Penrose’s 1965 singularity theorem: from geodesic incompleteness to cosmic censorship

Abstract

Supplementing earlier literature by e.g. Tipler et al. (1980), Israel (1987), Thorne (1994), Earman (1999), Senovilla and Garfinkle (2015), Curiel (2019) and Landsman (2021), I provide a historical and conceptual analysis of Penrose’s path-breaking 1965 singularity (or incompleteness) theorem. The emphasis is on the nature and historical origin of the assumptions and definitions used in—or otherwise relevant to—the theorem, as well as on the discrepancy between the (astro)physical goals of the theorem and its actual content: even if its assumptions are met, the theorem fails to prove the existence or formation of black holes. Penrose himself was well aware of this gap, which he subsequently tried to overcome with his visionary and influential cosmic censorship conjectures. Roughly speaking, to infer from (null) geodesic incompleteness that there is a “black” object one needs weak cosmic censorship, whereas in addition a “hole” exists (as opposed to a boundary of an extendible space-time causing the incompleteness of geodesics) if strong cosmic censorship holds.

1 Introduction

Penrose’s short paper ‘Gravitational collapse and space-time singularities’ from 1965 was surely one of the most important contributions to general relativity since Einstein had formulated the theory in 1915:

However, all bets were off concerning the possibility of a non-singular outcome to a collapse after the publication of a theorem by Roger Penrose (1965), which has claims to be considered the most influential development in general relativity in the 50 years after Einstein founded the theory. Penrose demonstrated that singularities were, after all, a generic feature of gravitational collapse and must appear soon after the formation of a trapped surface (a surface from which light cannot escape outwards). Although the theorem could offer no information about the nature of the singularity, there was a widespread belief that it would have an all-enveloping spacelike character, as in the Schwarzschild case, thus obstructing further development of the situation. Penrose’s paper was just as important for what it initiated as for what it accomplished. Powerful global techniques exploiting the causal structure of space-time had been introduced into the theory for the first time. (Israel [99, p. 253])

Penrose’s theorem said roughly this: Suppose that a star—any kind of star whatsoever—implodes so far that its gravity becomes strong enough to form an apparent horizon, that is, strong enough to pull outgoing light rays back inward. After this happens, nothing can prevent the gravity from growing so strong that it creates a singularity. Consequently (since black holes always have apparent horizons), every black hole must have a singularity inside itself. (Thorne [186, p. 227])

The singularity theorems constitute the first genuine post-Einsteinian content of classical gr, not foreseen in any way by Einstein (...) The global mathematical developments needed for the singularity theorems, and the ideas on incompleteness or trapping—and thus also their derived inferences—were not treated nor mentioned, either directly or indirectly, in any of Einstein’s writings. In 1965 grleft adolescence behind, emancipated from its creator, and became a mature physical theory full of vitality and surprises. (Senovilla and Garfinkle, [175, p. 2])

The comparably path-breaking series of papers—this time on the observational side—by the Event Horizon Telescope Collaboration (2019) on their famous image of the black hole in M87 opens as follows:

Black holes are a fundamental prediction of the theory of general relativity (gr; Einstein 1915). A defining feature of black holes is their event horizon, a one-way causal boundary in space-time from which not even light can escape (Schwarzschild 1916). The production of black holes is generic in gr  (Penrose (965)). (Event Horizon Telescope Collaboration [61], p. L1)

On the basis of this paper, to general acclaim Penrose won half of the 2020 Physics Nobel Prize:

for the discovery that black hole formation is a robust prediction of the general theory of relativity.

All records show that this is certainly how the (astro)physics community interpreted Penrose’s result.¹ In particular, it put an end to claims even by leading relativists to the effect that curvature singularities of the kind found in the static Schwarzschild solution from 1916 (and later in the Kerr solution from 1963), as well as in the dynamical Oppenheimer–Snyder solution from 1939, were artificial consequences of unphysical idealizations (such as spherical or axial symmetry) and would be absent in realistic models.

But let us recall that Penrose [148] actually proved that in the presence of a suitable curvature condition, a space-time (M, g) that contains a non-compact Cauchy surface as well as a trapped surface cannot be future null geodesically complete. Thus it is far from obvious how Penrose’s theorem relates to the above summaries by Thorne, the eht, and the Nobel Prize committee (Israel is more accurate; the discontinuity between Einstein and Penrose claimed by Senovilla and Garfinkle will be addressed).

Putting it bluntly, the main differences between Penrose’s actual theorem and its reputation are that:

• The theorem says nothing about event horizons, which form the “black” ingredient of a black hole;

• It is inconclusive about “singularities”, which should form the “hole” part of a black hole.

Hence the link between the theorem and black holes is opaque. What is missing is the further inference, perhaps under additional assumptions, from the existence of incomplete null geodesics to the existence of an event horizon, as well as an argument relating incompleteness to the existence of a singularity in some physically meaningful sense (such as a curvature singularity), as opposed to, for example, the sudden end of a geodesic being caused by extendibility of the space-time under consideration. Penrose [148] was well aware of both problems, see Sect. 3 below. Addressing the first had to wait for a good definition of an event horizon, which he provided in [151] and which led, in the same paper, to what is now called his weak cosmic censorship conjecture. Penrose subsequently addressed the second (extendibility) problem, albeit rather indirectly at first, by what we now call strong cosmic censorship.

Here is the plan of the remainder of this paper:

• Section 2 summarizes the status of singularities and singularity theorems in gr before 1965.²

• Section 3 goes through [148] in detail, paying special attention to what was and was not new.

• Section 4 resumes the discussion of singularities, including Penrose’s [154, 155] “localization” thereof.

• Section 5 discusses cosmic censorship mainly from the point of view of Penrose [151, 153,154,155].

• Section 6 explains the transition from his cosmic censorship conjectures to the versions that are currently used.

• Section 7 is an Epilogue, reflecting a little on scientific revolutions and historiography in the present context.

Let me stress, perhaps against first appearances, that this paper is intended to increase our admiration of Penrose’s work in gr, which—apart from equally deep and influential ideas on gravitational radiation and asymptotic flatness, spinor and twistor theory, black hole thermodynamics, and cosmology—may be seen as a sustained effort to accomplish what many people seem to believe his 1965 theorem already did.

2 Singularities and singularity theorems before 1965

The period from 1916 to 1965 is well covered by [47, 49, 51, 175, 181] but problems of interpretation remain. Like the history of gr itself [100, 101], I see a steady but bumpy road—this time from Einstein to Penrose—where ultimately the mathematician overruled the physicist and won.

Both Hilbert [96] and Einstein [55] saw singularities as points in space-time where some components $g_{\mu \nu }$ of the metric were ill-defined and/or $\det (g)=0$, with some attention paid to the role of the coordinate system. Such singularities were seen in the Schwarzschild solution from 1916 at $r=2m$, as well as in the de Sitter solution from 1917.³ Slightly rewritten by Einstein [55], de Sitter’s metric is

$$\begin{aligned} ds^2=dr^2+R^2\sin ^2\left( \frac{r}{R}\right) d\Omega ^2 -\cos ^2\left( \frac{r}{R}\right) dt^2, \end{aligned}$$

(2.1)

where $d\Omega ^2=d\theta ^2+\sin ^2\theta d\varphi ^2$, and R is related to the cosmological constant $\lambda =3/R^2$ (de Sitter [180]). Einstein’s understanding of singularities has (understandably) been questioned,⁴ but let me note that:

• Not just Einstein but even Hilbert [96], who was also an expert in gr, failed to recognize the apparent “singularity” of the Schwarzschild metric at $r=2m$ as a mere coordinate singularity.⁵

• Not just Einstein but even Weyl [191] saw $r=\pi R/2$ as a genuine singularity in de Sitter’s metric.⁶ But Einstein was not confused by the coordinate singularity at $r=0$ and did understand the need for different coordinate patches in order to define regularity (and hence singularity) of the metric:

  Furthermore, the continuity condition for the $g_{\mu \nu }$ and the $g^{\mu \nu }$ [including their first derivatives, as he says earlier] must not be interpreted to mean that it must be possible to choose coordinates such that the conditions are satisfied in the entire space. Obviously, one must only demand that for the neighborhood of every point one can select coordinates such that the continuity conditions are met within this neighborhood. [55, p. 271]; [57, p. 37].

