Insight into Gravitation via a Combination of Analytical and Numerical Methods
Astrophysics and Astronomy
Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
Final Report Abstract
Despite their elegant geometric origin, Einstein’s field equations of general relativity are extremely difficult to solve with pen and paper. For many interesting scenarios one has to resort to computer simulations. For instance, simulations of colliding binary black holes played an important role in the recent detection of gravitational radiation. However, beyond such astrophysical questions, numerical methods have applications to more fundamental problems in general relativity and geometric analysis. The aim of this project has been to explore such applications, find suitable formulations of the underlying equations and develop state of-the-art numerical methods. Three focus areas were identified: from the very large (global methods for the Einstein equations) to the very small (gravitational collapse), and problems beyond Einstein’s equations such as geometric flows. The standard approach to numerical relativity is based on a 3+1 ("space + time") decomposition of the Einstein equations. The slices of constant time are usually truncated at a finite distance. This raises the problem of specifying suitable boundary conditions, and there is no choice that is completely satisfactory from both a mathematical and physical point of view. Instead, we considered a setup where the slices extend all the way to infinity, more precisely to future null infinity—the asymptotic region of space-time that outgoing light rays approach. The coordinates are compactified such that infinity is mapped to a finite location in the computational domain. A numerical code based on this approach allowed us to determine exactly how matter fields decay at late times both at future null infinity and at a finite distance (so-called power-law tails). An article on this work addressed at a broader audience was featured in the 2014 Yearbook of the Max Planck Society. More recently, we have applied this method to the problem of black hole super-radiance. This term refers to the fact that black holes are not as black as their name suggests: under certain circumstances, radiation scattering off a black hole may return more energy than was sent in, thereby reducing the black hole mass and angular momentum or charge. We considered a charged scalar field scattering off a charged black hole in spherical symmetry. Our simulations are among the first that take into account the coupling of the matter field to gravity, which leads to dramatic changes of the black hole mass and charge during the scattering process. Space-time slices extending to future null infinity (so-called hyper-boloidal slices) are not suitable to represent an entire space-time, as one misses the region around spatial infinity (essentially our notion of “infinitely far away at the same moment of time”). In order to address this problem, we numerically implemented for the first time a gluing method, whereby prescribed initial data are truncated and extended to spatial infinity by the Schwarzschild metric, the simplest static black hole solution. The next step is to evolve these glued data to a first hyper-boloidal slice extending to future null infinity, and then to continue using the scheme described above. Hyper-boloidal evolution has also proved fruitful for gravitational collapse problems. A system that shows particularly rich behavior consists of the Einstein equations coupled to Yang-Mills matter fields. When such matter collapses to form a black hole, there are two outcomes with the Yang-Mills field in different vacuum states. We investigated in detail the threshold between these two end-states and moreover uncovered a surprising structure along the critical line separating the two states. Studies of gravitational collapse in the more demanding case of axi-symmetry have also been initiated, both for vacuum space-times and for collisionless matter (a model that is believed to describe well the evolution of galaxies). An example of a geometric flow we considered in this project arises in the static metric extension problem, where one specifies certain geometric boundary data on a 2-sphere and seeks a solution to the static Einstein equations in the exterior that induces these boundary data. In a suitably chosen coordinate system, the location of the boundary 2-sphere is not known beforehand. We have designed a nonlocal meancurvature-type flow that drives an initial guess to a surface satisfying the desired boundary conditions. This problem has important implications on the definition of a quasi-local mass in general relativity due to Robert Bartnik.
Publications
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2013. Hyperboloidal Einstein-matter evolution and tails for scalar and Yang-Mills fields Class. Quantum Grav. 30 095009
Rinne O and Moncrief V
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2014. Evolution of the Einstein equations to future null infinity Relativity and Gravitation: 100 Years after Einstein in Prague ed Bičák J and Ledvinka T. Springer Proceedings in Physics 157 199–206
Rinne O and Moncrief V
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2014. Formation and decay of Einstein-Yang-Mills black holes. Phys. Rev. D 90 124084
Rinne O
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2014. Numerical and Analytical Methods for Asymptotically Flat Spacetimes. Habilitation thesis, Freie Universität Berlin
Rinne O
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2014. Raumzeiten bis ins Unendliche rechnen. Jahrbuch der Max-Planck-Gesellschaft 2014
Rinne O
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Spectral approach to axisymmetric evolution of Einstein’s equations Proceedings of the Spanish Relativity Meeting 2014, ed. Cerda-Duran P et al. J. Phys.: Conf. Ser. 600 012060
Schell C and Rinne O
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2016. Constructing reference metrics on multicube representations of arbitrary manifolds. J. Comput. Phys. 313 31–56
Lindblom L and Taylor N W and Rinne O
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2016. Numerical construction of initial data for the Einstein equations with static extension to space-like infinity. Class. Quantum Grav. 33 075014
Doulis G and Rinne O
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2016. Superradiance of a charged scalar field coupled to the Einstein-Maxwell equations. Phys. Rev. D
Baake O and Rinne O
