $p$, its image contains $(-\mathrm{\infty },t(p)]$, in particular $c$; so some parameter lies on ${\mathrm{\Sigma }}_c$ and $p\in D^{+}({\mathrm{\Sigma }}_c)$. The case $t(p)\le c$ is the time reverse: every future-inextendible causal curve through $p$ has time range unbounded above and hence meets ${\mathrm{\Sigma }}_c$, so $p\in D^{-}({\mathrm{\Sigma }}_c)$. Therefore $D^{+}({\mathrm{\Sigma }}_c)\cup D^{-}({\mathrm{\Sigma }}_c)=A$, which is (v).

(Conclusion.) Parts (i)-(v) are steps 1.1, 1.2, 2.1, 2.2 and 2.3; this completes the example.

(The curve is a unit timelike geodesic.) Write $y(s):=(s,z_0)$ for the coordinate path of $\gamma$. Then $\dot{y}=(1,0,0,0)$ and $\ddot{y}=0$, so the geodesic equation holds by [L2] because the Christoffel symbols vanish; hence $\gamma$ is a geodesic with affine parameter $s$. Its tangent is ${\gamma }^{\mathrm{\prime }}(s)={\mathrm{\partial }}_0$, with $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1<0$ and $g({\gamma }^{\mathrm{\prime }},N)=-1<0$; so it is future unit timelike and the chart geodesic equation is $\ddot{y}=0$. This is (i).

(Affine and length computations.) By step 1.1, $\sqrt{-g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})}=1$ on $(0,1)$; the length of the segment over $[s_1,s_2]$ is therefore ${\int }_{s_1}^{s_2}1ds=s_2-s_1$, equal to the affine separation, and [L3] gives the metric-defined proper-time increment $(s_2-s_1)\mathrm{/}c$. This proves (ii).

(Maximal domain, limit point and conclusion.) By [L4] no geodesic of $M$ with the data of $\gamma$ extends the parameter interval beyond $(0,1)$, and $\gamma$ does not converge to a point of $M$ as $s\to 1$: its coordinate $x^0(\gamma (s))=s$ tends to $1$, while every point of $M$ has $x^0\in (0,1)$. Hence $\gamma$, restricted to the maximal geodesic domain $(0,1)$, is future-inextendible in the sense of Causal curves, chronological and causal reachability, and inextendibility clause (d) and its affine parameter is bounded above; the slab is timelike geodesically incomplete. By [L4] the slice ${\mathrm{\Sigma }}_{1\mathrm{/}2}$ is a Cauchy hypersurface and the model is globally hyperbolic, which is (iii) and (iv).

(The field is a smooth unit future timelike geodesic congruence.) On $O$ the function $\rho$ is smooth and strictly positive, and the components $W^a=(e^a-q^a)\mathrm{/}\rho$ are smooth; by [L2], $\eta (W,W)=-1$ and $W$ is future-directed relative to the time-orientation field $N={\mathrm{\partial }}_0$. It remains to verify the geodesic condition. Put $y:=e-q$, so $y^0=T-t$ and $y^i=-z^i$. From ${\rho }^2=(T-t{)}^2-\mathrm{\mid }z{\mathrm{\mid }}_{\mathrm{e}}^2$ one computes ${\mathrm{\partial }}_0\rho =-(T-t)\mathrm{/}\rho =y_0\mathrm{/}\rho$ and ${\mathrm{\partial }}_i\rho =-z^i\mathrm{/}\rho =y_i\mathrm{/}\rho$, hence ${\mathrm{\partial }}_b\rho =y_b\mathrm{/}\rho$ with $y_b={\eta }_{bc}{y}^c$; also ${\mathrm{\partial }}_b{y}^a=-{\delta }_b^a$. Therefore ${\mathrm{\partial }}_b{W}^a={\mathrm{\partial }}_b\left(\frac{y^a}{\rho }\right)=-\frac{{\delta }_b^a}{\rho }-\frac{y^a{y}_b}{{\rho }^3}=-\frac{{\delta }_b^a}{\rho }-\frac{W^a{W}_b}{\rho }\mathrm{.}$ Using $y^b{y}_b=\eta (y,y)=-{\rho }^2$, $W^b{\mathrm{\partial }}_b{W}^a=\frac{y^b}{\rho }(-\frac{{\delta }_b^a}{\rho }-\frac{y^a{y}_b}{{\rho }^3})=-\frac{y^a}{{\rho }^2}-\frac{y^a(y^b{y}_b)}{{\rho }^4}=-\frac{y^a}{{\rho }^2}+\frac{y^a}{{\rho }^2}=0.$ Since the metric-selected connection of the flat model has vanishing Christoffel symbols in this chart, ${\mathrm{\nabla }}_{W}W=0$; with [L3] the field $W$ is a smooth future unit timelike geodesic congruence. This proves (i).

