Mass Multipole Moments Research Articles

Erratum: Energy losses by gravitational radiation in inspiraling compact binaries to5/2post-Newtonian order [Phys. Rev. D54, 001417 (1996)

This paper derives the total power or energy loss rate generated in the form of gravitational waves by an inspiralling compact binary system to the five halves post-Newtonian (2.5PN) approximation of general relativity. Extending a recently developed gravitational-wave generation formalism valid for arbitrary (slowly-moving) systems, we compute the mass multipole moments of the system and the relevant tails present in the wave zone to 2.5PN order. In the case of two point-masses moving on a quasi-circular orbit, we find that the 2.5PN contribution in the energy loss rate is entirely due to tails. Relying on an energy balance argument we derive the laws of variation of the instantaneous frequency and phase of the binary. The 2.5PN order in the accumulated phase is significantly large, being grossly of the same order of magnitude as the previous 2PN order, but opposite in sign. However finite mass effects at 2.5PN order are small. The results of this paper should be useful when analyzing the data from inspiralling compact binaries in future gravitational-wave detectors like VIRGO and LIGO.

Post-1-Newtonian equations of motion for systems of arbitrarily structured bodies

We give a surface-integral derivation of post-1-Newtonian translational equations of motion for a system of arbitrarily structured bodies, including the coupling to all the bodies' mass and current multipole moments. The derivation requires only that the post-1-Newtonian vacuum field equations are satisfied in weak field regions between the bodies; the bodies' internal gravity can be arbitrarily strong. In particular, black holes are not excluded. The derivation extends previous results due to Damour, Soffel, and Xu (DSX) for weakly self-gravitating bodies in which the post-1-Newtonian field equations are satisfied everywhere. The derivation consists of a number of steps: (i) The definition of each body's current and mass multipole moments and center-of-mass world line in terms of the behavior of the metric in a weak field region surrounding the body. (ii) The definition for each body of a set of gravitoelectric and gravitomagnetic tidal moments that act on that body, again in terms of the behavior of the metric in a weak field region surrounding the body. For the special case of weakly self-gravitating bodies, our definitions of these multipole and tidal moments agree with definitions given previously by DSX. (iii) The derivation of a formula, for any given body, of the second time derivative of its mass dipole moment in terms of its other multipole and tidal moments and their time derivatives. This formula was obtained previously by DSX for weakly self-gravitating bodies. (iv) A derivation of the relation between the tidal moments acting on each body and the multipole moments and center-of-mass world lines of all the other bodies. A formalism to compute this relation was developed by DSX; we simplify their formalism and compute the relation explicitly. (v) The deduction from the previous steps of the explicit translational equations of motion, whose form has not been previously derived.

Time transfer and frequency shift to the order1/c4in the field of an axisymmetric rotating body

Within the weak-field, post-Newtonian approximation of the metric theories of gravity, we determine the one-way time transfer up to the order ${1/c}^{4},$ the unperturbed term being of order $1/c,$ and the frequency shift up to the order ${1/c}^{4}.$ We adapt the method of the world function developed by Synge to the Nordvedt-Will parametrized post-Newtonian (PPN) formalism. We get an integral expression for the world function up to the order ${1/c}^{3}$ and we apply this result to the field of an isolated, axisymmetric rotating body. We give a new procedure enabling us to calculate the influence of the mass and spin multipole moments of the body on the time transfer and the frequency shift up to the order ${1/c}^{4}.$ We obtain explicit formulas for the contributions of the mass, of the quadrupole moment and of the intrinsic angular momentum. In the case where the only PPN parameters different from zero are $\ensuremath{\beta}$ and \ensuremath{\gamma}, we deduce from these results the complete expression of the frequency shift up to the order ${1/c}^{4}.$ We briefly discuss the influence of the quadrupole moment and of the rotation of the Earth on the frequency shifts in the ESA's Atomic Clock Ensemble in Space mission.

