Finite Element Approximations of Scalar Fields for Near-Extremal Black Holes
Finding solutions to the highly nonlinear Einstein field equations, arising in General Relativity, is known to be quite intractable. Providing an analytical or numerical solution to the fully nonlinear model of gravity is thus no easy feat. Even the exact solutions, which are already known, are obtained using simplifying assumptions. To this end, we must employ a solution technique that simplifies the fully coupled nonlinear equations, whilst maintaining properties of the entire system.
Within the context of Black Hole Perturbation Theory, a linearization of the nonlinear problem always takes place initially. This allows for the investigation of the linear dynamics of generic small perturbations around a chosen black hole background. In regards to solutions that model Black Holes, two opposing solution techniques are commonly utilized. Often researchers proceed with analysis via the use of spherical harmonics. For our purposes, we will be proceeding with the second solution technique. We will choose a toy model that is derived to correctly maintain certain features of the background spacetime geometry. The solution of this model will be a scalar field that will give us clues as to the geometry of spacetime at various moments in time.
We will use a toy model that is a simplified partial differential equation (PDE), formed by the d’Alembertian operator acting on a scalar field. A closed form solution exists to this toy model for both Schwarzschild and Reissner-Nordström background metrics in the form of a Green’s function. The corresponding numerical solutions show that the time domain output exhibits three phases: prompt, ringing, and exponential decay or power law decay when plotted on a log scale. These homogeneous results, found via Finite Difference schemes, are ubiquitous within the Black Hole Perturbation Theory community. That is, the overall structure of the homogeneous time domain output, has been repeatedly verified.
In this research, we aim to show the same three phases of time domain output exist when in the presence of an added “cubic like” nonlinearity. That is, when we observe the solution at particular values of our spatial coordinate, we would expect to see a prompt, a ringing, and a power law decay region on a log scale. This would allow us to prove the stability of scalar fields which have regained a small amount of nonlinearity.
Finite Difference Methods (FDMs) are historically used in the field of numerical relativity for computations. This is due to their inherent simplicity for straightforward problems. They are intuitive to understand and relatively easy to implement. However, they quickly can develop hurdles with the addition of any complications in the problem setup, be it a complex domain, unstructured grid, or moving boundaries. Researchers have approached solving problems within the confines of black hole perturbation theory by making simplifying assumptions. Otherwise, the problem is just too hard to solve with the currently established mathematical toolbox. While this has been sufficient up to this point, in order to advance the field of black hole perturbation theory, we must begin to relax some of these assumptions to gain a closer understanding of the full, true problem of interest.
In light of the drawbacks of FDMs, we will obtain our numerics via a Galerkin Finite Element scheme. Although Finite Elements Methods (FEMs) haven’t been used historically in this field, they hold great promise in General Relativity due to the obvious advantages that FEMs hold over FDMs. These include but are not limited to a priori error bounds, more accurate results, ease of handling complicated geometries, and better resolution in between grid points. They also have the ability to handle complex bound- ary conditions with ease. Even in the case of a common Neumann boundary condition, the FDM requires the creation of ghost nodes. These, as we will see, are easily handled by the FEM. Thus, we will cast our problem into a variational framework, discretize appropriately, and then implement this utilizing an existing FEM open source software package, resulting in an easily shareable code base.