arXiv subjects

Florian Beyer

Publications and source records attributed to Florian Beyer.

At least 19 recordsLinked to original sources

arXiv2026

Big bang stability and isotropisation for the Einstein-scalar field equations in the ekpyrotic regime

It has been shown that, in spacetime dimensions $n\geq 3$, that the Kasner-scalar field solutions to the Einstein-scalar fields equations with potential $V_0 e^{-s \phi}$, where $s s_c$ and $V_0<0$. Such scalar field potentials are known in the literature as \textit{ekpyrotic}. In particular, we prove that the FLRW solutions to the Einstein-scalar field equations are nonlinearly stable to the past and terminate at a quiescent, crushing AVTD big bang singularity. A distinguishing property of these perturbed spacetimes is that they isotropise towards the big bang.

gr-qc↗

arXiv2026

The extremely-tilted fluid regime near asymptotically Kasner big bang singularities

In this paper, we solve the relativistic Euler equations with a linear barotropic equation of state on a large class of background spacetimes with Kasner big bang asymptotics. Building on previous work in the asymptotically non-tilted regime, which applies when the speed of sound of the fluid is large in comparison to the Kasner exponents, we now consider the asymptotically extremely-tilted regime when the speed of sound is small. We solve the Cauchy problem for the Euler equations towards the big bang singularity and prove, without any symmetry assumptions or smallness conditions on the Cauchy data, that the solutions exist globally in time provided the mean curvature of the initial hypersurface is sufficiently large. Finally, we prove that the solutions exhibit the asymptotics expected from standard heuristic arguments in this regime; in particular, fluid particles are driven towards the speed of light in the direction of the largest Kasner exponent as the big bang singularity is approached.

gr-qc↗

arXiv2024

Past instability of FLRW solutions of the Einstein-Euler-scalar field equations for linear equations of state $p=K\rho$ with $0 \leq K<1/3$

Using numerical methods, we examine, under a Gowdy symmetry assumption, the dynamics of nonlinearly perturbed FLRW fluid solutions of the Einstein-Euler-scalar field equations in the contracting direction for linear equations of state $p = K\rho$ and sound speeds $0\leq K<1/3$. This article builds upon the numerical work from \cite{BMO:2023} in which perturbations of FLRW solutions to the Einstein-Euler equations with positive cosmological constant in the expanding time direction were studied. The numerical results presented here confirm that the instabilities observed in \cite{BMO:2023,MarshallOliynyk:2022} for $1/3<K<1$, first conjectured to occur in the expanding direction by Rendall in \cite{Rendall:2004}, are also present in the contracting direction over the complementary parameter range $0\leq K<1/3$. Our numerical solutions show that the fractional density gradient of the nonlinear perturbations develop steep gradients near a finite number of spatial points and become unbounded towards the big bang. This behaviour, and in particular the characteristic profile of the fractional density gradient near the big bang, is strikingly similar to what was observed in the expanding direction near timelike infinity in the article \cite{BMO:2023}.

gr-qc↗

arXiv2024

Localized big bang stability for the Einstein-scalar field equations

We prove the nonlinear stability in the contracting direction of Friedmann-Lemaître-Robertson-Walker (FLRW) solutions to the Einstein-scalar field equations in $n\geq 3$ spacetime dimensions that are defined on spacetime manifolds of the form $(0,t_0]\times \mathbb{T}^{n-1}$, $t_0>0$. Stability is established under the assumption that the initial data is \textit{synchronized}, which means that on the initial hypersurface $Σ= \{t_0\}\times \mathbb{T}^{n-1}$ the scalar field $τ= \exp\bigl(\sqrt{\frac{2(n-2)}{n-1}}ϕ\bigr) $ is constant, that is, $Σ=τ^{-1}(\{t_0\})$. As we show that all initial data sets that are sufficiently close to FRLW ones can be evolved via the Einstein-scalar field equation into new initial data sets that are \textit{synchronized}, no generality is lost by this assumption. By using $τ$ as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form $M = \bigcup_{t\in (0,t_0]}τ^{-1}(\{t\})\cong (0,t_0]\times \mathbb{T}^{n-1}$, the perturbed FLRW solutions are asymptotically pointwise Kasner as $τ\searrow 0$, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at $τ=0$. An important aspect of our past stability proof is that we use a hyperbolic gauge reduction of the Einstein-scalar field equations. As a consequence, all of the estimates used in the stability proof can be localized and we employ this property to establish a corresponding localized past stability result for the FLRW solutions.

gr-qc↗

arXiv2023

Past stability of FLRW solutions to the Einstein-Euler-scalar field equations and their big bang singularites

