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Use the original PDF's spatial Ricci component $R_{\chi \chi }=a\ddot{a}+2{\dot{a}}^2+2$; the converted TeX's $2\dot{a}\ddot{a}$ is a transcription defect. Continue with $c=1$ and signature $(-,+,+,+)$. Tracing the vacuum Einstein field equations with a cosmological constant gives $R=4\Lambda$, and hence $R_{ab}=\Lambda g_{ab}$. The time component therefore gives $\ddot{a}=(\Lambda /3)a$. Substituting this into the spatial component givesThe left side is strictly positive, so $\Lambda >0$ is necessary for a real positive closed scale factor. Define $\ell =\sqrt{3/\Lambda }$. The acceleration equation has solution $a=A{e}^{t/\ell }+B{e}^{-t/\ell }$, and the first integral fixes $4AB={\ell }^2$. Taking the positive scale-factor branch and shifting the origin to the minimum givesThe universe contracts for $t<0$, has its nonzero minimum $a=\ell$ at $t=0$, and expands for $t>0$. The minimum is a smooth bounce, not a curvature singularity.

Introduce the dimensionless time $\tau =t/\ell$. Then the closed slicing of de Sitter spacetime isOne can identify it explicitly: in five-dimensional Minkowski space, set $X^0=\ell \sinh \tau$ and $X^I=\ell \cosh \tau n^I$, where ${\sum }_{I=1}^4(n^I{)}^2=1$. These coordinates satisfy $-(X^0{)}^2+{\sum }_I(X^I{)}^2={\ell }^2$, and their induced metric is the one above. Thus the solution is global de Sitter spacetime, with spherical spatial topology and both contracting and expanding parts.

The time entering the printed compactification formula is this rescaled time. WriteDirect differentiation and elementary trigonometry give $dT/d\tau =\mathrm{sech}\tau$, $\sin T=\tanh \tau$ and $\cos T=\mathrm{sech}\tau$. ThereforeMultiplying by $(\cos T/\ell {)}^2$ gives the metric of a finite-time slab of the Einstein static universe, often called the Einstein cylinder. This is a conformal transformation of the metric and does not assert that the transformed matter content remains vacuum. The coordinate ranges areThe angular-coordinate degeneracies at the poles are the usual regular spherical ones.

Radial null lines satisfy $d\chi =\pm dT$, so the Penrose diagram is the square displayed below. Every interior point represents an angular two-sphere. Its top and bottom are the spacelike conformal infinities ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$ at $T=\pi /2$ and $T=-\pi /2$; they are excluded from the physical spacetime and are not curvature singularities. The vertical sides $\chi =0$ and $\chi =\pi$ represent the regular north and south poles of the three-sphere, not spatial edges of the universe.

A particle horizon at an observer's current time bounds the comoving sources whose light could have reached that observer over the available past conformal interval. Here the cosmological past extends to $t=-\infty$, but its conformal duration is finite. For a comoving observer at $\chi =0$,Since $0<{\chi }_p<\pi$ at every finite time, a particle horizon exists. The corresponding null boundary in the diagram is $\chi =T+\pi /2$, emerging from the observer's past endpoint at ${\mathcal{I}}^{-}$. The antipodal point never has enough time to send a signal received at a finite proper time.

An observer event horizon bounds the events that can ever send a signal to the observer's future-inextendible worldline. The remaining conformal time fixes its distance:This too lies strictly between zero and $\pi$ at finite time, so the observer has a cosmological event horizon. Its null line ends at the observer's future endpoint on ${\mathcal{I}}^{+}$. Both horizon spheres have areal radius $a\sin \chi =\ell$, though their comoving angular positions differ except at $T=0$. These are the cosmological horizons in global de Sitter spacetime; changing the comoving observer relocates them by an isometry.

