The Characteristic Treatment of Black Holes

Abstract

The characteristic initial value problem has been successfully implemented as a robust computational algorithm (the PITT NULL CODE) to evolve 4-dimensional vacuum spacetimes. It has been applied to the calculation of gravitational waveforms emitted by black holes and to the event horizon structure in the merger of black holes. The characteristic code also has potential application to the binary black hole problem via Cauchy-characteristic matching. Because the event horizon is itself a characteristic hypersurface, it can be analyzed by characteristic techniques as a stand-alone object. We have developed an analytic conformal model of null hypersurfaces which gives new insight into the intrinsic geometry of the pairof-pants horizon found in the numerical simulation of the head-on collision of black holes and into the initially toroidal horizon found in the simulation of a collapsing, rotating cluster. Most studies of black hole formation and merger have been restricted to axisymmetry. However, axisymmetric horizons, like the Schwarzschild horizon, are non-generic. When applied to a non-axisymmetric horizon, the characteristic approach reveals substantially new features. In particular, coalescing black holes generically go through a toroidal phase before they become spherical. The conformal structure of the event horizon supplies part of the data for a simulation of the exterior space-time. This provides a new way to calculate the post-merger waveforms from a binary black hole inspiral. §1. Null cone evolution A longterm project 1) to develop the pioneering work of Bondi 2) and Penrose 3) into a computational algorithm for the characteristic initial value problem has recently culminated in a highly accurate,efficient and robust code — the PITT NULL CODE. 4) Because the evolution proceeds on a space-time foliation by null cones which are generated by the characteristic rays of the theory,this approach offers several advantages for numerical work. I will describe here the fruits of these investigations relevant to black hole physics. In null cone coordinates,Einstein’s equations reduce to propagation equations along the radial light rays,which can be integrated in hierarchical order for one variable at a time. This leads to a highly efficient characteristic marching algorithm. There is one complex evolution variable which describes the free degrees of freedom of the gravitational field and four auxiliary variables. A compactified grid,based upon Penrose’s conformal description of null infinity,removes the necessity of an artificial outgoing radiation condition and makes possible a rigorous description of geometrical quantities such as the Bondi mass and news function. The news function supplies both the true waveform and polarization of the gravitational radiation incident on a distant antenna. Furthermore the grid domain is exactly the region in which waves propagate,which is ideally efficient for the purpose of radiation studies. Since each