We establish, in spacetime dimensions $n\geq 3$, the nonlinear stability in the contracting direction of Friedmann-Lema\^itre-Robertson-Walker (FLRW) solutions to the Einstein-Euler-scalar field equations with linear equations of state $P=c_s^2 \rho$ for sounds speeds $c_s$ satisfying $1/(n-1)<c_s^2 < 1$. We further show that nonlinear perturbations of the FLRW solutions are asymptotically pointwise Kasner and terminate in crushing, asymptotically velocity term dominated (AVTD) big bang singularities characterised by curvature blow-up.

gr-qc↗

arXiv2023

Future instability of FLRW fluid solutions for linear equations of state $p=Kρ$ with $1/3 <K<1$

Using numerical methods, we examine the dynamics of nonlinear perturbations in the expanding time direction, under a Gowdy symmetry assumption, of FLRW fluid solutions to the Einstein-Euler equations with a positive cosmological constant $Λ>0$ and a linear equation of state $p = Kρ$ for the parameter values $1/3<K<1$. This paper builds upon the numerical work in \cite{Marshalloliynyk:2022} in which the simpler case of a fluid on a fixed FLRW background spacetime was studied. The numerical results presented here confirm that the instabilities observed in \cite{Marshalloliynyk:2022} are also present when coupling to gravity is included as was previously conjectured in \cite{Rendall:2004,Speck:2013}. In particular, for the full parameter range $1/3 < K <1$, we find that the density contrast of the nonlinear perturbations develop steep gradients near a finite number of spatial points and becomes unbounded there at future timelike infinity. This instability is of particular interest since it is not consistent with the standard picture for late time expansion in cosmology.

gr-qc↗

arXiv2022

Asymptotically hyperboloidal initial data sets from a parabolic-hyperbolic formulation of the Einstein vacuum constraints

In this paper, we continue our investigations of Rácz's parabolic-hyperbolic formulation of the Einstein vacuum constraints. Our previous studies of the asymptotically flat setting provided strong evidence for unstable asymptotics which we were able to resolve by introducing a certain modification of Rácz's parabolic-hyperbolic formulation. The primary focus of the present paper here is the asymptotically hyperboloidal setting. We provide evidence through a mixture of numerical and analytical methods that the asymptotics of the solutions of Rácz's parabolic-hyperbolic formulation is stable, and, in particular, no modifications are necessary to obtain solutions that are asymptotically hyperboloidal.

gr-qc↗

arXiv2022

Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes

We prove stability of the family of Kasner solutions within the class of polarized $T^2$-symmetric solutions of the vacuum Einstein equations in the contracting time direction with respect to an areal time foliation. All Kasner solutions for which the asymptotic velocity parameter $K$ satisfies $|K-1|>2$ are non-linearly stable, and all sufficiently small perturbations exhibit asymptotically velocity term dominated (AVTD) behavior and blow-up of the Kretschmann scalar.

gr-qc↗

arXiv2021

Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant

We prove the nonlinear stability of the asymptotic behavior of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarised $T^2$-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant $Λ$. This stability result generalizes the results proven in [3], which focus on the $Λ=0$ case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for $Λ=0$, the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarised $T^2$-symmetric vacuum solutions than those considered in [3] and [26]. Our results establish that the areal time coordinate takes all values in $(0, T_0]$ for some $T_0 > 0$, for certain families of polarised $T^2$-symmetric solutions with cosmological constant.

gr-qc↗

arXiv2020

Relativistic perfect fluids near Kasner singularities

We establish the existence of a stable family of solutions to the Euler equations on Kasner backgrounds near the singularity with the full expected asymptotic data degrees of freedom and no symmetry or isotropy restrictions. Existence is achieved through transforming the Euler equations into the form of a symmetric hyperbolic Fuchsian system followed by an application of a new existence theory for the singular initial value problem. Stability is shown to follow from the existence theory for the (regular) global initial value problem for Fuchsian systems that was developed in arXiv:1907.04071. In fact, for each solution in the family, we prove the existence of an open set of nearby solutions with the same qualitative asymptotics and show that any such perturbed solution agrees again with another solution of the singular initial value problem. All our results hold in the regime where the speed of sound of the fluid is large in comparison to all Kasner exponents. This is interpreted as the regime of stable fluid asymptotics near Kasner big bang singularities.

math.AP↗

arXiv2020

The Fuchsian approach to global existence for hyperbolic equations

We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions towards the singular time under a small initial data assumption. We then apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.

