Characteristic Evolution and Matching

Abstract

I review the development of numerical evolution codes for general relativity based upon the characteristic initial-value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D-axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black-hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black-hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.

Global aspects of radiation memory

“…This procedure can then be iterated to produce a solution provided the finite difference approximation converges. Such a convergent finite difference evolution algorithm has been successful for the analogous characteristic initial value problem for the Einstein equation [12]. The coupled Maxwell-Einstein equations have this same hierarchical structure as a set of hypersurface and dynamical equations which can be integrated sequentially in the radial direction [7].…”

Simple, explicitly time-dependent, and regular solutions of the linearized vacuum Einstein equations in Bondi-Sachs coordinates

“…The latter definition covers both commonly used normalizations, the traditional one q = 1 [10,15,26,27] and the numerical one q = 2 [4,14,28], which is employed in numerical relativity. The unit sphere metric, q AB , represented by the dyad is…”

“…(71)) make SPIN-2 an ideally suited testbed solution for simulations in the Bondi-Sachs framework and to test numerical wave extraction methods at null infinity, e.g. [20] describes the most recent process of such simulations (for others see [4]). …”

“…It is an ideal textbook solution allowing one to demonstrate, when working in the Bondi-Sachs frame work of General Relativity, important features, such as the regularity conditions at the vertices, the commonly used ð−formalism, and the subtleties at null infinity. In addition, since it describes all radiation multipoles, SPIN-2 is also well suited as a testbed solution for numerical relativity, when the Einstein field equations are solved in a Bondi-Sachs framework, (see [4] for a review). Thus it might be considered as a sibling, in null coordinates, of the well-known Teukolsky-Rinne solutions [5,6] employed in numerical relativity using the (3 + 1) formulation of General Relativity.…”

Black Holes on Supercomputers: Numerical Relativity Applications to Astrophysics and High-energy Physics

“…For generic fourdimensional spacetimes with no symmetry assumptions, the characteristic formalism results in a natural hierarchy of 2 evolution equations, 4 hypersurface equations relating variables on hypersurfaces of constant retarded (or advanced) time, as well as 3 supplementary and 1 trivial equations. A comprehensive overview of characteristic methods in numerical relativity is given in Winicour's Living Review article [62]. Although characteristic codes have been developed with great success in spacetimes with additional symmetry assumptions, evolutions of generic BH spacetimes face the problem of formation of caustics, resulting in a breakdown of the coordinate system; see [63] for a recent investigation.…”