Numerical calculations near spatial infinity

After describing in short some problems and methods regarding the smoothness of null infinity for isolated systems, I present numerical calculations in which both spatial and null infinity can be studied. The reduced conformal field equations based on the conformal Gauss gauge allow us in spherical symmetry to calculate numerically the entire Schwarzschild-Kruskal spacetime in a smooth way including spacelike, null and timelike infinity and the domain close to the singularity.Comment: 10 pages, 2 postscript figures, uses psfrag; to appear in the Proceedings of the Spanish Relativity Meeting (ERE 2006), Palma de Mallorca, Spain, 4-8 September 200

Intermediate behavior of Kerr tails

The numerical investigation of wave propagation in the asymptotic domain of Kerr spacetime has only recently been possible thanks to the construction of suitable hyperboloidal coordinates. The asymptotics revealed an apparent puzzle in the decay rates of scalar fields: the late-time rates seemed to depend on whether finite distance observers are in the strong field domain or far away from the rotating black hole, an apparent phenomenon dubbed "splitting". We discuss far-field "splitting" in the full field and near-horizon "splitting" in certain projected modes using horizon-penetrating, hyperboloidal coordinates. For either case we propose an explanation to the cause of the "splitting" behavior, and we determine uniquely decay rates that previous studies found to be ambiguous or immeasurable. The far-field "splitting" is explained by competition between projected modes. The near-horizon "splitting" is due to excitation of lower multipole modes that back excite the multipole mode for which "splitting" is observed. In both cases "splitting" is an intermediate effect, such that asymptotically in time strong field rates are valid at all finite distances. At any finite time, however, there are three domains with different decay rates whose boundaries move outwards during evolution. We then propose a formula for the decay rate of tails that takes into account the inter--mode excitation effect that we study.Comment: 16 page

A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime

We present a new approach to solve the 2+1 Teukolsky equation for gravitational perturbations of a Kerr black hole. Our approach relies on a new horizon penetrating, hyperboloidal foliation of Kerr spacetime and spatial compactification. In particular, we present a framework for waveform generation from point-particle perturbations. Extensive tests of a time domain implementation in the code {\it Teukode} are presented. The code can efficiently deliver waveforms at future null infinity. As a first application of the method, we compute the gravitational waveforms from inspiraling and coalescing black-hole binaries in the large-mass-ratio limit. The smaller mass black hole is modeled as a point particle whose dynamics is driven by an effective-one-body-resummed analytical radiation reaction force. We compare the analytical angular momentum loss to the gravitational wave angular momentum flux. We find that higher-order post-Newtonian corrections are needed to improve the consistency for rapidly spinning binaries. Close to merger, the subdominant multipolar amplitudes (notably the $m=0$ ones) are enhanced for retrograde orbits with respect to prograde ones. We argue that this effect mirrors nonnegligible deviations from circularity of the dynamics during the late-plunge and merger phase. We compute the gravitational wave energy flux flowing into the black hole during the inspiral using a time-domain formalism proposed by Poisson. Finally, a self-consistent, iterative method to compute the gravitational wave fluxes at leading-order in the mass of the particle is presented. For a specific case study with $\hat{a}$=0.9, a simulation that uses the consistent flux differs from one that uses the analytical flux by $\sim35$ gravitational wave cycles over a total of about $250$ cycles. In this case the horizon absorption accounts for about $+5$ gravitational wave cycles

Swarm-Based Optimization with Random Descent

We extend our study of the swarm-based gradient descent method for non-convex optimization, [Lu, Tadmor & Zenginoglu, arXiv:2211.17157], to allow random descent directions. We recall that the swarm-based approach consists of a swarm of agents, each identified with a position, ${\mathbf x}$, and mass, $m$. The key is the transfer of mass from high ground to low(-est) ground. The mass of an agent dictates its step size: lighter agents take larger steps. In this paper, the essential new feature is the choice of direction: rather than restricting the swarm to march in the steepest gradient descent, we let agents proceed in randomly chosen directions centered around -- but otherwise different from -- the gradient direction. The random search secures the descent property while at the same time, enabling greater exploration of ambient space. Convergence analysis and benchmark optimizations demonstrate the effectiveness of the swarm-based random descent method as a multi-dimensional global optimizer

