arXiv subjects

Jörg Frauendiener

Publications and source records attributed to Jörg Frauendiener.

At least 19 recordsLinked to original sources

arXiv2026

Fully global numerical evolutions of the linearised spin-2 equation

We study the linearised spin-2 field on conformally compactified Minkowski spacetime. Initial data are prescribed on past null infinity $\mathscr{I}^-$, and the system is evolved through the cylinder at spatial infinity to future null infinity $\mathscr{I}^+$. To avoid the development of logarithmic singularities and thereby preserve smooth peeling behaviour, we derive explicit conditions on the freely specifiable ingoing gravitational radiation $ψ_0$ at $\mathscr{I}^-$ that ensure regular initial data. For compactly supported data, we obtain necessary and sufficient integral conditions guaranteeing regularity and compact support of the full system. In the simplest cases, we construct examples of exact initial data, while for the general case, we present a numerical procedure that computes the remaining $ψ_k$ to obtain a full initial data set. We perform numerical evolutions using two gauges corresponding to compactification of either a neighbourhood of spatial infinity or the entire Minkowski spacetime, and demonstrate stable propagation of the system from $\mathscr{I}^-$ to $\mathscr{I}^+$. These results provide a step towards a framework for fully global simulations of gravitational radiation and clarify the role of initial data on $\mathscr{I}^-$ in determining asymptotic structure.

gr-qc↗

arXiv2026

A note on the conformal geometry of the 2-sphere

We discuss the geometry of the two-dimensional sphere. Our perspective focuses on the set of conformal factors that rescale a given metric to a round metric. We find that there exists a unique symmetric and trace-free tensor on the sphere which complements the Ricci tensor to a divergence-free symmetric tensor. Using this tensor, we derive a linear second-order PDE whose solution space carries a natural Minkowski structure. We show that there are in fact infinitely many such spaces, all related by isometries which can be thought of as being parametrised by ``4-vectors''. Finally, we show that every 2-sphere can be isometrically embedded in a Minkowski space in a canonical way.

gr-qc↗

arXiv2025

Fully non-linear gravitational wave simulations from past to future null-infinity

We present the first numerical simulations of asymptotically flat space-times whose computational domain includes past and future null-infinity. As an application, we explore the scattering of a gravitational wave in a black hole space-time. The study is conducted in the fully non-linear regime using an initial boundary value problem formulation of Friedrich's generalised conformal field equations. We calculate the Bondi energy, Bondi news and gravitational information at both past and future null-infinity and discuss their relationships.

gr-qc↗

arXiv2025

A numerical framework for studying asymptotic quantities

In this contribution we present an overview of our work on the numerical simulation of the perturbation of a black hole space-time by incoming gravitational waves. The formulation we use is based on Friedrich's general conformal equations which have the unique property that they allow access to the asymptotic region of an asymptotically regular space-time. In our approach we set up an initial boundary value problem on a finite boundary, which cleanly separates the initial conditions, a static black hole, from the perturbation, an incoming gravitational wave specified by a spin-2 function on the time-like boundary. The main advantage of this approach is that the finite boundary expands fast enough to reach null-infinity where the asymptotic properties can be studied. This provides, for the first time, a direct relationship between finite initial and boundary data and asymptotic quantities within one simulation. We discuss the possibilities and limitations of this approach.

gr-qc↗

arXiv2024

Quadrupole Moment of a Magnetically Confined Mountain on an Accreting Neutron Star in General Relativity

General relativistic corrections are calculated for the quadrupole moment of a magnetically confined mountain on an accreting neutron star. The hydromagnetic structure of the mountain satisfies the general relativistic Grad-Shafranov equation supplemented by the flux-freezing condition of ideal magnetohydrodynamics, as in previous calculations of the magnetic dipole moment. It is found that the ellipticity and hence the gravitational wave strain are up to $12\%$ greater than in the analogous Newtonian system. The direct contribution of the magnetic field to the nonaxisymmetric component of the stress-energy tensor is shown to be negligible in accreting systems such as low-mass X-ray binaries.

