We study gravitational perturbations of Schwarzschild spacetime by solving a hyperboloidal initial value problem for the Bardeen-Press equation. Compactification along hyperboloidal surfaces in a scri-fixing gauge allows us to have access to the gravitational waveform at null infinity in a general setup. We argue that this hyperboloidal approach leads to a more accurate and efficient calculation of the radiation signal than the common approach where a timelike outer boundary is introduced. The method can be generalized to study perturbations of Kerr spacetime using the Teukolsky equation.
In this paper we establish energy decay for solutions to the Klein-Gordon equation on the positive mass hyperboloidal anti-de Sitter Schwarzschild black hole, subject to Dirichlet, Neumann and Robin boundary conditions at infinity, for a range of the (negative) mass squared parameter. To do so we use vector field methods with a renormalised energy to avoid divergences that would otherwise appear in the energy integrals. For another region of the parameter space, we use the existence of negative energy solutions to demonstrate linear instability.
Charged Scalar Field at Future Null Infinity via Nonlinear Hyperboloidal Evolution
CENTRA, Departamento de Física do Instituto Superior Técnico (IST), Universidade de Lisboa（里斯本大学高级理论研究中心物理系技术研究所）；Instituto de Telecomunicações (IT), Universidade de Aveiro（阿威罗大学电信研究所）；Departament de Física, Universitat de les Illes Balears（巴利阿里群岛大学物理系）
We address the hyperboloidal initial value problem in the context of Numerical Relativity, motivated by its evolution on hyperboloidal slices: smooth spacelike slices that reach future null infinity, the "location" in spacetime where radiation is to be extracted. Our approach uses the BSSN and Z4 formulations and a time-independent conformal factor. The resulting system of PDEs includes formally diverging terms at null infinity. Here we discuss a regularized numerical scheme in spherical symmetry. A critical ingredient are the gauge conditions, which control the treatment of future null infini…
We present numerical solutions of the hyperboloidal initial value problem for a self-gravitating scalar field in spherical symmetry, using a variety of standard hyperbolic slicing and shift conditions that we adapt to our hyperboloidal setup. We work in the framework of conformal compactification, and study both evolutions that employ the preferred conformal gauge, which simplifies the formal singularities of our equations at null infinity, and evolutions without this simplification. In previous work we have used a staggered grid, which excludes null infinity, while now we include the option o…
A long-standing problem in numerical relativity is the satisfactory treatment of future null-infinity. We propose an approach for the evolution of hyperboloidal initial data in which the outer boundary of the computational domain is placed at infinity. The main idea is to apply the `dual foliation' formalism in combination with hyperboloidal coordinates and the generalized harmonic gauge formulation. The strength of the present approach is that, following the ideas of Zenginoglu, a hyperboloidal layer can be naturally attached to a central region using standard coordinates of numerical relativ…
We consider the hyperboloidal initial value problem in numerical relativity, motivated by the goal to evolve radiating compact objects such as black hole binaries with a numerical grid that includes null infinity. Unconstrained evolution schemes promise optimal efficiency, but are difficult to regularize at null infinity, where the compactified Einstein equations are formally singular. In this work we treat the spherically symmetry case, which already poses nontrivial problems and constitutes an important first step. We have carried out stable numerical evolutions with the generalized BSSN and…
One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a naive choice of evolved variables. Here we consider a system of wave equations that satisfy the weak-null condition and serve as a model system with similar nonlinearities to those present in the Einstein field equations in generalized harmonic gauge. We show that these equations can be explicitly regularized by a nonlinear change of variables. Working in sphe…
Comment on "Charged scalar field at future null infinity via nonlinear hyperboloidal evolution" [Phys. Rev. D {\bf 112}, 104053 (2025), arXiv:2506.15311]
The hyperboloidal initial value problem is addressed in the context of Numerical Relativity, motivated by its use of hyperboloidal slices - smooth spacelike slices that reach future null infinity, the "place" in spacetime where radiation is to be extracted. This is beneficial for studying the global properties of isolated systems and unambiguously extracting their gravitational radiation. The present approach implements the Einstein equations as a free evolution, using the BSSN and Z4 formulations (standard in current codes) expressed in terms of a conformally rescaled metric as suggested by P…
