Trapped surfaces and Penrose incompleteness — Examples

On the spherical-coordinate region $r>0$, the metric is $g=-(d{x}^0{)}^2+d{r}^2+r^2{\gamma }_{S^2}$. Hence $g(\mathrm{\ell },\mathrm{\ell })=g(k,k)=0$, $g(\mathrm{\ell },k)=-2$, and $g(\mathrm{\ell },N)=g(k,N)=-1$, so both normals are future-directed.

For angular coordinate fields $E_A$, the Christoffel formula [F4] gives ${\mathrm{\Gamma }}^B{}_{rA}=\frac{1}{2}(r^{-2}{\gamma }^{BC})(2r{\gamma }_{CA})=r^{-1}{\delta }_A^B$ and ${\mathrm{\Gamma }}^{\mu }{}_{0A}=0$. Therefore on $r=R$, ${\mathrm{\nabla }}_X\mathrm{\ell }=R^{-1}X$ and ${\mathrm{\nabla }}_{X}k=-R^{-1}X$ for every tangent vector $X$ to $S$.

By [F1], ${\chi }_{\mathrm{\ell }}=R^{-1}q$ and ${\chi }_k=-R^{-1}q$. Since $S$ is two-dimensional, tracing with $q$ gives ${\theta }_{\mathrm{\ell }}=2\mathrm{/}R>0$ and ${\theta }_k=-2\mathrm{/}R<0$. Thus the pair of strict negative inequalities required for trapping fails.

The positive expansion means this sphere does not supply the trapped-surface premise in [F5] or [F6]. The calculation makes no assertion about other surfaces in Minkowski space.

The map ${\mathrm{\Phi }}_H(\tau ,\omega )=(H^{-1}\sinh (H\tau ),H^{-1}\cosh (H\tau )\omega )$ is a smooth bijection from $\mathbb{R}\times S^3$ onto $M_H$. Its inverse is $\tau =H^{-1}\mathrm{arsinh}(H{X}_0)$ and $\omega =(X_1,X_2,X_3,X_4)\mathrm{/}\sqrt{H^{-2}+X_0^2}$, so it is a diffeomorphism. Pulling back the ambient metric $-d{X}_0^2+{\sum }_{i=1}^{4}d{X}_i^2$ gives $g=-d{\tau }^2+a^2{\gamma }_{S^3}$ with $a=H^{-1}\cosh (H\tau )>0$; hence $g$ is Lorentzian and $N={\mathrm{\partial }}_{\tau }$ is a smooth future timelike orientation. Let ${\mathrm{\Sigma }}_0=\{X_0=0\}$ and identify $z_{\omega }=(0,H^{-1}\omega )\in {\mathrm{\Sigma }}_0$. Then $\mathrm{\Psi }(\tau ,z_{\omega }):={\mathrm{\Phi }}_H(\tau ,\omega )$ is a diffeomorphism $\mathbb{R}\times {\mathrm{\Sigma }}_0\to M_H$ with $\mathrm{\Psi }(0,z)=z$; each orbit is future timelike, and $\tau \to \pm \mathrm{\infty }$ along its two ends prevents a limit point in $M_H$, so the orbit is inextendible. The unit sphere $S^3$ is compact by [F8] and connected because any two points are joined by a great-circle arc (with an intermediate point chosen for antipodal pairs).

