Schedule for: 19w5140 - Time-like Boundaries in General Relativistic Evolution Problems

Very often in physics the evolution systems we have to deal with are not purely hyperbolic, but contain also constraints and gauge freedoms. After fixing these gauge freedoms we obtain a new system with constraints which we want to solve subject to initial and boundary values. In particular, these values have to imply the correct propagation of constraints. In general, after fixing some reduction to a purely evolutionary system, this is asserting by computing by hand what is called the constraint subsidiary system, namely a system which is satisfied by the constraints quantities when the fields satisfy the reduced evolution system. If the subsidiary system is also hyperbolic then for the initial data case the situation is clear we need to impose the constraints on the initial data and then they will correctly propagated along evolution. For the boundary data we need to impose the constraint for all incoming constraint modes. These must be done by fixing some of the otherwise free boundary data, that is the incoming modes. Thus, there must be a relation between some of the incoming modes of the evolution system and all the incoming modes of the constraint subsidiary system. Under certain conditions on the constraints this relation is known and understood, but those conditions are very restrictive. In this talk we shall review the known results and discuss what is known so far for the general case and what are the open questions that still remain.

Force-free electrodynamics (FFE) describes a particular regime of magnetically dominated relativistic plasmas, which arise on several astrophysical scenarios of interest such as pulsars or active galactic nuclei. In those regimes, the electromagnetic field obeys a modified nonlinear version of Maxwell equations, while the plasma only accommodates to locally cancel out the Lorentz force. The aim of the present talk is to discuss the initial/boundary value formulation of FFE at some given astrophysical settings. We start by showing that, when restricted to the correct constraint submanifold, the system is symmetric hyperbolic: we introduce here a particular hyperbolization for the FFE equations [1], following a covariant approach due to R. Geroch [2]. Then, we analyze the characteristic structure of the resulting evolution system and use this information to construct appropriate boundary conditions [3,4]. In particular, we focus on the treatment to mimic the perfectly conducting surface of a neutron star, where incoming and outgoing physical modes needs to be combined on a very precise way. We shall illustrate this procedure with the simpler vacuum (linear) electrodynamics. Also, we discuss the methods employed to deal with the constraints of the theory at the boundaries. And finally, we show some results from our 3D numerical simulations based on this approach [4,5,6]. [1] F. Carrasco, O. Reula. PRD (93), 2016. DOI: 10.1103/PhysRevD.93.085013 [2] R. Geroch. “Partial differential equations of physics”. In General Relativity, pp. 19-60. Routledge, 1996. [3] F. Carrasco, O. Reula. PRD (96), 2017. DOI: 10.1103/PhysRevD.96.063006 [4] F. Carrasco, C. Palenzuela, O. Reula. PRD (98), 2018. DOI: 10.1103/PhysRevD.98.023010 [5] F. Carrasco, D. Viganò. C. Palenzuela, J. Pons. MNRAS Letters (484), 2019. DOI: 10.1093/mnrasl/slz016 [6] R. Cayuso, F. Carrasco, B. Sbarato, O. Reula. (arXiv:1905.00178), 2019.

A short summary of the well-posed IBVP for the vacuum Einstein field equations in harmonic coordinates formulated in Comm. Math. Phys. 289, 1099-1129 (2009) will be provided.

We describe ongoing work towards construction -via multidomain, modal, spectral methods- of helically symmetric spacetimes representing binary scenarios. Our approach starts with the helical reduction of the Einstein equations proposed by Beetle, Bromley, and Price. The talk will focus on unresolved issues related to boundary conditions (posed on the reduction of a timelike boundary) and enforcement of the harmonic gauge.

The breakthrough discovery of gravitational waves has given us a new sense to probe gravity in its most extreme, strong-field regime. Although general relativity (GR), our standard model of gravity, has passed this new stress-test, theory tells us that it is not the complete picture at high energy scales. Candidate theories aiming at resolving this conundrum typically involve couplings to additional fields or extensions to GR. Modeling the expected gravitational radiation in these beyond-GR theories enables us to search for -- or place novel observational bounds on -- deviations from our standard model. In this talk I will give an overview of the recent progress on simulating binary collisions in these situations as well as renewed mathematical challenges such as well-posedness of the underlying initial value formulation.