math.AP↗

arXiv2020

Explorations of the infinite regions of space-time

An important concept in Physics is the notion of an isolated system. It is used in many different areas to describe the properties of a physical system which has been isolated from its environment. The interaction with the `outside' is then usually reduced to a scattering process, in which incoming radiation interacts with the system, which in turn emits outgoing radiation. In Einstein's theory of gravitation isolated systems are modeled as asymptotically flat space-times. They provide the appropriate paradigm for the study of gravitational waves and their interaction with a material system or even only with themselves. In view of the emerging era of gravitational wave astronomy, in which gravitational wave signals from many different astrophysical sources are detected and interpreted, it is necessary to have a foundation for the theoretical and numerical treatments of these signals. Furthermore, from a purely mathematical point of view, it is important to have a full understanding of the implications of imposing the condition of asymptotic flatness onto solutions of the Einstein equations. While it is known that there exists a large class of asymptotically flat solutions of Einstein's equations, it is not known what the necessary and sufficient conditions at infinity are that have to be imposed on initial data so that they evolve into regular asymptotically flat space-times. The crux lies in the region near space-like infinity $i^0$ where incoming and outgoing radiation `meet'. In this paper we review the current knowledge and some of the analytical and numerical work that has gone into the attempt to understand the structure of asymptotically flat space-times near space-like and null-infinity.

gr-qc↗

arXiv2020

A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes

We consider a class of Fuchsian equations that, for instance, describes the evolution of compressible fluid flows on a cosmological spacetime. Using the method of lines, we introduce a numerical algorithm for the singular initial value problem when data are imposed on the cosmological singularity and the evolution is performed from the singularity hypersurface. We approximate the singular Cauchy problem of Fuchsian type by a sequence of regular Cauchy problems, which we next discretize by pseudo-spectral and Runge-Kutta techniques. Our main contribution is a detailed analysis of the numerical error which has two distinct sources, and our main proposal here is to keep in balance the errors arising at the continuum and at the discrete levels of approximation. We present numerical experiments which strongly support our theoretical conclusions. This strategy is finally applied to applied to compressible fluid flows evolving on a Kasner spacetime, and we numerically demonstrate the nonlinear stability of such flows, at least in the so-called sub-critical regime identified earlier by the authors.

gr-qc↗

arXiv2020

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

In this paper we continue earlier investigations of evolutionary formulations of the Einstein vacuum constraint equations originally introduced by Rácz. Motivated by the strong evidence from these works that the resulting vacuum initial data sets are generically not asymptotically flat we analyse the asymptotics of the solutions of a modified formulation by a combination of analytical and numerical techniques. We conclude that the vacuum initial data sets generated with this new formulation are generically asymptotically flat.

gr-qc↗

arXiv2019

Contracting asymptotics of the linearized lapse-scalar field sub-system of the Einstein-scalar field equations

We prove an asymptotic stability result for a linear coupled hyperbolic-elliptic system on a large class of singular background spacetimes in CMC gauge on the n-torus. At each spatial point these background spacetimes are perturbations of Kasner-like solutions of the Einstein-scalar field equations which are not required to be close to the homogeneous and isotropic case. We establish the existence of a homeomorphism between Cauchy data for this system and a set of functions naturally associated with the asymptotics in the contracting direction, which we refer to as asymptotic data. This yields a complete characterization of the degrees of freedom of all solutions of this system in terms of their asymptotics. Spatial derivative terms can in general not be fully neglected which yields a clarification of the notion of asymptotic velocity term dominance (AVTD).

gr-qc↗

arXiv2019

Numerical construction of initial data sets of binary black hole type using a parabolic-hyperbolic formulation of the vacuum constraint equations

In this paper we investigate the parabolic-hyperbolic formulation of the vacuum constraint equations introduced by R{á}cz with a view to constructing multiple black hole initial data sets without spin. In order to respect the natural properties of this configuration, we foliate the spatial domain with 2-spheres. It is then a consequence of these equations that they must be solved as an initial value problem evolving outwards towards spacelike infinity. Choosing the free data and the "strong field boundary conditions" for these equations in a way which mimics asymptotically flat and asymptotically spherical binary black hole initial data sets, our focus in this paper is on the analysis of the asymptotics of the solutions. In agreement with our earlier results, our combination of analytical and numerical tools reveals that these solutions are in general not asymptotically flat, but have a cone geometry instead. In order to remedy this and approximate asymptotically Euclidean data sets, we then propose and test an iterative numerical scheme.

gr-qc↗

arXiv2018

Analysis of a Bianchi-like equation satisfied by the Mars-Simon tensor

The Mars-Simon tensor (MST), which e.g. plays a crucial role to provide gauge invariant characterizations of the Kerr-NUT-(A)(dS) family, satisfies a Bianchi-like equation. In this paper we analyze this equation in close analogy to the Bianchi equation, in particular it will be shown that the constraints are preserved supposing that a generalized Buchdahl condition holds. This permits the systematic construction of solutions to this equation in terms of a well-posed Cauchy problem. A particular emphasis lies on the asymptotic Cauchy problem, where data are prescribed on a spacelike scri (i.e. for $Λ>0$). In contrast to the Bianchi equation, the MST equation is of Fuchsian type at scri, for which existence and uniqueness results are derived.

gr-qc↗

arXiv2017

Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities

We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus $T^3$ and, in this class, we solve the singular initial value problem for the Einstein-Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein-Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.

gr-qc↗