Counterfactually granting Einstein [55] that $r=\pi R/2$ is indeed a “singularity”—since he believed that ‘no choice of coordinates can remove this discontinuity’—the interesting point is his subsequent argument that it is ‘genuine’. Namely, he defines a point P to lie ‘in the finite domain when it can be connected by a curve with a fixed, chosen point $P_0$, so that the distance integral $\int _{P_0}^Pds$ has a finite value.’ He then shows that the (alleged) singularities at $r=\pi R/2$ lie ‘in the finite domain’, concluding that ‘the De Sitter solution has a genuine singularity’ (eine echte Singularität). This, then, he considers ‘a grave argument against the admissibility of [de Sitter’s] solution’. Thus Einstein’s logic seems crystal clear:

• He first defines precisely what he means by a singular space-time (or ‘solution’, as he calls it):

  • 1. There must be at least one point P where in all coordinate systems some component $g_{\mu \nu }$ or $g^{\mu \nu }$ (or its first derivative) fails to be continuous (this is the case, for example, if $\det (g)=0$);

  • 2. Such points P can be connected to “regular” points $P_0$ by some curve of finite proper length.

• Admissible space-times must be non-singular, i.e., either the metric is regular at all points P, or, if there are any singular points, these must lie at infinite (proper) distance from all regular points.

Criterion 1 is at odds with the modern notion of a (smooth) Lorentzian manifold, which cannot contain such points. But this was far from obvious at the time and still needed to be spelled out as late as 1963:⁷

the clue to clarity is to refuse ever to speak of a singularity but instead to phrase everything in terms of the properties of differentiable metric fields on manifolds. If one is given a manifold, and on it a metric which does not at all points satisfy the necessary differentiability requirements, one simply throws away all the points of singularity. The starting point for any further discussion is then the largest submanifold on which the metric is differentiable. This is done because there is not known any useful way of describing the singularities of a function except by describing its behavior at regular points near the singularity. The first problem then is to select a criteria which will identify in an intuitively acceptable way a “nonsingular space.” Evidently, differentiability is only a minimum prerequisite, since everything becomes differentiable when the singular points are discarded. The problem is rather to recognize the holes left in the space where singular (or even regular) points have been omitted. (Misner [132, p. 924])

I will discuss Misner’s paper below. Returning to Einstein [55], at least in a Whig interpretation of history his second criterion may be claimed to foreshadow geodesic incompleteness. It would have been more reasonable, though, if he had used causal geodesics instead of arbitrary curves; first deciding for causal curves on physical grounds and/or because this is enough for all interesting examples,⁸ his criterion does not even make sense if arbitrary (as opposed to geodesic) causal curves are allowed.⁹

Even in the absence of satisfactory definitions of a singular space-time, towards which little progress was made until the 1960s, one may look for “singularity theorems” prior to Penrose’s. But because it is not clear what is meant by a “singularity”, there is considerable leeway in identifying such theorems, ranging from accepting Genesis 1:1 to rejecting anything before Penrose [148] to deserve the name.¹⁰ 

For example, both Tipler et al. [187] and Earman [49] attribute the first singularity theorem to Tolman and Ward [188], whereas Senovilla and Garfinkle emphatically start with Raychaudhuri [159].

Prior to Robertson [165] and Walker [190] and Tolman and Ward [188] analyzed closed homogenous and isotropic universes filled with matter in the form of a perfect fluid with $\rho >0$ and $p\ge 0$ (and zero cosmological constant), to conclude that such models have a beginning in a state of zero volume and also recollapse after a state of maximal expansion. However, they did not conclude that there was a big bang, instead believing in some sort of cyclic process of repeated expansions, recollapses, and bounces.¹¹

Earman [49] reviews various analogous cosmological results from the 1930s, due to Tolman, Robertson, de Sitter, Synge, and Lemaître, which with hindsight may be classified as “singularity theorems”, in which “singularities” are defined in diverse ways, including zero volume (or radius), infinite density, as well as problematic expressions for the metric. Lemaître [119], which stands out in its astronomy, mathematics, and physics, was the first paper to use an energy condition.¹² But all these authors immediately give arguments why their results must be mathematical artifacts, typically by appealing to their unrealistic underlying idealizations or to unknown new physics. Even Lemaître [119] writes:

The matter has to find, though, a way of avoiding the vanishing of its volume. ([119], p. 678 of 1997 reprint)

Meanwhile, progress was also made on the astrophysical side.¹³ Light stars were found to retire as white dwarfs, in which nuclear burning has ended and inward gravitational pressure is stopped by a degenerate electron gas. In 1931, Chandrasekhar and Landau independently discovered that this only works for masses $M\le 1.46 M_{\odot }$ (I take the current value); heavier stars collapse into neutron stars (typically after a supernova explosion). But also these have an upper mass, as first suggested by Oppenheimer and Volkoff [141] ; the current upper bound is about $2.3 M_{\odot }$. Heavier stars collapse into a black hole.

Yet this conclusion, so obvious with hindsight, was resisted as much as the idea of a big bang, despite a paper by Oppenheimer and Snyder [142] in which the collapse process was explicitly described within gr, albeit under the assumption of spherical symmetry and an idealized equation of state.¹⁴ It was:

a manuscript that has strong claims to be considered the most daring and uncannily prophetic paper in the field. There is nothing in this paper which needs revision today. (Israel [99, pp. 226–227])

But there is a big difference between our perspective and the contemporary one (see also Sect. 7):

At the end of the 1930s not only was there no agreement on how to define singularities, there was not even a consensus about the status of singularities in the key test case, the Schwarzschild solution. (Earman [49, p. 240])

Restricting this summary to events directly relevant to Penrose’s paper, all authors agree that the next key paper was Raychaudhuri [159] ; as already mentioned, Senovilla and Garfinkle [175] even take this to contain the very first singularity theorem in gr. Raychaudhuri studies a universe filled with dust, i.e.¹⁵

$$\begin{aligned} T^{\mu \nu }=\rho u^{\mu }u^{\nu }, \end{aligned}$$

(2.2)

but allows $\rho$ to be an arbitrary positive function $\rho (t,\mathbf {x})$, as opposed to previous treatments where $\rho$ depends on t alone. Indeed, the goal of Raychaudhuri’s paper is clear from its abstract, which states that:

a simple change over to anisotropy without the introduction of spin does not solve any of the outstanding difficulties of isotropic cosmological models.¹⁶ (Raychaudhuri [159, p. 1123])

Instead of the parameter $\theta =\nabla _{\mu }u^{\mu }$ familiar from modern renditions of his work, Raychaudhuri uses the volume element $\sqrt{-\det (g)}$, or rather $G=(-\det (g))^{1/6}$, but these are related.¹⁷ For this reason, the famous equation for $\theta$ named after him, namely (assuming that $\nabla _uu=0$, which follows from (2.2))

$$\begin{aligned} \nabla _u\theta \equiv {\dot{\theta }}=-{\frac{1}{3}}\theta ^2-\sigma _{\mu \nu }\sigma ^{\mu \nu }+\omega _{\mu \nu }\omega ^{\mu \nu }-R_{\mu \nu }u^{\mu }u^{\nu }, \end{aligned}$$

(2.3)

cannot be found in his paper;¹⁸ instead, one finds an equivalent (but less transparent) second-order differential equation for G. Either way, if $R_{\mu \nu }u^{\mu }u^{\nu }\ge 0$ (which Raychaudhuri infers from the energy condition $\rho \ge 0$ and the Einstein equations) and $\omega _{\mu \nu }=0$ (i.e. if the flow is irrotational), then, since $\sigma _{\mu \nu }\sigma ^{\mu \nu }\ge 0$ for any congruence, (2.3) implies that if $\theta >0$ on some hypersurface orthogonal to the flow at some time, i.e. the universe is expanding, then $\theta \rightarrow \infty$ at some finite time in the past, which corresponds to $G\rightarrow 0$. Since $\rho \sqrt{-\det (g)}=\rho G^3$ is constant, this suggests $\rho \rightarrow \infty$, but G depends on the coordinates and $G\rightarrow 0$ could merely mark a singularity of the congruence. On Raychaudhuri’s assumption that matter moves along the timelike geodesics tangent to u, however, he does obtain a density and curvature singularity, adding that the time from such a ‘singular state’ to ‘the present state’ is a maximum for isotropic models, so that adding anisotropy to isotropic cosmological models even brings the initial singularity forward!¹⁹

