Numerical Relativity
Overview
Numerical relativity is the study of Einstein's equations of general relativity using
computer simulations. Groups of scientists around the world, like the group at the Goddard
Space Flight Center, are using supercomputer simulations based on Einstein's equations
to study how space and time change when very massive objects (like black holes and
neutron stars) collide in the universe. The focus of the Goddard group is on applying
numerical relativity simulations to model such mergers for gravitational wave and astrophysical
applications.
Numerical Simulation Techniques
We have developed a numerical simulation code with adaptive and fixed mesh refinement
(AMR/FMR) capabilities based on a general purpose AMR simulation package PARAMESH. Our approach
allows us to apply high resolution in the region near the black holes while moving the outer
boundary far from the region of interest. We currently use variations of the "BSSN"
formulation of Einstein's gravitational field equations, using second and higherorder
finite differencing techniques and with hyperbolictype gauge conditions. We interpret the
results of our simulations by applying an array of gaugeinvariant analyses including local
invariants, horizons, and asymptotic quantities.

Binary black hole gravitational radiation studies
Click image for movie.

A key motivation of our research effort is to provide theoretical support to ongoing and
future gravitational wave observations particularly for the LISA project, a joint venture between NASA and the European Space Agency (ESA).
Toward this, part of our effort is focused specifically on questions relating to developing
techniques for studying gravitational radiation from binary black hole systems. The studies
approach such questions as:
 How accurately can we simulate BBH radiation?
 How can we characterize "kicks" generated in asymmetric unequalmass BBH mergers?
 What can BBH radiation tell us about strong field interactions in simulations such as
spin and spinorbit interactions?
We are approaching these questions through simulations of approximately headon binary black
hole configurations with varies massratios and spins. These systems provide simple testbeds
to study phenomena expected in fully realistic configurations.

Inspiraling binary black holes
A key class of simulations directly relevant to astrophysical BBH configurations involves
black holes approaching each other on from a shrinking, approximately circular orbit. As we
develop our techniques we are expanding our work with inspiraling simulations designed for
both short runs with Lazarusstyle waveform estimation, and more costly direct simulations
of waveforms. Our simulations are presently focused on accurate radiation from nearISCO
initial configurations though we are pursuing techniques which we expect to be applicable to
longerlasting orbital configurations in the future.
Moving black holes
Generally, accurate simulations on binary black hole systems rely on fundamental techniques
which can accurately simulate a single moving black hole. The relevance of these studies is
increasing as we pursue increasingly longlasting BBH simulations. We use simulations of
the motion of a single black hole as a test bed arena to study such basic simulation
properties as constraint preservation and energy and momentum conservation. Using this test
bed we are investigating the performance of advances in gauge condition, higherorder finite
differencing, AMR, and modifications of the BSSN formalism.
Click here for a movie showing blackhole tracks from a recent long run (QC9).

Supercomputing 2005: Poster
View the Supercomputing 2005 poster (printable PDF)
Snapshot showing the field representing
gravitational wave generated from an inspirallying black hole binary.

We model the astrophysical coalescence of comparablemass, massive
black hole binaries for different mass ratios and spins and calculate
the resulting gravitational wave (GW) signatures. A key feature of
our work is the use of mesh refinement techniques to handle the wide
range of physical scales involved, from the black holes (~1M) to the
gravitational waves (~10100M), and to enable extraction of
gravitational waveforms in the wavezones.
Our methodology involves solving a closely coupled system of partial
differential equations with many variables on a very large
computational domain. Moreover, we use highly structured component
grids that are distributed across many processors, with a significant
amount of communication between processors. Our simulations typically
require hundreds of gigabytes and run on several hundred processors
for hundreds of hours.
Astrophysical motivation guides our simulations. The waveforms
determined by these simulations will be applied to analyzing and
interpreting observed GW data from the Laser Interferometer Space
Antenna (LISA) mission. Linear momentum loss due to asymmetrical
radiation of GW in the unequal mass mergers imparts 'kicks' to the
merger remnant. High kick velocities from such mergers have the
potential to strongly impact our understanding of how massive black
holes have developed over cosmological time scales.
Snapshots showing time sequence of the field representing the dynamics
of the merging black holes.


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