The search for an understanding of an energy source great enough to explain the gamma-ray burst (GRB) phenomenon has attracted much attention from the astrophysical community since its discovery. In this paper we extend the work of Asano and Fukuyama, and Salmonson and Wilson and analyze the off-axis contributions to the energy-momentum deposition rate (MDR) from the -" collisions above a rotating black hole/thin accretion disk system. Our calculations are performed by imaging the accretion disk at a specified observer using the full geodesic equations and calculating the cumulative MDR from the scattering of all pairs of neutrinos and antineutrinos arriving at the observer. Our results shed light on the beaming efficiency of GRB models of this kind. Although we confirm Asano and Fukuyama's conjecture as to the constancy of the beaming for small angles away from the axis, we find that the dominant contribution to the MDR comes from near the surface of the disk with a tilt of approximately /4 in the direction of the disk's rotation. We find that the MDR at large radii is directed outward in a conic section centered around the symmetry axis and is larger by a factor of 10-20 than the on-axis values. By including this off-axis disk source, we find a linear dependence of the MDR on the black hole angular momentum.
We reinvestigate the utility of time-independent constant mean curvature foliations for the numerical simulation of a single spherically-symmetric black hole. Each spacelike hypersurface of such a foliation is endowed with the same constant value of the trace of the extrinsic curvature tensor, K. Of the three families of K-constant surfaces possible (classified according to their asymptotic behaviors), we single out a sub-family of singularity-avoiding surfaces that may be particularly useful, and provide an analytic expression for the closest approach such surfaces make to the singularity. We then utilize a non-zero shift to yield families of K-constant surfaces which (1) avoid the black hole singularity, and thus the need to excise the singularity, (2) are asymptotically null, aiding in gravity wave extraction, (3) cover the physically relevant part of the spacetime, (4) are well behaved (regular) across the horizon, and (5) are static under evolution, and therefore have no "grid stretching/sucking" pathologies. Preliminary numerical runs demonstrate that we can stably evolve a single spherically-symmetric static black hole using this foliation. We wish to emphasize that this coordinatization produces K-constant surfaces for a single black hole spacetime that are regular, static and stable throughout their evolution. I. CONSTANT CRUNCH SURFACESIn this paper, we address a single question: Is there a numerically-viable coordinatization of a Schwarzschild black hole spacetime foliated by hypersurfaces of constant (not necessarily zero) mean extrinsic curvature? In other words, can we coordinatize the Schwarzschild spacetime with constant mean extrinsic curvature (T r(K) = constant) hypersurfaces so as to bound the growth of metric components and their gradients? We demonstrate here that the single shift freedom yields a spacetime metric that is static, and therefore bounds the growth in time of such gradients. A more complete analysis of the stability of our coordinatization, and a more thorough canvassing of the parameter space, will appear elsewhere.[1] Our foliation is consistent with that of Iriondo et al. [2], who provided a generic constant mean curvature (CMC) foliation of the Reissner-Nordström spacetime for the purpose of finding trapped surfaces. In this paper we focus on the utility of CMC slicings for the numerical simulation of black holes, in support of the emerging field of gravity-wave astrophysics.The trace of the extrinsic curvature tensor (T r(K) = K a a = K) at a point on a spacelike hypersurface measures the fractional rate of contraction of 3-volume along a unit normal to the surface. It represents the amount of "crunch" the 3-surface is experiencing at the point, at a given time. If all the observers throughout a spacelike hypersurface moving in time orthogonal to the surface experience the same amount of contraction per unit proper time, we say that the surface is a K-surface or a "constant crunch" surface. In this paper we examine foliations of a single sphericallysymmetric, static black h...
Abstract.Simplicial lattices provide an elegant framework for discrete spacetimes. The inherent orthogonality between a simplicial lattice and its circumcentric dual yields an austere representation of spacetime which provides a conceptually simple form of Einstein's geometric theory of gravitation. A sufficient understanding of simplicial spacetimes has been demonstrated in the literature for spacetimes devoid of all non-gravitational sources. However, this understanding has not been adequately extended to non-vacuum spacetime models. Consequently, a deep understanding of the diffeomorphic structure of the discrete theory is lacking. Conservation laws and symmetry properties are attractive starting points for coupling matter with the lattice. We present a simplicial form of the contracted Bianchi identity which is based on the E. Cartan moment of rotation operator. This identity manifest itself in the conceptuallysimple form of a Kirchhoff-like conservation law. This conservation law enables one to extend Regge Calculus to non-vacuum spacetimes and provides a deeper understanding of the simplicial diffeomorphism group.
