Hyperboloidal foliations and scri-fixing

In numerical studies of the initial value problem for test fields on asymptotically flat background spacetimes, one typically truncates the solution domain by introducing an artificial timelike outer boundary into the spacetime. The solution then is calculated on a finite, spatially compact domain. The boundary of this domain is not part of the physical problem. Therefore, one tries to construct boundary conditions that correspond to transparency of this artificial outer boundary. In addition, these boundary conditions are required to form a well posed initial boundary value problem (IBVP). In general, it is not possible to construct such boundary conditions. Spurious reflections occur from the outer boundary even for the simple case of the flat wave equation on a three-dimensional ball [50]. One therefore tries to minimize the amount of such spurious reflections in a manner that also ensures the wellposedness of the IBVP. Such boundary conditions are called non-reflecting or absorbing. It is, however, very difficult to construct them when the curvature of the background does not vanish or when non-linear terms appear in the equations. One needs to account for backscatter off curvature or self-interaction of the field near the boundary. A bad choice of boundary data can destroy relevant features of the solution. It has been shown, for example, that a certain choice of boundary data, commonly used in numerical relativity, destroys the polynomial tail behavior of solutions to wave equations on a Schwarzschild spacetime [8]. A further difficulty is related to the notion of radiation. In the study of radiative phenomena, one is interested in the behavior of solutions in the far-field zone. The electromagnetic field on flat space, for example, admits a global separation into near field terms and far field terms which simplifies the discussion of radiation considerably. When the background curvature does not vanish, such a separation can only be expected in the asymptotic limit to infinity in null directions. Therefore to study radiation accurately, one needs to calculate very large portions of spacetime. Using Cauchy-type foliations and truncating the computational domain, however, does not allow us to calculate such large portions of spacetime that astrophysically realistic distances can be modeled in a numerical calculation. Additional complications arise when one tries to construct absorbing boundary conditions for the Einstein equations [7, 19, 48, 51]. The treatment of the asymptotic region in numerical relativity is an important theoretical and practical problem as it effects the accuracy of the calculated radiation field [41, 49]. With increasing sensitivity of detectors and computer power, systematic errors in current numerical simulations due to wave extraction may restrict the predictable power of numerical relativity and blur the comparison between observed and calculated waveforms [5, 30]. A clean solution to the above mentioned difficulties is available on the level of geometry. The solution is to include null infinity in the computational domain. This geometric idea goes back to Penrose and has been known for a long time [42, 43]. Its implementation within the 3+1 approach, however, has been exceptionally difficult, numerically as well as analytically. In this article we discuss in detail the construction of suitable explicit gauges on asymptotically flat background spacetimes such that the geometric idea of null infinity can be incorporated in numerical calculations within the 3+1 approach. While we assume the spacetime to be given, a basic motivation is to extend the suggested methods to the treatment of the asymptotic region in numerical solutions to the Einstein equations. The article is organized as follows: In section II we introduce the basic concepts and explain the idea of scri-fixing. In section III we present an explicit construction of scri-fixing coordinates in spherical symmetry. This method is applied to Minkowski spacetime in section IV and to extended Schwarzschild spacetime in section V. Going beyond