(ed.). Gravitational waves (IOP, 2001)(422s).

394 Numerical relativityFigure 18.2. (a) Evolution of the log of the lapse α at r = 0 for the axisymmetricdata (4, 0, 0). The dashed/dotted/full curves represent simulations at low/medium/highresolution. (b) Evolution of the Riemann invariant J at r = 0. The wave disperses afterdynamic evolution, leaving flat space behind.forever. With this experience, we next try the case of an even stronger amplitudewave, which in this case will actually collapse on itself and form a black hole.In figure 18.3, we study the development of the data set (a = 6, c = 0.2,n = 1), a full 3D data set, and watch it collapse to form a black hole (the firstsuch 3D simulation). The figure also compares this black hole formation to resultsobtained with an axisymmetric data set. The system clearly collapses on itself andrapidly forms a black hole. The waveform extraction shows that the newly formedhole then rings at its quasinormal mode frequency. High quality images andmovies of these simulations can be found at http://jean-luc.aei-potsdam.mpg.de.These results are exciting examples of how numerical relativity can actas a laboratory to probe the nonlinear aspects of Einstein’s equations. Puregravitational waves are clearly a rich and exciting research area that allows oneto study Einstein’s equations as a nonlinear theory of physics. With these newcapabilities of accurate 3D evolution that can follow the implosion of wavesto a black hole, there is much more physics to study, including the structureof horizons, full 3D studies of critical phenomena, and much more. Further,this study of pure vacuum waves has helped us to understand the importance ofdeveloping and testing new formulations of Einstein’s equations for numericalpurposes. Without the new formulations, these results simply could not beobtained. Further, we have run literally hundreds of simulations like these in orderto determine which variation on the ‘BSSN’ families of formulations performbest. With this new knowledge, we turn to the problem of 3D black holes.18.7.2 Black holesHaving tested these new formulations of Einstein’s equations on the problem ofpure gravitational waves, we now apply what we have learned to the considerablymore complex problem of black hole evolutions. We first applied these newformulations to black hole spacetimes that have been very carefully tested in