We examine stability of summation by parts (SBP) numerical schemes that use hyperboloidal slices to include future null infinity in the computational domain. This inclusion serves to mitigate outer boundary effects and, in the future, will help reduce systematic errors in gravitational waveform extraction. We also study a setup with truncation error matching. Our SBP-Stable scheme guarantees energy-balance for a class of linear wave equations at the semidiscrete level. We develop also specialized dissipation operators. The whole construction is made at second order accuracy in spherical symmet…
In this paper we present a spectral decomposition of solutions to relativistic wave equations described on horizon penetrating hyperboloidal slices within a given Schwarzschild-black-hole background. The wave equa- tion in question is Laplace-transformed which leads to a spatial differential equation with a complex parameter. For initial data which are analytic with respect to a compactified spatial coordinate, this equation is treated with the help of the Mathematica-package in terms of a sophisticated Taylor series analysis. Thereby, all ingredients of the desired spectral decomposition aris…
Motivated by the need of a robust geometrical framework for the calculation of long, and highly accurate waveforms for extreme-mass-ratio inspirals, this work presents an extensive study of the hyperboloidal formalism for the Kerr spacetime and the Teukolsky equation. In a first step, we introduce a generic coordinate system foliating the Kerr spacetime into hypersurfaces of constant time extending between the black-hole horizon and future null infinity, while keeping track of the underlying degrees of freedom. Then, we express the Teukolsky equation in terms of these generic coordinates with …
We construct initial data corresponding to a single perturbed Kerr black hole in vacuum. These data are defined on specific hyperboloidal ("ACMC-") slices on which the mean extrinsic curvature K asymptotically approaches a constant at future null infinity scri+. More precisely, we require that K obeys the Taylor expansion K=K0 + s^4 where K0 is a constant and s describes a compactified spatial coordinate such that scri+ is represented by s=0. We excise the singular interior of the black hole and assume a marginally outer trapped surface as inner boundary of the computational domain. The moment…
Semilinear wave equations in homothetic hyperboloidal coordinates and tail decay
Institute for Physical Science and Technology, University of Maryland（马里兰大学物理科学与技术研究所）；Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena（耶拿大学理论物理研究所）；Department of Mathematics, Stockholm University（斯德哥尔摩大学数学系）
The Dirac equation governs the behaviour of spin-1/2 particles. The equation's separability into decoupled radial and angular differential equations is a crucial step in analytical and numerical computations of quantities like eigenvalues, quasinormal modes and bound states. However, this separation has been performed in co-ordinate systems that are not well-behaved in either limiting regions of $r \rightarrow r_{horizon}$, $r \rightarrow r_\infty$ or both. In particular, the extensively used Boyer-Lindquist co-ordinates contains unphysical features of spacetime geometry for both $r_{horizon}$…
Gravitational self-force theory is the leading approach for modeling 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 the order of the perturbative expansion. These sources fall into three broad classes: (i) distributional, (ii) worldtube, and (iii) unbounded support. The latter, in particular, is important for emerging second-order (in the mass ratio) calculations. Traditional frequency do…
The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably, a single formula works for an entire geodesic trajectory, even if it passes through turning points. Using this formula, we derive expressions for the proper and coordinate time along the geodesic.
We consider the Einstein-Maxwell-fluid constraint equations, and make use of the conformal method to construct and parametrize constant-mean-curvature hyperboloidal initial data sets that satisfy the shear-free condition. This condition is known to be necessary in order that a spacetime development admit a regular conformal boundary at future null infinity. We work with initial data sets in a variety of regularity classes, primarily considering those data sets whose geometries are weakly asymptotically hyperbolic, as defined in [arXiv:1506.03399]. These metrics are $C^{1,1}$ conformally compac…
Time-domain framework for the Teukolsky equation with a particle source using comoving hyperboloidal coordinates
School of Mathematical Sciences and STAG Research Centre, University of Southampton（南安普顿大学数学科学学院和STAG研究中心）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））；Center of Gravity, Niels Bohr Institute（尼尔斯·玻尔研究所重力中心）
Naked singularities form during the gravitational collapse of inhomogeneous matter clouds. The final nature of the singularity depends on the initial conditions of the matter properties and types of matter profiles. These naked singularities can also be divided into two types: null-like and timelike singularities. The spacelike singularity of the Schwarzschild black hole can be distinguished from the null and timelike naked singularity spacetimes. In light of this, we investigate the precession of timelike bound orbits in the null naked singularity spacetime, as well as tidal force effects and…
