The development of computational methods for solving Einstein's equations is motivated by the current deployment of gravitational wave detectors such as the ground-based LIGO and the future space-based LISA. To fully understand and analyze the signals and waveforms measured with such facilities it is essential that accurate, robust, and efficient computational tools be available for solving Einstein's equations over very long time scales. The research supported by this award will continue the development of the Spectral Einstein Code by the Caltech-Cornell numerical relativity collaboration. The high accuracy and efficiency of spectral methods could allow simulations over longer timescales than existing finite difference codes and could provide the additional accuracy needed for aspects of LIGO and LISA data analysis using currently available computer hardware.
This research will have a broad impact on our understanding of fundamental physics, in particular testing General Relativity for strong field situations like black holes. In addition, it will have a significant impact on the broader area of computational science. The computational techniques involved can be used to solve problems in many other areas, including fluid dynamics, meteorology, seismology, and astrophysics. Young researchers trained in these techniques are in great demand.
Introduction Just as a stone dropped into a pond will produce waves on the surface of the water, the motion of massive bodies (stars, planets, black holes,...) through space will produce gravitational waves (GWs), which are ripples in the curvature of spacetime that were predicted by Einstein. GWs have been indirectly detected and confirmed (one such observation led to the 1993 Nobel Prize in physics, and another, the BICEP2 experiment http://bicepkeck.org/, recently made newspaper headlines). However, GWs have not yet been directly detected on Earth, because they are so weak. Several experiments (such as LIGO, www.ligo.caltech.edu) will have the technology to detect GWs within a few years. One reason this is so exciting is that there are many objects in the universe (like black holes) that are hard to see with telescopes, but are expected to emit copious GWs. If the GWs from these objects can be "heard", we can learn much about them. A key target of the LIGO experiment is to detect GWs from two black holes that orbit each other and eventually collide, producing a larger black hole in the end. From these detected GWs, we wish to learn about the black holes and details of their collision. (This is analogous to someone standing at the edge of a pond, and figuring out what objects are in the middle of the pond [e.g. ducks swimming on the surface, stones being thrown in] by measuring the waves felt at the edge). To do this, it is necessary to compute in detail what the GWs from colliding black holes look like, so the results of these compuations can be compared with the actual GWs detected by LIGO. These computations can be done only by solving Einstein's equations (the set of equations written down by Einstein in 1915 to describe gravity) on a supercomputer. This project is concerned with techniques used to solve Einstein's equations. Our primary purpose is to provide predictions for LIGO for the GW signals emitted by orbiting, colliding black holes. A secondary purpose is to develop methods for solving Einstein's equations that can be used to study other interesting spacetimes, like the early universe or the spacetimes contained within black holes. Our group has pioneered the use of spectral methods for solving Einstein's equations: these methods provide superior accuracy and efficiency (in terms of how much computer time it takes to compute GWs) compared to standard methods used in the field. This award Much of the effort in this award went towards development of a computer code called SpEC, and the development and refinement of the methods and techniques used in this code. SpEC is now the most accurate and efficient code for computing black-hole binary systems. Thanks to this award and others, SpEC can now produce binary black hole computations with the highest accuracy, largest black hole spins, and largest number of orbits compared to other existing codes. We have published the largest catalog of many-orbit high-accuracy simulations to date. Through our numerical compuations, we have also studied the nonlinear dynamics of colliding black holes, and have gained new insights into the mechanisms of gravitational wave generation. We have developed a new description of strong-gravity phenomena based on "vortex" and "tendex" quantities, which are analogous to magnetic field lines. Besides black holes, we have also investigated the collisions of black holes and neutron stars; such collisions are also expected to emit GWs that will be detectable by LIGO. In addition to these black hole studies, we have significantly expanded the capabilities of the SpEC code with funding from this grant to allow us now to solve Einstein's equations on spacetimes having arbitrary topological structures. We have also developed new applications of spectral methods that can be used to determine the properties of the very highest density matter in the cores of neutron stars. Broader impacts: Numerical simulations of Einstein's equations are important for the success of LIGO and other GW detectors. The results of this research have been used and are being used to develop detection and data analysis techniques for LIGO. Results have been disseminated through publications, and computed GWs are publicly available at www.black-holes.org/waveforms. The new spectral methods developed with this award should provide the optimal way for measuring the properties of neutron star matter from astrophysical observations (when they become available). This award has partially supported one postdoc, and three graduate students, and has involved and undergraduate, thus contributing to future generations of leaders in science and technology.
