In this paper, we examine the efficiency of gravitational bremsstrahlung production in the process of head-on collision of two boosted Schwarzschild black holes. We construct initial data for the characteristic initial value problem in Robinson-Trautman space-times, which represent two instantaneously stationary Schwarzschild black holes in motion toward each other with the same velocity. The Robinson-Trautman equation is integrated for these initial data using a numerical code based on the Galerkin method. The resulting final configuration is a boosted black hole with Bondi mass greater than the sum of the individual masses of the individual initial black holes. Two relevant aspects of the process are presented. The first relates the efficiency ∆ of the energy extraction by gravitational wave emission to the mass of the final black hole. This relation is fitted by a 2049 Int. J. Mod. Phys. D 2008.17:2049-2064. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 04/12/15. For personal use only. 2050 R. F. Aranha et al.distribution function of nonextensive thermostatistics with entropic parameter q 1/2; the result extends and validates analysis based on the linearized theory of gravitational wave emission. The second aspect is a typical bremsstrahlung angular pattern in the early period of emission at the wave zone, a consequence of the deceleration of the black holes as they coalesce; this pattern evolves to a quadrupole form for later times.
We examine numerically the post-merger regime of two nonspining holes in non-head-on collisions in the realm of nonaxisymmetric Robinson-Trautman spacetimes. Characteristic initial data for the system are constructed and evolved via the Robinson-Trautman equation. The numerical integration is performed using a Galerkin spectral method which is sufficiently stable to reach the final configuration of the remnant black hole, when the gravitational wave emission ceases. The initial data contains three independent parameters, the ratio mass of the individual colliding black holes, their initial premerger infalling velocity and the incidence angle of collision 0 . The remnant black hole is characterized by its final boost parameter, rest mass and scattering angle. The motion of the remnant black hole is restricted to the plane determined by the directions of the two initial colliding black holes, characterizing a planar collision. The net momentum fluxes carried out by gravitational waves are confined to this plane. We evaluate the efficiency of mass-energy extraction, the total energy and momentum carried out by gravitational waves and the momentum distribution of the remnant black hole for a large domain of initial data parameters. Our analysis is based on the Bondi-Sachs four-momentum conservation laws. The process of mass-energy extraction is shown to be less efficient as the initial data departs from the head-on configuration. Head-on collisions ( 0 ¼ 0 o ) and orthogonal collisions ( 0 ¼ 90 ) constitute, respectively, upper and lower bounds to the power emission and to the efficiency of mass-energy extraction. On the contrary, head-on collisions and orthogonal collisions constitute, respectively, lower and upper bounds for the momentum of the remnant. Distinct regimes of gravitational wave emission (bursts or quiescent emission) are characterized by the analysis of the time behavior of the gravitational wave power as a function of . In particular, the net gravitational wave flux is nonzero for equal-mass colliding black holes in non-head-on collisions. The momentum extraction and the patterns of the momentum fluxes, as a function of the incidence angle, are examined. The relation between the incidence angle and the scattering angle closely approximates a relation for the inelastic collision of classical particles in Newtonian dynamics.
We examine the phase space dynamics of closed Friedmann-Robertson-Walker universes with a massive inflaton field, where the Friedmann equations contain additional terms arising from high energy corrections to cosmological scenarios. The model is based upon a Randall-Sundrum type of action, with an extra timelike dimension, and the corrections implement nonsingular bounces in the early evolution of the universe. In narrow windows of the parameter space of the models non-linear resonance phenomena of Kolmogorov-Arnold-Moser tori are shown to occur, leading to the destruction of tori that trap the inflaton. As a consequence the escape into inflation takes place. These resonance windows are labeled with an integer n ≥ 2, where n is related to the ratio of the frequencies in the scale factor to those in the scalar field degrees of freedom. We examine the constraints imposed by non-linear resonance in the physical domain of parameters of the model so that inflation may be realized. The larger the order n of the resonance, the stronger the gravitational interaction in the braneworld universe inflated from initial conditions connected with the resonance considered. We also discuss the structural stability of the resonance pattern, the complex dynamics arising in this pre-inflationary phase and some of its possible imprints in the physics of inflation.
We examine numerically the head-on collision of two boosted Schwarzschild black holes, in the realm of Robinson-Trautman spacetimes. Characteristic initial data for the system are constructed and the Robinson-Trautman equation is integrated for these data using a numerical code based on the Galerkincollocation method. The initial data already have a common horizon so that the evolution covers the postmerger regime up to the final configuration, when the gravitational wave emission ceases. In the nonlinear regime gravitational waves are emitted, extracting mass and linear momentum from the system. The final configuration is a boosted Schwarzschild black hole with rest mass larger than the masses of the two individual initial black holes, and with a smaller final boost parameter characterizing the recoil velocity of the remnant. The efficiency Á of the mass-energy extraction by gravitational waves is evaluated. The points ðÁ; yÞ, where y is the (normalized) rest mass of the remnant black hole, satisfy a nonextensive Tsallis distribution with entropic index q ' 1=2 for y & 12. Beyond y $ 12 the experimental points deviate from the distribution function and the efficiency presents an absolute maximum for the case of equally massive individual colliding black holes; the remnant has no recoil in this case. By using the Bondi mass formula we also evaluate the total energy E W carried out by gravitational waves as well as the radiative corrections to the efficiency. E W increases monotonically with y and the experimental points ðE W ; yÞ also satisfy a nonextensive Tsallis distribution but with q ' 2=3, up to y $ 14:2. Beyond this value the experimental points increase faster than the distribution function. For any initial infalling velocity v, the distribution of momentum of the remnant exhibits a maximum at 1 ¼ m ' 0:667, where 1 is related to the ratio of pre-merger rest masses, and has a one-to-one correspondence with y for fixed v. Two distinct regimes of gravitational wave emission can be characterized according to (i) 1 < m : bursts of gravitational bremsstrahlung; (ii) 1 > m : quiescent long time emission of gravitational waves. This picture is also sustained by the analysis of the time behavior of the power emitted (dE W =du).
We derive the Bondi–Sachs 4-momentum conservation laws that regulate the emission of gravitational waves in non-axisymmetric Robinson–Trautman spacetimes. Although Robinson–Trautman spacetimes describe the exterior vacuum gravitational field of a bounded system radiating gravitational waves, the Robinson–Trautman metrics do not satisfy the appropriate Bondi–Sachs boundary conditions formulated for radiating systems. We therefore derive the transformations between Robinson–Trautman and Bondi–Sachs coordinates which allow us to obtain some of the basic Bondi–Sachs physical quantities, namely the news functions and the Bondi–Sachs energy–momentum fluxes of the gravitational waves. These quantities are fundamental to the description of the radiative transfer processes involved in the generation and emission of gravitational waves by the system. The Bondi–Sachs 4-momentum conservation laws are covariant under the transformations of the Bondi–Metzner–Sachs group. The section of this paper containing the derivation of the coordinate transformations extends the work of von der Gönna and Kramer done for the case of axisymmetric Robinson–Trautman spacetimes. We discuss some possible applications of the conservation laws in numerical simulations of black hole systems emitting gravitational waves, and also future research of Bondi–Sachs conservation laws for twisting Robinson–Trautman-type spacetimes.
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