Adaptive computation of gravitational waves from black hole interactions

We construct a class of linear partial differential equations describing general perturbations of non-rotating black holes in 3D Cartesian coordinates. In contrast to the usual approach, a single equation treats all radiative $\ell -m$ modes simultaneously, allowing the study of wave perturbations of black holes with arbitrary 3D structure, as would be present when studying the full set of nonlinear Einstein equations describing a perturbed black hole. This class of equations forms an excellent testbed to explore the computational issues of simulating black spacetimes using a three dimensional adaptive mesh refinement code. Using this code, we present results from the first fully resolved 3D solution of the equations describing perturbed black holes. We discuss both fixed and adaptive mesh refinement, refinement criteria, and the computational savings provided by adaptive techniques in 3D for such model problems of distorted black holes.Comment: 16 Pages, RevTeX, 13 figure

A Reinvestigation of Moving Punctured Black Holes with a New Code

We report on our code, in which the moving puncture method is applied and an adaptive/fixed mesh refinement is implemented, and on its preliminary performance on black hole simulations. Based on the BSSN formulation, up-to-date gauge conditions and the modifications of the formulation are also implemented and tested. In this work we present our primary results about the simulation of a single static black hole, of a moving single black hole, and of the head-on collision of a binary black hole system. For the static punctured black hole simulations, different modifications of the BSSN formulation are applied. It is demonstrated that both the currently used sets of modifications lead to a stable evolution. For cases of a moving punctured black hole with or without spin, we search for viable gauge conditions and study the effect of spin on the black hole evolution. Our results confirm previous results obtained by other research groups. In addition, we find a new gauge condition, which has not yet been adopted by any other researchers, which can also give stable and accurate black hole evolution calculations. We examine the performance of the code for the head-on collision of a binary black hole system, and the agreement of the gravitational waveform it produces with that obtained in other works. In order to understand qualitatively the influence of matter on the binary black hole collisions, we also investigate the same head-on collision scenarios but perturbed by a scalar field. The numerical simulations performed with this code not only give stable and accurate results that are consistent with the works by other numerical relativity groups, but also lead to the discovery of a new viable gauge condition, as well as clarify some ambiguities in the modification of the BSSN formulation.Comment: 17 pages, 8 figures, accepted for publication in PR

Final State of Gregory-Laflamme Instability

We describe the behavior of a perturbed 5-dimensional black string subject to the Gregory-Laflamme instability. We show that the horizon evolves in a self-similar manner, where at any moment in the late-time development of the instability the horizon can be described as a sequence of 3-dimensional spherical black holes of varying size, joined by black string segments of similar radius. As with the initial black string, each local string segment is itself unstable, and this fuels the self-similar cascade to (classically) arbitrarily small scales; in the process the horizon develops a fractal structure. In finite asymptotic time, the remaining string segments shrink to zero-size, yielding a naked singularity. Since no fine-tuning is required to excite the instability, this constitutes a generic violation of cosmic censorship. We further discuss how this behavior is related to satellite formation in low-viscosity fluid streams subject to the Rayleigh-Plateau instability, and estimate the fractal dimension of the horizon prior to formation of the naked singularity.Comment: 27 pages, 6 Figures. Chapter of the book `Black Holes in Higher Dimensions' to be published by Cambridge University Press (editor: G. Horowitz

On the Azimuthal Stability of Shock Waves around Black Holes

Analytical studies and numerical simulations of time dependent axially symmetric flows onto black holes have shown that it is possible to produce stationary shock waves with a stable position both for ideal inviscid and for moderately viscous accretion disks. We perform several two dimensional numerical simulations of accretion flows in the equatorial plane to study shock stability against non-axisymmetric azimuthal perturbations. We find a peculiar new result. A very small perturbation seems to produce an instability as it crosses the shock, but after some small oscillations, the shock wave suddenly transforms into an asymmetric closed pattern, and it stabilizes with a finite radial extent, despite the inflow and outflow boundary conditions are perfectly symmetric. The main characteristics of the final flow are: 1) The deformed shock rotates steadily without any damping. It is a permanent feature and the thermal energy content and the emitted energy vary periodically with time. 2) This behavior is also stable against further perturbations. 3) The average shock is still very strong and well defined, and its average radial distance is somewhat larger than that of the original axially symmetric circular shock. 4) Shocks obtained with larger angular momentum exhibit more frequencies and beating phenomena. 5) The oscillations occur in a wide range of parameters, so this new effect may have relevant observational consequences, like (quasi) periodic oscillations, for the accretion of matter onto black holes. Typical time scales for the periods are 0.01 and 1000 seconds for black holes with 10 and 1 million solar mass, respectively.Comment: 15 pages, 7 figures, accepted by the Astrophysical Journa

On kk-Gons and kk-Holes in Point Sets

We consider a variation of the classical Erd\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position

Relationships between charge density response functions, exchange holes and localized orbitals

The charge density response function and the exchange hole are closely related to each other via the fundamental fluctuation-dissipation theorem of physics. A simple approximate model of the static response function is visually compared on several examples in order to demonstrate this relationship. This study is completed by illustrating the well-known isomorphism between the exchange hole and the square of the dominant localized orbital lying in the space region of the reference point of the exchange hole function. The implications of these relationships for the interpretation of common chemical concepts, such as delocalization, are discussed.Comment: 10 two-columns pages, including 3 figure