Florian Beyer

Introduction to dynamical horizons in numerical relativity

This paper presents a quasilocal method of studying the physics of dynamical black holes in numerical simulations. This is done within the dynamical horizon framework, which extends the earlier work on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds of marginal surfaces and study their time evolution. An important ingredient is the calculation of the signature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calculation of the black hole mass and angular momentum, which were previously defined for axisymmetric isolated horizons to dynamical situations. (iii) We calculate the source multipole moments of the black hole which can be used to verify that the black hole settles down to a Kerr solution. (iv) We also study the fluxes of energy crossing the horizon, which describes how a black hole grows as it accretes matter and/or radiation. We describe our numerical implementation of these concepts and apply them to three specific test cases, namely, the axisymmetric head-on collision of two black holes, the axisymmetric collapse of a neutron star, and a nonaxisymmetric black hole collision with nonzero initial orbital angular momentum.

Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T 2 -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 asymptotically velocity term dominated behavior in the neighborhood of their singularities and are polarized or half-polarized.

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.

Investigations of solutions of Einstein's field equations close to λ-Taub–NUT

We present investigations of a class of solutions of Einsteins field equations close to the family of λ-Taub–NUT spacetimes. The studies are done using a numerical code introduced by the author elsewhere. One of the main technical complications is due to the §-topology of the Cauchy surfaces. Complementing these numerical results with heuristic arguments, we are able to yield some first insights into the strong cosmic censorship issue and the conjectures by Belinskii, Khalatnikov and Lifschitz in this class of spacetimes. In particular, the current investigations suggest that strong cosmic censorship holds in this class. We further identify open issues in our current approach and point to future research projects.

Smooth Gowdy-symmetric generalized Taub–NUT solutions

We study a class of -Gowdy vacuum models with a regular past Cauchy horizon which we call smooth Gowdy-symmetric generalized Taub–NUT solutions. In particular, we prove the existence of such solutions by formulating a singular initial value problem with asymptotic data on the past Cauchy horizon. We prove that also a future Cauchy horizon exists for generic asymptotic data, and 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 -Gowdy models.

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 \({\mathcal{I}}^{+}\).

An exact smooth Gowdy-symmetric generalized Taub–NUT solution

In a recent paper (Beyer and Hennig 2012 Class. Quantum Grav. 29 245017), 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 -topology. For other parameter choices, the maximal globally hyperbolic region is singularity-free, but may contain ?false spikes?.

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 has so far been known to hold for solutions in areal coordinates only, is stable to perturbations of the coordinate systems. 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.

Self–gravitating fluid flows with Gowdy symmetry near cosmological singularities

ABSTRACT We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus T3 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.

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 the 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 Wandelts spectral decomposition algorithm to perform the mapping from physical to spectral space. The second is the application of Luscombe and Lubans 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.