A Cauchy horizon is the boundary of the domain determined by data on a chosen partial Cauchy hypersurface; beyond it an inextendible causal curve can avoid that initial surface. Each complete constant-$T$ three-sphere here is a Cauchy hypersurface. To see this directly, a causal curve in the conformal metric has spatial speed at most one when parameterized by $T$. On any finite interior time interval its path therefore has a limit in the complete compact three-sphere. If an inextendible curve stopped at an interior time, that limit would allow its continuation, a contradiction. Its time range consequently extends between both conformal ends, and monotonicity makes it cross any complete constant-$T$ sphere exactly once. Its domain of dependence is the entire physical spacetime.

Thus particle and cosmological event horizons exist for every comoving observer, while the complete global Cauchy slices have no Cauchy horizons. Horizons of a restricted static patch or a deliberately partial initial surface are relative to that restriction; they do not alter the global Cauchy conclusion.

There is only one zero of $f$, at $r=2m$, and the Kruskal–Szekeres coordinates give four quadrants. Both exterior quadrants have $r>2m$, while the future and past interior quadrants have $0<r<2m$. The tortoise coordinate is finite at $r=0$: for $q>0$, $d{r}_{\ast }/dr\sim -\sqrt{q}{r}^{1/2}/(2m)$, and for $q=0$ it is $-r/(2m)$. Throughout each interior $g^{rr}=f<0$, so constant-$r$ hypersurfaces are spacelike. The assumed curvature singularity at $r=0$ therefore forms a spacelike future boundary in the black-hole region and a spacelike past boundary in the white-hole region. At infinity, $f\to 1$ and $R/r\to 1$, giving two standard asymptotically flat ends.

The resulting Penrose diagram is the same four-region causal diagram as the maximal Schwarzschild spacetime:

Each exterior has its own past null infinity, future null infinity, spatial infinity and timelike infinities. The future and past event horizons intersect on the bifurcation surface. The diagram describes the eternal analytic solution; a spacetime formed by collapse need not contain its white-hole region or second exterior.

Unlike a nonextreme Reissner-Nordstrom black hole, this electrically charged solution has one horizon and spacelike singularities, with no inner Cauchy horizon. A nonextreme Reissner-Nordstrom solution has two distinct horizons, including an inner Cauchy horizon, and a timelike singularity; its ideal maximal extension contains an infinite sequence of exterior and interior blocks. Charge by itself does not determine this causal structure: the metric and field equations of the gravitational theory matter.

For $0<M<a$, the ring is a naked singularity and there is no event horizon shielding the extension's closed timelike curves. The weak cosmic censorship conjecture predicts that such superextreme Kerr geometries are not formed by generic collapse of regular isolated initial data satisfying the appropriate energy conditions. This is a conjectural physical restriction, not a proof from the exact metric alone.

For $M>a>0$, the closed timelike curves lie deep inside the ideal analytic extension, beyond the inner Cauchy horizon, so weak cosmic censorship does not itself exclude them. However, that inner horizon is unstable to perturbations, with mass inflation expected in realistic collapse. The strong cosmic censorship conjecture is the relevant obstruction to a regular continuation into the ideal chronology-violating region. Thus the exact Kerr extension exhibits such curves mathematically, but neither regular collapse nor the uniqueness theorem establishes that they occur in Nature.

The radial part of the Minkowski metric becomes $-dudv$, and $r=(v-u)/2$. With $u=\tan p$, $v=\tan q$ and the positive conformal factor $\Omega =2\cos p\cos q$, multiplication by ${\Omega }^2$ givesConsequently the Minkowski conformal compactification isThe spatial metric $d{R}^2+{\sin }^{2}Rd{\Sigma }^2$ is the round unit three-sphere. The extended unphysical metric is therefore that of the Einstein static universe, $\mathbb{R}\times S^3$ with unit spatial radius. Physical Minkowski spacetime occupies only the diamond-shaped causal patch of this cylinder, or a triangle after restricting to nonnegative spherical radius; it is not the whole cylinder.

