Mass Multipole Moments Research Articles

Multipole analysis on stationary massive vector and symmetric tensor fields with irreducible Cartesian tensors

The multipole expansions for massive vector and symmetric tensor fields in the region outside spatially compact stationary sources are obtained by using the symmetric and trace-free formalism in terms of the irreducible Cartesian tensors, and the closed-form expressions for the source multipole moments are provided. The expansions show a Yukawa-like dependence on the massive parameters of the fields, and the integrals of the stationary source multipole moments are all modulated by a common radial factor. For stationary massive vector field, there are two types of ‘magnetic’ multipole moments, among which one is the generalization of that of the magnetostatic field, and another, being an additional set of multipole moments of the stationary massive vector field, can not be transformed away. As to the stationary massive symmetric tensor field, its multipole expansion is presented when the trace of its spatial part is specified, where besides the counterparts of the mass and spin multipole moments of massless symmetric tensor field, three additional sets of multipole moments also appear. The multipole expansions of the tensor field under two typical cases are discussed, where it is shown that if the spatial part of the tensor field is trace-free, the monopole and dipole moments in the corresponding expansion will vanish.

Spherical harmonics representation of the gravitational phase shift

We investigate the general relativistic phase of an electromagnetic wave as it propagates in the gravitational field of the Earth, which is modeled as an isolated, weakly aspherical gravitating body. We introduce coordinate systems to describe light propagation in the Earth's vicinity along with the relevant coordinate transformations, and discuss the transformations between proper and coordinate times. We represent the Earth's gravitational field using Cartesian symmetric trace-free (STF) mass multipole moments. The light propagation equation is solvable along the trajectory of a light ray to all STF orders $\ensuremath{\ell}$. Although we focus primarily on the quadrupole ($\ensuremath{\ell}=2$), octupole ($\ensuremath{\ell}=3$), and hexadecapole ($\ensuremath{\ell}=4$) cases, our approach is valid to all orders. We express the STF moments via spherical harmonic coefficients of various degree and order, ${C}_{\ensuremath{\ell}k},{S}_{\ensuremath{\ell}k}$. The result is the gravitational phase shift expressed in terms of the spherical harmonics. These results are new. We also consider contributions due to tides and the Earth's rotation. We estimate the characteristic magnitudes of each term of the resulting overall gravitational phase shift. The terms assessed are either large enough to impact current-generation clocks or will become significant as future-generation clocks offer greater sensitivity. Our formulation is useful for many practical and scientific applications, including space-based time and frequency transfers, relativistic geodesy and navigation, as well as quantum communication links and space-based tests of fundamental physics.

Is a black hole shadow a reliable test of the no-hair theorem?

Capturing the image of the shadow cast by the event horizon of an illuminated black hole is, at the most basic level, an experiment of extreme light deflection in a strongly curved spacetime. As such, the properties of an imaged shadow can be used to probe the general relativistic Kerr nature of astrophysical black holes. As an example of this prospect, it is commonly asserted that a shadow can test the validity of the theory's famous ``no hair theorem'' for the black hole's mass and spin multipole moments. In this paper, we assess this statement by calculating the shadow's equatorial radius in spacetimes with an arbitrary multipolar structure and within a slow rotation approximation. We find that when moments higher than the quadrupole are taken into account, the shadow acquires a high degree of degeneracy as a function of the deviation from the Kerr multipole moments. The results of our analysis suggest that dark objects with strongly non-Kerr multipolar structure could nevertheless produce a Kerr-like shadow with its characteristic quasicircular shape.

Efficient trace-free decomposition of symmetric tensors of arbitrary rank

Symmetric trace-free tensors are used in many areas of physics, including electromagnetism, relativistic celestial mechanics and geodesy, as well as in the study of gravitational radiation and gravitational lensing. Their use allows integration of the relevant wave propagation equations to arbitrary order. We present an improved iterative method for the trace-free decomposition of symmetric tensors of arbitrary rank. The method can be used both in coordinate-free symbolic derivations using a computer algebra system and in numerical modeling. We obtain a closed-form representation of the trace-free decomposition in arbitrary dimensions. To demonstrate the results, we compute the coordinate combinations representing the symmetric trace-free (STF) mass multipole moments for rank 5 through 8, not readily available in the literature.

