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

376 Numerical relativityare the ability to prevent large shearing or drifting of coordinates during anevolution, and the ability to control the coordinate location of a physical object,for example, the horizon of a black hole. These considerations are discussedbelow. The development of appropriate shift conditions for full 3D evolution,for systems without symmetries, is an important research area that needs muchattention. Geometrical shift conditions that can be formulated without referenceto specific coordinate systems or symmetries seem to be desirable. The basic ideais to develop a condition that minimizes the stretching, shearing, and driftingof coordinates in a general way. A few examples have been devised whichpartially meet these goals, such as ‘minimal distortion’, ‘minimal strain’ andvariations [10], but much more investigations are needed. New gauge conditions,based on these earlier proposals, have recently been proposed but not yet testedin numerical simulations [28].It is important to emphasize that the lapse and shift only change the wayin which the slices are chosen through a spacetime and where coordinates arelaid down on every slice, and do not, in principle, affect any physical resultswhatsoever. They will affect the value of the metric quantities, but not the physicsderived from them. In this respect the freedom of choice in the lapse and shift isanalogous to the freedom of gauge in electromagnetic systems.On the other hand, it is also important to emphasize that proper choices oflapse and shift are crucial for the numerical construction of a spacetime in theEinstein theory of general relativity, in particular in a general 3D setting. In ageneral 3D simulation without symmetry assumption, there is no preferred choiceof the form of the metric (e.g., a diagonal 3-metric, or g θθ = r 2 as in sphericalsymmetry), hence forcing us to deal with the gauge degree of freedom in relativityin full. This, when coupled with the inevitable lower resolution in 3D simulations,often leads to development of coordinate singularities, when evolved without asophisticated choice of lapse and shift. Indeed the success of the ‘driver’ ideasuggested [87] that in order to obtain a stable evolution over a long timescale, itis important to ensure that the coordinate conditions used are not only suitablefor the geometry of the spacetime being evolved, but also that the conditionsthemselves are stable. That is, when the condition is perturbed, for example,by numerical inaccuracy, there is no long-term secular drifting. We regard theconstruction of an algorithm for choosing a suitable lapse and shift for a general3D numerical simulation to be one of the most important issues facing numericalrelativity at present.18.3.1 General relativistic hydrodynamicsIn order to make numerical relativity a tool for computational general relativisticastrophysics, it is important to combine numerical relativity with traditional toolsof computational astrophysics, and, in particular, relativistic hydrodynamics.While a large amount of 3D studies in numerical relativity have been devotedto the vacuum Einstein equations, the spacetime dynamics with a non-vanishing