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Spherical multipole moments

About: Spherical multipole moments is a research topic. Over the lifetime, 660 publications have been published within this topic receiving 19782 citations.


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TL;DR: The current state of the art on post-Newtonian methods as applied to the dynamics and gravitational radiation of general matter sources (including the radiation reaction back onto the source) and inspiralling compact binaries is presented.
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.

2,067 citations

Journal ArticleDOI
TL;DR: In this article, a unified notation for the multipole formalisms for gravitational radiation is presented, which includes scalar, vector, and tensor spherical harmonics used in the general relativity literature, including Regge-Wheeler harmonics, the symmetric, trace-free ("STF") tensors of Sachs and Pirani, the Newman-Penrose spin-weighted harmonics and the Mathews-Zerilli Clebsch-Gordan-coupled harmonics.
Abstract: This paper brings together, into a single unified notation, the multipole formalisms for gravitational radiation which various people have constructed. It also extends the results of previous workers. More specifically: Part One of this paper reviews the various scalar, vector, and tensor spherical harmonics used in the general relativity literature—including the Regge-Wheeler harmonics, the symmetric, trace-free ("STF") tensors of Sachs and Pirani, the Newman-Penrose spin-weighted harmonics, and the Mathews-Zerilli Clebsch-Gordan-coupled harmonics—which include "pure-orbital" harmonics and "pure-spin" harmonics. The relationships between the various harmonics are presented. Part One then turns attention to gravitational radiation. The concept of "local wave zone" is introduced to facilitate a clean separation of "wave generation" from "wave propagation." The generic radiation field in the local wave zone is decomposed into multipole components. The energy, linear momentum, and angular momentum in the waves are expressed as infinite sums of multipole contributions. Attention is then restricted to sources that admit a nonsingular, spacetime-covering de Donder coordinate system. (This excludes black holes.) In such a coordinate system the multipole moments of the radiation field are expressed as volume integrals over the source. For slow-motion systems, these source integrals are re-expressed as infinite power series in L / λ≡(size of source ) / (reduced wavelength of waves ). The slow-motion source integrals are then specialized to systems with weak internal gravity to yield (i) the standard Newtonian formulas for the multipole moments, (ii) the post-Newtonian formulas of Epstein and Wagoner, and (iii) post-post-Newtonian formulas. Part Two of this paper derives a multipole-moment wave-generation formalism for slow-motion systems with arbitrarily strong internal gravity, including systems that cannot be covered by de Donder coordinates. In this formalism one calculates, by any means, the source's instantaneous, near-zone, external gravitational field as a solution of the time-independent Einstein field equations. One then reads off of this near-zone field the source's instantaneous multipole moments; and one plugs those time-evolving moments into the standard radiation formulae given in Part One of this paper. As building blocks for this formalism, Part Two also does the following things: (1) In the linearized theory of gravity, for the vacuum exterior of an isolated system, it derives the general solution of the field equations (a result due to Sachs, Bergmann, and Pirani). (2) In full nonlinear general relativity, for the vacuum near-zone exterior of an isolated system, it derives the structure of the general solution of the Einstein field equations. That structure is expressed as a sum of products of multipole contributions. It also matches this near-zone field onto an outgoing-wave radiation field. (3) In full nonlinear general relativity, for the vacuum exterior of a stationary isolated system, (a) it presents a definition of multipole moments which meshes naturally with gravitational-wave theory; (b) it introduces the concept of "asymptotically Cartesian and mass centered" (ACMC) coordinate systems; and (c) it shows how to deduce the multipole moments of a source from the form of its metric in an ACMC coordinate system. As an example, the lowest few (l ≤ 3) multipole moments of the Kerr metric are computed.

1,253 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extend the theory of dipole moments in crystalline insulators to higher multipole moments, and describe the topological invariants that protect these moments.
Abstract: We extend the theory of dipole moments in crystalline insulators to higher multipole moments. In this paper, we expand in great detail the theory presented in Ref. 1, and extend it to cover associated topological pumping phenomena, and a novel class of 3D insulator with chiral hinge states. In quantum-mechanical crystalline insulators, higher multipole bulk moments manifest themselves by the presence of boundary-localized moments of lower dimension, in exact correspondence with the electromagnetic theory of classical continuous dielectrics. In the presence of certain symmetries, these moments are quantized, and their boundary signatures are fractionalized. These multipole moments then correspond to new SPT phases. The topological structure of these phases is described by "nested" Wilson loops, which reflect the bulk-boundary correspondence in a way that makes evident a hierarchical classification of the multipole moments. Just as a varying dipole generates charge pumping, a varying quadrupole generates dipole pumping, and a varying octupole generates quadrupole pumping. For non-trivial adiabatic cycles, the transport of these moments is quantized. An analysis of these interconnected phenomena leads to the conclusion that a new kind of Chern-type insulator exists, which has chiral, hinge-localized modes in 3D. We provide the minimal models for the quantized multipole moments, the non-trivial pumping processes and the hinge Chern insulator, and describe the topological invariants that protect them.

1,045 citations

Journal ArticleDOI
TL;DR: In this article, a formalism for analyzing a full-sky temperature and polarization map of the cosmic microwave background is presented, where temperature maps are analyzed by expanding over the set of spherical harmonics to give multipole moments of the two-point correlation function.
Abstract: We present a formalism for analyzing a full-sky temperature and polarization map of the cosmic microwave background. Temperature maps are analyzed by expanding over the set of spherical harmonics to give multipole moments of the two-point correlation function. Polarization, which is described by a second-rank tensor, can be treated analogously by expanding in the appropriate tensor spherical harmonics. We provide expressions for the complete set of temperature and polarization multipole moments for scalar and tensor metric perturbations. Four sets of multipole moments completely describe isotropic temperature and polarization correlations; for scalar metric perturbations one set is identically zero, giving the possibility of a clean determination of the vector and tensor contributions. The variance with which the multipole moments can be measured in idealized experiments is evaluated, including the effects of detector noise, sky coverage, and beam width. Finally, we construct coordinate-independent polarization two-point correlation functions, express them in terms of the multipole moments, and derive small-angle limits.

896 citations

Journal ArticleDOI
TL;DR: In this paper, a review of available information on molecular quadrupole and higher moments is presented and a theorem is proved which shows that only one independent scalar quantity is required to determine a molecular electric multipole tensor of rank p for molecules with an n-fold axis of symmetry where p < n.
Abstract: A summary and critical review of available information on molecular quadrupole and higher moments is presented. A theorem is also proved which shows that only one independent scalar quantity is required to determine a molecular electric multipole tensor of rank p for molecules with an n-fold axis of symmetry where p < n.

733 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202212
20211
20181
201718
201617