Finding apparent horizons in numerical relativity

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced H(h) function \H(\h) by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing \H(\h). Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum H(h) equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the \H(\h) Jacobian to be {\em much\/} more efficient than the usual numerical-perturbation method, and also much easier to implement than is commonly thought. When solving the (discrete) \H(\h) = 0 equations, we find that Newton's method generally converges very rapidly, although there are difficulties when the initial guess contains high-spatial-frequency errors. Using 4th~order finite differencing, we find typical accuracies for the horizon position in the 10^{-5} range for \Delta \theta = \frac{\pi/2}{50}

An explicit harmonic code for black-hole evolution using excision

We describe an explicit in time, finite-difference code designed to simulate black holes by using the excision method. The code is based upon the harmonic formulation of the Einstein equations and incorporates several features regarding the well-posedness and numerical stability of the initial-boundary problem for the quasilinear wave equation. After a discussion of the equations solved and of the techniques employed, we present a series of testbeds carried out to validate the code. Such tests range from the evolution of isolated black holes to the head-on collision of two black holes and then to a binary black hole inspiral and merger. Besides assessing the accuracy of the code, the inspiral and merger test has revealed that individual apparent horizons can touch and even intersect. This novel feature in the dynamics of the marginally trapped surfaces is unexpected but consistent with theorems on the properties of apparent horizons

Hyperbolic slicings of spacetime: singularity avoidance and gauge shocks

I study the Bona-Masso family of hyperbolic slicing conditions, considering in particular its properties when approaching two different types of singularities: focusing singularities and gauge shocks. For focusing singularities, I extend the original analysis of Bona et. al and show that both marginal and strong singularity avoidance can be obtained for certain types of behavior of the slicing condition as the lapse approaches zero. For the case of gauge shocks, I re-derive a condition found previously that eliminates them. Unfortunately, such a condition limits considerably the type of slicings allowed. However, useful slicing conditions can still be found if one asks for this condition to be satisfied only approximately. Such less restrictive conditions include a particular member of the 1+log family, which in the past has been found empirically to be extremely robust for both Brill wave and black hole simulations.Comment: 11 pages, revtex4. Change in acknowledgment

Adaptive grid methods for RLV environment assessment and nozzle analysis

Rapid access to highly accurate data about complex configurations is needed for multi-disciplinary optimization and design. In order to efficiently meet these requirements a closer coupling between the analysis algorithms and the discretization process is needed. In some cases, such as free surface, temporally varying geometries, and fluid structure interaction, the need is unavoidable. In other cases the need is to rapidly generate and modify high quality grids. Techniques such as unstructured and/or solution-adaptive methods can be used to speed the grid generation process and to automatically cluster mesh points in regions of interest. Global features of the flow can be significantly affected by isolated regions of inadequately resolved flow. These regions may not exhibit high gradients and can be difficult to detect. Thus excessive resolution in certain regions does not necessarily increase the accuracy of the overall solution. Several approaches have been employed for both structured and unstructured grid adaption. The most widely used involve grid point redistribution, local grid point enrichment/derefinement or local modification of the actual flow solver. However, the success of any one of these methods ultimately depends on the feature detection algorithm used to determine solution domain regions which require a fine mesh for their accurate representation. Typically, weight functions are constructed to mimic the local truncation error and may require substantial user input. Most problems of engineering interest involve multi-block grids and widely disparate length scales. Hence, it is desirable that the adaptive grid feature detection algorithm be developed to recognize flow structures of different type as well as differing intensity, and adequately address scaling and normalization across blocks. These weight functions can then be used to construct blending functions for algebraic redistribution, interpolation functions for unstructured grid generation, forcing functions to attract/repel points in an elliptic system, or to trigger local refinement, based upon application of an equidistribution principle. The popularity of solution-adaptive techniques is growing in tandem with unstructured methods. The difficultly of precisely controlling mesh densities and orientations with current unstructured grid generation systems has driven the use of solution-adaptive meshing. Use of derivatives of density or pressure are widely used for construction of such weight functions, and have been proven very successful for inviscid flows with shocks. However, less success has been realized for flowfields with viscous layers, vortices or shocks of disparate strength. It is difficult to maintain the appropriate mesh point spacing in the various regions which require a fine spacing for adequate resolution. Mesh points often migrate from important regions due to refinement of dominant features. An example of this is the well know tendency of adaptive methods to increase the resolution of shocks in the flowfield around airfoils, but in the incorrect location due to inadequate resolution of the stagnation region. This problem has been the motivation for this research

