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

Still newer formulations: towards a stable evolution system 369of the determinant of the 3-metric, and in those cases one can also identify gaugespeeds which can be different from light speed (harmonic slicing is one exampleof this family where the gauge speed corresponds to light speed). Some ofthese slicings, such as ‘1 + log’ [45], have been found to be very useful in 3Dnumerical evolutions. This information about the speed of gauge and physicalpropagation can be very helpful in understanding the system, and can also beuseful in developing numerical methods. Only extensive numerical studies willtell if the various hyperbolic formulations live up to their promise.Reula has recently reviewed, from the mathematical point of view, most ofthe recent hyperbolic formulations of the Einstein equations [46] (This article, inthe online journal Living Reviews in Relativity, will be periodically updated). Itis important to realize that the mathematical relativity field has been interestedin hyperbolic formulations of the Einstein equations for many years and somesystems that could have been suitable for numerical relativity were alreadypublished in the 1980s [30, 47]. However, these developments were generallynot recognized by the numerical relativity community until recently.18.3 Still newer formulations: towards a stable evolutionsystemSomewhere in between the standard ADM formulation and hyperbolicformulations are a class of formulations that have been getting significantattention lately, and which seem to be very promising and very stable.As discussed above, the 3D evolution of Einstein’s equations has proved verydifficult, with instabilities developing on rather short timescales, even in cases ofweakly gravitating, vacuum systems, such as low amplitude gravitational waves,as summarized in an important paper by Baumgarte and Shapiro [48]. In thiswork, it was shown how one can achieve highly improved stability by making afew key changes to the formulation of the ADM equations, most notably througha conformal decomposition and by rewriting certain terms in the 3D Ricci tensorto eliminate terms that spoil its elliptic nature. In fact, essentially these sametricks were already noticed a few years earlier by Shibata and Nakamura [49].Hence, we refer to these formulations collectively as ‘BSSN’ after the fourauthors. These subtle changes to the standard ADM formalism have a verypowerful stabilizing effect on the evolutions. Evolutions of weak waves thatwould develop instabilities and crash with the standard ADM formulation runmuch longer with the new system, and as shown in Alcubierre et al [50], the newsystem and variations allow for the first time the successful evolution of highlynonlinear gravitational waves to form a black hole in 3D while the standard ADMtreatment would fail well in advance of black hole formation. Further work by thePalma group, showed the deep connection between the BSSN formulations andthe Bona–Massó family of formulations [51], leading to the possibility of a fullyhyperbolic, very stable formulation that shares advantages from many sides.