(The flow lines.) For fixed $p\in O$ define $\gamma (s):=e-(1-s\mathrm{/}\rho (p))(e-p)$ for $s\in [0,\rho (p))$. Then $\gamma (0)=p$, $\gamma$ is affine in $s$ with derivative ${\gamma }^{\mathrm{\prime }}(s)=(e-p)\mathrm{/}\rho (p)=W(p)$, a constant vector; by [L1] straight coordinate lines are geodesics, so $s\mapsto \gamma (s)$ is a geodesic of the model with ${\gamma }^{\mathrm{\prime }}(s)=W(p)$. Its $\rho$-value is $\rho (\gamma (s))=\sqrt{-\eta (e-\gamma (s),e-\gamma (s))}$; since $e-\gamma (s)=(1-s\mathrm{/}\rho (p))(e-p)$, one gets $\rho (\gamma (s))=(1-s\mathrm{/}\rho (p))\rho (p)=\rho (p)-s$, which is positive for $s<\rho (p)$ and vanishes at $s=\rho (p)$; in particular $\gamma (s)\in O$ for $0\le s<\rho (p)$ and $\gamma (\rho (p))=e$ in the ambient affine space $A$. Also $W(\gamma (s))=(e-\gamma (s))\mathrm{/}\rho (\gamma (s))=(e-p)\mathrm{/}\rho (p)=\gamma \mathrm{\prime }(s)$, so it is an integral curve. Therefore $\gamma$ is the maximal forward flow line of $W$ through $p$, with $e$ as its limiting focal event at finite affine parameter $\rho (p)$. The same straight line continues for all real $s$ and is a geodesic of $A$ defined on all of $\mathbb{R}$ by [L1]. This proves (ii).

(The kinematic quantities.) By [L1] the connection is flat and $B_{ab}={\mathrm{\nabla }}_b{W}_a={\mathrm{\partial }}_b{W}_a$; from the computation in step 1.1, with $W_a=y_a\mathrm{/}\rho$, $B_{ab}={\mathrm{\partial }}_b\left(\frac{y_a}{\rho }\right)=-\frac{{\eta }_{ab}}{\rho }-\frac{y_a{y}_b}{{\rho }^3}=-\frac{1}{\rho }({\eta }_{ab}+W_a{W}_b)=-\frac{h_{ab}}{\rho }\mathrm{.}$ In particular $B$ is symmetric, so ${\omega }_{ab}=B_{[ab]}=0$, and $B_{ab}{W}^b=-h_{ab}{W}^b\mathrm{/}\rho =0$. For the expansion, [L4] and $g^{ab}{h}_{ab}=3$ give $\theta =g^{ab}{B}_{ab}=-\frac{1}{\rho }{g}^{ab}{h}_{ab}=-\frac{3}{\rho }$; then the shear is ${\sigma }_{ab}=B_{(ab)}-\frac{1}{3}\theta h_{ab}=-\frac{h_{ab}}{\rho }+\frac{1}{\rho }{h}_{ab}=0$. This proves (iii).

(The focusing bound is attained and the congruence does not extend.) Let $\gamma ={\gamma }_p$ be the flow line of step 2.1 and write $\theta (s):=\theta (\gamma (s))$. By step 2.2 and $\rho (\gamma (s))=\rho (p)-s$, $\theta (s)=-3\mathrm{/}(\rho (p)-s)$ for $0\le s<\rho (p)$. Hence ${\theta }_0:=\theta (p)=-3\mathrm{/}\rho (p)<0$ and $\frac{1}{-\theta (s)}=\frac{\rho (p)-s}{3}=\frac{1}{-{\theta }_0}-\frac{s}{3},$ so the focusing bound of [L5] holds with equality for $s_0=0$. To see directly that this congruence field cannot extend continuously across its focal event $e$, use the translation identification of tangent spaces of $A={\mathbb{R}}^4$ and set $v_1:={\mathrm{\partial }}_0{\mathrm{\mid }}_e$ and $v_2:=\sqrt{2}{\mathrm{\partial }}_0{\mathrm{\mid }}_e+{\mathrm{\partial }}_1{\mathrm{\mid }}_e$. The flat metric gives $\eta (v_i,v_i)=-1$ and $\eta (v_i,N_e)<0$ for $i=1,2$, while $v_1\mathrm{\ne }{v}_2$. Put $p_i:=e-v_i\in O$. Their flow lines have ${\gamma }_{p_i}(s)=e-(1-s)v_i$ for $0\le s<1$ and $W({\gamma }_{p_i}(s))=v_i$; both base curves tend to $e$ as $s\uparrow 1$, but the two limits of $W$ are distinct. A continuous extension of $W$ to $e$ would have to take both values $v_1$ and $v_2$, which is impossible. Thus this smooth congruence has no continuous, hence no smooth, extension through $e$; the individual affine geodesics still continue in $A$. This proves the local nonextension directly, independent of continuation beyond the region to which the hypotheses of [L5] apply.

