Miguel Alcubierre

Gauge conditions for long term numerical black hole evolutions without excision

We extend previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that are able to respond naturally to the spacetime dynamics. We show that the combination of excision and gauge conditions we use is able to drive highly distorted, rotating black holes to an almost static state at late times, with well behaved metric functions, without the need for any special initial conditions or analytically prescribed gauge functions. Further, we show for the first time that one can extract accurate waveforms from these simulations, with the full machinery of excision or no excision and dynamic gauge conditions. The evolutions can be carried out for long times, far exceeding the longevity and accuracy of even better resolved 2D codes. While traditional 2D codes show errors in quantities such as apparent horizon mass of over 100% by t ≈ 100M, and crash by t ≈ 150M, with our new techniques the same systems can be evolved for more than hundreds of M’s in full 3D with errors of only a few percent.

Towards a stable numerical evolution of strongly gravitating systems in general relativity: The conformal treatments

We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system. We develop an implementation of the conformal-traceless ~CT! approach that has improved stability properties in evolving weak and strong gravitational fields, and for both vacuum and spacetimes with active coupling to matter sources. Cases studied include weak and strong gravitational wave packets, black holes, boson stars and neutron stars. We show under what conditions the CT approach gives better results in 3D numerical evolutions compared to the ADM formulation. In particular, we show that our implementation of the CT approach gives more long term stable evolutions than ADM in all the cases studied, but is less accurate in the short term for the range of resolutions used in our 3D simulations.

Simple excision of a black hole in 3¿1 numerical relativity

We describe a simple implementation of black hole excision in 3+1 numerical relativity. We apply this technique to a Schwarzschild black hole with octant symmetry in Eddington-Finkelstein coordinates and show how one can obtain accurate, long-term stable numerical evolutions.

SYMMETRY WITHOUT SYMMETRY: NUMERICAL SIMULATION OF AXISYMMETRIC SYSTEMS USING CARTESIAN GRIDS

We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a three-dimensional Cartesian(x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule–width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)-coordinate finite differencing, while those in the y≠ 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves.

3D Grazing Collision of Two Black Holes

We present results for two colliding black holes (BHs), with angular momentum, spin, and unequal mass. For the first time, gravitational waveforms are computed for a grazing collision from a full 3D numerical evolution. The collision can be followed through the merger to form a single BH, and through part of the ringdown period of the final BH. The apparent horizon is tracked and studied, and physical parameters, such as the mass of the final BH, are computed. The total energy radiated in gravitational waves is shown to be consistent with the total initial mass of the spacetime and the apparent horizon mass of the final BH.

Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity

We study the stability properties of the standard ADM formulation of the 3+1 evolution equations of general relativity through linear perturbations of flat spacetime. We focus attention on modes with zero speed of propagation and conjecture that they are responsible for instabilities encountered in numerical evolutions of the ADM formulation. These zero speed modes are of two kinds: pure gauge modes and constraint violating modes. We show how the decoupling of the gauge by a conformal rescaling can eliminate the problem with the gauge modes. The zero speed constraint violating modes can be dealt with by using the momentum constraints to give them a finite speed of propagation. This analysis sheds some light on the question of why some recent reformulations of the 3+1 evolution equations have better stability properties than the standard ADM formulation.

Galactic collapse of scalar field dark matter

We present a scenario for core galaxy formation based on the hypothesis of scalar field dark matter. We interpret galaxy formation through the collapse of as calar field fluctuation. We find that a cosh potential for the self-interaction of the scalar field provides a reasonable scenario for the formation of a galactic core plus a remnant halo, which is in agreement with cosmological observations and phenomenological studies in galaxies. PACS numbers: 0425D, 9530S, 9535, 9862A, 9880 In the last few years, the quest concerning the nature of the dark matter in the universe has received much attention and has become of great importance for understanding the structure formation in the universe. Some candidates for dark matter have been discarded and some others have recently appeared. The standard candidates of the cold dark matter (CDM) model are axions and WIMP’S (weakly interacting massive particles), which are themselves not free of problems. Axions are massive scalar particles with no self interaction. In order for axions to be an essential component of the dark matter content of the universe, their mass should be m ∼ 10 −5 eV. With this axion mass, the scalar field collapses forming compact objects with masses of the order of Mcrit ∼ 0.6 m 2 m ∼ 10 −6 M� [1, 2], which corresponds to objects with the mass of a planet. Since the dark matte rm ass in galaxies is ten times higher than the luminous matter, we would need tenths of millions of such objects around the solar system, which is clearly not the case. On the other hand, there are many viable particles with nice features in super-symmetric theories, such a sW IMP’S, which behave just like standard CDM. However, a central debate nowadays is whether CDM can explain the observed scarcity of dwarf galaxies and the smoothness of the galactic-core matter densities, since high resolution numerical simulations with standard CDM predict an excess of dwarf galaxies and density

Black Hole Excision for Dynamic Black Holes

We extend the previous work on 3D black hole excision to the case of distorted black holes, with a variety of dynamic gauge conditions that respond naturally to the spacetime dynamics. We show that in evolutions of highly distorted, rotating black holes, the combination of excision and the gauge conditions we use is able to drive the coordinates to a frame in which the system looks almost static at late times. Further, we show for the first time that one can extract accurate wave forms from these simulations, with the full machinery of excision and dynamic gauge conditions. The evolutions can be carried out for a long time, far exceeding the longevity and accuracy of better resolved 2D codes.

Towards standard testbeds for numerical relativity

In recent years, many different numerical evolution schemes for Einsteins equations have been proposed to address stability and accuracy problems that have plagued the numerical relativity community for decades. Some of these approaches have been tested on different spacetimes, and conclusions have been drawn based on these tests. However, differences in results originate from many sources, including not only formulations of the equations, but also gauges, boundary conditions, numerical methods and so on. We propose to build up a suite of standardized testbeds for comparing approaches to the numerical evolution of Einsteins equations that are designed to both probe their strengths and weaknesses and to separate out different effects, and their causes, seen in the results. We discuss general design principles of suitable testbeds, and we present an initial round of simple tests with periodic boundary conditions. This is a pivotal first step towards building a suite of testbeds to serve the numerical relativists and researchers from related fields who wish to assess the capabilities of numerical relativity codes. We present some examples of how these tests can be quite effective in revealing various limitations of different approaches, and illustrating their differences. The tests are presently limited to vacuum spacetimes, can be run on modest computational resources and can be used with many different approaches used in the relativity community.

Gravitational collapse of gravitational waves in 3D numerical relativity.

We demonstrate that evolutions of three-dimensional, strongly non-linear gravitational waves can be followed in numerical relativity, hence allowing many interesting studies of both fundamental and observational consequences. We study the evolution of time-symmetric, axisymmetric and non-axisymmetric Brill waves, including waves so strong that they collapse to form black holes under their own self-gravity. An estimate for the critical amplitude for black hole formation in a particular interpolating family of initial data is obtained. The gravitational waves emitted in the black hole formation process are compared to those emitted in the head-on collision of two Misner black holes. PACS number~s!: 04.25.Dm, 04.30.Db, 95.30.Sf, 97.60.Lf