# Titles and Abstracts for Copenhagen 23

Titles and abstracts complementing the schedule of the Infinity on a Gridshell workshop in Copenhagen, July 10—13, 2023.

### Abhay Ashtekar

*Pennsylvania State University*

#### Bridging ${\cal I}^+$ and horizon through hyperboloid slices: Gravitational Wave Tomography

When black holes merge (or form by gravitational collapse), we have a common dynamical horizon whose geometry changes dramatically as it settles down to the final Kerr horizon. This evolution can be invariantly characterized by the dynamics of a set of multipoles. Unfortunately, the causal structure of space-time prevents the outside observers from directly witnessing it. However, *thanks to Einstein’s equations*, it is encoded in the profiles of gravitational waves observed at ${\cal I}^+$: Using hyperboloid slices to connect the two, one can introduce a transform that reconstructs the *late time* dynamics of the horizon geometry using gravitational waveforms. Just as one monitors changes in the internal structure of objects from outside using electromagnetic tomography, one can image the horizon dynamics using gravitational waves at infinity.

### Sebastiano Bernuzzi

*Friedrich Schiller University Jena*

#### The binary black hole mergers laboratory

The understanding of black hole mergers in general relativity heavily relies on numerical results in the test-mass limit to model the multipolar waveform and radiation reaction, the plunge dynamics, and quasi-normal-modes excitation. A successful approach combines perturbation theory (Teukolsky equation) with the effective-one-body dynamics for the test-body source. In this framework, hyperboloidal foliations are the key technical element that allows accurate numerical solutions of the time-domain (2+1) Teukolsky equation. The talk first reviews the methods and early results of this ““test-mass laboratory”” and then presents recent applications toward a faithful effective-one-body model for binary black holes on generic orbits.

### Florian Beyer

*University of Otago*

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

We discuss recent results regarding the asymptotics of solutions of certain evolutionary formulations of the Einstein-vacuum constraint equations based on previous work by Bartnik and Racz.

### Miguel Duarte

*Technical University of Lisbon*

#### Asymptotic systems in generalized harmonic gauge

Starting from a generalization of the good-bad-ugly model, we showed that the solutions to the Einstein field equations in generalized harmonic gauge (GHG) admit polyhomogeneous expansions near null infinity. This allows us to find out, under some assumptions on initial data, whether the peeling property holds for each gauge choice. In general, we find that it does not. However we find that the interplay between gauge and constraint addition can be exploited in order to recover peeling. The same method can be used to build a regularization of the Einstein field equations in GHG.

### Jörg Frauendiener

*University of Otago*

#### A global approach to the nonlinear perturbation of a black hole by gravitational waves

It is well known that gravitational waves interact non-linearly. This makes it difficult to describe them rigorously. The cleanest description is based on a certain conformal invariance of the Einstein equations — a fact established by R. Penrose and used by H. Friedrich to prove several significant global results for general relativistic space-times. The so-called conformal field equations implement this conformal invariance on the level of partial differential equations. They provide various well-posed initial (boundary) value problems for use in different situations. The talk will give a computational perspective on the non-linear interaction of gravitational waves with an initially static (and spherically symmetric) black hole. We will show how to kick it and spin it up. The emergence of quasinormal modes and the constancy of the Newman-Penrose ““constants”” will be discussed.

### Dejan Gajic

*Leipzig University*

#### Late-time tails and quasinormal modes

Hyperboloidal foliations form a convenient tool for determining precisely how radiation of energy through null infinity results in decay in time of waves propagating on asymptotically flat spacetimes. I will present two applications of hyperboloidal foliations in black hole dynamics: 1) proving the existence of inverse-polynomial tails in the late-time asymptotics of waves propagating on black hole backgrounds, and 2) characterizing quasinormal modes as eigenfunctions of the infinitesimal generator of time-translations with respect to hyperboloidal foliations.