We show that the smoothed-particle-hydrodynamics discretization technique is compatible with the principles of general relativity if the contact interactions are modeled by spatial smoothing functions in the local frame comoving with the fluid. We then develop a smoothed-particlehydrodynamics algorithm to model a non-self-gravitating isentropic fluid on a curved background.The equations of the fluid are discretized using a kernel whose spatial support is of a constant proper width in the local comoving frame. The one-dimensional relativistic shock problem is used for testing this algorithm. I. CONTACT FORCES AND THE LOCAL COMOVING FRAMEFurther insight into the dynamics of matter and fields in strong gravitational potentials requires, in part, truly three-dimensional relativistically covariant computer algorithms to reveal the complex motions and instabilities inherent to environments of this kind. Computer simulations that impose symmetries often times freeze out physical instabilities, and nonrelativistic codes both accumulate errors on the order of U /2c per time step and exclude essential driving mechanisms (e.g. , the gravitomagnetic force). An obvious astrophysical problem which calls for a fully three-dimensional treatment is a "generic" case of a thick accretion disk around a spinning black hole: ordinarily the rotation axis of the disk and of the hole will not coincide. Therefore, effects of the frame dragging (Bardeen-Petterson process') are important. Moreover, thick accretion tori exhibit nonaxisymmetric and hence inherently three-dimensional instabilities.The recently developed smoothed-particle-hydrodynamics (SPH) discretization technique provides a valuable tool for the analogous problems in the nonrelativistic regime, and it may be the only tractable approach to problems of this nature.However, every SPH code developed so far is valid only in the Newtonian limit. It is for this reason that we have developed a relativistically covariant formulation of SPH in this paper. It was not clear that a consistent relativistic treatment of Quid dynamics by the SPH technique was even possible. Our initial concerns stemmed from the fact that SPH modeled pressure gradients in a three-dimensional noncovariant fashion. However, we demonstrate that such a formalism is indeed possible. 8'e conclude that the relativistic hydrodynamical contact interactions are mediated by kernels whose supports reside in the local frame comoUing with the f?uid We test our th. esis by demonstrating via computer simulation that our algorithm reproduces shock propagation (to within 3% of the expected shock velocity) in a relativistic Riemann-shocktube experiment over an extreme range of effective temperatures (Fig. 3).A three-space model of the hydrodynamic contact forces by distributions does not lead to any contradictions with relativity. It does not assume in any way a superluminal speed of propagation of the interactions. The contact forces are due to the same kinetic processes that are responsible for the local thermodynamic equili...
LAGEOS is an accurately-tracked, dense spherical satellite covered with 426 retroreflectors. The tracking accuracy is such as to yield a medium term (years to decades) inertial reference frame determined via relatively inexpensive observations. This frame is used as an adjunct to the more difficult and data intensive VLBI absolute frame measurements. There is a substantial secular precession of the satellite's line of nodes consistent with the classical, Newtonian precession due to the non-sphericity of the earth. Ciufolini has suggested the launch of an identical satellite (LAGEOS-3) into an orbit supplementary to that of LAGEOS-1: LAGEOS-3 would then experience an equal and opposite classical precession to that of LAGEOS-1. Besides providing a more accurate real-time measurement of the earth's length of day and polar wobble, this paired-satellite experiment would provide the first direct measurement of the general relativistic frame-dragging effect. Of the five dominant error sources in this experiment, the largest one involves surface forces on the satellite, and their consequent impact on the orbital nodal precession. The surface forces are a function of the spin dynamics of the satellite. Consequently, we undertake here a theoretical effort to model the spin ndynamics of LAGEOS. In this paper we present our preliminary results. I. THE LAGEOS-3 MISSION.The Laser Geodynamic Satellite Experiment (LAGEOS-3) is a joint USAF, NASA, and ASI proposed program to measure, for the first time, a quasi-stationary property of the earth -its gravitational magnetic dipole moment (gravitomagnetism) as predicted by Einstein's theory of general relativity. This gravitomagnetic field causes local inertial frames to be dragged around with the earth at a rate proportional to the angular momentum of the earth, and inversely proportional to the cube of the distance from the center of the earth. Thus the line of nodes of the orbital plane of LAGEOS-3 precesses eastward at 32 mas/yr. Although in this example the frame dragging effect is small compared to the torque on the orbital plane due to the oblateness of the earth, it is an essential ingredient in the dynamics of accretion disks, binary systems, and other astrophysical phenomena [1].Today, almost eighty years after Einstein introduced his geometric theory of gravity, we have just begun to measure -to verify -his gravitation theory. Of no less stature than the "tide producing" −M/r 2 "electric component" of gravity is the inertial-frame defining "magnetic component" of gravitation −J/r 3 . To see this force in action: first, inject a satellite into a polar orbit about an earth-like mass idealized as not spinning with respect to the distant 1
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