Tidally induced multipole moments of a nonrotating black hole vanish to all post-Newtonian orders

The tidal Love numbers of a black hole vanish, and this is often taken to imply that the hole's tidally induced multipole moments vanish also. An obstacle to establishing a link between these statements is that the multipole moments of individual bodies are not defined in general relativity, when the bodies are subjected to a mutual gravitational interaction. In a previous publication [Phys. Rev. D 103, 064023 (2021)] I promoted the view that individual multipole moments can be defined when the mutual interaction is sufficiently weak to be described by a post-Newtonian expansion. In this view, a compact body is perceived far away as a skeletonized post-Newtonian object with a multipole structure, and the multipole moments can then be related to the body's Love numbers. I expand on this view, and demonstrate that all static, tidally induced, mass multipole moments of a nonrotating black hole vanish to all post-Newtonian orders. The proof rests on a perturbative solution to the Einstein-Maxwell equations that describes an electrically charged particle placed in the presence of a charged black hole. The gravitational attraction between particle and black hole is balanced by electrostatic repulsion, and the system is in an equilibrium state. The particle provides a tidal environment to the black hole, and the multipole moments vanish for this environment. I argue that the vanishing is robust, and applies to all slowly-varying tidal environments. The black hole's charge can be as small as desired (though not identically zero); by continuity, the multipole moments of an electrically neutral black hole will continue to vanish.

The current-type quadrupole moment and gravitational-wave mode (ℓ, m) = (2, 1) of compact binary systems at the third post-Newtonian order

Up to the third post-Newtonian (3PN) order, we compute (i) the current-type quadrupole moment of (non-spinning) compact binary systems, as well as (ii) the corresponding gravitational-wave mode (2, 1) (constituting a 3.5PN correction in the waveform). Moreover, at this occasion, (iii) we recompute and confirm the previous calculation of the mass-type octupole moment to 3PN order. The ultra-violet divergences due to the point-like nature of the source are treated by means of dimensional regularization. This entails generalizing the definition of the irreducible mass and current multipole moments of an isolated PN source in d spatial dimensions. In particular, we find that the current type moment has the symmetry of a particular mixed Young tableau and that, in addition, there appears a third type of moment which is however inexistent in 3 spatial dimensions.

Multipole expansion of gravitational waves: from harmonic to Bondi coordinates

We transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation from harmonic (de Donder) coordinates to radiative Newman-Unti (NU) coordinates. To linearized order, we obtain the NU metric as a functional of the mass and current multipole moments of the source, valid all-over the exterior region of the source. Imposing appropriate boundary conditions we recover the generalized Bondi-van der Burg-Metzner-Sachs residual symmetry group. To quadratic order, in the case of the mass-quadrupole interaction, we determine the contributions of gravitational-wave tails in the NU metric, and prove that the expansion of the metric in terms of the radius is regular to all orders. The mass and angular momentum aspects, as well as the Bondi shear, are read off from the metric. They are given by the radiative quadrupole moment including the tail terms.

The multipolar structure of fuzzballs

We extend and refine a general method to extract the multipole moments of arbitrary stationary spacetimes and apply it to the study of a large family of regular horizonless solutions to mathcal{N} = 2 four-dimensional supergravity coupled to four Abelian gauge fields. These microstate geometries can carry angular momentum and have a much richer multipolar structure than the Kerr black hole. In particular they break the axial and equatorial symmetry, giving rise to a large number of nontrivial multipole moments. After studying some analytical examples, we explore the four-dimensional parameter space of this family with a statistical analysis. We find that microstate mass and spin multipole moments are typically (but not always) larger that those of a Kerr black hole with the same mass and angular momentum. Furthermore, we find numerical evidence that some invariants associated with the (dimensionless) moments of these microstates grow monotonically with the microstate size and display a global minimum at the black-hole limit, obtained when all centers collide. Our analysis is relevant in the context of measurements of the multipole moments of dark compact objects with electromagnetic and gravitational-wave probes, and for observational tests to distinguish fuzzballs from classical black holes.

Multipole moments of compact objects with NUT charge: Theoretical and observational implications

We derive the multipole moments of the Kerr-NUT black hole spacetime using the Geroch-Hansen formalism, even though the spacetime is not asymptotically flat. Intriguingly, in the presence of the NUT charge, the absence of reflection symmetry about the equatorial plane, leads to mass and spin multipole moments of all orders, in stark contrast to Kerr-like spacetimes. This leads to a drastic departure of the multipolar structure of a compact object with NUT charge, whose implications for gravitational wave observations have been explored. Our analysis of multipole moments for the Kerr-NUT spacetime is also in tune with the Thorne's approach.

Probing the nature of central objects in extreme-mass-ratio inspirals with gravitational waves

We extend the work of Ryan on mapping the spacetime of the central object of an extreme mass-ratio inspiral (EMRI) by using gravitational waves (GWs) emitted by the system, which may be observed in future missions such as LISA. Whether the central object is a black hole or not can be probed by observing the phasing of these waves, which carry information about its mass and spin multipole moments. We go beyond the phase terms found by Ryan, which were obtained in the quadrupolar approximation of the point-particle limit, and derive terms up to the fifth post-Newtonian (PN) order. Since corrections due to horizon absorption (i.e., if the central object is a black hole) and tidal heating appear by that order, at 2.5PN and 5PN, respectively, we include them here. Corrections due to the motion of the central object, which was addressed only partially by Ryan, are included as well. Additionally, we obtain the contribution of the higher order radiative multipole moments. For the tidal interaction, our results have been derived in the approximation of the Newtonian tidal field. Therefore, in the potential for tidal field only the contribution due to the mass of the central object has been included as well. Using these results we argue that it might be possible for LISA to probe if the central object in an EMRI has a horizon or not. We also discuss how our results can be used to test the No-hair theorem from the inspiral phase of such systems.