Similar comments apply to Komar [105], whose Introduction is worth quoting as a sign of the times:

A principal success of the general theory of relativity in the realm of cosmology is given by the Friedmann solution of the field equations. This solution, which employs the assumptions that the universe is spacially isotropic and that the state of matter may be represented by incoherent dust, yields the result that the universe is not stationary, but is rather in a state either of expansion from a singular point in time (which would correspond to creation), or of contraction toward a singular point in time (which would correspond to annihilation). The question naturally arises whether such singular points are a consequence of the particular symmetry presupposed in Friedmann’s model, or whether perhaps for more general distributions of matter one need not expect instants of creation or annihilation of the universe. The purpose of this paper is to show that singularities in the solution of the field equations of general relativity are to be expected under very general hypotheses (enumerated specifically below), and in particular that the singular instant of creation (or annihilation) necessarily would occur at a finite time in the past (or future, respectively). (Komar [105, p. 104])

Replacing ‘cosmology’ by ‘gravitational collapse’ and ‘Friedmann’ by ‘Schwarzschild’, this purpose is hardly different from Penrose’s: it echoes the general spirit that one would like to go beyond the simplified singular models of the past and see if their singularities persist (which Komar shows they do).

In modern fashion, Komar just relies on energy assumptions, due to which he need not specify $T_{\mu \nu }$ explicitly—unlike Raychaudhuri [159], whom he does not cite.²⁰ These are: $T_{00}\ge 0$ and $T\ge 0$, and $T_{00}=0$ implies $T_{ij}=0$. But by fiat he forces his geodesic congruence to be the same as Raychaudhuri’s, including its identification with the matter flow lines. This then leads him to very similar conclusions.²¹

Both papers crossed mathematical work on relativistic fluid mechanics,²² which in turn crossed or included the study of spatially homogeneous (Bianchi) cosmologies initiated by Gödel [84], Taub [185] and the Hamburg group, summarized by Heckmann and Schücking [94]. The general goal was

to find singularity free [!] fluid filled Bianchi cosmological models. (Ellis [59, p. 2])

This research influenced all subsequent developments, where Raychaudhuri’s argument survived as a lemma for the mature singularity theorems, in whose proofs it is used to prove focusing of geodesics.

To preface the next phase involving the young relativists Misner, Penrose, and Hawking, I quote:

Wheeler and Sciama had a widespread impact on the development of the understanding of singularities, particularly through their students. Wheeler’s students included Misner, Shepley, Thorne, and Geroch, while Sciama’s included Ellis, Hawking, Carter, Rees, and Clarke. However, Sciama has claimed, on occasion, that his most important contribution to relativity has been in influencing Roger Penrose to work on the subject! (Ellis [187, p. 136])

The last pre-Penrose papers on singularities I mention are Misner [132] and Shepley ([176, 177]).²³ Once again, but for a different reason, it is instructive to quote from the Introduction of [176]:

Most cosmological models in general relativity have a point of singularity (...) which can be reached by a geodesic of finite total length from other points of the space-time manifold, where the metric is degenerate or otherwise irregular (for example, a point where a curvature scalar is infinite). All known dust-filled models, without cosmological constant, have singular points, and it may be conjectured that the presence of dustlike matter filling space always leads to a singularity. This paper shows that this conjecture is true for an important class of universes. (Shepley [176, p. 1403])

This shows that using geodesic incompleteness as a characterization of a singularity was familiar at the time, although e.g. Heckmann & Schucking [94] do not mention it.²⁴ But it was seen as insufficient:

completeness is vulgarly used as the definition of non-singularity. (Ryan & Shepley [167, p. 80])

Thus Misner [132] refines this idea. After the call to arms already quoted in Sect. 2, followed by a comparison with the Riemannian case, geodesic incompleteness is proposed as a necessary condition for a space-time to be singular, in which case it is sufficient that ‘some scalar polynomial in the curvature tensor and its covariant derivatives be unbounded on an (open) geodesic segment of finite length,’ since this implies inextendibility of the geodesic in question into some potential extension of the space-time.²⁵

Shepley [176] claims that Bianchi type IX universes filled with dust are singular even if the congruence of matter flow lines is rotating. But although he finds points where $\det (g)=0$, he shows neither geodesic incompleteness, nor divergence of the density or of some curvature scalar, admitting that ‘The question of what happens as the singular point approaches is presently under investigation’ (p. 1409).²⁶

3 Gravitational collapse and space-time singularities: exegesis

The brevity (2.5 pages) and brilliance of Penrose [148] makes almost every comma relevant, but let me try some restraint. Following a brief discussion of gravitational collapse, Penrose explains his motivation:

The question has been raised as to whether [the Schwarzschild] singularity is, in fact, simply a property of the high symmetry assumed. The matter collapses radially inwards to the single point at the center, so that a resulting space-time catastrophe there is perhaps not surprising. Could not the presence of perturbations which destroy the spherical symmetry alter the situation drastically? (...)

It will be shown that, after a certain critical condition has been fulfilled, deviations from spherical symmetry cannot prevent space-time singularities from arising. (Penrose [148, p. 57, 58])

Penrose does not strictly define what he means by a singularity, and cites none of the papers discussed in the previous section except [142]. But the examples he gives (namely gravitational collapse, Schwarzschild, and Kerr) suggest that he has curvature singularities in mind. Indeed, the preamble to his theorem shows that he does not identify singularities with incomplete geodesics:²⁷

The existence of a singularity can never be inferred, however, without an assumption such as completeness [i.e. inextendibility] for the manifold under consideration.                          [148] (p. 58)]

Similarly, in a list of four possibilities to avoid singularities despite his theorem,²⁸ Penrose includes:

(c) The space-time manifold is incomplete [i.e. extendible].²⁹ (p. 58)

Geodesic completeness first appears in the statement of the theorem, in which it is a reductio ad absurdum assumption, see below, whose negation as a way out of the ensuing contradiction is left to the reader and is nowhere defined as a singularity. The logic of the above two quotes therefore seems to be that inextendibility of the manifold, as opposed to a ‘physical’ singularity, could be the cause of geodesic incompleteness and as such has to be excluded to make room for a curvature singularity as the physically relevant cause of geodesic incompleteness. This logic foreshadows strong cosmic censorship; cf. Sect. 5.

We now come to the theorem, which states that the following assumptions are together inconsistent:

Theorem 1

For any space-time (M, g), the following assumptions are contradictory:³⁰

• 1. A space-time is a four-dimensional time-orientable Lorentzian manifold;³¹

• 2. (M, g) is future null geodesically complete;

• 3. M contains a non-compact Cauchy surface $C_3$;

• 4. One has $(-R_{\mu \nu }+{\frac{1}{2}}g_{\mu \nu } R)t^{\mu }t^{\nu }\ge 0$ for any timelike vector t (at any point);³²

• 5. There exists a trapped surface in M.

Penrose [148] defines a Cauchy surface by the property that every inextendible timelike or null geodesic meets it.³³ This specific concept of a Cauchy surface in gr seems to have originated with Penrose, who therefore, with Leray [120], Choquet-Bruhat [12] and Geroch [76] should be included on the list of architects of global hyperbolicity in gr.³⁴ Here is a picture from 1965 in the master’s hand:

Drawing of a (would-be) Cauchy (hyper)surface, taken from Penrose [146], p. 218.

The concept of a trapped surface was also introduced by Penrose [148] himself, defined as a

closed, spacelike, two-surface $T^2$ with the property that the two systems of null geodesics which meet $T^2$ orthogonally converge locally in future directions at $T^2$. (p. 58)

Qua idea and definition trapped surfaces belong to the most innovative and lasting contributions of [148].³⁵ In the presence of a radial coordinate r, as in the Schwarzschild, Reissner–Nordström, and Kerr solutions, Penrose’s convergence condition is equivalent to the gradient $\nabla r$ being timelike, which in the Schwarzschild solution happens for $r<2m$, and which in the other two (subcritical) cases is the case at least for a while after crossing the event horizon. The convergence condition can be stated in terms of the null hypersurface C generated by the future directed null congruence emanating from any spacelike surface $T^2$. Pick a tetrad $(e_1,e_2, L_+, L_-)$ with $(e_1,e_2)$ spacelike and tangent to C, $L_{\pm }$ null and tangent (and hence orthogonal) to C, such that $g(e_i,e_j)= \delta _{ij}$, $g(e_i,L_{\pm })=0$ for $i,j=1,2$, $g(L_{\pm },L_{\pm })=0$, and $g(L_+,L_-)=-1$, all defined on C. With $\theta _{\pm }=\nabla _{\mu }L_{\pm }^{\mu }$, the surface $T^2$ is trapped iff $\theta _{\pm }<0$ throughout $T^2$ for both signs (whereas in Minkowski space-time one of $\theta _{\pm }$ is $>0$ whilst the other $\theta _{\mp }$ is $<0$).

