Richard H. Price

Black Holes: The Membrane Paradigm

The physics of black holes is explored in terms of a membrane paradigm which treats the event horizon as a two-dimensional membrane embedded in three-dimensional space. A 3+1 formalism is used to split Schwarzschild space-time and the laws of physics outside a nonrotating hole, which permits treatment of the atmosphere in terms of the physical properties of thin slices. The model is applied to perturbed slowly or rapidly rotating and nonrotating holes, and to quantify the electric and magnetic fields and eddy currents passing through a membrane surface which represents a stretched horizon. Features of tidal gravitational fields in the vicinity of the horizon, quasars and active galalctic nuclei, the alignment of jets perpendicular to accretion disks, and the effects of black holes at the center of ellipsoidal star clusters are investigated. Attention is also given to a black hole in a binary system and the interactions of black holes with matter that is either near or very far from the event horizon. Finally, a statistical mechanics treatment is used to derive a second law of thermodynamics for a perfectly thermal atmosphere of a black hole.

A design overview of an eccentric-motion electrostatic microactuator (the wobble motor)

Abstract A description is given of the design, fabrication and analysis of an electrostatically-driven micro-actuator in which a planetary-motion rotor rolls inside a cylindrically shaped stator cavity. The design has four primary advantages: (1) the motor geometry and the rolling motion enable very small gaps to be achieved, which are accurate and stable, and across which electrostatic forces act, leading to high forces on the rotor; (2) relative motion is achieved by rolling rather than sliding, thus obviating the concern over internal friction; (3) higher output torques can be traded for lower rotor speeds, due to immediate planetary reduction; and (4) the power output should be higher than for systems constructed using two-dimensional silicon fabrication approaches, since woble motor lengths are not limited by such fabrication methods. The stator segment recruitment logic can range from simple, open-loop stepping to full servo-controlled commutation using rotor position sensors. Two-dimensional analytical and finite-element simulations that estimate motor torque generated by electrostatic fields have been used to determine the influece of: (1) rotor and stator radii; (2) stator segment angular width and position with respect to the contact point; and (3) dielectric properties and dimensions (e.g., insulator thickness on rotor) of motor materials. Dynamic modelling is being used in the comparison of predicted and observed motor behavior, and for the study of the effects of stator segment recruitment logic. A number of eccentric-motion micromotors constructed via different fabrication techniques, have been operated. Electro-discharge machining (EDM) is the fabrication method of choice for the prototypes presently used for experimental studies. Typical rotor diameters for the EDM motor are about 500 μm, with lengths of 500 μm. Motor operation has been achieved with commutation rates in excess of 120 000 r.p.m.

The wobble motor: an electrostatic, planetary-armature, microactuator

A variety of micromotor concepts has been evaluated, with the wobble motor (WM) approach being one of those selected for extensive study. Various WM configurations have been analytically evaluated using finite-element methods and closed-form solutions. Important performance characteristics have been estimated such as stall torque and free speed, and alternate strategies for motor control have been examined. Five motor configurations and a variety of silicon-based and non-silicon-based fabrication techniques have been examined in detail. In exploratory exercises, motors have been fabricated using direct mechanical assembly, electro-discharge machining (EDM), cylindrical photolithographic etching, and coextrusion of metal and plastic. The EDM approach was selected as the motor alternative which could function as an experimental testbed. Fifteen EDM WMs have been constructed and utilized for experimental purposes. Experiments aimed at generating simple preliminary data have been conducted, and results compare reasonably to analytical studies.<<ETX>>

Nonexistence of conformally flat slices of the Kerr spacetime

Initial data for black hole collisions are commonly generated using the Bowen-York approach based on conformally flat 3-geometries. The standard (constant Boyer-Lindquist time) spatial slices of the Kerr spacetime are not conformally flat, so that use of the Bowen-York approach is limited in dealing with rotating holes. We investigate here whether there exist foliations of the Kerr spacetime that are conformally flat. We limit our considerations to foliations that are axisymmetric and that smoothly reduce in the Schwarzschild limit to slices of constant Schwarzschild time. With these restrictions, we show that no conformally flat slices can exist.

