Tag - Black holes

Mihalis Dafermos: Extremal and near-extremal black holes

27th March, 2024

Extremal (maximally rotating or maximally charged) and near-extremal black holes are of intense interest both for real astrophysics and in the context of fashionable speculations in high energy physics. They remain perhaps the most misunderstood objects in classical general relativity. In this talk, I will first introduce extremal black holes to a general mathematical audience. I will then discuss the stability problem for extremal (and near-extremal) black holes and describe a new conjectural picture of the moduli space of solutions of the Einstein equations describing gravitational collapse.

Tanja Hinderer: Analytical methods in general relativity

24th January, 2024

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Vojtech Witzany: Motion of spinning test particles in black hole space-times

22nd January, 2024

Many interesting gravitational-wave sources can be effectively described in a large-mass ratio expansion. The leading effect of the lighter secondary in such sources can be read off from the dynamics of a spinning test particle in the space-time of the heavy primary. I will discuss the integrability and exact solution of motion of spinning test particles in black hole space-times, focusing on bound motion but also mentioning some results on scattering.

Lucile Cangemi: General spin Compton amplitudes for Kerr black holes

22nd January, 2024

Classical observables for Kerr black hole dynamics can be constructed from scattering amplitudes. Elegant three-point spin-s amplitudes exist for Kerr black holes, however constructing the corresponding four-point Compton amplitudes is an open problem. In this talk, I will discuss the origin of the Kerr three-point amplitudes from a higher-spin theory perspective. Guided by higher-spin constraints and classical-limit analysis, I will propose quantum and classical tree-level Compton amplitudes relevant for Kerr to all orders in spin. I will also comment on upcoming results for scattering observables that require the classical Compton amplitude as input.

Anton Ilderton: Strong-field and non-perturbative amplitudes

22nd January, 2024

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Frans Pretorius: Insights into the rotating black hole interior from numerical solutions

31st May, 2023

The last couple of decades has seen tremendous progress in numerical solution of the Einstein field equations for regions of spacetime exterior to black hole horizons. For reasons I will briefly discuss, the corresponding advances have not been of much help for the problem of the black hole interior, in particular for black holes formed from gravitational collapse outside of spherical symmetry.

We are thus still left having to appeal to simplified scenarios to try to gain some insight into this problem. In that regard, I will present results form numerical studies of rotating black holes formed from scalar field collapse in asymptotically Anti de-Sitter spacetime, in 2+1 dimensions, with circular symmetry imposed. Despite the simplicity of this model, the interior exhibits rich phenomenology that I will describe in the talk.

Marius Junge: Complexity and dynamics in finite and infinite dimension

26th May, 2023

In recent joint work with Yidong Chen, we discovered spectral gap estimates and concentration inequalities for for dynamics with few generators. Some of these estimates are dimension free and then can be used to feed in the recent theory of complexity initiated by Lloyd and Jaffe, and adapted more recently for specific resources. The goal is to find a viable theory of complexity which holds in type II1 and III1 von Neumann algebras, both of which come naturally in quantum field theory and Witten's take on black holes.

Sergiu Klainerman: On the mathematical theory of black holes

16th October, 2017

The gravitational waves detected recently by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which, generically, solutions of the initial value problem of the Einstein vacuum equations approach asymptotically, in any compact region, a Kerr black hole. Though the conjecture is so very easy to formulate and happens to be consistent with astrophysical observations as well as numerical experiments, its proof is far beyond our current mathematical understanding, let alone available techniques techniques. In fact even the far simpler and fundamental question of the stability of the Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of new geometric and analytic techniques which I will try to outline in my lectures.