William L. Burke

Bio: William L. Burke is an academic researcher. The author has contributed to research in topics: Differential form & Gravitational field. The author has an hindex of 2, co-authored 2 publications receiving 168 citations.

Gravitational Radiation Damping of Slowly Moving Systems Calculated Using Matched Asymptotic Expansions

Abstract: This paper treats the slow‐motion approximation for radiating systems as a problem in singular perturbations. By using the method of matched asymptotic expansions, we can construct approximations valid both in the near zone and the wave zone. The outgoing‐wave boundary condition applied to the wave‐zone expansion leads, by matching, to a unique and easily calculable radiation resistance in the near zone. The method is developed and illustrated with model problems from mechanics and electromagnetism; these should form a useful and accessible introduction to the method of matched asymptotic expansions. The method is then applied to the general relativistic problem of gravitational radiation from gravitationally bound systems, where a significant part of the radiation can be attributed to nonlinear terms in the expansion of the metric. This analysis shows that the formulas derived from the standard linear approximation remain valid for gravitationally bound systems. In particular, it shows that, according to general relativity, bodies in free‐fall motion do indeed radiate. These results do not depend upon any definition of gravitational field energy.

Manifestly parity invariant electromagnetic theory and twisted tensors

Abstract: We develop here the calculus of twisted tensors and in particular twisted differential forms, treating them as tensors with complementary orientations. These geometrical objects give us the proper language for electromagnetic theory in a 3‐space plus time representation. The parity properties of the fields are simplified and many graphical illustrations are given.

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.

Abstract: The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail eects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

Gravitational radiation reaction to a particle motion

Abstract: A small mass particle traveling in a curved spacetime is known to trace a background geodesic in the lowest order approximation with respect to the particle mass. In this paper, we discuss the leading order correction to the equation of motion of the particle, which presumably describes the effect of gravitational radiation reaction. We derive the equation of motion in two different ways. The first one is an extension of the well-known formalism by DeWitt and Brehme developed for deriving the equation of motion of an electrically charged particle. Constructing the conserved rank two symmetric tensor, and integrating it over the interior of the world tube surrounding the orbit, we derive the equation of motion. Although the calculation in this approach is straightforward, it contains less rigorous points. In contrast to the electromagnetic case, in which there are two different charges, i.e., the electric charge and the mass, the gravitational counterpart has only one charge. This fact prevents us from using the same renormalization scheme that was used in the electromagnetic case. In order to overcome this difficulty, we put an ansatz in evaluating the integral ofthe conserved tensor on a three spatial volume which defines the momentum of the small particle. To make clear the subtlety in the first approach, we then consider the asymptotic matching of two different schemes, i.e., the internal scheme in which the small particle is represented by a spherically symmetric black hole with tidal perturbations and the external scheme in which the metric is given by small perturbations on the given background geometry. The equation of motion is obtained from the consistency condition of the matching. We find that in both ways the same equation of motion is obtained. The resulting equation of motion is analogous to that derived in the electromagnetic case. We discuss implications of this equation of motion. PACS number(s): 04.30.Db, 04.25.-g

Black Hole Binary Dynamics from the Double Copy and Effective Theory

Abstract: We describe a systematic framework for computing the conservative potential of a compact binary system using modern tools from scattering amplitudes and effective field theory. Our approach combines methods for integration and matching adapted from effective field theory, generalized unitarity, and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. With these methods we derive the third post-Minkowskian correction to the conservative two-body Hamiltonian for spinless black holes. We describe in some detail various checks of our integration methods and the resulting Hamiltonian.