Thirty years later, Penrose recalled how he suddenly got the idea in the late autumn of 1964:³⁶

My conversation with Robinson stopped momentarily as we crossed a side road, and resumed again at the other side.³⁷ Evidently, during those few moments an idea occurred to me, but then the ensuing conversation blotted it from my mind! Later in the day, after Robinson had left, I returned to my office. I remember having an odd feeling of elation that I could not account for. I began going through in my mind all the various things that had happened to me during the day, in an attempt to find what it was that had caused this elation. After eliminating numerous inadequate possibilities, I finally brought to mind the thought that I had had while crossing the street.³⁸ [186, p. 227]

The proof of the singularity theorem displays some of the unprecedented causal and topological techniques Penrose had developed in the preceding years in connection with gravitational radiation [136, 145, 147]. It is very similar to all later reformulations I am aware of:³⁹

• Assumptions 1, 2, 4, and 5 imply (directly) that $\partial I^+(T^2)$ (which Penrose calls $B^3$) is compact;

• Assumptions 1 and 3 imply (by contradiction) that $\partial I^+(T^2)$ is non-compact.

The argument for the first point is that the null Raychaudhuri equation for $\theta _{\pm }$ (which is due to Penrose, who writes $\rho =-{\frac{1}{2}}\theta _{\pm }$ for both cases) and the boundary condition $\theta _{\pm }<0$ give a focal point on each null geodesic ruling C. It is left to the reader to realize that $\partial I^+(T^2)=J^+(T^2)\backslash I^+(T^2)$, which follows from global hyperbolicity (which makes $J^+(T^2)$ closed), and that beyond each focal point the null geodesic enters $I^+(T^2)$ and hence leaves $\partial I^+(T^2)$, in order to conclude with Penrose that $\partial I^+(T^2)$ is compact.⁴⁰

Penrose’s homotopical argument for the second point seems correct but has not survived;⁴¹ the most rigorous proof I know of (see e.g. [71]) is obtained by combining the following results:⁴²

• 1. Any achronal boundary such as $\partial I^+(T^2)$ is a (locally Lipschitz) topological hypersurface.

• 2. Any compact achronal topological hypersurface in a globally hyperbolic space-time is a Cauchy surface; hence by Geroch [76] Property 7, any Cauchy surface in this space-time is compact.

4 Intermezzo: singularities revisited and refined

In his PhD Thesis, Hawking writes without any further ado (or mention of inextendibility):

any model must have a singularity, that is, it cannot be a geodesically complete $C^1$, piecewise $C^2$ manifold. (Hawking [89, Sect. 4.1])

And similarly in the paper announcing his singularity theorem, citing Misner [132] at this point:

Space-time is said to be singularity free if it is time-complete (all timelike geodesics can be extended to arbitrary length) and if the metric is a $C^2$ tensor field. (Hawking [90, p. 444])

In his subsequent Adams Prize Essay,⁴³ Hawking explicitly assumes inextendibility and proposes to:

take timelike and lightlike geodesic incompleteness as our definition of a singularity of space-time. (Hawking [91, Sect. 6.1])

All of this was directly inspired by Penrose [148], but it seems to have been Hawking, rather than Penrose, who first advocated causal geodesic incompleteness as the definition of a space-time singularity. In a later defense of this definition, Hawking and Ellis make both a physical and a pragmatic point:⁴⁴

Timelike geodesic incompleteness has an immediate physical significance in that it presents the possibility that there could be freely moving observers or particles whose histories did not exist after (or before) a finite interval of proper time. This would appear to be an even more objectionable feature than infinite curvature and so it seems appropriate to regard such a space as singular. (...) The advantage of taking timelike and/or null incompleteness as being indicative of the presence of a singularity is [also] that on this basis one can establish a number of theorems about their occurrence. (Hawking & Ellis [93, p. 258])

As we have seen, crucial nuance is lost in doing so. Although Geroch initially used practically the same definition as Hawking, also citing Misner [132], his subsequent analysis [74] is more critical:

• (a) there is no widely accepted definition of a singularity in general relativity;

• (b) each of the proposed definitions is subject to some inadequacy. (Geroch [74, p. 526])

This paper is still worth reading, as is its more technical successor [77]; see also [3, 26, 31, 32, 48, 173].

Almost 60 years later, none of the problems that arose from identifying space-time singularities with incomplete causal geodesics has been resolved: timelike, null (i.e. lightlike), causal, and general geodesic incompleteness are all inequivalent to each other; each is in turn inequivalent to incompleteness with respect to curves with either arbitrary or bounded acceleration; geodesically incomplete manifolds (of any kind) exist without curvature singularities; global hyperbolicity is compatible with geodesic incompleteness; geodesic completeness is compatible with a lack of global hyperbolicity; et cetera.

Penrose [154] gave a new definition of a singularity (i.e. Definition 2 below) ‘which is not quite the same as that suggested by the singularity theorems’ (p. 85). It has two novel and distinguishing features:

• 1. It is perspectival, in that not just the singularity matters but also its relationship to points which it can influence. This is crucial for understanding Penrose’s mature idea of cosmic censorship (Sect. 5).

• 2. Inextendible timelike curves replace incomplete causal geodesics as marks of singularities.

The change from ‘causal’ to ‘timelike’ is innocent,⁴⁵ but the change from incomplete geodesics to inextendible curves is not (this change is required by the proof of Theorem 5 below).⁴⁶ The key difference lies in the possibility that an inextendible causal curve may go off to infinity and yet count as a singularity in the new sense, i.e. as a ‘locally naked’ one.⁴⁷ For example, even $M={{\mathbb {R}}}^4$ carrying plane gravitational waves, which is geodesically complete and strongly causal but not globally hyperbolic, will now count as singular,⁴⁸ as does anti-de Sitter.⁴⁹ Consequently, Definition 2 below accepts more singularities than the singularity theorems, and excluding these new singularities therefore certainly excludes all old-school singularities: it will be an unnecessarily strong condition. However, in strongly causal space-times that are asymptotically flat at null infinity the right intuition to think about the “new” singularities remains: causal curves that either crash into a curvature singularity or reach the edge of an extendible space-time.⁵⁰

Penrose states his new definition in the language of terminal indecomposable sets [80], but this language can be avoided. A simpler account is as follows:

• 1. Suppose that (M, g) is past distinguishing, i.e., $I^-(x)=I^-(y)$ implies $x=y$. This property allows us to exchange properties of points x for properties of their timelike pasts $I^-(x)$.⁵¹

• 2. By definition, if $z\in I^-(x)$ then z can signal to x, or influence x; we may say that z is naked for x. For the next step, we note that $z\in I^-(x)$ is equivalent to the property $I^-(z)\subset I^-(x)$.

• 3. If z is the endpoint of some future-directed timelike curve c, then $I^-(z)\subset I^-(x)$ if and only if

  $$\begin{aligned} I^-(c)\subset I^-(x). \end{aligned}$$

  (4.1)

• 4. Crucially, this inclusion is also defined if c has no endpoint (i.e. is inextendible). This suggests:

Definition 2

A future-directed future-inextendible causal curve c in M is naked for $x\in M$ if (4.1) holds.

For example, in Minkowski space-time, for fixed x take $z\in I^-(x)$ and remove z. Then any fd future-inextendible timelike curve c whose endpoint would have been z satisfies (4.1).

