Mass Multipole Research Articles

Mass octupole and current quadrupole corrections to gravitational wave emission from close hyperbolic encounters

Abstract In this paper, we study the next-to-leading order corrections in the mass multipole expansion, i.e. the mass octupole and current quadrupole, to gravitational wave production by close hyperbolic encounters of compact objects. We find that the signal is again, as in the simple quadrupole case, a burst event with the majority of the released energy occurring during the closest approach. In particular, we investigate the relative contribution to the power, both in the time and frequency domains, and total energy emitted by each order in the mass multipole expansion in gravitational waves. To do so, we include in the quadrupole term its first order post-Newtonian correction, giving this a contribution to the power of the same order as that of the mass octupole and the current quadrupole. We find specific configurations of systems where these corrections could be important and should be taken into account when analysing burst events.

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.

Multipole moments on the common horizon in a binary-black-hole simulation

We construct the covariantly defined multipole moments on the common horizon of an equal-mass, nonspinning, quasicircular binary-black-hole system. We see a strong correlation between these multipole moments and the gravitational waveform. We find that the multipole moments are well described by the fundamental quasinormal modes at sufficiently late times. For each nonzero multipole moment with $\ensuremath{\ell}\ensuremath{\le}6$, at least two fundamental quasinormal modes of different $\ensuremath{\ell}$ are detectable in the best model. These models provide faithful estimates of the true mass and spin of the remnant black hole. We also show that by including overtones, the $\ensuremath{\ell}=m=2$ mass multipole moment admits an excellent quasinormal-mode description at all times after the merger. This demonstrates the perhaps surprising power of perturbation theory near the merger.

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.

Static magnetic dipole solutions of Einstein–Maxwell equations from the soliton solutions of a Kerr object

Using the solution generating techniques of Das and Chaudhuri (Pramana J. Phys. 40, 277 (1993); Pramana J. Phys. 58, 449 (2002)), static magnetovac solutions of Einstein–Maxwell equations in general relativity are constructed from the stationary gravitational soliton solutions of Einstein's field equations corresponding to a Kerr object. The techniques followed in the present paper have not been much used in the literature so far although the results are obtained in a straightforward way. The stationary gravitational two-soliton solutions of Einstein's field equations for a Kerr object are generated using the soliton technique of Belinskii and Zakharov (Sov. Phys. JETP 48, 985 (1978); Sov. Phys. JETP 50, 1 (1979)). In the next step, following the procedure of Das and Chaudhuri, mentioned above, static magnetovac solutions of Einstein–Maxwell field equations corresponding to the generated stationary two-soliton solutions of the Kerr object are constructed. The solutions are found to be well behaved at spatial infinity and contain monopole, dipole, and other higher mass multipoles. The mass and the dipole moment of the source are evaluated. It is shown that by redefining some constants appearing in the solutions, Bonnor's (Z. Phys. 190, 444 (1966)) magnetic dipole solutions are faithfully reproduced.

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.

Black holes lessons from multipole ratios

We explain in detail how to calculate the gravitational mass and angular momentum multipoles of the most general non-extremal four-dimensional black hole with four magnetic and four electric charges. We also calculate these multipoles for generic supersymmetric four-dimensional microstate geometries and multi-center solutions. Both for Kerr black holes and BPS black holes many of these multipoles vanish. However, if one embeds these black holes in String Theory and slightly deforms them, one can calculate an infinite set of ratios of vanishing multipoles which remain finite as the deformation is taken away, and whose values are independent of the direction of deformation. For supersymmetric black holes, we can also compute these ratios by taking the scaling limit of multi-center solutions, and for certain black holes the ratios computed using the two methods agree spectacularly. For the Kerr black hole, these ratios pose strong constraints on the parameterization of possible deviations away from the Kerr geometry that should be tested by future gravitational wave interferometers.

Post-Newtonian Lagrangian of an N -body system with arbitrary mass and spin multipoles

The present paper derives the post-Newtonian Lagrangian of translational motion of N arbitrary-structured bodies with all mass and spin multipoles in a scalar-tensor theory of gravity. The multipoles depend on time and evolve in accordance with their own dynamic equations of motion. The Lagrangian is retrieved from the post-Newtonian equations of motion by solving the inverse problem of the Lagrangian mechanics and generalizes a well-known Lagrangian of pole-dipole-quadrupole massive particles to the particles of higher multipolarity. Analytic treatment of the higher-order multipole contributions is important for more rigorous computation of gravitational waveform of inspiralling compact binaries at the latest stage of their orbital evolution before merger when tidal and rotational deformations of stars are no longer small and rapidly change in time. The Lagrangian of an N-body system with arbitrary mass and spin multipoles is instrumental for formulation of the post-Newtonian conservation laws of energy, momenta and the integrals of the center of mass.

