S-Duality at the Black Hole Threshold in Gravitational Collapse

We study gravitational collapse of the axion/dilaton field in classical low energy string theory, at the threshold for black hole formation. A new critical solution is derived that is spherically symmetric and continuously self-similar. The universal scaling and echoing behavior discovered by Choptuik in gravitational collapse appear in a somewhat different form. In particular, echoing takes the form of SL(2,R) rotations (cf. S-duality). The collapse leaves behind an outgoing pulse of axion/dilaton radiation, with nearly but not exactly flat spacetime within it.Comment: 8 pages of LaTeX, uses style "revtex"; 1 figure, available in archive, or at ftp://ftp.itp.ucsb.edu/figures/nsf-itp-95-15.ep

Critical Exponents and Stability at the Black Hole Threshold for a Complex Scalar Field

This paper continues a study on Choptuik scaling in gravitational collapse of a complex scalar field at the threshold for black hole formation. We perform a linear perturbation analysis of the previously derived complex critical solution, and calculate the critical exponent for black hole mass, $\gamma \approx 0.387106$. We also show that this critical solution is unstable via a growing oscillatory mode.Comment: 15 pages of latex/revtex; added details of numerics, in press in Phys Rev D; 1 figure included, or available by anonymous ftp to ftp://ftp.itp.ucsb.edu/figures/nsf-itp-95-58.ep

Galerkin Method in the Gravitational Collapse: a Dynamical System Approach

We study the general dynamics of the spherically symmetric gravitational collapse of a massless scalar field. We apply the Galerkin projection method to transform a system of partial differential equations into a set of ordinary differential equations for modal coefficients, after a convenient truncation procedure, largely applied to problems of turbulence. In the present case, we have generated a finite dynamical system that reproduces the essential features of the dynamics of the gravitational collapse, even for a lower order of truncation. Each initial condition in the space of modal coefficients corresponds to a well definite spatial distribution of scalar field. Numerical experiments with the dynamical system show that depending on the strength of the scalar field packet, the formation of black-holes or the dispersion of the scalar field leaving behind flat spacetime are the two main outcomes. We also found numerical evidence that between both asymptotic states, there is a critical solution represented by a limit cycle in the modal space with period $\Delta u \approx 3.55$.Comment: 9 pages, revtex4, 10 ps figures; Phys. Rev. D, in pres

Critical behavior and scaling in vacuum axisymmetric gravitational collapse

We report a second example of critical behavior in gravitational collapse. Collapse of axisymmetric gravitational wave packets is computed numerically for a one-parameter family of initial data. A black hole first appears along the sequence at a critical parameter value p*. As with spherical scalar-field collapse, a power law is found to relate black-hole mass (the order parameter) and critical separation: MBHp-p*β. The critical exponent is β0.37, remarkably close to that observed by Choptuik. Near-critical evolutions produce echoes from the strong-field region which appear to exhibit scaling

Reading off gravitational radiation waveforms in numerical relativity calculations: Matching to linearized gravity

Two methods are described, both based on the use of multipole moments in linearized gravity, to read off gravitational radiation waveforms during numerical relativity simulations. In the first, matching is made at a finite radius in the weak-field exterior of a strong-field source to an analytic template developed via an infinitesimal gauge transformation from a general solution to the vacuum weak-field equations. The matching procedure allows the asymptotic waveforms to be separated from the confusing influences of the sources (e.g., black hole, neutron star, collapsing stellar core) stationary moments, the waves near-zone field, and gauge dependencies in the metric. This is achieved by computing the multipole-moment amplitudes of the gravitational field with a set of surface integrals of the metric over one (or more) coordinate two-sphere(s). The two-surface(s) need not be placed far out in the local wave zone, nor does the method require the existence of a deep near zone (i.e., the source need not be slow motion). The procedure is demonstrated through its application to two standard axisymmetric numerical relativity gauges (quasi-isotropic and radial). The second matching approach uses a surface integral over components of the Riemann tensor to eliminate gauge effects. The near-zone field is separated off as in the previous method. This latter technique may be applicable to problems in any gauge

Gauge-invariant treatment of gravitational radiation near the source: Analysis and numerical simulations

We discuss a procedure based on the use of multipole-moment expansions for matching numerical solutions for the gravitational-radiation field around a compact source to linear analytic solutions. Gauge-invariant perturbation theory is used to generate even- and odd-parity matching equations for each spherical harmonic order l, m. This technique determines asymptotic wave forms, valid in the local wave zone, from the numerically evolved fields in a weak-field annular region surrounding the isolated source. The separation of the wave form from near-zone and residual gauge effects is demonstrated using fully general-relativistic simulations of relativistic stars undergoing nonradial pulsation

Universality in axisymmetric vacuum collapse

Evidence of universality is observed in the critical behavior of axisymmetric vacuum gravitational collapse. The threshold of black hole formation in the future development of time-antisymmetric initial data is found numerically and compared to previous results based on ingoing pulses of gravitational waves. The power-law behavior of the black hole mass is again found near the critical point and the critical exponent value β0.36 is consistent with our previous determination despite stark differences in the two sets of initial data. Similar evidence of universality is exhibited by the scaling factor Δ of the echoes in the gravitational field produced in the central region of collapse

Trapping a geon: Black hole formation by an imploding gravitational wave

We describe the formation of a black hole via the implosion of an axisymmetric gravitational wave. Finite difference simulations of the vacuum Einstein equations are used to obtain these results. The initial data consist of nearly linear solutions to the vacuum constraint equations that represent even-parity, ingoing wave packets with quadrupole angular dependence. A black hole is demonstrated to form as a result of imploding a wave packet with a sufficiently large value of a strength parameter, 2Mp=1.06>crit0.80, where 2 is the radial width of the wave packet and Mp denotes its mass. Black hole formation is verified by observing (i) the exponential collapse of the central value of the lapse function , (ii) the formation of a trapped region and marginally outer-trapped surfaces, and (iii) the emission of quasi-normal-mode radiation. For the =1.06 case, just over 2% of the mass emerges in normal-mode radiation

Critical Collapse of the Massless Scalar Field in Axisymmetry

We present results from a numerical study of critical gravitational collapse of axisymmetric distributions of massless scalar field energy. We find threshold behavior that can be described by the spherically symmetric critical solution with axisymmetric perturbations. However, we see indications of a growing, non-spherical mode about the spherically symmetric critical solution. The effect of this instability is that the small asymmetry present in what would otherwise be a spherically symmetric self-similar solution grows. This growth continues until a bifurcation occurs and two distinct regions form on the axis, each resembling the spherically symmetric self-similar solution. The existence of a non-spherical unstable mode is in conflict with previous perturbative results, and we therefore discuss whether such a mode exists in the continuum limit, or whether we are instead seeing a marginally stable mode that is rendered unstable by numerical approximation.Comment: 11 pages, 8 figure

The Singularity Threshold of the Nonlinear Sigma Model Using 3D Adaptive Mesh Refinement

Numerical solutions to the nonlinear sigma model (NLSM), a wave map from 3+1 Minkowski space to S^3, are computed in three spatial dimensions (3D) using adaptive mesh refinement (AMR). For initial data with compact support the model is known to have two regimes, one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by removing the assumption of spherical symmetry. The evolutions suggest that the spherically symmetric critical solution remains an intermediate attractor separating the two end states.Comment: 5 pages, 5 figures, 1 table; To be published in Phys. Rev. D.; Added discussion of initial data; Added figure and reference