It is well known that gravitational waves interact in a non-linear way. This makes it difficult to describe them rigorously. The cleanest description is based on a certain conformal invariance of the Einstein equations — a fact which was established by R. Penrose and was used by H. Friedrich to prove several important 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 plane gravitational waves and also present preliminary results of a simulation of the behaviour of an initially spherically symmetric black hole under the impact of a gravitational wave burst.

In the absence of time-like boundary, the classical initial value problem in GR verifies a geometric uniqueness property. In particular, isometric Cauchy data leads to (maximal globally hyperbolic) developments which are isometric. While there exists several well-posed formulation of the initial boundary value problem in GR, no such geometric uniqueness is known. This important issue was put forward by Helmut Friedrich. It is relevant not only for the local initial boundary value problem, but also for more global aspects, due to the possible breakdown of gauge choices. I will review the mathematical analysis of the initial boundary value problem in GR, with an emphasis on various aspects relevant to the geometric uniqueness problem. If time (and progress) permits, I will present work in progress with Grigorios Fournodavlos concerning an approach to the initial boundary value problem based on the wave equation satisfied by the second fundamental form of a foliation with prescribed mean curvature.

The numerical computation of gravitational radiation emitted from binary black hole systems has been pivotal in the detection and characterization of 10 binary black hole astrophysical events since Sept. 14, 2015. However, the drive for more accurate waveforms remains, especially when computing higher-order spherical harmonic modes. The numerical relativity simulations currently used for LIGO / VIRGO detections solve Einstein’s field equations on a finite computational domain with an outer boundary. In order to obtain a unique Cauchy evolution, it is necessary to impose boundary conditions which should yield a well posed problem and ideally, be completely transparent to the physical problem on the unbounded domain. Short of achieving this ideal, one can try to develop so-called absorbing boundary conditions which form a well-posed problem and insure that only a very small amount of spurious gravitational radiation is reflected from the outer boundary into the computational domain. In this talk, I discuss the implementation of a hierarchy of such boundary conditions which hopefully will improve the accuracy of binary black hole numerical simulations.

Exploring large computational models is generally a complex task, requiring the efficient inspection of high-dimensional parameter spaces. This is exemplified, e.g., by the task of Identifying which combinations of binary-black-hole spin and mass ratio values lead to specific gravitational wave profiles, or which cosmological matter distribution produces a certain weak-lensing signature. The challenges involved are similar to those encountered in system identification and nonlinear control theory, where the behaviour of a system as a function of its configuration variables is reconstructed based on the direct, optimal sampling of the space of all possible system trajectories. In this talk, I will describe the application of an exploration technique, proposed to probe rare solutions of certain ODEs [1], to the exploration of null geodesics in simple spacetimes. I will also discuss the complementarity of this method to other model exploration techniques which are widely used in General Relativity, such as reduced-order modelling. [1] J. Tailleur and J. Kurchan, Probing rare physical trajectories with Lyapunov Weighted Dynamics, Nature Physics, 3:203 EP –, 02 (2007)

The mathematical study of the metric produced by compactly supported perfect fluid mass distributions governed by the Einstein-Euler equations is now developing. The main difficulty of the study comes from the situation that the boundary between the mass and the vacuum requires mathematically delicate treatise. This so called `physical vacuum boundary' is a kind of free boundary. The spherically symmetric evolution of the metric near spherically symmetric equilibria, say, solutions of the Tolman-Oppenheimer-Volkoff equation, is studied in [1] by application of the Nash-Moser theorem. The metric can be matched to the exterior Schwarrzschild metric in the vacuum region, but it turns out to be the boundary is of class C^2 if and only if the metric is static. The existence of the axially symmetric metric of slowly rotating mass under the weak gravitational field has been established in [2] in a bonded region containing the support of the density. But the global continuation of the metric beyond the bounded region has not yet been done. Actually we are not sure whether the matching can be done using the Kerr metric, and, what on earth, we are not sure whether the metric can be continued as asymptotically flat one or not. Much less the problem of time evolution near this rotating stationary metric is completely open. [1] T. Makino, Kyoto J. Math., 56 (2016), 243-282. DOI 10.1215/21562261-3478880. [2] T. Makino, J. Math. Physics., 59(2018), 102502. DOI 10.1063/1.5026133.

