arXiv subjects

Annegret Y. Burtscher

Publications and source records attributed to Annegret Y. Burtscher.

6 recordsLinked to original sources

arXiv2019

Initial data and black holes for matter models

To observe the dynamic formation of black holes in general relativity, one essentially needs to prove that closed trapped surfaces form during evolution from initial data that do not already contain trapped surfaces. We discuss the recent development of the construction of such admissible initial data for matter models. In addition, we extend known results for the Einstein equations coupled to perfect fluids in spherical symmetry and with linear equation of state to unbounded domains. Polytropic equations of state and regularity issues with the direct application of the singularity theorems in general relativity are discussed briefly.

math.AP↗

arXiv2019

On the asymptotic behavior of static perfect fluids

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear and polytropic-type equations of state with index n>5. In order to capture the asymptotic behavior we introduce a notion of scaled quasi-asymptotic flatness, which encodes a form of asymptotic conicality. In particular, these spacetimes are asymptotically simple.

gr-qc↗

arXiv2017

Hyperboloidal evolution and global dynamics for the focusing cubic wave equation

The focusing cubic wave equation in three spatial dimensions has the explicit solution $\sqrt{2}/t$. We study the stability of the blowup described by this solution as $t \to 0$ without symmetry restrictions on the data. Via the conformal invariance of the equation we obtain a companion result for the stability of slow decay in the framework of a hyperboloidal initial value formulation. More precisely, we identify a codimension-1 Lipschitz manifold of initial data leading to solutions which converge to Lorentz boosts of $\sqrt{2}/t$ as $t\to\infty$. These global solutions thus exhibit a slow nondispersive decay, in contrast to small data evolutions.

math.AP↗

arXiv2015

Length structures on manifolds with continuous Riemannian metrics

It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as a differentiable exponential map etc. are available nor (generalized) curvature bounds are imposed. In this paper we advance low-regularity Riemannian geometry by investigating general length structures on manifolds that are equipped with Riemannian metrics of low regularity. We generalize the length structure by proving that the class of absolutely continuous curves induces the standard metric space structure. The main result states that the arc-length of absolutely continuous curves is the same as the length induced by the metric. For the proof we use techniques from the analysis of metric spaces and employ specific smooth approximations of continuous Riemannian metrics. We thus show that when dealing with lengths of curves, the metric approach for low-regularity Riemannnian manifolds is still compatible with standard definitions and can successfully fill in for lack of differentiability.

math.DG↗

arXiv2014

The formation of trapped surfaces in spherically-symmetric Einstein-Euler spacetimes with bounded variation

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein-Euler equations of general relativity. We formulate the initial value problem in Eddington-Finkelstein coordinates and prescribe spherically symmetric data on a characteristic initial hypersurface. We introduce here a broad class of initial data which contain no trapped surfaces, and we then prove that their Cauchy development contains trapped surfaces. We therefore establish the formation of trapped surfaces in weak solutions to the Einstein equations. This result generalizes a theorem by Christodoulou for regular vacuum spacetimes (but without symmetry restriction). Our method of proof relies on a generalization of the "random choice" method for nonlinear hyperbolic systems and on a detailed analysis of the nonlinear coupling between the Einstein equations and the relativistic Euler equations in spherical symmetry.

gr-qc↗

arXiv2011

New energy inequalities for tensorial wave equations on spacetimes that satisfy a one-sided bound

We consider several tensorial wave equations, specifically the equations of Maxwell, Yang-Mills, and Weyl fields, posed on a curved spacetime, and we establish new energy inequalities under certain one-sided geometric conditions. Our conditions restrict the lapse function and deformation tensor of the spacetime foliation, and turn out to be a one-sided and integral generalization of conditions recently proposed by Klainerman and Rodnianski as providing a continuation criterion for Einstein's field equations of general relativity. As we observe it here for the first time, one-sided conditions are sufficient to derive energy inequalities for certain tensorial equations, provided one takes advantage of some algebraic properties enjoyed by the natural energy functionals associated with the equations under consideration. Our method especially applies to the Bel-Robinson energy for Weyl fields, and our inequalities control the growth of the energy in a uniform way, with implied constants depending on the one-sided geometric bounds, only.

gr-qc↗