Gravitational Theory

The study of conformal methods in Relativity evolves from the need to understand gravitational radiation and isolated systems within the theory of gravitation. In this concept, the ultimate destination, called future null-infinity, of gravitational waves is taken into consideration.

Another research area, where conformal methods come into play, is the accurate numerical calculation of general relativistic elliptic boundary value problems that arise in the description of axisymmetric and stationary matter configurations as well as in the context of the initial data problem for the dynamical evolution of binary systems. In the combination with multi-domain pseudo-spectral techniques, conformal methods permit the exact adaptation of the numerical domain boundaries to the physical boundaries, given by the situation under consideration (e.g. a yet to be determined shape of a neutron star).

In recent years, the realm of Numerical relativity has seen tremendous development. The computation of more than 10 orbits of inspiraling binary black holes has become feasible, and the emitted gravitational waves have been studied in great detail. In the upcoming years, the research in this area will be concerned with refined studies to explore the available parameter space. In particular, large (or even extremely large) mass ratios of the two constituents are of interest, as such binary objects are among the most relevant sources for detection of gravitational waves.

This sophisticated exploration requires, especially in view of large mass ratios, the development of novel numerical algorithms that go beyond the techniques which are used at the present stage. In particular, very high accuracy would be desirable as well as specifically adapted methods, in order to deal with the issues of the resolution of strong gradient regimes in the vicinity of the small mass constituent.

The research activities of our group addresses these points in terms of a combination of three aspects:

  1. The incorporation of future null infinity ("Conformal infinity").
    The outer computational boundary is placed at "future null-infinity".This boundary contains all points which are approached asymptotically by null rays (light rays and gravitational waves) which can escape to infinity. Hence this concept permits the complete investigation of the outgoing radiation and is therefore an important ingredient in the development of novel numerical algorithms.
    The inclusion of future null infinity is performed by an appropriate conformal compactification, through which this boundary is put at the exterior boundary of a finite computational domain. As a consequence, Einstein's field equations are considered on hyperboloidal slices which are space-like in the interior of this domain but become asymptotically null at the outer boundary. In this setting, the field equations split into constraint and evolution equations, which both have to be treated according to the specific mathematical and geometric conditions given at future null infinity.

  2. The use of pseudo-spectral methods
    Pseudo-spectral methods have the remarkable capability of providing exponential convergence rate when the underlying problem admits a smooth solution. They have been used widely for the solution of elliptic equations in relativity, in particular for the construction of equilibrium models of rotating neutron stars, fluid rings and central-black-hole-fluid-ring systems, as well as for the highly accurate computation of binary initial data. In the context of dynamical binary black hole evolutions, pseudo-spectral methods applied to the spatial directions yield the most accurate wave forms to date.
    An important requirement for the applicability of pseudo-spectral methods is the regularity of the underlying solution. In the context of future null infinity it has been shown that (i) regular initial data can be constructed on hyperboloidal slices, and (ii) the time evolution of such data remains regular there.
    In order to realize extremely accurate solutions, we use pseudo-spectral expansions with respect to spatial directions. In particular, constraint equations on the initial hyperboloidal slice are going to be treated in terms of pseudo-spectral methods. But also for the time evolution equations we intend to expand the field quantities, considered in the spatial directions, with respect to a pseudo-spectral scheme.

  3. The introduction of specifically adapted coordinates related to conformal mappings.
    In a number of contributions in relativity, pseudo-spectral methods have proven to work particularly well when considered in specifically adapted coordinate systems. The computational domain may or may not possess specific boundary points at which the underlying field equations degenerate. We explore the applicability of coordinate transformations related to conformal mappings (known from complex analysis) and their compatibility with pseudo-spectral methods in the treatment of the Einstein equations on hyperboloidal slices. As a specific feature, the coordinates to be used in the schemes are adapted to both future null infinity and inner boundaries like black hole horizons such that they coincide exactly with numerical domain boundaries.

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