In this talk, I will discuss a new approach to establishing the well-posedness of the relativistic Euler equations for liquid bodies in vacuum. The approach is based on a wave formulation of the relativistic Euler equations that consists of a system of non-linear wave equations in divergence form together with a combination of acoustic and Dirichlet boundary conditions. The equations and boundary conditions of the wave formulation differs from the standard one by terms proportional to certain constraints, and one of the main technical problems to overcome is to show that these constraints propagate, which is necessary to ensure that solutions of the wave formulation determine solutions to the Euler equations with vacuum boundary conditions. During the talk, I will describe the derivation of the wave equation and boundary conditions, the origin of the constraints, and how one shows that the constraints propagate. Time permitting, I will also discuss how energy estimates can be obtained from this new formulation paying particular attention to the role of the acoustic boundary conditions.

Anti-de Sitter spacetime has a peculiar property that differences of any two normal mode frequencies for its linearized perturbations are integer in appropriate units. As a result, there is a profusion of resonances, and arbitrarily small nonlinearities may produce arbitrarily large effects, provided that one waits long enough. This situation is closely paralleled by nonlinear Schroedinger equations in harmonic traps, studied in relation to the physics of cold atomic gases and connected to the AdS dynamics via a nonrelativistic limit. At leading order, weakly nonlinear long-time dynamics of this sort is captured by the corresponding resonant Hamiltonian systems. I'll describe the derivation of this resonant approximation and its remarkable structures, including manifestations of turbulent behaviors, as well as symmetry enhancements and explicit analytic solutions.

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary? In this talk, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following question: when can a symmetry on the conformal boundary be extended into the interior? This is mostly joint work with Gustav Holzegel (Imperial College London).

I will present a recent result on solutions to the massive linear wave equation $\Box_g \psi - \mu \psi =0$ on the interior of Reissner--Nordström--AdS black holes. This is motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS black holes with negative cosmological constant $\Lambda <0$. Our main result shows that linear waves arising from a spacelike hypersurface with Dirichlet (reflecting) boundary conditions imposed at infinity remain bounded in the interior and can be extended continuously beyond the Cauchy horizon. This result is surprising because in contrast to black hole backgrounds with non-negative cosmological constant, the decay of $\psi$ in the exterior region for asymptotically AdS black holes is only logarithmic (cf. polynomial ($\Lambda =0$) and exponential ($\Lambda >0$)).

When investigating quantum or semiclassical phenomena in a general relativistic context one usually resorts to quantum field theory in curved spacetime. However, traditional quantization methods rely on foliating spacetime into spacelike hypersurfaces or on fixing asymptotic boundary conditions in time. This is a serious limitation if for the problem of interest no suitable Cauchy hypersurfaces exist and/or if boundary conditions are naturally given on timelike hypersurfaces, either at finite locations or asymptotically. I will outline methods of quantization adapted to such situations and the conceptual insights on which they are based. If time permits I will comment on how this may contribute to the search for a quantum theory of gravity.

For linear perturbations of black hole spacetimes, the ringdown is described by quasinormal modes. Quasinormal modes are resonance states, defined by ingoing and outgoing radiation conditions at the horizon and infinity. Their wavefunctions do not lie in a Hilbert space, and they do not in general form a complete basis. In particular, there is a priori no obvious inner product under which they are orthonormal. In contrast to normal modes, this limits efforts to carry out higher order perturbation theory in terms of quasinormal modes. In this work, we show that for type D spacetimes with a t-φ reflection symmetry, gravitational quasinormal modes are in fact orthogonal. We work in terms of Weyl scalars. On this space we define a symmetric bilinear form ⟨⟨·, ·⟩⟩ in terms of the symplectic form for the Teukolsky equation on a Cauchy surface Σ and the t-φ reflection operator. (Our bilinear form is motivated by earlier work of Leung, Liu, and Young [Phys. Rev. A 49, 3057 (1994)].) The bilinear form is complex-linear in both entries, it is independent of the precise choice of Σ, and the time-evolution operator is symmetric with respect to ⟨⟨·, ·⟩⟩. It follows that quasinormal modes are orthogonal with respect to ⟨⟨·, ·⟩⟩ and have finite norm. We also relate the bilinear form on Weyl scalars to bilinear forms on the spaces of metric perturbations and Hertz potentials. The formula for a quasinormal mode excitation coefficient emerges naturally as the projection of initial data onto the quasinormal mode, however this projector also extends to data consisting of quasinormal modes. By projecting with the bilinear form, our goal is to develop a framework for higher order black hole perturbation theory in terms of quasinormal modes.