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Florian Beyer

Publications and source records attributed to Florian Beyer.

At least 37 records · Page 2Linked to original sources
arXiv2017

Asymptotics of solutions of a hyperbolic formulation of the constraint equations

In this paper we consider the hyperbolic formulation of the constraints introduced by Rácz. Using the numerical framework recently developed by us we construct initial data sets which can be interpreted as nonlinear perturbations of Schwarzschild data in Kerr-Schild coordinates and investigate their asymptotics. Our results suggest that, unless one finds a way to exploit the freedom to pick the free part of the initial data in some suitable way, generic initial data sets obtained by this method may violate fundamental asymptotic conditions.

gr-qc↗
arXiv2017

The numerical initial boundary value problem for the generalized conformal field equations

In this paper we study a numerical implementation for the initial boundary value formulation for the generalized conformal field equations. We propose a formulation which is well suited for the study of the long-time behaviour of perturbed exact solutions such as a Schwarzschild or even a Kerr black hole. We describe the derivation of the implemented equations which we give in terms of the space-spinor formalism. We discuss the conformal Gauss gauge, and a slight generalization thereof which seems to be particularly useful in the presence of boundaries. We discuss the structure of the equations at the boundary and propose a method for imposing boundary conditions which allow the correct number of degrees of freedom to be freely specified while still preserving the constraints. We show that this implementation yields a numerically well-posed system by testing it on a simple case of gravitational perturbations of Minkowski space-time and subsequently with gravitational perturbations of Schwarzschild space-time.

gr-qc↗
arXiv2017

Criticality of inhomogeneous Nariai-like cosmological models

In this paper, we construct and study solutions of Einstein's equations in vacuum with a positive cosmological constant which can be considered as inhomogeneous generalizations of the Nariai cosmological model. Similar to this Nariai spacetime, our solutions are at the borderline between gravitational collapse and de-Sitter-like exponential expansion. Our studies focus in particular on the intriguing oscillatory dynamics which we discover. Our investigations are carried out both analytically (using heuristic mode analysis arguments) and numerically (using the numerical infrastructure recently introduced by us).

gr-qc↗
arXiv2016

A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges

We establish the existence of smooth vacuum Gowdy solutions, which are asymptotically velocity term dominated (AVTD) and have T3-spatial topology, in an infinite dimensional family of generalized wave gauges. These results show that the AVTD property, which is known to hold for solutions in areal coordinates, is stable to perturbations with respect to the gauge source functions. Our proof is based on an analysis of the singular initial value problem for the Einstein vacuum equations in the generalized wave gauge formalism, and provides a framework which we anticipate to be useful for more general spacetimes.

gr-qc↗
arXiv2016

Numerical solutions of Einstein's equations for cosmological spacetimes with spatial topology S3 and symmetry group U(1)

In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to Einstein's equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einstein's equations obtained in the generalized wave map gauge formalism combined with Geroch's symmetry reduction, and focus on cosmological spacetimes with spatial S3-topologies and symmetry groups U(1) or U(1) x U(1). We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained in [7].

gr-qc↗
arXiv2014

An exact smooth Gowdy-symmetric generalized Taub-NUT solution

In a recent paper (Beyer and Hennig, 2012 [9]), we have introduced a class of inhomogeneous cosmological models: the smooth Gowdy-symmetric generalized Taub-NUT solutions. Here we derive a three-parametric family of exact solutions within this class, which contains the two-parametric Taub solution as a special case. We also study properties of this solution. In particular, we show that for a special choice of the parameters, the spacetime contains a curvature singularity with directional behaviour that can be interpreted as a "true spike" in analogy to previously known Gowdy symmetric solutions with spatial T3-topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain "false spikes".

gr-qc↗
arXiv2014

Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics

Many applications in science call for the numerical simulation of systems on manifolds with spherical topology. Through use of integer spin weighted spherical harmonics we present a method which allows for the implementation of arbitrary tensorial evolution equations. Our method combines two numerical techniques that were originally developed with different applications in mind. The first is Huffenberger and Wandelt's spectral decomposition algorithm to perform the mapping from physical to spectral space. The second is the application of Luscombe and Luban's method, to convert numerically divergent linear recursions into stable nonlinear recursions, to the calculation of reduced Wigner d-functions. We give a detailed discussion of the theory and numerical implementation of our algorithm. The properties of our method are investigated by solving the scalar and vectorial advection equation on the sphere, as well as the 2+1 Maxwell equations on a deformed sphere.

physics.comp-ph↗
arXiv2013

Graceful exit from inflation for minimally coupled Bianchi A scalar field models

We consider the dynamics of Bianchi A scalar field models which undergo inflation. The main question is under which conditions does inflation come to an end and is succeeded by a decelerated epoch. This so-called "graceful exit" from inflation is an important ingredient in the standard model of cosmology, but is, at this stage, only understood for restricted classes of solutions. We present new results obtained by a combination of analytical and numerical techniques.

gr-qc↗
arXiv2013

The Spin-2 Equation on Minkowski Background

The linearised general conformal field equations in their first and second order form are used to study the behaviour of the spin-2 zero-rest-mass equation on Minkowski background in the vicinity of space-like infinity.