5 Cosmic censorship: Penrose

Let us now return to Penrose [148]. In Sect. 3 I already noted that his emphasis on the inextendibility (or, as he put it, completeness) of the underlying space-time shows that Penrose was fully aware that (null) geodesic incompleteness alone was insufficient to point to a curvature (or “physical”) singularity, i.e., a “hole”. Similarly, Penrose realized that the “black” in our (later) term “black hole” was as yet lacking:⁵²

Whether or not “visible” singularities inevitably arise under appropriate circumstances is an intriguing question not covered by the present discussion.             (Penrose [148, footnote 9])

This was taken up in his magisterial and very influential paper ‘Gravitational collapse: The role of general relativity’ [151], in which Penrose introduced what we now call weak cosmic censorship:⁵³

We are thus presented with what is perhaps the most fundamental question of general-relativistic collapse theory, namely: does there exist a “cosmic censor” who forbids the appearance of naked singularities, clothing each one in an absolute event horizon? In one sense, a “cosmic censor” can be shown not to exist. For it follows from a theorem of Hawking that the “big bang” singularity is, in principle, observable. But it is not known whether singularities observable from outside will ever arise in a generic collapse which starts off from a perfectly reasonable nonsingular initial state. (Penrose [151, p. 1162])

For a mathematician like Penrose, this conjecture (here phrased as a question) relied on a precise definition of an “absolute event horizon”. Like his 1965 singularity theorem, also this concept used mathematical ideas that Penrose had initially developed for gravitational waves, especially his notion of null infinity ${{\mathscr {I}}}$, pronounced “scri” [144, 145, 150]. Perhaps because [151] is mainly addressed to physicists, he relegates this pretty important matter to a footnote, which reads:⁵⁴

In a general space-time with a well-defined external future infinity, the absolute event horizon would be defined as the boundary of the union of all timelike curves which escape to this external future infinity. In the terminology of [150], if M is a weakly asymptotically simple space-time, for example, then the absolute event horizon in M is ${\dot{I}}_-({{\mathscr {I}}}^+)$ [$=\partial I^-({{\mathscr {I}}}^+)$]. (Penrose [151, footnote 3, p. 1146 of the 2002 reprint])

Penrose leaves it to the reader to extract the mathematical definition of a black hole from this; and as in the case of the definition of a singularity, it was again Hawking who was subsequently more explicit:

A black hole on a spacelike surface [S] is defined to be a connected component of the region of the surface bounded by the event horizon [i.e. $M\backslash I^-({{\mathscr {I}}}^+)\cap S$].(Hawking [92, Abstract, p. 152])

Later (p. 156) he explains that this is a region ‘from which there is no escape to ${{\mathscr {I}}}^+$’. This definition is repeated in Hawking & Ellis [93], Sect. 9.2, and is still standard,⁵⁵ usually in the simple form $M\backslash I^-({{\mathscr {I}}}^+)$.

However, since there can be holes (in the sense of curvature singularities) that are not black in having no event horizons,⁵⁶ and there may be event horizons that do not cover any kind of singularity,⁵⁷ it might be better to call (a connected component of) $M\backslash I^-({{\mathscr {I}}}^+)$ a black object instead of a black hole. In any case, his definition of an event horizon led Penrose to a more precise definition of weak cosmic censorship:

I first review what is now the “standard” picture of gravitational collapse to a black hole—which I shall refer to as the establishment viewpoint. The picture is, in fact, a very good one, with a remarkable degree of internal consistency. But it is worth emphasizing that, among other things, this picture does depend on a very big assumption, that of cosmic censorship. In my own view, this assumption might well be false in suitable circumstances. I believe that the consequences of possible violations of cosmic censorship are certainly worth considering seriously (...) In effect, this principle states that naked singularities do not develop out of an initially nonsingular state (...). A naked singularity may be said to exist in a space-time M if there is a well-defined external infinity (that is, ${{\mathscr {I}}}^+$, if M is weakly asymptotically simple) from which timelike curves may be drawn into the past that terminate on the singularity. I shall not make precise, here, the concept of “terminating on a singularity” but refer instead to Hawking’s concept of asymptotic predictability.⁵⁸ This is a precise condition on a space-time M that may be taken as a statement that M accords with a form of cosmic censorship. Imposing such a condition, we deduce the existence of an absolute event horizon E, which encloses but does not intersect the singular region. This horizon E may be defined as the boundary of the past of the external infinity. (Penrose [153, p. 125])

The first part is tongue in cheek, since the “establishment” included—or even was—Penrose himself! The ‘form of cosmic censorship’ probably refers to Hawking’s [92] theorem to the effect that any future trapped surface lies entirely within a black hole.⁵⁹ Penrose [153] goes on to develop the Penrose inequality, which he saw as a test for cosmic censorhip he expected to fail (which it has not, so far).⁶⁰

The challenge of making ‘the concept of “terminating on a singularity”’ more precise was taken up in [154], see also [155], and led to Definition 2 above. A definition of cosmic censorship should then specify the nature of the points x relative to which “singular” curves c are naked. Hence it would have been natural if Penrose had proposed e.g. the following definition (cf. footnote 54):⁶¹

Definition 3

Let (M, g) be a strongly causal space-time that is asymptotically flat at null infinity.

• 1. An N-naked singularity in M with respect to some region of exposure $N\subset {\hat{M}}$ is a future-inextendible causal curve c in M that is naked for some point $x\in N$, i.e., the pair (c, x) satisfies $I^-(c)\subset I^-(x)$.

• 2. The N-weak cosmic censorship conjecture states that a “physically reasonable” space-time (M, g) contains no N-naked singularities, i.e., no curve c in M and point $x\in N$ as specified in 1. exist.

Paraphrasing Geroch & Horowitz [79], p. 274, the set N represents the region of space-time (or possibly of its conformal completion ${\hat{M}}$) ‘in which observers could detect that their universe is singular.’ This definition is quite subtle and excludes for example that the big bang or the big crunch are seen as naked singularities (and the ensuing cosmos as violating weak cosmic censorship): for the big bang, which we can “see”, no such curve c exists (any relevant c is past inextendible), whereas for the big crunch no x would exist. See Geroch & Horowitz [79], pp. 274–277 and Penrose [155], §12.3.2.⁶²

Various choices of the region N have (often implicitly) been proposed in the literature, for example:

• $N=I^-({{\mathscr {I}}}^+)$, i.e. the complement of the black hole region in M (my interpretation of [151]);

• $N=I^-({{\mathscr {I}}}^+)\cap I^+({{\mathscr {I}}}^-)\equiv \mathscr {D}(M,{{\mathscr {I}}})$, i.e. the domain of outer communication(s) in M;⁶³

• $N=\mathscr {D}(M,{{\mathscr {I}}})\cup {{\mathscr {I}}}$ with ${{\mathscr {I}}}={{\mathscr {I}}}^+\cup {{\mathscr {I}}}^-$ [157];

• $N=J^-({{\mathscr {I}}}^+)\cap J^+(S)$, where S is some partial Cauchy surface in M.⁶⁴

• $N= J^-({{\mathscr {I}}}^+)$ [187];⁶⁵. See also footnote 73.

• $N={{\mathscr {I}}}^+$ [79];⁶⁶

• $N=M$ [154, 155], see below (strong cosmic censorship); this is defined without any need for asymptotic flatness.

The first and last choices seem optimal in having a clean characterization via Theorems 4 and 5. First:

Theorem 4

Let (M, g) be a strongly causal space-time that is asymptotically flat at null infinity. Then (M, g) contains no naked singularities with respect to $I^-({{\mathscr {I}}}^+)$ iff $I^-({{\mathscr {I}}}^+)$ is globally hyperbolic.

This is proved in the same way as Penrose’s Theorem 5 below, adding a case distinction $x\in I^-({{\mathscr {I}}}^+)$ and $x\notin I^-({{\mathscr {I}}}^+)$ that does not upset the result. If one accepts the choice $N=I^-({{\mathscr {I}}}^+)$, Theorem 4 might of course serve to define weak cosmic censorship as global hyperbolicity of $I^-({{\mathscr {I}}}^+)$. Except for $N=M$ there seems to be no complete analogue of Theorem 4 for the other choices of N; at best, there is a one-sided implication to global hyperbolicity of N. In any case, I side with Earman (private communication) that one may regard weak cosmic censorship as a family of conjectures, one for each relevant region N.

