664 research outputs found
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 3-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 \neq 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.Comment: 17 pages, 4 figure
Three dimensional numerical relativity: the evolution of black holes
We report on a new 3D numerical code designed to solve the Einstein equations
for general vacuum spacetimes. This code is based on the standard 3+1 approach
using cartesian coordinates. We discuss the numerical techniques used in
developing this code, and its performance on massively parallel and vector
supercomputers. As a test case, we present evolutions for the first 3D black
hole spacetimes. We identify a number of difficulties in evolving 3D black
holes and suggest approaches to overcome them. We show how special treatment of
the conformal factor can lead to more accurate evolution, and discuss
techniques we developed to handle black hole spacetimes in the absence of
symmetries. Many different slicing conditions are tested, including geodesic,
maximal, and various algebraic conditions on the lapse. With current
resolutions, limited by computer memory sizes, we show that with certain lapse
conditions we can evolve the black hole to about , where is the
black hole mass. Comparisons are made with results obtained by evolving
spherical initial black hole data sets with a 1D spherically symmetric code. We
also demonstrate that an ``apparent horizon locking shift'' can be used to
prevent the development of large gradients in the metric functions that result
from singularity avoiding time slicings. We compute the mass of the apparent
horizon in these spacetimes, and find that in many cases it can be conserved to
within about 5\% throughout the evolution with our techniques and current
resolution.Comment: 35 pages, LaTeX with RevTeX 3.0 macros. 27 postscript figures taking
7 MB of space, uuencoded and gz-compressed into a 2MB uufile. Also available
at http://jean-luc.ncsa.uiuc.edu/Papers/ and mpeg simulations at
http://jean-luc.ncsa.uiuc.edu/Movies/ Submitted to Physical Review
Exploring nonlocal observables in shock wave collisions
We study the time evolution of 2-point functions and entanglement entropy in
strongly anisotropic, inhomogeneous and time-dependent N=4 super Yang-Mills
theory in the large N and large 't Hooft coupling limit using AdS/CFT. On the
gravity side this amounts to calculating the length of geodesics and area of
extremal surfaces in the dynamical background of two colliding gravitational
shockwaves, which we do numerically. We discriminate between three classes of
initial conditions corresponding to wide, intermediate and narrow shocks, and
show that they exhibit different phenomenology with respect to the nonlocal
observables that we determine. Our results permit to use (holographic)
entanglement entropy as an order parameter to distinguish between the two
phases of the cross-over from the transparency to the full-stopping scenario in
dynamical Yang-Mills plasma formation, which is frequently used as a toy model
for heavy ion collisions. The time evolution of entanglement entropy allows to
discern four regimes: highly efficient initial growth of entanglement, linear
growth, (post) collisional drama and late time (polynomial) fall off.
Surprisingly, we found that 2-point functions can be sensitive to the geometry
inside the black hole apparent horizon, while we did not find such cases for
the entanglement entropy.Comment: 28 pp, 9 figs; v2: updated references, changed color bars in Figure 2
and Figure
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
Braneworld dynamics with the BraneCode
We give a full nonlinear numerical treatment of time-dependent 5d braneworld
geometry, which is determined self-consistently by potentials for the scalar
field in the bulk and at two orbifold branes, supplemented by boundary
conditions at the branes. We describe the BraneCode, an algorithm which we
designed to solve the dynamical equations numerically. We applied the BraneCode
to braneworld models and found several novel phenomena of the brane dynamics.
Starting with static warped geometry with de Sitter branes, we found
numerically that this configuration is often unstable due to a tachyonic mass
of the radion during inflation. If the model admits other static configurations
with lower values of de Sitter curvature, this effect causes a violent
re-structuring towards them, flattening the branes, which appears as a lowering
of the 4d effective cosmological constant. Braneworld dynamics can often lead
to brane collisions. We found that in the presence of the bulk scalar field,
the 5d geometry between colliding branes approaches a universal, homogeneous,
anisotropic strong gravity Kasner-like asymptotic, irrespective of the
bulk/brane potentials. The Kasner indices of the brane directions are equal to
each other but different from that of the extra dimension.Comment: 38 pages, 10 figure
Implementation of standard testbeds for numerical relativity
We discuss results that have been obtained from the implementation of the
initial round of testbeds for numerical relativity which was proposed in the
first paper of the Apples with Apples Alliance. We present benchmark results
for various codes which provide templates for analyzing the testbeds and to
draw conclusions about various features of the codes. This allows us to sharpen
the initial test specifications, design a new test and add theoretical insight.Comment: Corrected versio
Three-dimensional simulations of distorted black holes; 1, Comparison with axisymmetric results
We consider the numerical evolution of black hole initial data sets, consisting of single black holes distorted by strong gravitational waves, with a full 3D, nonlinear evolution code. These data sets mimic the late stages of coalescing black holes. We compare various aspects of the evolution of axisymmetric initial data sets, obtained with this 3D code, to results obtained from a well established axisymmetric code. In both codes we examine and compare the behavior of metric functions, apparent horizon properties, and waveforms, and show that these dynamic black holes can be accurately evolved in 3D. In particular we show that with present computational resources and techniques, the process of excitation and ringdown of the black hole can be evolved, and one can now extract accurately the gravitational waves emitted from the 3D Cartesian metric functions, even when they carry away only a small fraction () of the rest mass energy of the system. Waveforms for both the and compared with axisymmetric calculations. In addition to exploring the physics of distorted black hole data sets, and showing the extent to which the waves can be accurately extracted, these results also provide important testbeds for all fully nonlinear numerical codes designed to evolve black hole spacetimes in 3D, whether they use singularity avoiding slicings, apparent horizon boundary conditions, or other evolution methods
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