### Edgar Gasperin

*University of Lisbon*

#### Polyhomogeneity of spin-0 fields and the good-bad-ugly system

A model system of equations that serves as a model for the Einstein field equation in generalised harmonic gauge called the good-bad-ugly system is studied in the region close to null and spatial infinity in Minkowski spacetime. This analysis is performed using H. Friedrich’s cylinder construction at spatial infinity and defining suitable conformally rescaled fields. The results are translated to the physical set up to investigate the relation between the polyhomogeneous expansions arising from the analysis of linear fields using the spatial-infinity-cylinder framework and those obtained through a heuristic method based on Hormander’s asymptotic system.

### Shalabh Gautam

*International Centre for Theoretical Sciences or ICTS-TIFR*

#### Compactified Hyperboloidal Evolution in Numerical Relativity

One symmetric hyperbolic formulation of the Einstein Field Equations (EFEs) is in generalized harmonic gauge. The choice of gauge is generally related to the coordinates used to cast the EFEs within a spacetime foliation. A naive choice of gauge adapted to hyperboloidal coordinates may not be the most optimal way to solve the EFEs on these slices. In this talk, we shall discuss a choice of compactified hyperboloidal coordinates that not only are physically motivated but also facilitate mapping future-null infinity onto a finite computational grid. We decouple our choice of gauge from this choice of coordinates to maintain the hyperbolicity of the EFEs via the dual-foliation formalism. We introduce a numerical scheme that assures stability for a class of linear hyperbolic systems on these slices. Finally, we discuss the possibility of extending this numerical scheme for studying initial value problems for the EFEs on these slices.

### Lidia Gomes da Silva

*Queen Mary University of London*

#### Hyperboloidal numerical algorithm for self-force applications in the time domain

In the 2030s LISA is due to be launched, providing the opportunity to extract physics from stellar objects and systems that would not otherwise be possible, among which are EMRIs. Unlike previous sources detected at LIGO, these sources can be simulated using an accurate computation of the gravitational self-force. Whereas the field has seen outstanding progress in the frequency domain, metric reconstruction and self-force calculations are still an open challenge in the time domain. Such computations would not only further corroborate frequency domain calculations/models but also allow for full self-consistent evolution of the orbit under the effect of the self-force . Given we have a priori information about the local structure of the discontinuity at the particle, we will show how we can construct discontinuous spatial and temporal discretizations by operating on discontinuous Lagrange and Hermite interpolation formulae and hence recover higher order accuracy.

We show how this technique in conjunction with well-suited conformal (hyperboloidal slicing) and numerical (discontinuous time symmetric) methods can provide a relatively simple method of lines numerical recipe approach to the problem. We will start by reviewing our algorithm at the light of the wave equation and finally share our results for both the scalar and gravitational self-force cases for a particle in a circular orbit on a Schwarzschild background. In the spirit of the workshop we focus on highlighting important intermediary steps and results singular to using hyperboloidal foliations hopefully addressing outstanding concerns and further motivate its implementation in present and future gravitational TD codes as a simpler and highly accurate alternative to boundary conditions.

In collaboration with: Rodrigo Panosso Macedo, Jonathan Thompson, Juan Valiente Kroon, Oliver Long, Leor Barack and Leanne Durkan.

### Sascha Husa

*University of the Balearic Islands*

#### Status and open issues for numerical relativity as a tool for waveform modelling

Accurate numerical relativity simulations are a crucial ingredient for waveform models that allow to identify the sources of gravitational wave signals, and the further development of numerical relativity methods is essential for gravitational wave astronomy. In this talk I will give a brief overview about how numerical relativity simulations are used in waveform modelling, and I will review the status of numerical relativity techniques and compare with the accuracy requirements of current and future detectors. I will place this discussion into the context of the ambitions of the hyperboloidal approach, and I will in particular discuss issues of wave extraction and Cauchy-characteristic methods.