Gravitational multipole moments from Noether charges

We define the mass and current multipole moments for an arbitrary theory of gravity in terms of canonical Noether charges associated with specific residual transformations in canonical harmonic gauge, which we call multipole symmetries. We show that our definition exactly matches Thorne’s mass and current multipole moments in Einstein gravity, which are defined in terms of metric components. For radiative configurations, the total multipole charges — including the contributions from the source and the radiation — are given by surface charges at spatial infinity, while the source multipole moments are naturally identified by surface integrals in the near-zone or, alternatively, from a regularization of the Noether charges at null infinity. The conservation of total multipole charges is used to derive the variation of source multipole moments in the near-zone in terms of the flux of multipole charges at null infinity.

Third post-Newtonian gravitational waveforms for compact binary systems in general orbits: Instantaneous terms

We compute the instantaneous contributions to the spherical harmonic modes of gravitational waveforms from compact binary systems in general orbits up to the third post-Newtonian order. We further extend these results for compact binaries in quasi-elliptical orbits using the 3PN quasi-Keplerian representation of the conserved dynamics of compact binaries in eccentric orbits. Using the multipolar post-Minkowskian formalism, starting from the different mass and current type multipole moments, we compute the spin weighted spherical harmonic decomposition of the instantaneous part of the gravitational waveform. These are terms which are functions of the retarded time and do not depend on the history of the binary evolution. Together with the hereditary part, which depends on the binary's dynamical history, these waveforms form the basis for construction of accurate templates for the detection of gravitational wave signals from binaries moving in quasi-elliptical orbits.

Tidal deformations of spinning black holes in Bowen–York initial data

We study the tidal deformations of the shape of a spinning black hole horizon due to a binary companion in the Bowen–York initial data set. We use the framework of quasi-local horizons and identify a black hole by marginally outer trapped surfaces. The intrinsic horizon geometry is specified by a set of mass and angular-momentum multipole moments and , respectively. The tidal deformations are described by the change in these multipole moments caused by an external perturbation. This leads us to define two sets of dimensionless numbers, the tidal coefficients for and , which specify the deformations of a black hole with a binary companion. We compute these tidal coefficients in a specific model problem, namely the Bowen–York initial data set for binary black holes. We restrict ourselves to axisymmetric situations and to small spins. Within this approximation, we analytically compute the conformal factor, the location of the marginally trapped surfaces, and finally the multipole moments and the tidal coefficients.

The gravitational time delay in the field of a slowly moving body with arbitrary multipoles

We calculate the time delay of light in the gravitational field of a slowly moving body with arbitrary multipoles (mass and spin multipole moments) by the Time-Transfer-Function (TTF) formalism. The parameters we use, first introduced by Kopeikin for a gravitational source at rest, make the integration of the TTF very elegant and simple. Results completely coincide with expressions from the literature. The results for a moving body (with constant velocity) with complete multipole-structure are new, according to our knowledge.

Testing the existence of regions of stable orbits at small radii around black hole candidates

Black hole candidates in X-ray binary systems and at the centers of galaxies are expected to be the Kerr black holes of general relativity, but the actual nature of these objects has yet to be verified. In this paper, we consider the possibility that they are exotic compact objects and we describe their exterior gravitational field with a subclass of the Manko-Novikov metrics, which are exact solutions of the vacuum Einstein's equations and can describe the spacetime geometry around bodies with arbitrary mass-multipole moments. We point out that around a Manko-Novikov object there may exist many disconnected nonplunging regions at small radii, with no counterpart in the Kerr background, and that their existence may be tested. For instance, in the presence of an accretion disk, they may be filled by the accreting gas, forming a ring structure that might remind the one of the rings of Saturn. We suggest that the existence of these regions may have a clear observational signature in the waveform of the gravitational radiation emitted by an extreme-mass-ratio inspiral: in the last stage of the inspiral, the waveform would be the combination of ``regular chirps,'' produced when the small object orbits in one of the nonplunging regions, and ``bursts,'' which are released when the small object jumps from a nonplunging region to another one at smaller radii. Our conclusions are supported by some numerical calculations of trajectories in the geodesic approximation, in which a particle plunges from the innermost stable circular orbit and then seems to get trapped in the potential well at smaller radii.