Let $\gamma (s)=(\tau (s),\omega (s))$ be any future causal curve. At almost every parameter its tangent is nonzero future causal, so $\dot{\tau }>0$ and $a(\tau {)}^2\mathrm{\mid }\dot{\omega }{\mathrm{\mid }}_{{\gamma }_{S^3}}^2\le {\dot{\tau }}^2$. Since $a(\tau )\ge H^{-1}$ and the round metric on $S^3$ is induced from ${\mathbb{R}}^4$, every angular component satisfies $\mathrm{\mid }{\dot{\omega }}^j(s)\mathrm{\mid }\le H\dot{\tau }(s)$ almost everywhere. By [F12], for any compact subinterval $[s_1,s_2]$ and each coordinate $j$, $\mathrm{\mid }{\omega }^j(s_2)-{\omega }^j(s_1)\mathrm{\mid }\le {\int }_{s_1}^{s_2}\mathrm{\mid }{\dot{\omega }}^j(s)\mathrm{\mid }ds\le H(\tau (s_2)-\tau (s_1))\mathrm{.}$ The function $\tau$ is strictly increasing, since its derivative is positive almost everywhere on every nondegenerate subinterval. If its future endpoint were a finite $b$, then $\tau (s)\uparrow b$ and the four component estimates make each ${\omega }^j(s)$ Cauchy as $s$ approaches the endpoint. Completeness of $\mathbb{R}$ gives a limit ${\omega }_b\in {\mathbb{R}}^4$, and continuity of the Euclidean norm gives $\mathrm{\mid }{\omega }_b\mathrm{\mid }=1$, so ${\omega }_b\in S^3$. Hence $\gamma (s)\to (b,{\omega }_b)$ in $M_H$, contradicting the definition of future-inextendibility; the vertical curve $\tau \mapsto (\tau ,{\omega }_b)$ also exhibits a future timelike continuation. The same estimates at a finite past endpoint give a past limit and contradict past-inextendibility. Thus every inextendible causal curve has $\tau$-range $\mathbb{R}$. Continuity and strict monotonicity make every slice meet it exactly once. Each slice is smooth and spacelike with metric $a(\tau {)}^2{\gamma }_{S^3}$; it is compact and connected by [F1], [F8], hence is a compact connected Cauchy hypersurface.

On $0<\chi <\pi$, write $\omega =(\cos \chi ,\sin \chi \sigma )$ with $\sigma \in S^2$, so $g=q+r^2{\gamma }_{S^2}$ for $q=-d{\tau }^2+a^{2}d{\chi }^2$ and $r=a\sin \chi$. Directly from the two-dimensional Christoffel symbols ${\mathrm{\Gamma }}_{\chi \chi }^{\tau }=a{a}^{\mathrm{\prime }}$ and ${\mathrm{\Gamma }}_{\tau \chi }^{\chi }=a^{\mathrm{\prime }}\mathrm{/}a$, one gets $\mathrm{Ric}(q{)}_{\tau \tau }=-a^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}a$, $\mathrm{Ric}(q{)}_{\chi \chi }=a{a}^{\mathrm{\prime }\mathrm{\prime }}$, ${\mathrm{Hess}}_{\tau \tau }r=a^{\mathrm{\prime }\mathrm{\prime }}\sin \chi$, ${\mathrm{Hess}}_{\chi \chi }r=-a(1+a^{\mathrm{\prime }2})\sin \chi$, $z={\cos }^2\chi -a^{\mathrm{\prime }2}{\sin }^2\chi$, and ${\mathrm{\Delta }}_{q}r=-a^{\mathrm{\prime }\mathrm{\prime }}\sin \chi -(1+a^{\mathrm{\prime }2})\sin \chi \mathrm{/}a$.

Let $X(\lambda )$ be an affinely parametrized null geodesic on the hyperboloid. The tangential projection of the ambient flat derivative is torsion-free and compatible with the induced metric: projecting $D_{U}V$ removes only a normal component, which is orthogonal to every tangent field, while the ambient derivative preserves the metric and torsion. By uniqueness of the metric-selected connection, this projected derivative is the geodesic connection. Hence $X^{\mathrm{\prime }\mathrm{\prime }}(\lambda )$ is normal to the hyperboloid and is a scalar multiple $f(\lambda )X(\lambda )$. Differentiating $⟨X,X⟩=H^{-2}$ twice gives $⟨X,X^{\mathrm{\prime }\mathrm{\prime }}⟩=-⟨X^{\mathrm{\prime }},X^{\mathrm{\prime }}⟩=0$; because $⟨X,X⟩=H^{-2}>0$, $f=0$. Therefore $X(\lambda )=X_0+(\lambda -{\lambda }_0)V$ with constant null $V$ and $⟨X_0,V⟩=0$. This entire affine line remains on the hyperboloid, so every null geodesic extends through every finite affine parameter; in particular all future null geodesics are complete.