(No singularity.) By [L1] the curvature and the Ricci tensor of the model vanish identically, every maximal geodesic of the model is a straight coordinate line with domain all of $\mathbb{R}$, and at $s=\rho (p)$ all flow lines pass through the single event $e$; the field $W$ is not defined at $e$ because $\rho (e)=0$. Hence the finite-time breakdown of the congruence in step 3.1 is exactly the focal crossing of the flow lines at $e$, while the spacetime and each individual geodesic continue; the example therefore shows that the local focusing theorem's conclusion is sharp and must be read as a statement about the congruence, not as a singularity. This proves (v) and completes the example.

(The witness is a future timelike curve.) By [L1] $\gamma$ is a smooth mapping into $M$ with velocity ${\gamma }^{\mathrm{\prime }}(s)=d{q}_{o+s{e}_0}({\mathrm{\partial }}_0)$; the local isometry identity gives $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1<0$ and $g({\gamma }^{\mathrm{\prime }},N)=\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1<0$, so every tangent is future unit timelike and $\gamma$ is a future timelike curve in the class of Causal curves, chronological and causal reachability, and inextendibility.

(The witness is closed and nonconstant on one period.) By [L1] $\gamma (s+L)=q(o+s{e}_0+L{e}_0)=q(o+s{e}_0)=\gamma (s)$ because $L{e}_0\in \mathrm{\Lambda }$. It is nonconstant on $[0,L]$: if $\gamma (s)=\gamma (0)$ for some $0<s<L$, then $s{e}_0\in \mathrm{\Lambda }$, so $s=nL$ for some nonzero integer $n$, which is impossible for $0<s<L$. Hence $\gamma {\mathrm{\mid }}_{[0,L]}$ is a nonconstant future timelike curve with equal endpoints.

(Conclusion.) By step 2.1 and [L2] the model $(M,g,N)$ is not chronological: it contains a closed future timelike curve. Since $M$ is flat and time-oriented by [L1], this refutes the displayed statement. It also shows that a model with a closed timelike curve is not causal, and by The flat cylinder model: closed timelike curves, total viciousness and no Cauchy set clause (d) it admits no Cauchy set at all.

(The set is spacelike.) By [L3] the set $\mathrm{\Sigma }$ is an embedded three-dimensional submanifold of $A$ with tangent space spanned by ${\mathrm{\partial }}_1,{\mathrm{\partial }}_2,{\mathrm{\partial }}_3$, on which $\eta =\mathrm{diag}(1,1,1)$ is positive definite; hence $\mathrm{\Sigma }$ is a spacelike embedded hypersurface.

(An inextendible causal curve missing it.) Define $u:\mathbb{R}\to A$, $u(s):=x^{-1}(s,z_0)$. This is a smooth curve with constant tangent ${\mathrm{\partial }}_0=N$, which satisfies $\eta ({\mathrm{\partial }}_0,{\mathrm{\partial }}_0)=-1<0$ and is future-directed by clause (c) of The flat model: Minkowski space R^4, its standard Lorentz form, inertial charts and cone sets, so $u$ is a future timelike (hence causal) curve; its $t$-coordinate is the identity of $\mathbb{R}$, so it has no point limit at either infinite parameter end: such a limit would force its continuous real time coordinate to have a finite limit. Thus $u$ is inextendible by Causal curves, chronological and causal reachability, and inextendibility clause (d). But $t(u(s))=s$ and $z(u(s))=z_0$ with $\mathrm{\mid }{z}_0{\mathrm{\mid }}_{\mathrm{e}}=2>1$, so $u(s)\mathrm{\notin }\mathrm{\Sigma }$ for every $s$; the parameter set $\{s:u(s)\in \mathrm{\Sigma }\}$ is empty and in particular not a singleton.

(Conclusion.) By step 1.2 there is an inextendible causal curve that meets $\mathrm{\Sigma }$ in no parameter value; by [L4] the set $\mathrm{\Sigma }$ is not a Cauchy set, and by step 1.1 it is a spacelike embedded hypersurface, so it is not a Cauchy hypersurface. Since the ambient model is globally hyperbolic by [L1], this refutes the displayed statement: in a globally hyperbolic model a spacelike hypersurface need not be Cauchy.