The complete coordinate ranges and joint restrictions areThe polar angles have the usual coordinate degeneracies at their poles and at $r=0$. In the extended Einstein static universe one may let $T\in \mathbb{R}$ and $0\le R\le \pi$, but these are not the ranges of the original physical patch. In its interior $\Omega >0$. The line $R=0$ is the regular centre $r=0$, not a component of infinity.

The conformal completion has future timelike infinity $i^{+}=(\pi ,0)$, past timelike infinity $i^{-}=(-\pi ,0)$, and spacelike infinity $i^0=(0,\pi )$. The open sloping boundaries arenamely future null infinity and past null infinity. Each ordinary interior point of the radial diagram represents a two-sphere, and each point on either open null boundary represents a two-sphere of null directions.

A timelike geodesic is an inertial worldline $\mathbf{x}=\mathbf{b}+\mathbf{\beta }t$ with $|\mathbf{\beta }|<1$. As $t\to +\infty$, both $t-r$ and $t+r$ tend to $+\infty$, giving $p,q\to \pi /2$ and endpoint $i^{+}$. At past infinity both tend to $-\infty$, giving $i^{-}$. These endpoints are reached at infinite physical proper time, although at finite conformal coordinate time $T$. A null geodesic has $r\sim |t|$. In its outgoing future, $u$ approaches a finite retarded time while $v\to +\infty$, giving an endpoint on ${\mathcal{I}}^{+}$; in its incoming past, $v$ has a finite limit and $u\to -\infty$, giving an endpoint on ${\mathcal{I}}^{-}$. A spacelike geodesic is a straight line with spatial tangent larger than its temporal tangent, so $u\to -\infty$, $v\to +\infty$ at either end and both endpoints are $i^0$ in the compactified radial description. All these physical geodesics are complete. Only the unparametrized null geodesics are necessarily preserved as geodesics of the unphysical metric; timelike and spacelike images need not be its geodesics.

For horizons, distinguish a property of the spacetime from a property of an observer or of an initial-data hypersurface. Complete inertial observers have no particle horizon: in the flat cosmological slicing $a(t)=1$, the past light-travel distance ${\int }_{-\infty }^{t_0}dt$ is infinite. There is no finite-age initial singularity truncating communication between comoving observers. This statement does not say that an event receives signals from spacelike-separated events at the same time.

There is also no black-hole event horizon, because every event can send a future-directed null ray to future null infinity. Nevertheless accelerated timelike observers can have both future and past observer event horizons. For an entire observer worldline $\gamma$, define them by $H_{\gamma }^{+}=\partial I^{-}(\gamma )$ and $H_{\gamma }^{-}=\partial I^{+}(\gamma )$, using its complete future and past respectively. An inertial observer's chronological past and future of the whole worldline are both all of Minkowski spacetime. By contrast, for the uniformly accelerated worldline in a Cartesian spatial direction,its null coordinates obey $t-x=-a^{-1}{e}^{-a\tau }<0$ and $t+x=a^{-1}{e}^{a\tau }>0$. An event with $t-x<0$ can signal to a sufficiently late point of this worldline, whereas an event with $t-x\ge 0$ cannot; finite transverse displacements do not change that conclusion as $t+x\to \infty$. Similarly the worldline can signal to precisely the events with $t+x>0$. Thus its two observer event horizons areThey are the Rindler horizons, and their accessible common region is $x>|t|$. They do not indicate a curvature singularity or a failure of global predictability.

Finally Minkowski spacetime is a globally hyperbolic spacetime: every inextendible causal curve crosses each complete spacelike plane $t=t_0$ exactly once. Indeed $t$ is strictly monotone along future-directed causal curves, and the speed bound $|d\mathbf{x}/dt|\le 1$ prevents an inextendible curve from ending at finite $t$ by escaping to spatial infinity. Its time range is therefore all of $\mathbb{R}$. These planes are Cauchy hypersurfaces with full domain of dependence, so there are no Cauchy horizons associated with complete Cauchy data.