Self-force via Green functions and worldline integration

A compact object moving in curved spacetime interacts with its own gravitational field. This leads to both dissipative and conservative corrections to the motion, which can be interpreted as a self-force acting on the object. The original formalism describing this self-force relied heavily on the Green function of the linear differential operator that governs gravitational perturbations. However, because the global calculation of Green functions in non-trivial black hole spacetimes has been an open problem until recently, alternative methods were established to calculate self-force effects using sophisticated regularization techniques that avoid the computation of the global Green function. We present a method for calculating the self-force that employs the global Green function and is therefore closely modeled after the original self-force expressions. Our quantitative method involves two stages: (i) numerical approximation of the retarded Green function in the background spacetime; (ii) evaluation of convolution integrals along the worldline of the object. This novel approach can be used along arbitrary worldlines, including those currently inaccessible to more established computational techniques. Furthermore, it yields geometrical insight into the contributions to self-interaction from curved geometry (back-scattering) and trapping of null geodesics. We demonstrate the method on the motion of a scalar charge in Schwarzschild spacetime. This toy model retains the physical history-dependence of the self-force but avoids gauge issues and allows us to focus on basic principles. We compute the self-field and self-force for many worldlines including accelerated circular orbits, eccentric orbits at the separatrix, and radial infall. This method, closely modeled after the original formalism, provides a promising complementary approach to the self-force problem.Comment: 18 pages, 9 figure

Hyperboloidal data and evolution

We discuss the hyperboloidal evolution problem in general relativity from a numerical perspective, and present some new results. Families of initial data which are the hyperboloidal analogue of Brill waves are constructed numerically, and a systematic search for apparent horizons is performed. Schwarzschild-Kruskal spacetime is discussed as a first application of Friedrich's general conformal field equations in spherical symmetry, and the Maxwell equations are discussed on a nontrivial background as a toy model for continuum instabilities.Comment: 11 pages, 9 figures. To appear in the Proceedings of the Spanish Relativity Meeting (ERE 2005), Oviedo, Spain, 6-10 Sept 200

A conformal approach to numerical calculations of asymptotically flat spacetimes

This thesis is concerned with the development and application of conformal techniques to numerical calculations of asymptotically flat spacetimes. The conformal compactification technique enables us to calculate spatially unbounded domains, thereby avoiding the introduction of an artificial timelike outer boundary. We construct in spherical symmetry an explicit scri-fixing gauge, i.e. a conformal and a coordinate gauge in which the spatial coordinate location of null infinity is independent of time so that no resolution loss in the physical part of the conformal extension appears. Going beyond spherical symmetry, we develop a method to include null infinity in the computational domain. With this method, hyperboloidal initial value problems for the Einstein equations can be solved in a scri-fixing general wave gauge. To study spatial infinity, we discuss the conformal Gauss gauge and the reduced general conformal field equations from a numerical point of view. This leads us to the first numerical calculation of the entire Schwarzschild-Kruskal solution including spatial, null and timelike infinity and the domain close to the singularity. After developing a three dimensional, frame based evolution code with smooth inner and outer boundaries we calculate a radiative axisymmetric vacuum solution in a neighbourhood of spatial infinity represented as a cylinder including a piece of null infinity. In this context, a certain component of the rescaled Weyl tensor representing the radiation field is calculated unambiguously with respect to an adapted tetrad at null infinity

Effective source approach to self-force calculations

Numerical evaluation of the self-force on a point particle is made difficult by the use of delta functions as sources. Recent methods for self-force calculations avoid delta functions altogether, using instead a finite and extended "effective source" for a point particle. We provide a review of the general principles underlying this strategy, using the specific example of a scalar point charge moving in a black hole spacetime. We also report on two new developments: (i) the construction and evaluation of an effective source for a scalar charge moving along a generic orbit of an arbitrary spacetime, and (ii) the successful implementation of hyperboloidal slicing that significantly improves on previous treatments of boundary conditions used for effective-source-based self-force calculations. Finally, we identify some of the key issues related to the effective source approach that will need to be addressed by future work.Comment: Invited review for NRDA/Capra 2010 (Theory Meets Data Analysis at Comparable and Extreme Mass Ratios), Perimeter Institute, June 2010, CQG special issue - 22 pages, 8 figure

Numerical investigation of the late-time Kerr tails

The late-time behavior of a scalar field on fixed Kerr background is examined in a numerical framework incorporating the techniques of conformal compactification and hyperbolic initial value formulation. The applied code is 1+(1+2) as it is based on the use of the spectral method in the angular directions while in the time-radial section fourth order finite differencing, along with the method of lines, is applied. The evolution of various types of stationary and non-stationary pure multipole initial states are investigated. The asymptotic decay rates are determined not only in the domain of outer communication but along the event horizon and at future null infinity as well. The decay rates are found to be different for stationary and non-stationary initial data, and they also depend on the fall off properties of the initial data toward future null infinity. The energy and angular momentum transfers are found to show significantly different behavior in the initial phase of the time evolution. The quasinormal ringing phase and the tail phase are also investigated. In the tail phase, the decay exponents for the energy and angular momentum losses at future null infinity are found to be smaller than at the horizon which is in accordance with the behavior of the field itself and it means that at late times the energy and angular momentum falling into the black hole become negligible in comparison with the energy and angular momentum radiated toward future null infinity. The energy and angular momentum balances are used as additional verifications of the reliability of our numerical method.Comment: 33 pages, 12 figure