### José Luis Jaramillo

*University of Burgundy*

#### Quasi-Normal Mode perturbations: a Hyperboloidal Approach

Assessing the structural stability of Quasi-Normal Mode (QNM) frequencies of self-gravitating compact objects is an important open problem in general relativity and its extensions, given the fundamental role that QNMs play in diverse areas of gravitational physics. A particular avenue to this problem is potentially provided by Kato’s perturbation theory of linear operators. However, such an approach requires first to cast QNMs in terms of a proper eigenvalue problem, in particular with proper (finite norm) QNM eigenfunctions in a Hilbert space. The hyperboloidal approach to (black hole) perturbations provides a geometric approach to such a characterization of QNMs in terms of the spectrum of a non-selfadjoint operator. In this talk we discuss some of the concepts and tools employed in this context, imported from the spectral theory of non-selfadjoint operators. Focus will be placed on the notion of pseudospectrum, paying special attention to the open issues needed to be addressed before any sound conclusion can be reached about QNM instability.

### Jason Joykutty

*University of Cambridge*

#### Zero-damped Modes and Nearly Extremal Horizons

Quasinormal modes are the gravitational wave analogue to the overtones heard after striking a bell. They dominate the signal observed during the ringdown phase after a dynamical event and are characterised by complex frequencies, which encode oscillation and exponential decay in time. As horizons become extremal, various computations (both analytic and numerical) have shown that in many cases, there exists a sequence of frequencies which become purely oscillatory in the limit and which cluster on a line in the complex plane. These are zero-damped modes and are conjectured to exist generically for near-extremal horizons. In this talk, we shall discuss rigorous results that can be obtained toward resolving this question by using a hyperboloidal foliation of the background spacetime.

### Benjamin Leather

*Max Planck Institute for Gravitational Physics*

#### Hyperboloidal methods for self-force calculations

Gravitational self-force theory is the leading approach for modelling gravitational wave emission from small mass-ratio compact binaries. This method perturbatively expands the metric of the binary in powers of the mass ratio. The source for the perturbations depends on the orbital configuration, calculational approach, and/or the order in which the perturbative expansion is carried. These sources fall into three broad classes: distributional, extended and compact, and non-compact. The latter, in particular, is important for emerging second-order in the mass ratio calculations. Traditional frequency domain approaches employ the variation of parameters method and compute the perturbation on constant time slices of the spacetime with numerical boundary conditions supplied at a finite radius from series expansions of the asymptotic behaviour. This approach has been very successful, but the boundary conditions calculations are tedious, and the approach is not well suited to non-compact sources where homogeneous solutions must be computed at all radii. In this talk, I outline an alternative approach where horizon-penetrating hyperboloidal slices foliate the spacetime. Further compactifying the coordinates along these slices allows for simple treatment of the boundary conditions. We implement this approach with a multi-domain spectral solver with analytic mesh refinement and present results for the scalar-field self-force on circular orbits as an example problem. We find the method works efficiently for all three classes of sources encountered in self-force calculations and has some distinct advantages over the traditional approach.

### Philippe LeFloch

*Sorbonne University*

#### Evolution of self-gravitating massive fields: nonlinear stability, asymptotic behavior, and singularities

I will present recent developments on the dynamics of self-gravitating massive fields. In collaboration with Yue Ma, I established the global nonlinear stability of self-gravitating Klein-Gordon fields in the near-Minkowski regime, when the metric enjoys only low decay at spacelike infinity. To this purpose we developed the Euclidean-Hyperboloidal Foliation Method, as we call it, which applies to coupled systems of nonlinear wave and Klein-Gordon equations. In collaboration with Bruno Le Floch (LHEP, Sorbonne) and T.C. Nguyen (Paris), I introduced the localized Seed-to-Solution Method and constructed classes of asymptotically Euclidean solutions to the Einstein constraints under prescribed asymptotic conditions at infinity. Our solutions are localized at the super-harmonic rates in cone-like domains at spacelike infinity. In collaboration with B. Le Floch and G. Veneziano (CERN, Geneva), I studied the bouncing behavior in the vicinity of gravitational singularity and proposed the notion of singularity scattering maps in order to describe the junction conditions. Blog: philippelefloch.org

### Oliver Long

*Max Planck Institute for Gravitational Physics*

#### Hyperbolic self-force calculations within a hyperboloidal framework

Information from unbound self-force calculations can inform ongoing efforts to construct an accurate description of the general-relativistic binary dynamics across all mass ratios. With this motivation in mind, we present the state-of-the-art calculations of the scalar self-force correction to the scattering angle and highlight the limitations of the current methods. We give details of an ongoing effort to construct a spectral 1+1D code which utilises a hyperboloidal foliation which aims to drastically increase the accuracy of the calculations and discuss the extension of the scheme to the gravitational case.