A Cauchy horizon is, however, defined relative to a specified initial surface. Under a literal existence reading of “admits”, non-Cauchy initial surfaces can have such horizons even in flat spacetime. For example the edgeless spacelike hyperboloid ${\Sigma }_a:t=\sqrt{a^2+r^2}$ has $D({\Sigma }_a)=\{t>r\}$. To see this, inside the future light cone $t-r$ and $t+r$ are positive and nondecreasing on future-directed causal curves. Their product $t^2-r^2$ increases past $a^2$ on every future-inextendible curve from below the hyperboloid; every past-inextendible curve from above it crosses down through $a^2$ before leaving the cone. Outside the cone there is an inextendible null curve avoiding the hyperboloid. The future light cone itself is its past Cauchy horizon. The time-reversed hyperboloid gives a future Cauchy horizon. This is the dependence of a Cauchy horizon on the initial hypersurface, not an intrinsic loss of predictability of full Minkowski spacetime. Thus a categorical claim that no choice of partial initial surface can ever have a Cauchy horizon would be too strong.

The future domain of dependence $D^{+}(\Sigma )$ consists of points through which every past-inextendible causal curve meets the partial Cauchy hypersurface $\Sigma$. Data on $\Sigma$ determine evolution there for suitable hyperbolic equations. Its future boundary is the Cauchy horizon $H^{+}(\Sigma )$; one precise definition is $\overline{D^{+}(\Sigma )}\setminus I^{-}(D^{+}(\Sigma ))$.

For a nonextremal Reissner-Nordstrom spacetime, $0<|Q|<M$, the metric function is $F=1-2M/r+Q^2/r^2$, with $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The outer event horizon lies at $r_{+}$. The future trapped region leads toward the inner null surface $r_{-}$, a future Cauchy horizon for bridge data on the illustrated $\Sigma$. Beyond it some past causal curves arrive from other analytic blocks or timelike singularities without meeting $\Sigma$. The CP diagram repeats these analytic blocks; $r=0$ is a timelike curvature singularity.

Near the inner horizon, arbitrarily late exterior radiation is compressed into finite infaller proper time. Its frequency is exponentially blueshifted, with scale $e^{|{\kappa }_{-}|v}$, so even a decaying flux can have diverging measured stress. Generic ingoing and outgoing perturbations produce mass inflation, growing internal mass and curvature. The smooth RN Cauchy horizon is expected to be unstable to back-reaction. This is a generic instability argument, not a universal theorem from every specially tuned test trajectory; the approximation of a harmless test particle becomes inconsistent.

One standard form of the Penrose singularity theorem assumes a time-oriented globally hyperbolic spacetime with a noncompact Cauchy hypersurface, the null convergence condition $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$, and a closed future trapped surface. It concludes future null geodesic incompleteness: some future-inextendible null geodesic has a finite upper endpoint of its affine parameter.

The focusing mechanism is the Null Raychaudhuri equation. The normal generators have zero null twist, soAn initial ${\theta }_0<0$ therefore gives a conjugate point to a spacelike surface within affine distance at most $2/|{\theta }_0|$, assuming the generator can be continued that far. Such a generator ceases to lie on the achronal boundary after its first focal point. Compactness of the trapped surface supplies a uniform bound for all normalized initial null normals. Future completeness would consequently make its future boundary compact. Projection along timelike curves to a connected Cauchy hypersurface is injective on the achronal boundary and has open image. Compactness makes the image closed as well; the nonempty image must therefore be the entire hypersurface, contradicting its noncompactness. This explains why the global assumptions supplement local focusing.

For Reissner-Nordstrom spacetime, the Maxwell stress-energy tensor satisfies the null energy condition; the Einstein field equations imply the required null convergence condition. The spheres in the band just found are closed and trapped. Apply the theorem to a maximal Cauchy development with a noncompact Cauchy hypersurface and containing one such sphere. That globally hyperbolic spacetime must be future null-geodesically incomplete.

