Singularities, Black Holes, and Cosmic Censorship: A Tribute to Roger Penrose

Abstract

In the light of his recent (and fully deserved) Nobel Prize, this pedagogical paper draws attention to a fundamental tension that drove Penrose’s work on general relativity. His 1965 singularity theorem (for which he got the prize) does not in fact imply the existence of black holes (even if its assumptions are met). Similarly, his versatile definition of a singular space–time does not match the generally accepted definition of a black hole (derived from his concept of null infinity). To overcome this, Penrose launched his cosmic censorship conjecture(s), whose evolution we discuss. In particular, we review both his own (mature) formulation and its later, inequivalent reformulation in the pde literature. As a compromise, one might say that in “generic” or “physically reasonable” space–times, weak cosmic censorship postulates the appearance and stability of event horizons, whereas strong cosmic censorship asks for the instability and ensuing disappearance of Cauchy horizons. As an encore, an “Appendix” by Erik Curiel reviews the early history of the definition of a black hole.

Black holes and their horizons in semiclassical and modified theories of gravity

“…To describe what the end results of these scenarios entail we have to clarify the terminology. There is no unanimously agreed upon definition of a black hole, 38,39 and the different alternatives for gravitational collapse leave us with a variety of notions for what is meant by this concept. For objects with a horizon it is useful to adapt the terminology of Ref.…”

“…2. (ii) A philosophical justification of the second assumption is based on the principle 39,61 that "no effect can be counted as a genuine physical effect if it disappears when the idealizations are removed". Consequently, in order for a horizon to be considered a genuine physical object rather than merely a useful mathematical tool, it must form in finite time according to a distant observer, and there should be some potentially observable consequences of this formation.…”

“…However, in this case Eq. (39) indicates that the apparent horizon cannot be reached in a finite time t of Bob.…”

“…This geodesic incompleteness means that some of its inextendible timelike or null geodesics, future or past directed, have a finite proper length or a finite canonical parameter. 43,60,74,78 (A refinement, 39,58 introduced to eliminate trivial examples of incompleteness, also requires the absence of the spacetime manifold extension M → M that produces an extendible image γ ∈ M).…”

“…58,60,74,78,248 The theorems themselves do not establish the existence of physically relevant curvature singularites. 39,43 Moreover, a curvature singularity does not imply geodesic incompleteness. 60,249 Nevertheless, coupled with additional input, such as various forms of the cosmic censorship conjecture, the end results are physically relevant singularities.…”