An introduction to local Black Hole horizons in the 3+1 approach to General Relativity

We present an introduction to dynamical trapping horizons as quasi-local models for black hole horizons, from the perspective of an Initial Value Problem approach to the construction of generic black hole spacetimes. We focus on the geometric and structural properties of these horizons aiming, as a main application, at the numerical evolution and analysis of black hole spacetimes in astrophysical scenarios. In this setting, we discuss their dual role as an "a priori" ingredient in certain formulations of Einstein equations and as an "a posteriori" tool for the diagnosis of dynamical black hole spacetimes. Complementary to the first-principles discussion of quasi-local horizon physics, we place an emphasis on the "rigidity" properties of these hypersurfaces and their role as privileged geometric probes into near-horizon strong-field spacetime dynamics.Comment: 37 pages, 5 figures. Notes prepared for the course at the 2011 Shanghai Asia-Pacific School and Workshop on Gravitation (Shanghai Normal University, February 10-14, 2011

Finite VEVs from a Large Distance Vacuum Wave Functional

We show how to compute vacuum expectation values from derivative expansions of the vacuum wave functional. Such expansions appear to be valid only for slowly varying fields, but by exploiting analyticity in a complex scale parameter we can reconstruct the contribution from rapidly varying fields.Comment: 39 pages, 16 figures, LaTeX2e using package graphic

Bounds on area and charge for marginally trapped surfaces with cosmological constant

We sharpen the known inequalities $A \Lambda \le 4\pi (1-g)$ and $A\ge 4\pi Q^2$ between the area $A$ and the electric charge $Q$ of a stable marginally outer trapped surface (MOTS) of genus g in the presence of a cosmological constant $\Lambda$. In particular, instead of requiring stability we include the principal eigenvalue $\lambda$ of the stability operator. For $\Lambda^{*} = \Lambda + \lambda > 0$ we obtain a lower and an upper bound for $\Lambda^{*} A$ in terms of $\Lambda^{*} Q^2$ as well as the upper bound $Q \le 1/(2\sqrt{\Lambda^{*}})$ for the charge, which reduces to $Q \le 1/(2\sqrt{\Lambda})$ in the stable case $\lambda \ge 0$. For $\Lambda^{*} < 0$ there remains only a lower bound on $A$. In the spherically symmetric, static, stable case one of the area inequalities is saturated iff the surface gravity vanishes. We also discuss implications of our inequalities for "jumps" and mergers of charged MOTS.Comment: minor corrections to previous version and to published versio

PT-symmetry from Lindblad dynamics in a linearized optomechanical system

We analyze a lossy linearized optomechanical system in the red-detuned regime under the rotating wave approximation. This so-called optomechanical state transfer protocol provides effective lossy frequency converter (quantum beam-splitter-like) dynamics where the strength of the coupling between the electromagnetic and mechanical modes is controlled by the optical steady-state amplitude. By restricting to a subspace with no losses, we argue that the transition from mode-hybridization in the strong coupling regime to the damped-dynamics in the weak coupling regime, is a signature of the passive parity-time (PT) symmetry breaking transition in the underlying non-Hermitian quantum dimer. We compare the dynamics generated by the quantum open system (Langevin or Lindblad) approach to that of the PT-symmetric Hamiltonian, to characterize the cases where the two are identical. Additionally, we numerically explore the evolution of separable and correlated number states at zero temperature as well as thermal initial state evolution at room temperature. Our results provide a pathway for realizing non-Hermitian Hamiltonians in optomechanical systems at a quantum level

Painlevé-II approach to binary black hole merger dynamics: universality from integrability

The binary black hole merger waveform is both simple and universal. Adoptingan effective asymptotic description of the dynamics, we aim at accounting forsuch universality in terms of underlying (effective) integrable structures.More specifically, under a ``wave-mean flow'' perspective, we propose that fastdegrees of freedom corresponding to the observed waveform would be subject toeffective linear dynamics, propagating on a slowly evolving background subjectto (effective) non-linear integrable dynamics. The Painlev\'e property of thelatter would be implemented in terms of the so-called Painlev\'e-IItranscendent, providing a structural link between i) orbital (in particular,EMRI) dynamics in the inspiral phase, ii) self-similar solutions of non-lineardispersive Korteweg-de Vries-like equations (namely, the `modified Korteweg-deVries' equation) through the merger and iii) the matching with the isospectralfeatures of black hole quasi-normal modes in late ringdown dynamics. Moreover,the Painlev\'e-II equation provides also a `non-linear turning point' problem,extending the linear discussion in the recently introduced Airy approach tobinary black hole merger waveforms. Under the proposed integrabilityperspective, the simplicity and universality of the binary black hole mergerwaveform would be accounted to by the `hidden symmetries' of the underlyingintegrable (effective) dynamics. In the spirit of asymptotic reasoning, andconsidering Ward's conjecture linking integrability and self-dual Yang-Millsstructures, it is tantalizing to question if such universal patterns wouldreflect the actual full integrability of a (self-dual) sector of generalrelativity, ultimately responsible for the binary black hole waveform patterns.<br