gr-qc↗
arXiv2013

Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes

We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T2-symmetric solutions to the vacuum Einstein equations, which exhibit AVTD (asymptotically velocity term dominated) behavior in the neighborhood of their singularities and are polarized or half-polarized.

gr-qc↗
arXiv2013

Linearized gravitational waves near space-like and null infinity

Linear perturbations on Minkowski space are used to probe numerically the remote region of an asymptotically flat space-time close to spatial infinity. The study is undertaken within the framework of Friedrich's conformal field equations and the corresponding conformal representation of spatial infinity as a cylinder. The system under consideration is the (linear) zero-rest-mass equation for a spin-2 field. The spherical symmetry of the underlying background is used to decompose the field into separate non-interacting multipoles. It is demonstrated that it is possible to reach null-infinity from initial data on an asymptotically Euclidean hyper-surface and that the physically important radiation field can be extracted accurately on $\scri^+$.

gr-qc↗
arXiv2012

Numerical space-times near space-like and null infinity. The spin-2 system on Minkowski space

In this paper we demonstrate for the first time that it is possible to solve numerically the Cauchy problem for the linearisation of the general conformal field equations near spacelike infinity, which is only well-defined in Friedrich's cylinder picture. We have restricted ourselves here to the "core" of the equations - the spin-2 system - propagating on Minkowski space. We compute the numerical solutions for various classes of initial data, do convergence tests and also compare to exact solutions. We also choose initial data which intentionally violate the smoothness conditions and then check the analytical predictions about singularities. This paper is the first step in a long-term investigation of the use of conformal methods in numerical relativity.

gr-qc↗
arXiv2012

Quasi-linear symmetric hyperbolic Fuchsian systems in several space dimensions

We establish existence and uniqueness results for the singular initial value problem associated with a class of quasilinear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.

gr-qc↗
arXiv2012

Smooth Gowdy symmetric generalized Taub-NUT solutions

We study a class of S3 Gowdy vacuum models with a regular past Cauchy horizon which we call smooth Gowdy symmetric generalized Taub-NUT solutions. In particular, we prove existence of such solutions by formulating a singular initial value problem with asymptotic data on the past Cauchy horizon. The result of our investigations is that a future Cauchy horizon exists for generic asymptotic data. Moreover, we derive an explicit expression for the metric on the future Cauchy horizon in terms of the asymptotic data on the past horizon. This complements earlier results about S2xS1 Gowdy models.

gr-qc↗
arXiv2011

Second-order hyperbolic Fuchsian systems. Asymptotic behavior of geodesics in Gowdy spacetimes

Recent work by the authors led to the development of a mathematical theory dealing with `second--order hyperbolic Fuchsian systems', as we call them. In the present paper, we adopt a physical standpoint and discuss the implications of this theory which provides one with a new tool to tackle the Einstein equations of general relativity (under certain symmetry assumptions). Specifically, we formulate the `Fuchsian singular initial value problem' and apply our general analysis to the broad class of vacuum Gowdy spacetimes with spatial toroidal topology. Our main focus is on providing a detailed description of the asymptotic geometry near the initial singularity of these inhomogeneous cosmological spacetimes and, especially, analyzing the asymptotic behavior of timelike geodesics ---which represent the trajectories of freely falling observers --- and null geodesics. In particular, we numerically construct Gowdy spacetimes which contain a black hole--like region together with a flat Minkowski--like region. By using the Fuchsian technique, we investigate the effect of the gravitational interaction between these two regions and we study the unexpected behavior of geodesic trajectories within the intermediate part of the spacetime limited by these two regions.

gr-qc↗
arXiv2011

Second-order hyperbolic Fuchsian systems and applications

We introduce a new class of singular partial differential equations, referred to as the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First, we establish a general existence theory of solutions with asymptotic behavior prescribed on the singularity, which relies on a new approximation scheme, suitable also for numerical purposes. Second, this theory is applied to the (vacuum) Einstein equations for Gowdy spacetimes, and allows us to recover, by more direct arguments, well-posedness results established earlier by Rendall and collaborators. Another main contribution in this paper is the proposed approximation scheme, which we refer to as the Fuchsian numerical algorithm and is shown to provide highly accurate numerical approximations to the singular initial value problem. For the class of Gowdy spacetimes, the numerical experiments presented here show the interest and efficiency of the proposed method and demonstrate the existence of a class of Gowdy spacetimes containing a smooth, incomplete, and non-compact Cauchy horizon.

gr-qc↗
arXiv2011

Second-order hyperbolic Fuchsian systems. General theory

We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a class of equations in which hyperbolicity is not assumed and we construct asymptotic solutions of arbitrary order. Second, for the proposed class of second-order hyperbolic Fuchsian systems, we establish the existence of solutions with prescribed asymptotic behavior on the singularity. Our proof is based on a new scheme which is also suitable to design numerical approximations. Furthermore, as shown in a follow-up paper, the second-order Fuchsian framework is appropriate to handle Einstein's field equations for Gowdy symmetric spacetimes and allows us to recover (and slightly generalize) earlier results by Rendall and collaborators, while providing a direct approach leading to accurate numerical solutions. The proposed framework is also robust enough to encompass matter models arising in general relativity.

gr-qc↗