The r-modes of rotating fluids

An analysis of the toroidal modes of a rotating fluid, by means of the dierential equations of motion, is not readily tractable. A matrix representation of the equations on a suitable basis, however, simplies the problem considerably and reveals many of its intricacies. Let be the angular velocity of the star and ('; m) be the two integers that specify a spherical harmonic function. One readily nds the followings: 1) Because of the axial symmetry of equations of motion, all modes, including the toroidal ones, are designated by a denite azimuthal number m. 2) The analysis of equations of motion in the lowest order of shows that Coriolis forces turn the neutral toroidal motions of ('; m) designation of the non-rotating fluid into a sequence of oscillatory modes with frequencies 2m='(' + 1). This much is common knowledge. One can say more, however. a) Under the Coriolis forces, the eigendisplacement vectors remain purely toroidal and carry the identication ('; m). They remain decoupled from other toroidal or poloidal motions belonging to dierent ''s. b) The eigenfrequencies quoted above are still degenerate, as they carry no reference to a radial wave number. As a result the eigendisplacement vectors, as far as their radial dependencies go, remain indeterminate. 3) The analysis of the equation of motion in the next higher order of reveals that the forces arising from asphericity of the fluid and the square of the Coriolis terms (in some sense) remove the radial degeneracy. The eigenfrequencies now carry three identications (s; '; m), say, of which s is a radial eigennumber. The eigendisplacement vectors become well determined. They still remain zero order and purely toroidal motions with a single ('; m) designation. 4) Two toroidal modes belonging to ' and '2 get coupled only at the 2 order. 5) A toroidal and a poloidal mode belonging to ' and '1, respectively, get coupled but again at the 2 order. Mass and mass-current multipole moments of the modes that are responsible for the gravitational radiation, and bulk and shear viscosities that tend to damp the modes, are worked out in much detail.

Aperture multipole moments from weak gravitational lensing

The projected mass of a gravitational lens inside (circular) apertures can be derived from the measured shear inside an annulus which is caused by the tidal field of the deflecting mass distribution. Here we show that also the multipoles of the two-dimensional mass distribution can be derived from the shear in annuli. We derive several expressions for these mass multipole moments in terms of the shear, which allow large flexibility in the choice of a radial weight function. In contrast to determining multipole moments from weak-lensing mass reconstructions, this approach allows to quantify the signal-to-noise ratio of the multipole moments directly from the observed galaxy ellipticities, and thus to estimate the significance of the multipole detection. Radial weight functions can therefore be chosen such as to optimize the significance of the detection given an assumed radial mass profile. Application of our formulae to numerically simulated clusters demonstrates that the quadrupole moment of realistic cluster models can be detected with high signal-to-noise ratio S/N; in about 85 per cent of the simulated cluster fields S/N >~ 3. We also show that the shear inside a circular annulus determines multipole moments inside and outside the annulus. This is relevant for clusters whose central region is too bright to allow the observation of the shear of background galaxies, or which extend beyond the CCD. We also generalize the aperture mass equation to the case of `radial' weight functions which are constant on arbitrarily-shaped curves which are not necessarily self-similar.

Energy losses by gravitational radiation in inspiraling compact binaries to 5/2 post-Newtonian order.

This paper derives the total power or energy loss rate generated in the form of gravitational waves by an inspiraling compact binary system to the $\frac{5}{2}$ post-Newtonian (2.5PN) approximation of general relativity. Extending a recently developed gravitational-wave generation formalism valid for arbitrary (slowly moving) systems, we compute the mass multipole moments of the system and the relevant tails present in the wave zone to 2.5PN order. In the case of two point masses moving on a quasicircular orbit, we find that the 2.5PN contribution in the energy loss rate is entirely due to tails. Relying on an energy balance argument we derive the laws of variation of the instantaneous frequency and phase of the binary. The 2.5PN order in the accumulated phase is significantly large, being grossly of the same order of magnitude as the previous 2PN order, but opposite in sign. However, finite mass effects at 2.5PN order are small. The results of this paper should be useful when analyzing the data from inspiraling compact binaries in future gravitational-wave detectors such as VIRGO and LIGO.