The Penrose theorem at a Cauchy horizon needs care: the full maximal analytic Reissner-Nordstrom spacetime has inner Cauchy horizons and is not globally hyperbolic. It does not satisfy every hypothesis of the displayed theorem. In the exact solution some incomplete geodesics of the globally hyperbolic development reach a smoothly extendible Cauchy horizon in finite affine parameter. The theorem asserts incompleteness of the development, not that every such endpoint is a curvature singularity. The separate curvature singularity at $r=0$ does not justify silently dropping the theorem's global hypothesis.

The inner horizon in the nonextreme Reissner-Nordstrom spacetime diagram is a Cauchy horizon. Radiation from exterior perturbations can approach it at arbitrarily late advanced time and undergo unbounded blueshift. Here its positive blueshift scale is ${\kappa }_{-}=(r_{+}^2-r_{-}^2)/r_{-}^3$. The inner-horizon blueshift coordinate behaves as $V_{-}\propto -e^{-{\kappa }_{-}v}$, soEven a decaying power-law tail can therefore generate divergent local flux. With counterstreaming radiation, backreaction produces mass inflation and curvature growth, rather than the smooth inner horizon of the exact solution.

This supports the strong cosmic censorship conjecture: a generically perturbed maximal Cauchy development is expected to lose the smooth extension across its Cauchy horizon, restoring predictability in the appropriate regularity class. The exact smooth inner horizon is unstable, not a robust failure of deterministic evolution. Curvature divergence supports an obstruction to a $C^2$ metric extension; it does not by itself exclude every continuous metric extension. The regularity class is part of the conjecture, and the blueshift argument is evidence, not a general theorem proving it.

Use the subextremal four-dimensional Reissner-Nordstrom spacetime, with $0<|Q|<M$. Its static radial function isTake the time-symmetric two-ended bridge through the outer bifurcation surface. Its initial data have $K_{ij}=0$ andon each $r\ge r_{+}$ end. The bridge is smooth: in proper radial distance $s$, $r-r_{+}=f^{\prime }(r_{+})s^2/4+O(s^4)$. Each asymptotically flat end is an infinite Riemannian distance away. Hence the spatial Riemannian manifold is complete and admits no proper same-dimensional smooth isometric extension as a connected spatial manifold.

Its maximal Cauchy development includes the two exteriors and the adjacent future and past regions between $r_{+}$ and $r_{-}$. It ends at inner Cauchy horizons, not at a curvature singularity. Since the simple root at $r_{-}$ is removable in horizon-penetrating coordinates and all curvature invariants are finite there, the exact solution extends across these Cauchy horizons. The extension is no longer globally determined by the given initial data.

The shaded region is the maximal Cauchy development; dashed upper and lower null edges are inner Cauchy horizons. The displayed neighboring diamonds illustrate smooth continuation, rather than the entire infinite extension.

The strong cosmic censorship conjecture concerns generic admissible initial data, in a specified extension regularity. Exact charged spherical data are exceptional. Perturbations can produce mass inflation at the inner Cauchy horizon, obstructing suitably regular extensions; the precise conjecture depends on the matter model and whether extensions are required to be $C^0$, $C^2$, or another regularity. Thus this exact extendible example does not refute a generic strong cosmic censorship conjecture.

For an Einstein-Maxwell example the gravitational triple $(\Sigma ,h,K)$ must be accompanied by electromagnetic initial data. One may take zero magnetic field and the smooth radial electric flux $Q/r^2$ through the bridge. The charges at the two ends have opposite signs when measured with outward normals. This is an example in the electrovacuum theory, not vacuum initial data.

Relative to a chosen asymptotically flat end, define the black hole region bywhere ${\mathcal{I}}^{+}$ is that end's future null infinity in a conformal completion. It consists of events unable to send a future causal signal to that infinity; its boundary is the corresponding future event horizon.

