Bypassing stability conditions and curing logarithmic singularities
By Jörg Frauendiener and Jörg Hennig
Assume you want to model a general relativistic spacetime. Due to the annoying limitations of conventional computers, like finite memory and processing speed, it is tempting to focus on a finite portion of the spacetime. Then, without waiting endlessly, one can obtain an approximate description of this portion. One just has to choose a suitable numerical method and solve the field equations for the metric at some set of grid-points. While this approach is standard, it introduces unpleasant problems. Firstly, the set of equations needs to be complemented with boundary conditions at the outer edges of this finite portion, in order to obtain a complete mathematical problem. This, however, is quite unphysical as usually no information about the actual behaviour at such an artificial boundary is available. Consequently, spurious gravitational radiation enters the numerical domain. Secondly, if one is interested in accurately describing gravitational waves, one should recall that these are only well-defined at infinity. Hence it is desirable to extend the simulation up to infinity.
A promising way out is the concept of conformal compactification, the detailed analysis of which was initiated by Roger Penrose in the 1960s. The basic idea is to map an infinite spacetime to a finite domain and to rescale the metric to obtain a conformal metric that is regular even at infinity. Actually, there is a formulation of the Einstein equations adapted to this setting: Helmut Friedrich’s conformal field equations. Numerically, one can then reach infinity with a finite number of grid-points; just the distances between neighbouring points get larger and larger. While this seems to solve the above problems, a new difficulty is introduced. In traditional numerical methods, PDEs are solved by using the information at some point in time to construct the approximations at the next instance. Unfortunately, the size of the time step is restricted by the CFL condition, which is necessary for numerical stability. As a consequence, smaller and smaller step-sizes are required as one approaches infinity. Hence, we are again in the unpleasant situation that infinitely many steps are necessary.
Our solution is to abandon the usual “time-marching” approach and to use a method that computes the metric at all grid-points simultaneously. In this way, no CFL condition is required. This can be achieved with a fully pseudospectral scheme. Generally, in a (pseudo)spectral method, the unknown functions are approximated by finite linear combinations of certain basis functions. This is often applied in space and supplemented with finite-difference time-marching. With the fully pseudospectral scheme, however, we use expansions both in space and time. This allows us to solve dynamical problems with high accuracy, usually close to machine accuracy, and to reach infinity with just a moderate number of grid-points. The high accuracy has the advantage that it is possible to numerically discern the character of singularities that may arise.
In order to test how well this works, we study the example of the conformally invariant wave equation on the background of a Schwarzschild black hole spacetime. This equation has the interesting property that the leading terms are very similar to those of Friedrich’s conformal field equations. Hence this is a first step towards solving the full Einstein equations.
Particularly interesting is the question of how regular the solutions are at infinity, specifically at space-like infinity. In a conformally compactified spacetime, space-like infinity is usually thought of as a “point” from which the future and past null null infinities emerge. Hence the late incoming radiation and the early outgoing radiation must somehow communicate via space-like infinity, and it is desirable to better understand this process. Mathematically, this point introduces certain problems as the equations degenerate there. In order to better resolve the behaviour of the fields, one can “blow up” space-like infinity to an entire cylinder. With this formulation of the problem, the wave equation can be numerically solved up to infinity, where the cylinder and future null infinity become outer boundaries. Since they arise from the geometry of the situation, they are natural boundaries. This has the consequence that no artificial boundary conditions are required there.
Our numerical experiments show that the solutions can be obtained up to infinity, but they only have a limited regularity there: they suffer from certain logarithmic singularities. In some cases, the singularities are sufficiently weak, and highly accurate solutions are still possible. This behaviour can be influenced by the initial data. In other situations, however, they are so strong that the numerics breaks down. What can we do in this case? The trick is to appropriately stretch the spacetime near infinity to smooth out the singularities. A clever coordinate transformations allows us to transform singular solutions into smooth solutions. After this transformation, highly accurate numerics is again possible!
In other words, one just needs to find the right mathematical formulation of a problem, in order to remove all the obstacles that are put into the long way to infinity.
About the authors:
Jörg Frauendiener (Chair of Applied Mathematics) and Jörg Hennig (Senior Lecturer) are researchers and teachers at the Department of Mathematics and Statistics at the University of Otago, Dunedin, New Zealand.
Read the full article “Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. III: nonspherical Schwarzschild waves and singularities at null infinity” in Classical and Quantum Gravity: here.
This work is licensed under a Creative Commons Attribution 3.0 Unported License.