Generalizations of the Kerr and Kerr-Newman metrics possessing an arbitrary set of mass-multipole moments

The authors present in a concise analytical form two asymptotically flat metrics describing the superposition of the Kerr solution with an arbitrary static vacuum Weyl field which differ in their angular momentum distributions. They are then used for the construction of two asymptotically flat generalizations of the Kerr-Newman spacetime possessing the full set of mass-multipole moments able to describe the exterior gravitational field of a charged rotating arbitrary axisymmetric mass.

Multipole analysis for electromagnetism and linearized gravity with irreducible Cartesian tensors.

The relativistic time-dependent multipole expansion for electromagnetism and linearized gravity in the region outside a spatially compact source has been obtained directly using the formalism of irreducible Cartesian (i.e., symmetric trace-free) tensors. In the electromagnetic case, our results confirm the validity of the results obtained earlier by Campbell, Macek, and Morgan using the Debye potential formalism. However, in the more complicated linearized gravity case, the greater algebraic transparence of the Cartesian multipole approach has allowed us to obtain, for the first time, fully correct closed-form expressions for the time-dependent mass and spin multipole moments (the results of Campbell et al. for the mass moments turning out to be incorrect). The first two terms in the slow-motion expansion of the gravitational moments are explicitly calculated and shown to be equivalent to earlier results by Thorne and by Blanchet and Damour.

Multipole Moments in General Relativity —Static and Stationary Vacuum Solutions—

In the first part of this paper, we review the most important properties of the axisymmetric and stationary solutions of Einstein's vacuum field equations, the solution generating techniques developed in the last few years the relativistic definitions of multipole moments, and their use for obtaining and investigating static and stationary axisymmetric vacuum solutions. In the second part, we derive the general static axisymmetric vacuum solution in prolate spheroidal coordinates. Finally, this solution with all mass multipole moments is generalized to include a further parameter related to the rotation of the mass. This stationary solution can be used in order to describe the exterior gravitational field of a rotating deformed mass.

Celestial coordinate reference systems in curved space-time

In the weak-field and slow-motion approximation of the General Relativity theory the relativistic theory of construction of nonrotating harmonic coordinate reference systems is developed. The general case of an isolated astronomicalN-body system is considered. The bodies are assumed to consist of a perfect fluid and to possess any number of the internal mass and current multipole moments characterizing the internal structure and own gravitational field of the bodies. The description of the coordinate reference systems and gravitational field is realized by the specific forms of the metric tensor. A metric is determined by the method of the Post-Newtonian Approximations (PNA) from the inhomogeneous Einstein equations under the harmonic coordinate condition. We have obtained two different specific forms of a metric, which are related to the inertial and quasiinertial coordinate reference systems. The metric in inertial coordinates is a near-zone solution of the Einstein equations for anN-body system. The metric in quasi-inertial coordinates is a solution of Einstein equations in the body's neighborhood, which is a world tube surrounding the body under consideration and extending up to another nearest body. The coordinate transformation between the inertial and quasiinertial reference systems is derived by a matching of both solutions in the body's neighborhood. A new method is proposed for construction of the body's proper reference system. This coordinate system has an origin which moves along a nongeodesic (in the general case) worldline of the body's center of inertia. The proper reference system is used for derivation of the Newtonian equations of translational and rotational motion of the body. The equations give us exhaustive information about the nonlinear Newtonian interaction between gravitational fields of the bodies in terms of internal mass multipole moments. Finally, the coordinate transformation between the inertial and proper reference systems is discussed in the first PNA.

Multipole moments in general relativity

It is proved that a stationary space-time for which all the angular momentum multipole moments vanish is static and that a static space-time for which all the mass multipole moments vanish is flat.

Vibrations of non-spherical nuclei

The vibrational levels of non-spherical nuclei are studied by a semi-classical method analogous to the cranking model for rotation. The method of Green's one-particle functions is used to describe the superfluidity of nuclei. A dispersion equation is obtained for determining the eigenvalues of frequencies of the proper vibrations of the mass multipole moments. The mass coefficient for multipole vibrations, obtained with the help of this equation, contains an additional term which makes it possible to derive the hydrodynamic limit.