Session B13 - Numerical Relativity and Classical Gravity.
ORAL session, Saturday morning, April 28
Room 12-13, Renaissance Hotel$

[B13.001] Axisymmetric Gravitational Collapse

A stable gravitational collapse code is constructed under the assumption of axisymmetry in an effort to study black hole critical collapse of various matter sources. A unigrid, Crank-Nicholson scheme is implemented with a multigrid elliptic solver to evolve a scalar field along with electric and magnetic fields. Stable and convergent evolutions for weak and strong field are found. For the strong field regime, black hole excision succeeds in yielding long-lasting, stable evolutions in the presence of a black hole. Studies of the threshold of black hole formation (critical phenomena) await the implementation of adaptive mesh refinement (AMR).

[B13.002] Relativistic Hydrodynamics with AMR

Modeling astrophysical sources of gravitational radiation requires fully 3-D numerical simulations. Due to the range of scales involved, adaptive mesh refinement (AMR) is a critical component for successful modeling. We present a progress report on the development of a fully general relativistic hydrodynamics code for modeling such sources. Results from 3-D runs involving shocks using AMR will be shown.

[B13.003] Three-dimensional Adaptive Evolutions of Strong Gravitational Waves

General relativistic research is entering a new era with the installation of several worldwide gravitational wave (GW) observatories based on laser interferometers. These include ground-based detectors such as LIGO, VIRGO, GEO, and TAMA and a space-based detector LISA.

Theoretical challenges for numerical relativists include calculating accurate waveforms generated by the sources likely to be detected by these GW observatories. These waveforms will greatly enhance the successful detection and interpretation of the signals.

The final goal of our research program is to calculate waveforms for one such candidate--inspiralling binary neutron star system. The full Einstein equations in 3-D must be solved to follow both the dynamics of the binaries from initial inspiral to final merger, and the generation and propagation of gravitational waves into the wave zone.

One of the crucial requirements for this kind of simulation is AMR (Adaptive Mesh Refinement). My talk is based on work-in-progress that solves the vacuum Einstein equations with strong gravitational waves as initial data. This problem constitutes a first step towards full simulations and allows us to test our AMR code without involving the complexity of hydrodynamics. 2-level AMR runs show that the fine grid tracks the features of the gravitational waves well.

[B13.004] Isolated horizons in numerical relativity

The notion of isolated horizons was recently introduced to describe the near horizon geometry of a black hole. We use this framework to extract invariant physical information from the numerical simulations of black hole collisions. In particular we calculate the mass and angular momentum only using quantities intrinsic to the horizon. To extract this physical information, we do not need to embed the spacetime in a Kerr solution. This is a significant advantage because one does not know what Kerr parameters to use or how the physical geometry is to approach Kerr.

[B13.005] Generic T^2 Symmetric Cosmologies: A Precision Numerical Laboratory for Local Mixmaster Dynamics

Unlike the velocity dominated special case of vacuum Gowdy cosmological spacetimes, generic T^2 symmetric (twisted Gowdy) models appear to exhibit local mixmaster dynamics (LMD) (of an unusual type). Numerical simulations of these models are sufficiently accurate to allow precise comparison between predicted quantitative signatures of LMD and the actual behavior of the solutions. Implications for more general cosmological spacetimes will be discussed.

[B13.006] Adaptive Mesh Refinement for Collapsing Cosmological Spacetimes

Gravitational wave amplitudes in collapsing cosmological spacetimes are characterized by narrowing spiky features which challenge any fixed spatial resolution computer simulation. This points to the possible usefulness of adaptive mesh refinement (AMR). Using Godunov's method in combination with the Structured Adaptive Mesh Refinement Algorithm Infrastructure (SAMRAI) of Lawrence Livermore National Laboratory, a fully-functional code using AMR has been developed and applied to the collapse of twisted Gowdy cosmologies to explore their local mixmaster dynamics.

[B13.007] Numerical simulation of generic singularities

This talk is on a numerical simulation of the approach to the singularity with a scalar field as the matter. The simulation is done using a 3+1 code that uses harmonic coordinates. The asymptotic behavior of the metric is found. Since the metric has no symmetries, it is expected that the behavior found is the generic behavior of singularities with this type of matter.

[B13.008] Numerical evolution of constant mean curvature foliations of single black hole spacetimes

We investigate the convergence and long-term stability properties of numerical evolutions of single black holes spacetimes using time independent foliations constructed from hypersurfaces of constant mean curvature. We focus our attention on foliations that are asymptotic to future null infinity because of their potential to facilitate radiation extraction in more complicated settings. We consider both, foliations with and without singularity avoidance properties. We discuss the extension and possible advantages of these foliations in the numerical evolutions of multiple black holes systems.