Fix ${\tau }_0<0$ and set $S_{{\tau }_0}=\{\tau ={\tau }_0,\chi =\pi \mathrm{/}2\}$. This is a smooth embedded equatorial $S^2$ with positive induced metric $q_S=a({\tau }_0{)}^2{\gamma }_{S^2}$ and is compact by [F1], [F8]. Its future null normals $\mathrm{\ell }={\mathrm{\partial }}_{\tau }+a^{-1}{\mathrm{\partial }}_{\chi }$ and $k={\mathrm{\partial }}_{\tau }-a^{-1}{\mathrm{\partial }}_{\chi }$ satisfy $g(\mathrm{\ell },\mathrm{\ell })=g(k,k)=0$, $g(\mathrm{\ell },k)=-2$, and $g(\mathrm{\ell },N)=g(k,N)=-1$, so they are normalized future null normals by [F7].

By [F6] and [F9], the assumed Axiom of Choice supplies Countable Choice and Dependent Choice, so the existence of ${\mathrm{\Sigma }}_0$ makes $M_H$ globally hyperbolic.

Since $a^{\mathrm{\prime }\mathrm{\prime }}=H^{2}a$ and $1+a^{\mathrm{\prime }2}=H^2{a}^2$, [F2] and step 1.3 give ${\mathrm{Ric}}_{\tau \tau }=-3H^2$, ${\mathrm{Ric}}_{\chi \chi }=3H^2{a}^2$, ${\mathrm{Ric}}_{\tau \chi }=0$, and ${\mathrm{Ric}}_{ij}=3H^2{r}^2{\gamma }_{ij}$. Thus $\mathrm{Ric}=3H^{2}g$ on the regular spherical chart; both tensors are smooth on the global hyperboloid, so equality extends by continuity to the two poles. Consequently $\mathrm{Ric}(v,v)=3H^{2}g(v,v)=0$ for every null vector.

The mixed Christoffel symbols in [F2] give ${\mathrm{\nabla }}_X{\mathrm{\partial }}_{\tau }=(a^{\mathrm{\prime }}\mathrm{/}a)X$ and ${\mathrm{\nabla }}_X{\mathrm{\partial }}_{\chi }=({\mathrm{\partial }}_{\chi }r\mathrm{/}r)X=(\cot \chi )X$ for every angular tangent vector $X$. Thus at $\chi =\pi \mathrm{/}2$, ${\chi }_{\mathrm{\ell }}=(a^{\mathrm{\prime }}\mathrm{/}a)q_S$ and ${\chi }_k=(a^{\mathrm{\prime }}\mathrm{/}a)q_S$. Tracing in two dimensions gives ${\theta }_{\mathrm{\ell }}={\theta }_k=2a^{\mathrm{\prime }}\mathrm{/}a=2H\tanh (H{\tau }_0)<0$. So $S_{{\tau }_0}$ is future trapped by [F3].

The spacetime is globally hyperbolic by step 2.1, has the compact trapped surface of step 2.3, satisfies the null Ricci condition by step 2.2, and is future null complete by step 1.4. By [F9] and the assumed AC, the choice hypotheses of [F10] also hold, so [F10] gives compactness of $E^{+}(S_{{\tau }_0})$, consistently with the compact Cauchy slices in step 1.2.

By [F13], applied to the complete timelike product flow $\mathrm{\Psi }$ from step 1.1 and the compact slice ${\mathrm{\Sigma }}_0$ from step 1.2, every smooth spacelike Cauchy hypersurface of $M_H$ is diffeomorphic to ${\mathrm{\Sigma }}_0$ and hence compact. The model satisfies AC, global hyperbolicity, the compact trapped-surface, and null-Ricci hypotheses of [F11], but has no connected noncompact smooth spacelike Cauchy hypersurface. It is therefore outside the theorem’s scope while remaining future null complete; the noncompact-Cauchy condition cannot simply be omitted.