But Definition 3 is ahistorical! Instead, Penrose [154, 155] immediately moved on, arguing that:⁶⁷

It seems to me to be comparatively unimportant whether the observer himself can escape to infinity. Classical general relativity is a scale-invariant theory, so if locally naked singularities occur on a very tiny scale, they should also, in principle, occur on a very large scale in which a ‘trapped’ observer could have days or even years to ponder upon the implications of the uncertainties introduced by the observations of such a singularity. (...) Indeed, for inhabitants of recollapsing closed universes (as possibly we ourselves are) there is no ‘infinity’, so the question of being locally ‘trapped’ is one of degree rather than principle. It would seem, therefore, that if cosmic censorship is a principle of Nature, it should be formulated in such a way as to preclude such locally naked singularities (Penrose [155, p.619]).

Thus ‘locally naked singularities’ are defined via Definition 3 (assuming only strong causality) by choosing $N=M$. Then the key result (which a posteriori justifies his use of curves instead of geodesics) is due to Penrose [154, 155]:

Theorem 5

A strongly causal space-time has no locally naked singularities iff it is globally hyperbolic.

Proof.⁶⁸ I first prove the implication “global hyperbolicity $\Rightarrow$ no locally naked singularities” by contradiction. Suppose that (M, g) is globally hyperbolic and that (4.1) holds for some c and x. Take y on c and then take a future-directed sequence $(y_n)$ of points on c, with $y_0=y$. Because of (4.1) this sequence lies in $J^+(y)\cap J^-(x)$, which is compact by assumption. Hence $(y_n)$ has a limit point z in $J^+(y)\cap J^-(x)$.

Now define curves $(\gamma _n)$ as the segments of c from y to $y_n$. By Theorem 2.53 in [131], case (i), which is a sharp version of the limit curve lemma of causal theory going back to [152], these curves have a uniform limit $\gamma$. Its arc length (as measured by an auxiliary complete Riemannian metric) is, on the one hand, infinite (since c is endless and hence has infinite arc length, which is approached as the $y_n$ move up along c). But on the other hand it is finite, since $\gamma$ ends at z (and fd continuous causal curves have finite arc length iff they have an endpoint). Hence (4.1) cannot be true for any c and x.

The (contrapositive) proof of the converse implication relies on the following lemma:⁶⁹

Lemma 6

Let (M, g) be a space-time, let $S\subset M$ be closed and achronal, and let $x,y\in M$.

• 1. If $y\in \mathrm {int}(D^-(S))$, then $J^+(y)\cap J^-(S)$ is compact. In particular, taking $S=\partial I^-(x)$ and assuming $y\in I^-(x)$, it follows that $J^+(y)\cap J^-(x)$ is compact.

• 2. We have $\mathrm {int}(D^-(S))=I^-(S)\cap I^+(D^-(S))$.

To prove the converse direction of Theorem 5, assume that (M, g) is not globally hyperbolic. Then, under the assumption of strong causality, there are x, y for which $J^-(x)\cap J^+(y)$ is not compact.⁷⁰ We may assume that $y\in I^-(x)$. Part 1 of Lemma 6 gives $y\notin \mathrm {int}(D^-(\partial I^-(x)))$. Part 2 gives some $y'\in I^-(x)$ with $y'\notin D^-(\partial I^-(x))$, so that, by definition of $D^-$, there exists some fd future-inextendible curve c from $y'$ that avoids $\partial I^-(x)$. Since $y'\in I^-(x)$, this curve does lie in $I^-(x)$, and hence (4.1) holds.⁷¹$\square$

Definition 7

The strong cosmic censorship conjecture states that no “physically reasonable” space-time (M, g) contains locally naked singularities, i.e. pairs (c, x) as in Definition 2 such that $I^-(c)\subset I^-(x)$. Equivalently, by Theorem 5 the conjecture states that any “physically reasonable” space-time is globally hyperbolic.

In this formulation (unlike many others), strong cosmic censorship (as defind by the last bullet above) implies weak cosmic censorship (as defind by the first bullet).⁷²

Under the assumptions of Penrose’s singularity theorem one can prove an analogue of Hawking’s [92] theorem on the invisibility of trapped surfaces (cf. [93, Proposition 9.2.1]):

Theorem 8

Under the assumptions of Theorem 1 and the additional clause that (M, g) be asymptotically flat at null infinity, any future trapped surface must lie entirely within the “black region” $M\backslash I^-({{\mathscr {I}}}^+)$.

I just sketch the proof, as the details are similar to Hawking’s. If the trapped surface $T^2$ were to (partly) lie in $I^-({{\mathscr {I}}}^+)$, then also part of $\partial I^+(T^2)$ lies in $I^-({{\mathscr {I}}}^+)$. Hence some of the lightlike geodesics $\gamma$ ruling $\partial I^+(T^2)$ with past endpoint on $T^2$ would reach ${{\mathscr {I}}}^+$ and hence have infinite length. But the definition of a trapped surface excludes this, as in the proof of Theorem 1, since it forces each $\gamma$ to be incomplete. $\square$

Since this forces $M\backslash I^-({{\mathscr {I}}}^+)$ to be non-empty, there must be an event horizon $\partial I^-({{\mathscr {I}}}^+)$, which by the same argument must cover not only any trapped surface but also all ensuing incomplete null geodesics.

6 Cosmic censorship: from Penrose to pde

Theorem 8 bridges part of the gap between Penrose’s 1965 singularity theorem and its intended application to black holes, in that in the asymptotically flat setting an event horizon comes for free. However, according to this analysis (whose centerpiece is Penrose’s Theorem 5), strong cosmic censorship (à la Penrose, cf. Definition 7) is in fact one of the assumptions of the singularity theorem, so that its implication of weak cosmic censorship (once again in the sense of Penrose, i.e. defined by taking $N=I^-({{\mathscr {I}}}^+)$ in Definition 3) is hardly surprising. And we are also left with the issue of the (in)extendibility of space-time.

With hindsight, it would have been natural if, one the basis of [148], Penrose had defined:⁷³

• Weak cosmic censorship as the existence and stability of event horizons (in case of singularities);

• Strong cosmic censorship as the nonexistence or instability of Cauchy horizons (in all cases).

This would have directly addressed the above gap. In the first (weak) case this is indeed what he did, but he subsequently replaced this by the version of strong cosmic censorship that really interested him.⁷⁴ However, via a detour explained below, the latter eventually did lead to the second bullet point.

The current versions of both cosmic censorship conjectures are based on the initial-value approach or pde approach to gr, whose starting point may be taken to be the celebrated theorem of Choquet-Bruhat and Geroch [16].⁷⁵ This theorem states that for each initial data triple (S, h, K) satisfying the vaccum constraints, where (S, h) is a complete 3d Riemannian manifold carrying an additional symmetric 2-tensor K, there exists a maximal globally hyperbolic development or mghd (M, g, i), where (M, g) is a space-time satisfying the (vacuum) Einstein equations and $i:S\hookrightarrow M$ is an embedding such that g returns the initial data (h, K) on i(S), in that $i^*g=h$ and K is the extrinsic curvature of $i(S)\subset M$. The space-time (M, g) contains i(S) as a Cauchy surface and hence is globally hyperbolic (as the name mghd  suggests). Moreover, the triple (M, g, i) is unique up to (unique) time-orientation-preserving isometries fixing the Cauchy surface,⁷⁶ i.e. for any other mghd $(M',g',i')$ there exists an isometry $\psi :M'\rightarrow M$ that preserves time orientation and satisfies $\psi \circ i'=i$. This theorem has given rise to an ideology about gr in which:

• All valid assumptions about gr are assumptions about initial data (S, h, K).

• All valid questions about gr  are questions about “the” mghd (M, g, i) of these data.

This pde-based program sometimes gives a different perspective from the Penrose–Hawking–Geroch mathematical approach to gr  originating in the 1960s, in which typically larger (usually maximally extended) space-times are studied. In particular, in the pde approach one would at first sight ask if some mghd (M, g, i) obtained from “physically reasonable” initial data satisfies strong cosmic censorship.

By Theorem 5 this comes down to the question if (M, g) is globally hyperbolic. But this is true by construction! Hence Penrose’s clear and distinct formulation of strong cosmic censorship seems problematic in a paradoxical way:

• 1. Global hyperbolicity of the mghd (M, g, i) of any initial data set (S, h, K) is automatic.