“I Is Someone Else”: Constituting the Extended Mind’s Fourth Wave, with Hegel

We seek to constitute the extended mind’s fourth wave, socially distributed group cognition, and we do so by thinking with Hegel. The extended mind theory’s first wave invokes the parity principle, which maintains that processes that occur external to the organism’s skin should be considered mental if they are regarded as mental when they occur inside the organism. The second wave appeals to the complementarity principle, which claims that what is crucial is that these processes together constitute a cognitive system. The first two waves assume that cognitive systems have well-defined territories or boundaries, and that internal and external processes do not switch location. The third wave rejects these assumptions, holding instead that internal processes are not privileged, and internal and external processes can switch, and that processes can be distributed among individuals. The fourth wave would advocate socially distributed group cognition. Groups are deterritorialized collective agents; they are ineliminatively and irreducibly real, they have mental states. Individuals constitute groups, but groups also constitute individuals. What counts as an individual and a group is a function of the level of analysis. And they are conflicted

Quantifying Selection Bias in Cross-Sectional Studies of Ovarian Hormones

Most studies of ovarian hormones in adult women collect data from a cross-sectional sample of participants meeting various selection criteria including not having been pregnant or breastfeeding for several months. Although this approach is intended to eliminate the effects of these factors on hormonal variation, it introduces a selection bias of unknown magnitude: in a non-contracepting population, those women with the highest fecundity are more likely to be either pregnant or lactating, and so not included in a study sample. Thus a cross-sectional sample disproportionately represents women with the lowest fecundity (and potentially the lowest hormone levels). Here we present a preliminary evaluation of the magnitude of this selection bias, focusing on progesterone ($P$) levels near the luteal peak. We use data from Project REPA, a longitudinal study of reproductive functioning in rural Bolivians, recruited without regard to reproductive status (Vitzthum, Spielvogel, and Thornburg \textit{Proceedings of the U.S. National Academy of Sciences/} 101, 1443 (2004)). Drawing from 542~non-conception cycles in 144~women, we construct simulated cross-sectional samples meeting various inclusion criteria and compare their anovulation rates and progesterone levels.National Science Foundation, University of California, Binghamton University, Indiana Universit

Accurate Evolutions of Orbiting Binary Black Holes

We present a detailed analysis of binary black hole evolutions in the last orbit and demonstrate consistent and convergent results for the trajectories of the individual bodies. The gauge choice can significantly affect the overall accuracy of the evolution. It is possible to reconcile certain gauge-dependent discrepancies by examining the convergence limit. We illustrate these results using an initial data set recently evolved by Brügmann et al. [Phys. Rev. Lett. 92, 211101 (2004)]. For our highest resolution and most accurate gauge, we estimate the duration of this data set's last orbit to be approximately 59MADM

Entropic force in black hole binaries and its Newtonian limits

We give an exact solution for the static force between two black holes at the turning points in their binary motion. The results are derived by Gibbs' principle and the Bekenstein-Hawking entropy applied to the apparent horizon surfaces in time-symmetric initial data. New power laws are derived for the entropy jump in mergers, while Newton's law is shown to derive from a new adiabatic variational principle for the Hilbert action in the presence of apparent horizon surfaces. In this approach, entropy is strictly monotonic such that gravity is attractive for all separations including mergers, and the Bekenstein entropy bound is satisfied also at arbitrarily large separations, where gravity reduces to Newton's law. The latter is generalized to point particles in the Newtonian limit by application of Gibbs' principle to world-lines crossing light cones.Comment: Accepted for publication in Phys. Rev.