Colliding black holes: The Close limit

The problem of the mutual attraction and joining of two black holes is of importance as both a source of gravitational waves and as a testbed of numerical relativity. If the holes start out close enough that they are initially surrounded by a common horizon, the problem can be viewed as a perturbation of a single black hole. We take initial data due to Misner for close black holes, apply perturbation theory and evolve the data with the Zerilli equation. The computed gravitational radiation agrees with and extends the results of full numerical computations.

Late-time behavior of stellar collapse and explosions. II. Nonlinear evolution.

We compare the predictions of linearized theory for the radiation produced in the collapse of a spherically symmetric scalar field with a full numerical integration of the Einstein equations. We find power-law tails and quasinormal ringing remarkably similar to predictions of linearized theory even in cases where nonlinearities are crucial. We also show that power-law tails develop even when the collapsing scalar field fails to produce a black hole.

The wobble motor: design, fabrication and testing of an eccentric-motion electrostatic microactuator

The basic characteristics and advantages of microelectromechanical systems, machines constructed of small moving subelements which have characteristic dimensions in the range of about 0.5 to 500 mu m, are reviewed; design problems are considered; and particular attention is given to the wobble motor (WM) developed by S.C. Jacobsen et al. (1989) using electrodischarge machining (EDM). Fifteen EDM WMs have been constructed and utilized for experimental purposes. WM explorations have demonstrated that micro systems can be quite easily constructed by means other than etching silicon. Experiments aimed at generating simple preliminary data have been conducted, and results compare reasonably to analytical studies. Results reported include stall torque measurements, speed measurements, torque-speed measurements, and wear measurements.<<ETX>>

Modelling considerations for electrostatic forces in electrostatic microactuators

Abstract Electrostatic force generation may offer distinct advantages over more familiar magnetostatics at size scales approaching microns. The fabrication of very small electrostatic actuators is becoming technologically feasible, but is extremely difficult, so that mathematical modelling of actuator designs is likely to be very important in the advancement of this technology. Modelling involves difficulties not only in finding solution (typically numerical) to a mathematical problem, but more important, it requires that the mathematical problem be well formulated. This in turn requires an understanding of, and intuition for, what electrostatic effects are likely to be revelant, as well as an appreciation for the behavior of materials in electrostatic interactions and for the impact on other machine components (bearing, loads, etc). The well-established lore of magnetostatics is not of much use as a guide in this task for several reasons: magnetic materials tend to be either highly permeable (i.e. ferromagnetic) or to have no magnetic effect. By contrast, there are no electrostatically inert materials; the relative dielectric constant e of any solid (of normal density) is of order two or greater, and thus any solid element of an electrostatic configuration has a significant influence on the field. Also, the sources of magnetic fields, currents or magnetization, can be specified with some confidence, while the sources of the electrostatic field, electric charge and polarization, are much more elusive and subject to change. It is the purpose of this paper to point out some of the effects that must be taken into account if a mathematical model is to give an adequate representation of the behavior of an actualy system. To do this we sketch a brief list of the types of electrostatic elements and interactions (conductors, dielectrics, compensated and uncompensated electrets, ferroelectrics, image forces, dielectrophretic forces, etc.) and use this list as background for discussing some electrostatic effects that may be important in the design or modelling of microactuators. For some of these effects, applicable results are reported from experimental investigations carried out with both a small (several hundred micron scale) electrostatically actuated device (‘SCOFSS’) built to study aspects of microelectromechanical design and of control via electrostatic actuation, and the ‘Wobble Motor’, a successful electrostatic microactuator.

Head-on collisions of black holes: The particle limit

We compute gravitational radiation waveforms, spectra, and energies for a point particle of mass

Optimizing the collegiate preparticipation physical evaluation.

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