[B13.009] Solving the Constraint Equations of General Relativity

Posing initial data for general relativity requires solving the so-called constraint equations, a set of coupled partial differential equations. Work towards a new solver for these constraint equations is presented. The new code is based on the pseudospectral collocation method. Ultimately it will solve the full coupled initial data equations with no simplifying assumptions.

[B13.010] Numerical Simulations of Black Holes with Excision

We present results from numerical simulations of single black holes in three dimensions using the Maya code. We focus on how to numerically solve the Einstein's field equations in three dimensions written in an ADM like system including the ramifications that the equations and gauge choice have on numerical stability. One of the key ingredients in simulating a black hole in a long term stable manner is excising the black hole's singularity. Two of the design features of the Maya code are flexibility and readability. These features are exploited in our excision algorithm design. This talk includes a discussion of the excision scheme accompanied by results from numerically solving the ADM equations for a black hole while excising the singularity.

[B13.011] Gravitation, Symmetry and Undergraduates

This talk will discuss "Project Petrov" Which is designed to investigate gravitational fields with symmetry. Project Petrov represents a collaboration involving physicists, mathematicians as well as graduate and undergraduate math and physics students. An overview of Project Petrov will be given, with an emphasis on students' contributions, including software to classify and generate Lie algebras, to classify isometry groups, and to compute the isometry group of a given metric.

[B13.012] Internal-structure dependent coefficients in the post-Newtonian equations of motion

The post-3-Newtonian equation of motion for "point particles" recently derived by Damour, Jaranowski and Schafer and by Blanchet and Faye contains a term whose numerical coefficient has not yet been determined by the methods used to date to derive the equations of motion. One might speculate that the coefficient depends on the internal structure of the bodies, so that, for example, it would have different values for polytropic stellar models with different polytropic indices. We show here that this is not the case -- the coefficient must take the same value for all spherically symmetric bodies -- and that such internal-structure dependent coefficients cannot arise at orders below the 5th post Newtonian order. The argument is based on an earlier, matched asymptotic expansion based analysis of the interaction of the orbital motion and internal degrees of freedom in binary stellar systems.

[B13.013] Fast travel in spherically symmetric geometries

We investigate spherically symmetric configurations that allow the fastest possible travel between two spatial points. The end points are chosen unambiguously by fixing the metric to be "Schwarzschild" for values of the area-radius coordinate r bigger than R. Thus, the end points of the path are in the "Schwarzschild" external region. We demand also that such geometries should satisfy the weak energy condition (WEC). The result shows that, provided the WEC is satisfied, the fastest travel is achieved by setting the metric in the interior (r<R) equal to that of Minkowski space. Then, the mass that enters the Schwarszchild metric is concentrated in a thin shell at r=R.

[B13.014] Neutron Star Structure In The Presence of Scalar Fields

Motivated by the possible presence of scalar fields on astrophysical scales, suggested by the recent measurement of the deceleration parameter by distance supernovae surveys, we present models of neutron star structure under the assumption that a scalar field makes a significant contribution to the stress energy momentum tensor, in addition to that made by the normal matter. To that end we solve the coupled Einstein - scalar field - hydrostatic balance equations to compute the effect of the presence of the scalar field on the neutron star structure. We find that the presence of the scalar field does change the structure of the neutron star, especially in cases of strong coupling between the scalar field and the matter density. We present the neutron star radius as a function of the matter--scalar field coupling constant for different values of the neutron star central density. The presence of the scalar field does affect both the maximum neutron star mass and its radius, the latter increasing with the value of the above coupling constant. Our results may be testable with the recent timing observations of accreting neutron stars.

[B13.015] Weak Field Limit of a Scale Invariant Yang-Mills Theory of Gravity

A theory of gravity where the action is taken to be in Yang-Mills form ("square of the Riemann tensor") is discussed. If we demand that the spin-connection field be treated as an "ordinary" gauge field, then it must remain invariant under scale transformations. This implies invariance of the Riemann tensor and the gravitational action, but the torsion tensor transforms inhomogeneously. To construct a scale invariant action for a scalar field requires terms quadratic in the torsion, but this action is not uniquely determined by the demand of scale invariance since it is only the vector ("trace") part of the torsion which transforms inhomogeneously, the remaining pieces being invariant. This ambiguity is lifted by the further demand that in the weak field limit the field equations reduce to the weak field limit of Einstein's equation. Thus we obtain a classically viable alternative to Einsteinian gravity, but whose construction as a Yang-Mills theory may ameliorate the notoriously difficult problem of quantization.

Part B of program listing