Cauchy-characteristic Matching

This paper gives a detailed pedagogic presentation of the central concepts underlying a new algorithm for the numerical solution of Einstein’s equations for gravitation. This approach incorporates the best features of the two leading approaches to computational gravitation, carving up spacetime via Cauchy hypersurfaces within a central worldtube, and using characteristic hypersurfaces in its exterior to connect this region with null infinity and study gravitational radiation. It has worked well in simplified test problems, and is currently being used to build computer codes to simulate black hole collisions in 3-D. 1. Preamble Throughout his career in theoretical gravitational physics, Vishu has been interested in questions of black holes and gravitational radiation. The subject of his useful early study of radiation from binary systems [1] is now at 2 NIGEL T. BISHOP ET AL. the forefront of research as the principal target for the first exciting experimental measurements of the LIGO project. His fundamental studies of the properties of black hole excitations and radiation [2] used the formidable technology of the mid 1960’s, i.e. Regge-Wheeler perturbation theory, to analytically extract the crucial physical result that black holes were stable. In his continuing studies of the interaction of black holes with gravitational radiation [3], he first demonstrated the phenomenon that was later to be called normal mode excitations of black holes, and noted that the frequency of the emitted radiation carried with it key information which could be used to determine the mass of the invisible black hole. These issues are still at the heart of current research three decades later. Today, the technology to study these questions has become even more formidable, requiring large groups of researchers to work hard at developing computer codes to run on the massively parallel supercomputers of today and tomorrow. This has caused gravitational theory to enter into the realm of “big science” already familiar to experimental physics, with large, geographically distributed collaborations of scientists engaged on work on expensive, remote, central facilities. The goal of this modern work is to understand the full details of black hole collisions, the fundamental two-body problem for this field. Recent developments in this area are very encouraging, but it may well take another three decades until all the riches of this subject are mined. In this paper, we will present all the gory details of how the best current methods in computational gravitation can be forged into a single tool to attack this crucial problem, one which is currently beyond our grasp, but perhaps not out of our reach. 2. Introduction Although Einstein’s field equations for gravitation have been known for the past 80 years, their complexity has frustrated attempts to extract the deep intellectual content hidden beneath intractable mathematics. The only tool with potential for the study of the general dynamics of time-dependent, strongly nonlinear gravitational fields appears to be computer simulation. Over the past two decades, two alternate approaches to formulating the specification and evolution of initial data for complex physical problems have emerged. The Cauchy (also known as the ADM or “3 + 1”) approach foliates spacetime with spacelike hypersurfaces. Alternatively, the characteristic approach uses a foliation of null hypersurfaces. Each scheme has its own different and complementary strengths and weaknesses. Cauchy evolution is more highly developed and has demonstrated good ability to handle relativistic matter and strong fields. However, Cauchy-characteristic matching 3 it is limited to use in a finite region of spacetime, and so it introduces an outer boundary where an artificial boundary condition must be specified. Characteristic evolution allows the compactification of the entire spacetime, and the incorporation of future null infinity within a finite computational grid. However, in turn, it suffers from complications due to gravitational fields causing focusing of the light rays. The resultant caustics of the null cones lead to coordinate singularities. At present, the unification of both of these methods [4] appears to offer the best chance for attacking the fundamental two-body problem of modern theoretical gravitation: the collision of two black holes. The basic methodology of the new computational approach called Cauchycharacteristic matching (CCM), utilizes Cauchy evolution within a prescribed world-tube, but replaces the need for an outer boundary condition by matching onto a characteristic evolution in the exterior to this world-tube, reaching all the way out to future null infinity. The advantages of this approach are: (1) Accurate waveform and polarization properties can be computed at null infinity; (2) Elimination of the unphysical outgoing radiation condition as an outer boundary condition on the Cauchy problem, and with it all accompanying contamination from spurious backreflections, consequently helping to clarify the Cauchy initial value problem. Instead, the matching approach incorporates exactly all physical backscattering from true nonlinearities; (3) Production of a global solution for the spacetime; (4) Computational efficiency in terms of both the grid domain and algorithm. A detailed assessment of these advantages is given in Sec. 3. The main modules of the matching algorithm are: − The outer boundary module which sets the grid structures. − The extraction module whose input is Cauchy grid data in the neighborhood of the world-tube and whose output is the inner boundary data for the exterior characteristic evolution. − The injection module which completes the interface by using the exterior characteristic evolution to supply the outer Cauchy boundary condition, so that no artificial boundary condition is necessary. Details of the Cauchy and characteristic codes have been presented elsewhere. In this paper, we present only those features necessary to discuss the matching problem. 3. Advantages of Cauchy-characteristic matching (CCM) There are a number of places where errors can arise in a pure Cauchy computation. The key advantage of CCM is that there is tight control over the errors, which leads to computational efficiency in the following sense. For a given target error ε, what is the amount of computation required for 4 NIGEL T. BISHOP ET AL. CCM (denoted by ACCM ) compared to that required for a pure Cauchy calculation (denoted by AWE)? It will be shown that ACCM/AWE → O as ε → O, so that in the limit of high accuracy CCM is by far the most efficient method. In CCM a “3 + 1” interior Cauchy evolution is matched to an exterior characteristic evolution at a world-tube of constant radius R. The important point is that the characteristic evolution can be rigorously compactified, so that the whole spacetime to future null infinity may be represented on a finite grid. From a numerical point of view this means that the only error made in a calculation of the gravitational radiation at infinity is that due to the finite discretization h; for second-order algorithms this error is O(h2). The value of the matching radius R is important, and it will turn out that for efficiency it should be as small as possible. The difficulty is that if R is too small then caustics may form. Note however that the smallest value of R that avoids caustics is determined by the physics of the problem, and is not affected by either the discretization h or the numerical method. On the other hand, the standard approach is to make an estimate of the gravitational radiation solely from the data calculated in a pure Cauchy evolution. The simplest method would be to use the raw data, but that approach is too crude because it mixes gauge effects with the physics. Thus a substantial amount of work has gone into methods to factor out the gauge effects and to produce an estimate of the gravitational field at null infinity from its behavior within the domain of the Cauchy computation [5, 6, 7]. We will call this method waveform extraction, or WE. The computation is performed in a domain D, whose spatial cross-section is finite and is normally spherical or cubic. Waveform extraction is computed on a world-tube Γ, which is strictly in the interior of D, and which has a spatial cross-section that is spherical and of radius rE . While WE is a substantial improvement on the crude approach, it has limitations. Firstly, it disregards the effect, between Γ and null infinity, of the nonlinear terms in the Einstein equations; the resulting error will be estimated below. Secondly, there is an error, that appears as spurious wave reflections, due to the inexact boundary condition that has to be imposed at ∂D. However, we do not estimate this error because it is difficult to do so for the general case; and also because it is in principle possible to avoid it by using an exact artificial boundary condition (at a significant computational cost). The key difference between CCM and WE is in the treatment of the nonlinear terms between Γ and future null infinity. WE ignores these terms, and this is an inherent limitation of a perturbative method (even if it is possible to extendWE beyond linear order, there would necessarily be a cutoff at some finite order). Thus our strategy for comparing the computational efficiency of CCM andWE will be to find the error introduced into WE from Cauchy-characteristic matching 5 ignoring the nonlinear terms; and then to find the amount of computation needed to control this error. 3.1. ERROR ESTIMATE IN WE As discussed earlier, ignoring the nonlinear terms between Γ (at r = rE) and null infinity introduces an error, which we estimate using characteristic methods. The Bondi-Sachs metric is ds = − (