Carter-like constants of motion in the Newtonian and relativistic two-center problems

In Newtonian gravity, a stationary axisymmetric system admits a third, Carter-like constant of motion if its mass multipole moments are related to each other in exactly the same manner as for the Kerr black-hole spacetime. The Newtonian source with this property consists of two point masses at rest a fixed distance apart. The integrability of motion about this source was first studied in the 1760s by Euler. We show that the general relativistic analog of the Euler problem, the Bach–Weyl solution, does not admit a Carter-like constant of motion, first, by showing that it does not possess a non-trivial Killing tensor, and secondly, by showing that the existence of a Carter-like constant for the two-center problem fails at the first post-Newtonian order.

A new method for computing quasi-local spin and other invariants on marginally trapped surfaces

We accurately compute the scalar 2-curvature, Weyl scalars, associated quasi-local spin, mass and higher multipole moments on marginally trapped surfaces in numerical 3+1 simulations. To determine the quasi-local quantities, we introduce a new method which requires a set of invariant surface integrals, allowing for surface grids of a few hundred points only. The new technique circumvents solving the Killing equation and is also an alternative to approximate Killing vector fields. We apply the method to a perturbed non-axisymmetric black hole ringing down to Kerr and compare the quasi-local spin with other methods that use Killing vector fields, coordinate vector fields, quasi-normal ringing and properties of the Kerr metric on the surface. Interesting is the agreement with the spin of approximate Killing vector fields during the phase of perturbed axisymmetry. Additionally, we introduce a new coordinate transformation, adapting spherical coordinates to any two points on the sphere such as the two minima of the scalar 2-curvature on axisymmetric trapped surfaces.

Tidal dynamics of extended bodies in planetary systems and multiple stars

<i>Context. <i/>With the discovery during the past decade of a large number of extrasolar planets orbiting their parent stars at distances lower than 0.1 astronomical unit (and the launch and the preparation of dedicated space missions such as CoRoT and KEPLER), with the position of inner natural satellites around giant planets in our Solar System and with the existence of very close but separated binary stars, tidal interaction has to be studied carefully.<i>Aims. <i/>This interaction is usually studied with a punctual approximation for the tidal perturber. The purpose of this paper is to examine the step beyond this traditional approach by considering the tidal perturber as an extended body. To achieve this, we studied the gravitational interaction between two extended bodies and, more precisely, the interaction between mass multipole moments of their gravitational fields and the associated tidal phenomena.<i>Methods. <i/>We use cartesian symmetric trace free tensors, their relation with spherical harmonics and Kaula's transform enables us to analytically derive the tidal and mutual interaction potentials, as well as the associated disturbing functions in extended body systems.<i>Results. <i/>The tidal and mutual interaction potentials of two extended bodies are derived. In addition, the external gravitational potential of such a tidally disturbed extended body is obtained, using the Love number theory, as well as the associated disturbing function. Finally, the dynamical evolution equations for such a system are given in their more general form without any linearization. We also compare, under a simplified assumption, this formalism to the punctual case. We show that the non-punctual terms have to be taken into account for strongly deformed perturbers (<i>J<i/><sub>2<sub/> <i>≥<i/> 10<sup>-2<sup/>) in very close systems (<i>a<i/><sub>B<sub/>/<i>R<i/><sub>B<sub/><i>≤<i/>5).<i>Conclusions. <i/>We show how to derive the dynamical equations for the gravitational and tidal interactions between extended bodies and associated dynamics. The conditions for applying this formalism are given.

Publisher’s Note: Influence of mass multipole moments on the deflection of a light ray by an isolated axisymmetric body [Phys. Rev. D77, 044029 (2008)

Future space astrometry missions are planned to measure positions and/or parallaxes of celestial objects with an accuracy of the order of the microarcsecond. At such a level of accuracy, it will be indispensable to take into account the influence of the mass multipole structure of the giant planets on the bending of light rays. Within the parametrized post-Newtonian formalism, we present an algorithmic procedure enabling to determine explicitly this influence on a light ray connecting two points located at a finite distance. Then we specialize our formulae in the cases where 1) the light source is located at space infinity, 2) both the light source and the observer are located at space infinity. We examine in detail the cases where the unperturbed ray is in the equatorial plane or in a meridian plane.

Influence of mass multipole moments on the deflection of a light ray by an isolated axisymmetric body

Future space astrometry missions are planned to measure positions and/or parallaxes of celestial objects with an accuracy of the order of the microarcsecond. At such a level of accuracy, it will be indispensable to take into account the influence of the mass multipole structure of the giant planets on the bending of light rays. Within the parametrized post-Newtonian formalism, we present an algorithmic procedure enabling us to determine explicitly this influence on a light ray connecting two points located at a finite distance. Then we specialize our formulas in the cases where (1) the light source is located at space infinity, (2) both the light source and the observer are located at space infinity. We examine in detail the cases where the unperturbed ray is in the equatorial plane or in a meridian plane.0.