### Charalampos Markakis

*Queen Mary University of London*

#### Symmetric integration of the 1+1 Teukolsky equation on hyperboloidal foliations of Kerr spacetimes

This talk outlines a fast, high-precision time-domain solver for scalar, electromagnetic, and gravitational perturbations on hyperboloidal foliations of Kerr space-times. Time-domain Teukolsky equation solvers have typically used explicit methods, which numerically violate Noether symmetries and are Courant-limited. These restrictions can limit the performance of explicit schemes when simulating long-time extreme mass ratio inspirals, expected to appear in the LISA band for 2-5 years. We outline work on using symmetric (exponential, Padé, or Hermite) integrators, which are unconditionally stable and known to preserve certain Noether symmetries and phase-space volume. For linear hyperbolic equations, these implicit integrators can be cast in explicit form, making them well-suited for long-time evolution of black hole perturbations. The 1+1 modal Teukolsky equation is discretized in space using polynomial collocation methods and reduced to a linear system of ordinary differential equations coupled via mode-coupling arrays and discretized differential operators. Hyperboloidal methods, combined with a matricization technique, allow us to cast the mode-coupled system in a form amenable to a method-of-lines framework, which dramatically simplifies numerical implementation and enables efficient parallelization on CPU and GPU architectures. We present tests of our numerical code on late-time tails of Kerr spacetime perturbations in the sub-extremal and extremal cases.

### Marica Minucci

*Queen Mary University of London*

#### The Maxwell-scalar field system near spatial infinity

In this talk, I will discuss how Friedrich’s representation of spatial infinity can be used to study asymptotic expansions of the Maxwell-scalar field system near spatial infinity. In this framework, spatial infinity is represented by a cylinder connecting the endpoints of past and future null infinity, namely the critical sets. This formulation allows us to relate the behaviour of the fields at the critical sets with their initial value on a fiduciary Cauchy hypersurface. The main objective of this analysis is to understand the effects of the non-linearities of the system on the regularity of solutions and polyhomogeneous expansions at null infinity and, in particular, at the critical sets. The main result of this analysis is that the non-linear interaction makes both fields more singular at the conformal boundary than what is seen when the fields are non-interacting. In particular, there is a whole new class of logarithmic terms in the asymptotic expansions, which depend on the coupling constant between the Maxwell and scalar fields.

### Todd Oliynyk

*Monash University*

#### Wave equations and the bounded weak null condition

In this talk, we will discuss systems of semilinear wave equations in 3+1 dimensions whose associated asymptotic equations admit bounded solutions, which is a weak null condition that we refer to as the bounded weak null condition. After reviewing prior results on wave equations satisfying special cases of this weak null condition, we will present a new approach to establishing the existence of solutions to these system of wave equations near spatial infinity that requires no additional assumptions beyond the boundedness of solutions to the associated asymptotic equation, which is in the spirit of the weak null conjecture by Lindblad and Rodnianski. As we will discuss in some detail during the talk, the proof relies on a cylinder at infinity construction and formulating the wave equations as a Fuchsian system of equations. Time permitting, we will discuss open problems and future directions.

### Christian Peterson

*University of Lisbon*

#### Towards Hyperboloidal Numerical Relativity in Generalized Harmonic Gauge

The main usefulness of hyperboloidal numerical methods is the clean capture of the radiation signal at future null infinity. To do so, the asymptotic decay of the evolved fields has to be known a priori to get regular variables throughout the whole numerical grid. In this talk I will present 3+1-dimensional numerical results of a model that captures the asymptotics of General Relativity in Generalized Harmonic Gauge for a particular choice of metric variables that have a clean asymptotic hierarchy. Moreover, I will discuss the status of spherical evolutions in Numerical Relativity within this scenario and give a perspective on full 3D hyperboloidal simulations in GR.

