Lectures on Linear Stability of Rotating Black Holes

These lecture notes are concerned with linear stability of the non-extreme Kerr geometry under perturbations of general spin. After a brief review of the Kerr black hole and its symmetries, we describe these symmetries by Killing fields and work out the connection to conservation laws. The Penrose process and superradiance effects are discussed. Decay results on the long-time behavior of Dirac waves are outlined. It is explained schematically how the Maxwell equations and the equations for linearized gravitational waves can be decoupled to obtain the Teukolsky equation. It is shown how the Teukolsky equation can be fully separated to a system of coupled ordinary differential equations. Linear stability of the non-extreme Kerr black hole is stated as a pointwise decay result for solutions of the Cauchy problem for the Teukolsky equation. The stability proof is outlined, with an emphasis on the underlying ideas and methods.Comment: 25 pages, LaTeX, 3 figures, lectures given at first DOMOSCHOOL in July 2018, minor improvements (published version

A Simple Theory of Every 'Thing'

One of the criteria to a strong principle in natural sciences is simplicity. This paper claims that the Free Energy Principle (FEP), by virtue of unifying particles with mind, is the simplest. Motivated by Hilbert’s 24th problem of simplicity, the argument is made that the FEP takes a seemingly mathematical complex domain and reduces it to something simple. More specifically, it is attempted to show that every ‘thing’, from particles to mind, can be partitioned into systemic states by virtue of self-organising symmetry break, i.e. self-entropy in terms of the balance between risk and ambiguity to achieve epistemic gain. By virtue of its explanatory reach, the FEP becomes the simplest principle under quantum, statistical and classical mechanics conditions

Disordered systems and Burgers' turbulence

Talk presented at the International Conference on Mathematical Physics (Brisbane 1997). This is an introduction to recent work on the scaling and intermittency in forced Burgers turbulence. The mapping between Burgers' equation and the problem of a directed polymer in a random medium is used in order to study the fully developped turbulence in the limit of large dimensions. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer, correlated on large distances. A replica symmetry breaking solution of the polymer problem provides the full probability distribution of the velocity difference $u(r)$ between points separated by a distance $r$ much smaller than the correlation length of the forcing. This exhibits a very strong intermittency which is related to regions of shock waves, in the fluid, and to the existence of metastable states in the directed polymer problem. We also mention some recent computations on the finite dimensional problem, based on various analytical approaches (instantons, operator product expansion, mapping to directed polymers), as well as a conjecture on the relevance of Burgers equation (with the length scale playing the role of time) for the description of the functional renormalisation group flow for the effective pinning potential of a manifold pinned by impurities.Comment: Latex, 11 page