Diophantine approximation as Cosmic Censor for Kerr–AdS black holes

The purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant $$\Lambda <0$$ Λ < 0 and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations $$\psi $$ ψ of Kerr–AdS solving $$\Box _g\psi -\frac{2}{3}\Lambda \psi =0$$ □ g ψ - 2 3 Λ ψ = 0 with reflecting boundary conditions imposed at infinity. Understanding the behavior of $$\psi $$ ψ at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass $${\mathfrak {m}} = M \sqrt{-\Lambda }$$ m = M - Λ and angular momentum $${\mathfrak {a}} = a \sqrt{-\Lambda }$$ a = a - Λ satisfy a certain non-Diophantine condition, then perturbations $$\psi $$ ψ arising from generic smooth initial data blow up $$|\psi |\rightarrow +\infty $$ | ψ | → + ∞ at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for $$\Lambda \ge 0$$ Λ ≥ 0 —in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters $${\mathfrak {m}}, {\mathfrak {a}}$$ m , a for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking–Reall bound. On the other hand, we conjecture that for a set of parameters $${\mathfrak {m}}, {\mathfrak {a}} $$ m , a which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon $$|\psi |\le C$$ | ψ | ≤ C . This suggests that the validity of the $$C^0$$ C 0 -formulation of Strong Cosmic Censorship for $$\Lambda <0$$ Λ < 0 may change in a spectacular way according to the notion of genericity imposed.

Strong cosmic censorship in near-extremal Kerr-Sen-de Sitter spacetime

In this work, we first calculate equations of motion for particles in the Kerr-Sen-de Sitter black hole spacetime. Then, in the eikonal regime, we analytically obtain the quasi-normal resonant modes of massless neutral scalar field perturbation and find its imaginary part to be characterized by the surface gravity of a near-extremal Kerr-Sen-de Sitter black hole with the Cauchy horizon approaching the event horizon. We further show that the Penrose strong cosmic censorship conjecture is thus respected in this spacetime with dilaton scalar field and axion pseudoscalar field.

Vacuum Semiclassical Gravity Does Not Leave Space for Safe Singularities

General relativity predicts its own demise at singularities but also appears to conveniently shield itself from the catastrophic consequences of such singularities, making them safe. For instance, if strong cosmic censorship were ultimately satisfied, spacetime singularities, although present, would not pose any practical problems to predictability. Here, we argue that under semiclassical effects, the situation should be rather different: the potential singularities which could appear in the theory will generically affect predictability, and so one will be forced to analyse whether there is a way to regularise them. For these possible regularisations, the presence and behaviour of matter during gravitational collapse and stabilisation into new structures will play a key role. First, we show that the static semiclassical counterparts to the Schwarzschild and Reissner–Nordström geometries have singularities which are no longer hidden behind horizons. Then, we argue that in dynamical scenarios of formation and evaporation of black holes, we are left with only three possible outcomes which could avoid singularities and eventual predictability issues. We briefly analyse the viability of each one of them within semiclassical gravity and discuss the expected characteristic timescales of their evolution.

Diving inside a hairy black hole

We investigate the interior of the Einstein-Gauss-Bonnet charged black-hole with scalar hair. We find a variety of dynamical epochs, with the particular important feature that the Cauchy horizon is not present. This makes the violation of the no-hair theorem a possible tool to understand how might the strong cosmic censorship conjecture work.

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

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.