astro-ph.HE↗

arXiv2023

Magnetically Confined Mountains on Accreting Neutron Stars in General Relativity

The general relativistic formulation of the problem of magnetically confined mountains on neutron stars is presented, and the resulting equations are solved numerically, generalising previous Newtonian calculations. The hydromagnetic structure of the accreted matter and the subsequent magnetic burial of the star's magnetic dipole moment are computed. Overall, it is observed that relativistic corrections reduce the hydromagnetic deformation associated with the mountain. The magnetic field lines are curved more gently than in previous calculations, and the screening of the dipole moment is reduced. Quantitatively, it is found that the dimensionless dipole moment ($m_{\rm d}$) depends on the accreted mass ($M_{\rm a}$) as $m_{\rm d} = -3.2\times10^{3}M_{\rm a}/M_\odot + 1.0$, implying approximately three times less screening compared to the Newtonian theory. Additionally, the characteristic scale height of the mountain, governing the gradients of quantities like pressure, density, and magnetic field strength, reduces by approximately $40\%$ for an isothermal equation of state.

astro-ph.HE↗

arXiv2023

The non-linear perturbation of a black hole by gravitational waves. III. Newman-Penrose constants

In this paper we continue our study of the non-linear response of a Schwarzschild black hole to an ingoing gravitational wave by computing the Newman-Penrose (NP) constants. The NP constants are five complex, supertranslation-invariant quantities defined on null infinity $\mathscr{I}^+$ and although put forward in the 60's, they have never been computed in a non-stationary setting. We accomplish this through a numerical implementation of Friedrich's generalized conformal field equations whose semi-global evolution yields direct access to $\mathscr{I}+$. Generalizations of the NP constants' integral expressions are made to allow their computation in a more general gauge that better suits the output of a numerical evolution. Canonical methods of fixing inherent degrees of freedom in their definitions are discussed. The NP constants are then computed for a variety of different ingoing wave profiles in axisymmetry, and then with no symmetry assumptions in 3+1 for which all five are non-zero.

gr-qc↗

arXiv2022

The non-linear perturbation of a black hole by gravitational waves. II. Quasinormal modes and the compactification problem

Recently, Friedrich's Generalized Conformal Field Equations (GCFE) have been implemented numerically and global quantities such as the Bondi energy and the Bondi-Sachs mass loss have been successfully calculated directly on null-infinity. Although being an attractive option for studying global quantities by way of local differential geometrical methods, how viable are the GCFE for study of quantities arising in the physical space-time? In particular, how long can the evolution track phenomena that need a constant proper physical timestep to be accurately resolved? We address this question by studying the curvature oscillations induced on the Schwarzschild space-time by a non-linear gravitational perturbation. For small enough amplitudes, these are the well approximated by the linear quasinormal modes, where each mode rings at a frequency determined solely by the Schwarzschild mass. We find that the GCFE can indeed resolve these oscillations, which quickly approach the linear regime, but only for a short time before the compactification becomes ``too fast'' to handle numerically.

gr-qc↗

arXiv2022

New Horizons for Fundamental Physics with LISA

The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of GWs can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas.

gr-qc↗

arXiv2021

A new look at the Bondi-Sachs energy-momentum

How does one compute the Bondi mass on an arbitrary cut of null infinity $\scri$ when it is not presented in a Bondi system? What then is the correct definition of the mass aspect? How does one normalise an asymptotic translation computed on a cut which is not equipped with the unit-sphere metric? These are questions which need to be answered if one wants to calculate the Bondi-Sachs energy-momentum for a space-time which has been determined numerically. Under such conditions there is not much control over the presentation of $\scri$ so that most of the available formulations of the Bondi energy-momentum simply do not apply. The purpose of this article is to provide the necessary background for a manifestly conformally invariant and gauge independent formulation of the Bondi energy-momentum. To this end we introduce a conformally invariant version of the GHP formalism to rephrase all the well-known formulae. This leads us to natural definitions for the space of asymptotic translations with its Lorentzian metric, for the Bondi news and the mass-aspect. A major role in these developments is played by the "co-curvature", a naturally appearing quantity closely related to the Gauß curvature on a cut of~$\scri$.

gr-qc↗

arXiv2021

The non-linear perturbation of a black hole by gravitational waves. I. The Bondi-Sachs mass loss