For $0<\alpha <r_0/2$, $h>0$ and $f<0$ on $r_{-}<r<r_{+}$. The gradient of $r$ has norm $g^{ab}{\nabla }_{a}r{\nabla }_{b}r=f$, so it is timelike there. Use the future extension regular in the ingoing coordinates above, reached by an exterior future ingoing null geodesic with $\dot{r}=-E\sqrt{h}<0$. This determines $\nabla r$ to be future timelike in the inter-horizon block. Every nonzero future causal tangent $V$ therefore obeysAt the future outer horizon the gradient becomes future null and gives the corresponding one-way inequality $dr/d\lambda \le 0$. No future signal can cross that horizon outward to the selected exterior. The future inter-horizon block is consequently within $\mathcal{B}$; a future-directed path can leave it only through the inner horizon, not return through the outer horizon to the selected infinity.

This statement requires a branch and an end. The maximal extension also contains time-reversed inter-horizon blocks, where future-directed curves have increasing $r$ and emerge into an exterior: they are white holes, not the selected black hole region. Thus the unqualified claim for every copy of $r_{-}<r<r_{+}$ in an arbitrary extension is false. Analytic continuation past a Cauchy horizon can also introduce other asymptotic ends; the displayed definition explicitly refers to the chosen component of infinity.

To sketch the causal structure, suppress the spacelike $\psi$ circle by its orbit-space projection. The induced three-dimensional metric tensor isThe causal projection along a spacelike circular fiber is exact: any causal curve projects to a $q$-causal curve, and every base curve has a lift $d\psi =\Omega dt$ with exactly its base norm. The diagrams therefore represent this two-dimensional orbit space, not a constant-$\psi$ slice of the three-dimensional submanifold.

There are three nonextremal block types. For $r>r_{+}$, $r_{\ast }$ spans the full real line from the outer horizon to infinity, producing an exterior diamond. For $r_{-}<r<r_{+}$, $r_{\ast }$ spans the full line with the time and space roles reversed; these diamonds contain the future black-hole or past white-hole branches. For $0<r<r_{-}$, $r_{\ast }$ is finite at zero and diverges at the inner horizon, producing a static half-diamond with a timelike singular edge. The supplied Kretschmann scalar diverges as $384{\alpha }^4{r}_0^4/r^{12}$ at $r=0$, and $f>0$ on the inner side, so that edge is a genuine timelike curvature singularity. In contrast, both horizon radii are regular in horizon-adapted coordinates.

The finite strip shows two exterior levels, a future trapped block, an inner static level with timelike singularities, and the next emerging block. Continuing through the inner Cauchy horizons repeats this pattern upward and downward in the maximal analytic extension. The inner boundary is not a spacelike Schwarzschild-type singularity. Such an ideal extension need not describe a physical collapse past its unstable inner horizon.

At $\alpha =r_0/2$, put $R=r_0/\sqrt{2}$. Then $r_{+}=r_{-}=R$, $f=(r^2-R^2{)}^2/r^4$, and the interval $r_{-}<r<r_{+}$ is empty. The lapse is positive on both sides, so there is no inter-horizon trapped diamond. The horizon is a degenerate Killing horizon, with $\kappa =0$. In the exterior $r_{\ast }\to -\infty$ as $r\downarrow R$, while inside $r_{\ast }\to +\infty$ as $r\uparrow R$; more precisely its leading pole is $-\sqrt{2}{R}^2/[4(r-R)]$. Static proper distance to the horizon is infinite. Future ingoing rays still cross it at finite affine parameter, as the regular ingoing metric tensor shows.

The extremal sketch gives the exterior diamond and singular interior half-diamond, with the complete gluing prescription: the exterior future horizon attaches to the interior past horizon; the interior future horizon attaches to the past horizon of another exterior. Repeating these attachments gives the maximal extension. The marked throat endpoints are conformal ideal endpoints, not bifurcation points of the spacetime. This block representation avoids incorrectly treating the extremal geometry as two transverse horizons with a collapsed trapped region.

The zeros of $f$ give three causal structures.

Thus the cases are subextremal, extremal, and overextremal according as $M>Q$, $M=Q$, and $M<Q$.