Stephen R. Lau

Canonical quasilocal energy and small spheres

Consider the definitionE of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard ‘‘small-sphere limit,’’ first considered by Horowitz and Schmidt in their examination of Hawking’s quasilocal mass. By the term small sphere we mean a cut S(r), level in an affine radius r, of the light cone Np belonging to a generic spacetime point p .A s a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface S spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero point of the energy. For the small-sphere limit, we argue that the correct zero point is obtained via a ‘‘light cone reference,’’ which stems from a certain isometric embedding of S(r) into a genuine light cone of Minkowski spacetime. Choosing this zero point, we find the following results: ~i! in the presence of matter E5 4 3 pr 3 @T mnu m u n #u p1O(r 4 ) and ~ii! in vacuo E5 1 90 r 5 @T mnlku m u n u l u k #u p1O(r 6 ). Here, u m is a unit, future-pointing, timelike vector in the tangent space at p ~which defines the choice of affine radius !; Tmn is the matter stress-energy-momentum tensor; Tmnlk is the Bel-Robinson gravitational super stress-energymomentum tensor; and u p denotes ‘‘restriction to p.’’ Hawking’s quasilocal mass expression agrees with the results ~i! and ~ii! up to and including the first non-trivial order in the affine radius. The non-vacuum result~i! has the expected form based on the results of Newtonian potential theory. @S0556-2821~99!02904-5#

Center of mass integral in canonical general relativity

Abstract For a two-surface B tending to an infinite-radius round sphere at spatial infinity, we consider the Brown–York boundary integral HB belonging to the energy sector of the gravitational Hamiltonian. Assuming that the lapse function behaves as N∼1 in the limit, we find agreement between HB and the total Arnowitt–Deser–Misner energy, an agreement first noted by Braden, Brown, Whiting, and York. However, we argue that the Arnowitt–Deser–Misner mass-aspect differs from a gauge invariant mass-aspect by a pure divergence on the unit sphere. We also examine the boundary integral HB corresponding to the Hamiltonian generator of an asymptotic boost, in which case the lapse N∼xk grows like one of the asymptotically Cartesian coordinate functions. Such a two-surface integral defines the kth component of the center of mass for (the initial data belonging to) a Cauchy surface Σ bounded by B. In the large-radius limit, we find agreement between HB and an integral introduced by Beig and o Murchadha as an improvement upon the center-of-mass integral first written down by Regge and Teitelboim. Although both HB and the Beig– o Murchadha integral are naively divergent, they are in fact finite modulo the Hamiltonian constraint. Furthermore, we examine the relationship between HB and a certain two-surface integral which is linear in the spacetime Riemann curvature tensor. Similar integrals featuring the curvature appear in works by Ashtekar and Hansen, Penrose, Goldberg, and Hayward. Within the canonical 3+1 formalism, we define gravitational energy and center of mass as certain moments of Riemann curvature.

Rapid evaluation of radiation boundary kernels for time-domain wave propagation on blackholes: theory and numerical methods

For scalar, electromagnetic, or gravitational wave propagation on a fixed Schwarzschild blackhole background, we describe the exact nonlocal radiation outer boundary conditions (ROBC) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the ROBC is based on Laplace and spherical-harmonic transformation of the Regge-Wheeler equation, the PDE governing the wave propagation, with the resulting radial ODE an incarnation of the confluent Heun equation. For a given angular index I the ROBC feature integral convolution between a time-domain radiation boundary kernel (TDRK) and each of the corresponding 21+1 spherical-harmonic modes of the radiation field. The TDRK is the inverse Laplace transform of a frequency-domain radiation kernel (FDRK) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ODE. We develop several numerical methods for examining the frequency dependence of both the outgoing solution and the FDRK. Using these methods we numerically implement the ROBC in a follow-up article. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom (AGH), save for one key difference. Whereas AGH had the usual armamentarium of analytical results (asymptotics, order recursion relations, bispectrality) for Bessel functions at their disposal, what we need to know about Heun functions must be gathered numerically as relatively less is known about them.

Analytic structure of radiation boundary kernels for blackhole perturbations

Exact outer boundary conditions for gravitational perturbations of the Schwarzschild metric feature integral convolution between a time-domain boundary kernel and each radiative mode of the perturbation. For both axial (Regge–Wheeler) and polar (Zerilli) perturbations, we study the Laplace transform of such kernels as an analytic function of (dimensionless) Laplace frequency. We present numerical evidence indicating that each such frequency-domain boundary kernel admits a “sum-of-poles” representation. Our work has been inspired by Alpert, Greengard, and Hagstrom’s analysis of nonreflecting boundary conditions for the ordinary scalar wave equation.

