Diophantine Approximation as Cosmic Censor for AdS Black Holes

The present thesis reveals a novel connection of Diophantine approximation arising from small divisors to general relativity, more precisely, the Strong Cosmic Censorship conjecture. The main results provide theorems which resolve a linear scalar analog of the Strong Cosmic Censorship conjecture in general relativity for Λ < 0. The proofs are intimately tied to small divisors and the resolution crucially depends on suitable Diophantine conditions. A further ingredient is the novel scattering theory on black hole interiors established at first. The thesis consists of three parts. In the first part we develop a scattering theory for the linear wave equation on the interior of Reissner–Nordström black holes. The main result shows the existence, uniqueness and asymptotic completeness of finite energy scattering states on the interior of Reissner–Nordström. The past and future scattering states are represented as suitable traces of the solution ψ on the bifurcate event and Cauchy horizons. Finally, we prove that, in contrast to the above, on the Reissner–Nordström–(Anti-)de Sitter interior, there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass. This part is joint work with Yakov Shlapentokh-Rothman (Princeton University). The second and third parts are motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes. We consider smooth linear perturbations governed by the conformal wave equation on Reissner–Nordström–AdS and Kerr–AdS black holes, respectively. We prescribe initial data on a spacelike hypersurface of a Reissner–Nordström–AdS and Kerr–AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity. It was known previously by work of Holzegel–Smulevici that such waves only decay at a sharp logarithmic rate (in contrast to the polynomial rate in the asymptotically flat regime) in the black hole exterior. In view of this slow decay, the question of uniform boundedness or blow-up at the Cauchy horizon in the black hole interior (and thus the validity of the linear scalar analog of the C0-formulation of the Strong Cosmic Censorship conjecture) has remained up to now open. In the second part of the thesis, we answer the question of uniform boundedness in the affirmative for Reissner–Nordström–AdS: We show that |ψ| ≤ C in the black hole interior. In this setting, this corresponds to the statement that the linear scalar analog of the C0-formulation of Strong Cosmic Censorship is false. The proof follows a new approach, combining physical space estimates with Fourier based estimates exploited in the scattering theory developed in the first part. In the third part of the thesis, we show that |ψ| → ∞ at the Cauchy horizon of Kerr–AdS if the dimensionless black hole parameters mass and angular momentum satisfy certain Diophantine properties. This is in stark contrast to the second part as well as previous works on Strong Cosmic Censorship for Λ ≥ 0. In particular, as a result of the Diophantine conditions, we show that these resonant black hole parameters form a Baire-generic but Lebesgue-exceptional subset of parameters below the Hawking–Reall bound. On the other hand, we conjecture that, as is the case for Reissner–Nordström–AdS, linear waves remain bounded at the Cauchy horizon |ψ| ≤ C for a set of black hole parameters which is Baire-exceptional but Lebesgue-generic. This means that the answer to the above question concerning uniform boundedness or blow-up on the Kerr–AdS interior is either negative or affirmative depending on the parameters considered. Thus, in this setting, the validity of the linear scalar analog of the C0 -formulation of Strong Cosmic Censorship depends in an unexpected way on the notion of genericity imposed

A Long Way From Home: Transatlantic Sea Star Migration

For a marine biologist strolling along the eastern coast of South America, finding some sea stars (popularly called starfish, and technically members of the class Asteroidea of the phylum Echinodermata) is not all that unusual. However, when closer inspection reveals the sea stars to be ones that are also found on the shores of the western coast of Africa, the marine biologist now faces an interesting question: How did these sea stars come to be so far from home? Originally published in Consortiumand used with permission

Gravitational collapse to extremal black holes and the third law of black hole thermodynamics

We construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry which are exactly isometric to extremal Reissner-Nordstr\"om after a finite advanced time along the event horizon. Moreover, in each of these examples the apparent horizon of the black hole coincides with that of a Schwarzschild solution at earlier advanced times. In particular, our result can be viewed as a definitive disproof of the "third law of black hole thermodynamics." The main step in the construction is a novel $C^k$ characteristic gluing procedure, which interpolates between a light cone in Minkowski space and a Reissner-Nordstr\"om event horizon with specified charge to mass ratio $e/M$. Our setup is inspired by the recent work of Aretakis-Czimek-Rodnianski on perturbative characteristic gluing for the Einstein vacuum equations. However, our construction is fundamentally nonperturbative and is based on a finite collection of scalar field pulses which are modulated by the Borsuk-Ulam theorem.Comment: Final version, 46 pages + references, 14 figure

Every Data Point Counts: Political Elections in the Age of Digital Analytics

Synthesizing the investigative research and cautionary messages from experts in the fields of technology, political science, and behavioral science, this project explores the ways in which digital analytics has begun to influence the American political arena. Historically, political parties have constructed systems to target voters and win elections. However, rapid changes in the field of technology (such as big data, artificial intelligence, and the prevalence of social media) threaten to undermine the integrity of elections themselves. Future political campaigns will utilize profiling to micro-target individuals in order to manipulate and persuade them with hyper-personalized political content. Most dangerously, the average American voter does not understand how these technologies will influence elections. This research project simplifies the complex technological and sociopolitical landscape in order to educate the public on the issue. Even more important than understanding the problem is arming people with practical solutions to combat this intimidating problem. Therefore, this project utilizes the language of visual design to turn something complex and intimidating into something approachable and empowering. This academic and creative contribution aims to ensure that American elections remain free and fair

Extremal black hole formation as a critical phenomenon

In this paper, we prove that extremal black holes arise on the threshold of gravitational collapse. More precisely, we construct smooth one-parameter families of smooth, spherically symmetric solutions to the Einstein-Maxwell-Vlasov system which interpolate between dispersion and collapse and for which the critical solution is an extremal black hole. Physically, these solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space from past infinity. Depending on the precise value of the parameter, we show that the Vlasov matter either disperses due to the combined effects of angular momentum and electromagnetic repulsion, or undergoes gravitational collapse. At the critical value of the parameter, an extremal Reissner-Nordstr\"om black hole is formed. No naked singularities occur as the extremal threshold is crossed. We call this critical phenomenon extremal critical collapse and the present work constitutes the first rigorous result on the black hole formation threshold in general relativity.Comment: 91 pages + references, 16 figure

A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes

We develop a scattering theory for the linear wave equation □ψ=0 on the interior of Reissner-Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution ψ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies ω and ℓ. This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants Λ, there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein-Gordon equation with conformal mass on the (anti-) de Sitter-Reissner-Nordström interior

Diophantine approximation as Cosmic Censor for Kerr–AdS black holes

AbstractThe 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 &lt; 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 &lt; 0 may change in a spectacular way according to the notion of genericity imposed.</jats:p