• 2. Global hyperbolicity of maximally extended (black hole) space-times is often too strong.⁷⁷

In what they call the ‘evolutionary approach’ to cosmic censorship, Geroch & Horowitz [79], Sect. 5.4, show a way out and en passant sketch the modern pde versions of both the weak and strong case. As we shall see, their version of strong cosmic censorship can actually be seen as a special case of Penrose’s.

Their proposal for weak cosmic censorship is based on their earlier refinement of Penrose’s notion of asymptotic flatness at null infinity [78]. Using the notation of footnote 54, assuming that the scaling function $\Omega$ satisfies ${\hat{\Delta }}\,\Omega =0$, as can always be achieved by a redefinition of $\Omega$, the extra condition is that the null geodesics ruling the null hypersurface ${{\mathscr {I}}}^{\pm }$ are complete.⁷⁸

Consider now the following statement: for any asymptotically flat initial-data set, topologically ${{\mathbb {R}}}^3$, its maximal evolution [i.e. mghd] is an asymptotically flat spacetime [at null infinity] (i.e. it even satisfies completeness of null infinity). This statement, we claim, captures a sense of cosmic censorship. That the initial-data set be topologically ${{\mathbb {R}}}^3$ ensures that the evolution is not singular already on S; that S be asymptotically flat ensures that one deals with isolated systems, and in particular that any singular behavior of the evolution must be due to the system itself and not external influences. Suppose, then, that this maximal evolution were singular, say to the future of S. It certainly cannot be nakedly singular for the future of S, for S must be a Cauchy surface for its evolution. Furthermore, the statement asserts that this evolution must be sufficiently large that it includes the entire asymptotic regime, so in particular asymptotic observers can live out their entire lives within this maximal evolution. What this statement means, then, is that asymptotic observers will forever be unaffected by any singular behavior of the spacetime. But this is a version of cosmic censorship. We have essentially just re-expressed our earlier statement (that the closure of N have empty intersection with ${{\mathscr {I}}}^+$), but now in a way that avoids the counterexample obtained by removing a point from Minkowski spacetime. (Geroch & Horowitz [79, pp. 285–286])

In heuristic terms this means that, if we disregard exceptional initial conditions, no singularities are observed from infinity, even though observations from infinity are allowed to continue indefinitely. (Christodoulou [18, p. A26])

This statement of weak cosmic censorship is exemplary from a pde point if view: it starts with an assumption on initial data and formulates a question about the ensuing mghd. However, like the definition of a black hole it depends on the concept of null infinity, which is “prescient” and may be seen as an undesirable idealization (cf. [33], and footnote 55). Thus [18] reformulated the above statement of weak cosmic censorship in such a way that the idealization ${{\mathscr {I}}}^+$ no longer occurs.⁷⁹

Either way, this turns the Penrosian version of weak cosmic censorship on its head! For whereas his version states that outgoing signals from a black hole singularity are blocked by a (future) event horizon $H_+(\mathscr {E})$, the new version is about incoming (null) signals: the further these are away from $H_+(\mathscr {E})$, the longer it takes them to enter $H_+(\mathscr {E})$, and in the limit (i.e., in the original formulation, at null infinity) this takes infinitely long, making future null infinity ${{\mathscr {I}}}^+$ complete. Yet there is a connection with Penrose: lack of global hyperbolicity of $I^-({{\mathscr {I}}}^+)$ gives a partial Cauchy surface in M a Cauchy horizon which cuts off ${{\mathscr {I}}}^+$. This is clear from simple examples like Schwarzschild for $m<0$, Reissner–Nordström at $e^2>m^2$, or Kerr at $a^2>m^2$; see e.g. [42] and [112], Sect. 10.6.

The status of the weak cosmic censorship conjecture(s) in any reasonable form is open.⁸⁰ One especially problematic issue is the status of the genericity conditions that define “physically reasonable” space-times (or initial conditions) and hence the range of applicability of the conjecture(s). For example, the physical relevance of genericity conditions typically used in the mathematical literature, which are evidently based on pde techniques, has been questioned [87, Sect. 3.4] .

The current (pde) version of strong cosmic censorship also goes back to [79]:

Consider, for motivation, an initial-data set whose maximal evolution is extendible to the future of S (...) This extended spacetime cannot, by definition of the maximal evolution, have S as a Cauchy surface. That is, from a point p in the extension there must exist a maximally extended past-directed timelike curve which cannot be assigned a past endpoint, and which fails to meet S. In this rather mild sense the extended spacetime must be nakedly singular. One might therefore imagine formulating cosmic censorship as the assertion that every maximal evolution [i.e. mghd] is inextendible, i.e. that, once the maximal evolution is completed, it is not possible to add any ‘extra regions’ as vantage points from which observers could detect that their spacetime is singular to the future of S. (Geroch & Horowitz [79, pp. 286–287])

Interestingly, they immediately refine this suggestion by giving the example of a small spacelike disk S in Minkowski spacetime $\mathbb {M}$, whose mghd is its domain of dependence D(S) (which is a double cone), which is clearly extendible to all of $\mathbb {M}$. To avoid cases like that, they end up with the following:

Conjecture 5.2. For p any point in any extension of the maximal evolution of any non-compact initial data set S, $I^-(p)\cap S$ has non-compact closure. [79, pp. 288]

But this refinement seems not to have survived, perhaps because the models on which the conjecture is tested already exclude trivial (counter)examples like the one mentioned. Thus we now simple have:

The appropriate notion of cosmic censorship (...) is that the generic solution to Einstein’s equations is globally hyperbolic, i.e., that the maximal Cauchy development of a generic initial data set is inextendible. (Moncrief [134, p. 88])

The strong cosmic censorship conjecture says that ‘most’ spacetimes developed as solutions of Einstein’s equations from prescribed initial data cannot be extended outside of their maximal domains of dependence.⁸¹(Chruściel, Isenberg, & Moncrief [24, p. 1671])

Conjecture 17.1 (Strong Cosmic Censorship). For generic initial data for Einstein’s equations, the mghd  is inextendible. (Ringström [162, p. 188])

Conjecture 3.5 (Strong cosmic censorship). For generic asymptotically flat vacuum data sets, the maximal Cauchy development (M, g) is inextendible as a suitably regular Lorentzian manifold. (Dafermos [40, p. 11])

If the extension satisfies the Einstein equations this is a special case of Penrose’s formulation [155], applied to a judicious choice of space-time! To see this, recall the following concept [21]:

Definition 9

A development of initial data (S, h, K) satisfying the vacuum constraints is a triple (M, g, i), where (M, g) is a space-time solving the vacuum Einstein equations and $i:S\rightarrow M$ is a spacelike embedding such that $i^*g=h$ and i(S) has extrinsic curvature K in M. Such a development is called maximal if it has no extension $(M',g')$ that also satisfies the vacuum Einstein equations.⁸²

Now apply Penrose’s strong cosmic censorship to such a maximal development, i.e. require it to be globally hyperbolic (cf. Theorem 5). The connection with inextendibility is then easily made:

Proposition 10

A maximal development of initial data (S, h, K) is globally hyperbolic (with Cauchy surface i(S)) if and only if the mghd  of (S, h, K) is inextendible as a solution to the vacuum Einstein equations. In that case, “the” maximal development coincides with “the” mghd (up to isomorphism).

Proof

The set of isometry classes [M, g, i] of Cauchy developments (M, g, i) of given initial data (S, h, K) is partially ordered, and the mghd $[M_t,g_t,i_t]$ is its top element. Hence if some maximal development $(M_m,g_m,i_m)$ is globally hyperbolic, then $[M_m,g_m,i_m]\le [M_t,g_t,i_t]$. On the other hand, since $(M_t,g_t,i_t)$ is a solution and $(M_m,g_m,i_m)$ is maximal also the converse holds, so $(M_m,g_m,i_m)\cong (M_t,g_t,i_t)$. $\square$

Therefore, Penrose’s conjecture applied to the maximal development à la [21] states that for “generic” initial data the mghd is inextendible as a solution to the vacuum Einstein equations. The tension that Penrose’s formulation may have given rise to reflected the ambiguity about the space-time his condition of global hyperbolicity is supposed to apply to: it seems that in the 1960s and 1970s people had maximally extended solutions in mind,⁸³ see e.g. the Penrose diagrams in [93], whereas from the 1990s onward the mghd in the pde picture had become the space-time of choice.