Rapid evaluation of radiation boundary kernels for time-domain wave propagation on black holes: implementation and numerical tests*

For scalar, electromagnetic, or gravitational wave propagation on a fixed Schwarzschild black hole background, we consider the exact nonlocal radiation outer boundary conditions (ROBC) appropriate for a spherical outer boundary of finite radius enclosing the black hole. Such boundary conditions feature temporal integral convolution between each spherical harmonic mode of the wave field and a time-domain radiation kernel (TDRK). For each orbital angular integer l the associated TDRK is the inverse Laplace transform of a frequency-domain radiation kernel (FDRK). Drawing upon theory and numerical methods developed in a previous article, we numerically implement the ROBC via a rapid algorithm involving approximation of the FDRK by a rational function. Such an approximation is tailored to have relative error e uniformly along the axis of imaginary Laplace frequency. Theoretically, e is also a long-time bound on the relative convolution error. Via study of one-dimensional radial evolutions, we demonstrate that the ROBC capture the phenomena of quasinormal ringing and decay tails. We also consider a three-dimensional evolution based on a spectral code, one showing that the ROBC yield accurate results for the scenario of a wave packet striking the boundary at an angle. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom.

Boundary conditions and quasilocal energy in the canonical formulation of all (1+1) models of gravity

Abstract Within a first-order framework, we comprehensively examine the role played by boundary conditions in the canonical formulation of a completely general two-dimensional gravity model. Our analysis particularly elucidates the perennial themes of mass and energy. The gravity models for which our arguments are valid include theories with dynamical torsion and so-called generalized dilaton theories (GDTs). Our analysis of the canonical action principle (i) provides a rigorous correspondence between the most general first-order two-dimensional Einstein–Cartan model (ECM) and GDT and (ii) allows us to extract in a virtually simultaneous manner the “true degrees of freedom” for both ECMs and GDTs. For all such models, the existence of an absolutely conserved (in vacuo) quantity C is a generic feature, with (minus) C corresponding to the black-hole mass parameter in the important special cases of spherically symmetric four-dimensional general relativity and standard two-dimensional dilaton gravity. The mass C also includes (minimally coupled) matter into a “universal mass function.” We place particular emphasis on the (quite general) class of models within GDT possessing a Minkowski-like groundstate solution (allowing comparison betweenC and the Arnowitt–Deser–Misner mass for such models).

Discontinuous Galerkin method for the spherically reduced Baumgarte-Shapiro-Shibata-Nakamura system with second-order operators

We present a high-order accurate discontinuous Galerkin method for evolving the spherically reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system expressed in terms of second-order spatial operators. Our multidomain method achieves global spectral accuracy and longtime stability on short computational domains. We discuss in detail both our scheme for the BSSN system and its implementation. After a theoretical and computational verification of the proposed scheme, we conclude with a brief discussion of issues likely to arise when one considers the full BSSN system.

Persistent junk solutions in time-domain modeling of extreme mass ratio binaries

In the context of metric perturbation theory for nonspinning black holes, extreme mass ratio binary systems are described by distributionally forced master wave equations. Numerical solution of a master wave equation as an initial boundary value problem requires initial data. However, because the correct initial data for generic-orbit systems is unknown, specification of trivial initial data is a common choice, despite being inconsistent and resulting in a solution which is initially discontinuous in time. As is well known, this choice leads to a burst of junk radiation which eventually propagates off the computational domain. We observe another potential consequence of trivial initial data: development of a persistent spurious solution, here referred to as the Jost junk solution, which contaminates the physical solution for long times. This work studies the influence of both types of junk on metric perturbations, waveforms, and self-force measurements, and it demonstrates that smooth modified source terms mollify the Jost solution and reduce junk radiation. Our concluding section discusses the applicability of these observations to other numerical schemes and techniques used to solve distributionally forced master wave equations.

IMEX evolution of scalar fields on curved backgrounds

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time-steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.

Spinors and the reference point of quasilocal energy

This paper investigates the relationship between the quasilocal energy of Brown and York and certain spinorial expressions for gravitational energy constructed from the Witten--Nester integral. A key feature of the Brown--York method for defining quasilocal energy is that it allows for the freedom to assign the reference point of the energy. When possible, it is perhaps most natural to reference the energy against flat space, i.e. assign flat space the zero value of energy. It is demonstrated that the Witten--Nester integral when evaluated on solution spinors to the Sen--Witten equation (obeying appropriate boundary conditions) is essentially the Brown--York quasilocal energy with a reference point determined by the Sen--Witten spinors. For the case of round spheres in the Schwarzschild geometry, these spinors determine the flat-space reference point. A similar viewpoint is proposed for the Schwarzschild-case quasilocal energy of Dougan and Mason.