100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

158 P. LagunaWith the generalized harmonic formulation one looses the explicit geometrodynamicspoint of view; that is, one no longer deals with geometricalquantities directly associated with the foliation (i.e. spatial metric, extrinsiccurvature, lapse, shift). Instead, one works directly with the spacetimemetric components. Is this something to worry about? The answer is likelyno. After all, most of the 3+1 hyperbolic formulations introduce auxiliaryvariables devoid of any geometrical interpretation. Also, there are othernon-3+1 formulations such as the characteristic formulation that have alsoenjoyed relative success. Unfortunately, because of the difficulties that thecharacteristic formulation has on handling caustics, characteristic codesthese days are mostly used in connection with gravitational wave extraction,namely Cauchy-Characteristic matching.There are signs that the next revolution in codes will be the generalizedharmonic codes. Preliminary work by Pretorius 48 indicates that these codeshave great potential. He has able to carry out eccentric binary black holeorbits without encountering instabilities or the need of introducing a largenumber of fine tuned parameters. It is too early to say if the generalizedharmonic codes are indeed the silver bullet against instabilities. What isclear is that they are quickly positioning as the leading contestant.3. Black Hole Excision: Space-time SurgeryWhat makes black hole evolutions unique is the presence of black hole singularities.These are not simple singularities, as the gravitational interaction1/r 2 -singularities in N-body simulations, singularities that are amenableto smoothing. In modelling black holes, the singularities are not known apriori. They are intimately related to the solution one seeks. It would seemthat there is no way around it! Not quite, the Cosmic Censorship conjecturesstates that these singularities will be inside a horizon, hidden fromexternal observers. If we are able to track the existence of a horizon, onecan in principle exclude the singularity from the physically relevant part ofthe computational domain.There are two generic approaches to handle black hole singularities innumerical evolutions. The oldest approach was to avoid the black hole singularities.This is possible because of the freedom to foliate the space-timearbitrarily. An example of a slicing condition that avoids crushing into thesingularities is the maximal slicing, defined by the condition that the traceof the extrinsic curvature vanishes. 57 Slicing conditions that avoid singularitiessuffer, however, from what is known as grid stretching; that is, the