However, the strong cosmic censorship conjecture as stated by all authors quoted above is stronger than the one suggested by Proposition 10, in that even extensions that do not satisfy the Einstein equations are excluded. Indeed, any space-time solves these equations for some energy-momentum tensor, however unfamiliar, and one has no idea what kind of (strange) matter is contained in an extension. But the Einstein equations should at least be defined:⁸⁴ to this end one may either consider extensions in which the metric is $C^2$ (i.e. the borderline case where Einstein equations make sense as classical pdes), or allow $C^0$ metrics for which the associated Christoffel symbols are locally $L^2$ (which is the least regular case in which the metric can still be defined as a weak solution to Einstein’s equations).⁸⁵

Sensitivity to the precise formulation of genericity conditions (which from a pde point of view define which initial data are deemed “physically reasonable” and from a traditional gr  perspective do so for space-times) is another issue;⁸⁶ already the Kerr metric, in which strong cosmic censorship fails for all parameter values (as long as $a\ne 0$ and $m\ne 0$), shows that genericity conditions are necessary. Such counterexamples made it especially courageous of Penrose to conjecture strong cosmic censorship; but of course he had good arguments. His key observation, indeed one of his most prophetic insights, was first published in 1968 (i.e. before even weak cosmic censorship had been formulated by him):

There is a further difficulty confronting our observer who tries to cross [the Cauchy horizon] $H_+(\mathscr {H})$. As he looks out at the universe he is “leaving behind,” he sees, in one final flash, as he crosses $H_+(\mathscr {H})$, the entire later history of the rest of his “old universe.” If, for example, an unlimited amount of matter eventually falls into the star then presumably he will be confronted with an infinite density of matter along “$H_+(\mathscr {H})$”. Even if only a finite amount of matter falls in, it may not be possible in generic situations to avoid a curvature singularity in place of $H_+(\mathscr {H})$. This is at present an open question. But it may be, that the place to look for curvature singularities is in this region rather than (or as well as?) at the “center.” (Penrose [150, p. 222])

Here one should realize that for Penrose $H_+(\mathscr {H})$ is a (future) Cauchy horizon (relative to some partial Cauchy surface S) in some “large” (typically maximally extended) space-time, whereas from a pde point of view it is the boundary of the mghd  of the corresponding initial data on S, if it has one; that is, if it is extendible (in a suitable regularity class). Either way, the conjectured “blueshift instability” of $H_+(\mathscr {H})$ has been confirmed in a large number of studies and remains the key to strong cosmic censorship.⁸⁷

Failure of strong cosmic censorship is often taken to imply a failure of determinism in gr.⁸⁸ The idea is that the (classical) world—including the (physical degrees of freedom of the) gravitational field itself—is governed by hyperbolic partial differential equations whose initial data should be given on a hypersurface S and whose solutions should thereby be determined on its domain of dependence D(S) [29]. From Penrose’s point of view, if a (typically maximally extended) space-time $(M',g')$ fails to be globally hyperbolic, then, taking some partial Cauchy surface $S'$, neither the part $M' \backslash D(S')\ne \emptyset$ of space-time behind the Cauchy horizon of $S'$ itself, nor things happening within space-time behind this horizon, are determined by initial data on $S'$. From the pde point of view, although any mghd (M, g) is globally hyperbolic with Cauchy surface $S\subset M$, if (M, g) is extendible with extension $(M',g')$, then by definition S fails to be a Cauchy surface for $M'\supset M$ and we are back to Penrose.

If the extension $(M',g')$ satisfies the Einstein equations and is globally hyperbolic, which is logically possible, then the lack of determinism is merely apparent: although $(M',g')$ is not determined by the initial data on S originally expected to do so, it is determined by new initial data on some Cauchy surface $S'\subset M'$, which may be accessible to physicists in $M'$. But in interesting examples the mghd(M, g) typically has extensions $(M',g')$ that fail to be either globally hyperbolic or unique (up to isometry), or both.⁸⁹ In those cases the ensuing violation of determinism seems real (though hard to interpret).

In any case, although extendibility of the mghd does seem to give rise to some form of indeterminism in gr, so that (contrapositively) determinism necessitates strong cosmic censorship in the pde sense of inextendibility of the mghd, the latter is not sufficient for determinism. The indeterminism of someone falling into a black hole singularity is perfectly well compatible with global hyperbolicity, as the Schwarzschild solution shows. More generally, in classical (mathematical) physics indeterminism may come from either a lack of uniqueness of solutions or from a lack of existence thereof; the latter includes incompleteness of solutions, i.e. non-existence after (or before) some finite time. Strong cosmic censorship secures uniqueness of solutions of the Einstein equations (whose existence is given by pde theory à la Choquet-Bruhat), but, as we have seen, it fails to imply existence (i.e. for all times) of solutions of the geodesic equation [46]. Hence at the classical level strong cosmic censorship does not imply determinism.⁹⁰ But ironically, determinism might be rescued by quantum mechanics: because of the unitarity of their time-evolution quantum objects cannot disappear into a singularity, and hence they form standing waves. These only become sources of indeterminism if they undergo measurement.⁹¹

7 Epilogue

Prior to the 1960s spacetime singularities were regarded as a minor embarrassment for GTR. They constituted an embarrassment because it was thought by Einstein and others that a true singularity, a singularity in the fabric of spacetime itself, was an absurdity. But the embarrassment was a minor one that could be swept under the rug; for the then known models of GTR containing singularities all embodied very special and physically unrealistic features. Two developments forced a major shift in attitude. First, the observation of the cosmic low temperature blackbody radiation lent credence to the notion that our universe originated in a big bang singularity.⁹² Second, and even more importantly, a series of theorems due principally to Stephen Hawking and Roger Penrose indicated that, according to GTR, singularities cannot be relegated to the distant past because under quite general conditions they can be expected to occur both in cosmology and in the gravitational collapse of stars. Thus, singularities cannot be swept under the rug; they are, so to speak, woven into the pattern of the rug. Of course, these theorems might have been taken as turning what was initially only a minor embarrassment into a major scandal. Instead, what occurred in some quarters was a 180$^\circ$ reorientation in point of view: singularities were no longer relegated to obscurity; rather they were to be recognized as a central feature of the GTR, a feature which called attention to a new aspect of reality that was neglected in all previous physical theories, Newtonian and special relativistic alike.

(Earman [47, p. 65])

Thus the way the scientific community came to believe in both black holes and a big bang in the 1960s, following 50 years of denial in the light of considerable evidence right from the start of general relativity, seems nothing short of a scientific revolution. In Kuhnian parlance, the singularity theorems of Penrose and Hawking from 1965–1966 launched the underlying “paradigm shift”. But here is a critical note:

Kuhn has much of interest to say about normal science, about the way in which a scientific community is united by a set of practices. But what Kuhn failed to articulate is that practices are just that—practices. They need not be, in general, statements in which scientists (implicitly or explicitly) believe, and this for two main reasons. First, what unites a scientific community need not be a set of beliefs. Shared beliefs are much less common than shared practices. (...) Second, beliefs, in themselves, cannot explain the scientific process. Statements lead on to statements only in the logical plane. Historically, people must intervene to get one statement from the other. (Netz [135, p. 2])

Much like ancient Greek mathematics (which is what Netz writes about), singularities in gr provide a fascinating interplay between practices and beliefs. With hindsight this interplay was almost paradoxical:

• Before 1965, practices like Friedman [66, 67] or Lemaître [118, 119] in cosmology, and Oppenheimer & Snyder [142] in astrophysics, which on sober judgement should have caused a paradigm shift, failed to do so, because they clashed with what the leading scientists then believed.

• After 1965 the opposite happened: at the level of practice, for the reasons explained in this paper (namely that it is inconclusive exactly about the subject it should be about, namely black holes) Penrose’s theorem would not have been able to launch a revolution. But what the relevant scientific community believed about the theorem, culminating in his 2020 Nobel Prize, carried the day.

Finally, in the context of the history of mathematics, Grattan-Guiness [85] calls attention to the historical actors’ knowledge or ignorance of their ignorance. Restricting ourselves to the two main actors:

Did Einstein [55] know that he did not know what a singular space-time was? The evidence is mixed (see Sect. 2): he combined egregious mistakes (in which he was not alone) with stunning foresight.

Was Penrose [148] aware of the gap between his theorem and its intended application to black holes? Here the evidence is clear: he was, and he subsequently did everything he could to overcome it.