The excitation of a black hole by infalling matter or radiation has been studied for a long time, mostly in linear perturbation theory. In this paper we study numerically the response of a Schwarzschild black hole to an incoming gravitational wave pulse. We present a numerically well-posed initial boundary value problem for the generalized conformal field equations in which a wave profile for the ingoing wave is specified at an outer time-like boundary which then hits an initially static and spherically symmetric black hole. The non-linear interaction of the black hole with the gravitational wave leads to scattered radiation moving back out. The clean separation between initial state and incoming radiation makes this setup ideal to study scattering problems. The use of the conformal field equations allows us to trace the response of the black hole to null infinity where we can read off the scattered gravitational waves and compute the Bondi-Sachs mass and the gravitational flux through $\mathscr{I}$. In this way we check the Bondi-Sachs mass loss formula directly on null infinity. We also comment on comparisons with quasinormal modes.

gr-qc↗

arXiv2021

Can gravitational waves halt the expansion of the universe?

We numerically investigate the propagation of plane gravitational waves in the form of an initial boundary value problem with de Sitter initial data. The full non-linear Einstein equations with positive cosmological constant $λ$ are written in the Friedrich-Nagy gauge which yields a wellposed system. The propagation of a single wave and the collision of two with colinear polarization are studied and contrasted with their Minkowskian analogues. Unlike with $λ=0$, critical behaviours are found with $λ>0$ and are based on the relationship between the wave profile and $λ$. We find that choosing boundary data close to one of these bifurcations results in a "false" steady state which violates the constraints. Simulations containing (approximate) impulsive wave profiles are run and general features are discussed. Analytic results of Woodard and Tsamis, which describe how gravitational waves could affect an expansion rate at an initial instance of time, are explored and generalized to the entire space-time. Finally we put forward boundary conditions that, at least locally, slow down the expansion considerably for a time.

gr-qc↗

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

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

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↗

arXiv2019

COFFEE -- An MPI-parallelized Python package for the numerical evolution of differential equations

COFFEE (ConFormal Field Equation Evolver) is a Python package primarily developed to numerically evolve systems of partial differential equations over time using the method of lines. It includes a variety of time integrators and finite differencing stencils with the summation-by-parts property, as well as pseudo-spectral functionality for angular derivatives of spin-weighted functions. Some additional capabilities include being MPI-parallelisable on a variety of different geometries, HDF data output and post processing scripts to visualize data, and an actions class that allows users to create code for analysis after each timestep.

cs.MS↗

arXiv2019

Gravitational waves and the Sagnac effect

Light propagating in opposite directions around the same loop in general shows a relative phase shift when recombined. This phenomenon is known as the Sagnac effect after Georges Sagnac who, in 1913, demonstrated with an interferometer on a rotating table that the phase shift depended on the angular velocity of the table. In previous work we have given a very general formula for the Sagnac effect, valid in full general relativity. The relativistic effect not only contains the `classical' contribution from the rotation of the laboratory but also contributions due its acceleration and due to incoming gravitational waves. Here, we point out a major consequence of this gravitational effect which may have implications for third generation gravitational wave detectors. We describe an `antenna' design which picks out specific components of the Weyl tensor describing the incident gravitational waves.

gr-qc↗

arXiv2019

Numerical initial data deformation exploiting a gluing construction: I. Exterior asymptotic Schwarzschild

In this work a new numerical technique to prepare Cauchy data for the initial value problem (IVP) formulation of Einstein's field equations is presented. Directly inspired by the exterior asymptotic gluing (EAG) result of Corvino (2000) our (pseudo)-spectral scheme is demonstrated under the assumption of axisymmetry so as to fashion composite Hamiltonian constraint satisfying initial data featuring internal binary black holes (BBH) as glued to exterior Schwarzschild initial data in isotropic form. The generality of the method is illustrated in a comparison of the ADM mass of EAG initial data sets featuring internal BBHs as modelled by Brill-Lindquist and Misner data.In contrast to the recent work of Doulis and Rinne (2016), and Pook-Kolb and Giulini (2018) we do not make use of the York-Lichnerowicz conformal framework to reformulate the constraints.

gr-qc↗