### Adam Pound

*University of Southampton*

#### Hyperboloidal slices and Bondi gauges in second-order self-force theory

In second-order self-force theory, two types of infrared divergences arise due to the large-r behavior of the source terms in the second-order field equations. Both types are associated with the slow evolution of the binary system, but they have different underlying causes and very different manifestations. One type can be straightforwardly eliminated by adopting asymptotically null slicing. The second type, associated with hereditary dynamics and gravitational-wave memory, persists regardless of the choice of slicing. This more pernicious type can be eliminated by combining asymptotically null slicing with a Bondi-type gauge condition on the metric perturbation, enforcing that the background slices remain asymptotically null in the perturbed spacetime.

### István Rácz

*Wigner Research Centre for Physics*

#### On the Constructions of Hyperboloidal Initial Data

The regularity of the solutions of the constraint equations on hyperboloidal initial data surfaces is discussed. Some numerical results are also presented which support a recent argument of Beyer and Ritchie concerning the use of the parabolic-hyperbolic form of the constraints.

### Alex Vañó-Viñuales

*University of Lisbon*

#### Status of 3D free hyperboloidal conformal numerical evolutions

Gravitational wave radiation is only unambiguously defined at future null infinity. A convenient way to include it in numerical relativity simulations is via hyperboloidal foliations, which can be tackled using conformal compactification. I will describe one such implementation in the BSSN / conformal Z4 formulations. Spherically symmetric numerical simulations evolved with this method and dealing with black hole data will be presented. I will also give an update on ongoing work in 3D evolutions.

### Manas Vishal

*University of Massachusetts Dartmouth*

#### A discontinuous Galerkin method for the distributionally-sourced s=0 Teukolsky equation

The upcoming space-borne gravitational wave detector Laser Interferometer Space Antenna (LISA) is primarily sensitive to Extreme Mass Ratio Inspirals (EMRI) where the mass ratio between two black holes is higher than $10^5$. For the matched filtering process, we need a highly accurate template wave bank. In this talk, I will describe a Discontinuous Galerkin (DG) method for simulating waveforms from EMRI systems. We reduce the Teukolsky equation, which governs the behavior of EMRIs, to a set of coupled 1+1D wave equations and apply the DG method to it with a delta source term, acting like the secondary black hole in an EMRI system. Unlike other numerical schemes, our DG method can exactly incorporate the point particle behavior of the smaller black hole in the form of a delta function. Due to the spectral convergence properties of the scheme, our efficient method generates highly accurate waveforms in a very short time as compared to other methods. We have also introduced the hyperboloidal layers in our time domain solver that gives us access to the solution at null infinity. We verify our computation by computing Price tail power laws and scalar energy fluxes at null infinity.

### Barry Wardell

*University College Dublin*

#### Metric perturbations of Kerr spacetime in Lorenz gauge

Perturbations of Kerr spacetime are typically studied with the Teukolsky formalism, in which a pair of gauge invariant components of the perturbed Weyl tensor are expressed in terms of separable modes that satisfy ordinary differential equations. However, for certain applications it is desirable to construct the full metric perturbation in the Lorenz gauge, in which the linearized Einstein field equations take a manifestly hyperbolic form. Directly solving the Lorenz gauge equations in Kerr spacetime is challenging for two reasons: (i) unlike the Teukolsky equation, the Lorenz gauge equations are not known to admit separable solutions; (ii) the equations for the ten components of the metric perturbation are coupled. In this talk, I will present a formalism in which the Lorenz gauge metric perturbation is obtained from a set of six decoupled and separable solutions to the spin-2, spin-1 and spin-0 Teukolsky equations. The formalism is ideally suited to hyperboloidal methods, which have been shown to provide a highly-efficient approach to solving Teukolsky equations. As a demonstration of the approach, I will give results for the Lorenz-gauge gravitational self-force problem in Kerr spacetime. This talk is based on work with Sam Dolan, Chris Kavanagh and Leanne Durkan.