Covariant equations of motion of extended bodies with arbitrary mass and spin multipoles

This paper employs the post-Newtonian approximations of scalar-tensor theory of gravity along with the Cartesian STF tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies with arbitrary distribution of mass and velocity. We assume that spacetime can be covered by a global coordinate chart which is Minkowskian at infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field of each body is parametrized by an infinite set of the body's mass and spin multipoles and tidal multipoles of external N-1 bodies. The origin of the local coordinates is set moving along accelerated worldline of the center of mass of the body by an appropriate choice of the internal and external dipole moments of its gravitational field. Translational and rotational equations of motion are derived by integrating microscopic equations of matter and applying the method of asymptotic matching. The matching is also used for separating the post-Newtonian self-field effects from the external gravitational environment and for constructing the effective background spacetime manifold. It allows us to present the equations of translational and rotational motion of each body in covariant form by making use of the Einstein principle of equivalence. Our approach significantly generalizes the Mathisson-Papapetrou-Dixon covariant equations of motion with regard to the number of body's multipoles and the post-Newtonian terms taken into account. The equations of translational and rotational motion derived in the present paper include the infinite set of mass and spin multipoles of the bodies and can be used for much more accurate prediction of orbital dynamics of extended bodies in inspiraling binary systems before they merge.

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.

Theory of post-Newtonian radiation and reaction

We address issues with extant formulations of dissipative effects in the effective field theory (EFT) which describes the post-Newtonian (PN) inspiral of two gravitating bodies by (re)formulating several parts of the theory. Novel ingredients include gauge invariant spherical fields in the radiation zone; a system zone which preserves time reversal such that its violation arises not from local odd propagation but rather from interaction with the radiation sector in a way which resembles the balayage method; 2-way multipoles to perform zone matching within the EFT action; and a double-field radiation-reaction action which is the non-quantum version of the Closed Time Path formalism and generalizes to any theory with directed propagators including theories which are defined by equations of motion rather than an action. This formulation unifies the treatment of outgoing radiation and its reaction force. We demonstrate the theory in the scalar, electromagnetic and gravitational cases by economizing the following: the expression for the radiation source multipoles; the derivation of the leading outgoing radiation and associated reaction force such that it is maximally reduced to mere multiplication; and the derivation of the gravitational next to leading PN order. In fact we present a novel expression for the +1PN correction to all mass multipoles. We introduce useful definitions for multi-index summation, for the normalization of Bessel functions and for the normalization of the gravito-magnetic vector potential.

Black hole stereotyping: induced gravito-static polarization

We discuss the black hole effective action and define its static subsector. We determine the induced gravito-static polarization constants (electric Love numbers) of static black holes (Schwarzschild) in an arbitrary dimension, namely the induced mass multipole as a result of an external gravitational field. We demonstrate that in 4d these constants vanish thereby settling a disagreement in the literature. Yet in higher dimensions these constants are non-vanishing, thereby disproving (at least in d>4) speculations that black holes have no effective couplings beyond the point particle action. In particular, when l/(d-3) is half integral these constants demonstrate a (classical) renormalization flow consistent with the divergences of the effective field theory. In some other cases the constants are negative indicating a novel non-spherical instability. The theory of hypergeometric functions plays a central role.

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.

Relativistic theory of tidal Love numbers

In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.

MULTIPOLE FORMULAE FOR GRAVITATIONAL LENSING SHEAR AND FLEXION

ABSTRACT The gravitational lensing equations for convergence, potential, shear, and flexion are simple in polar coordinates and separate under a multipole expansion once the shear and flexion spinors are rotated into a “tangential” basis. We use this to investigate whether the useful monopole aperture–mass shear formulae generalize to all multipoles and to flexions. We re-derive the result of Schneider and Bartelmann that the shear multipole m at radius R is completely determined by the mass multipole at R, plus specific moments Q (m) in and Q (m) out of the mass multipoles internal and external, respectively, to R. The m ⩾ 0 multipoles are independent of Q out. But in contrast to the monopole, the m < 0 multipoles are independent of Q in. These internal and external mass moments can be determined by shear (and/or flexion) data on the complementary portion of the plane, which has practical implications for lens modeling. We find that the ease of E/B separation in the monopole aperture moments does not generalize to m ≠ 0: the internal monopole moment is the only nonlocal E/B discriminant available from lensing observations. We have also not found practical local E/B discriminants beyond the monopole, though they could exist. We show also that the use of weak-lensing data to constrain a constant shear term near a strong-lensing system is impractical without strong prior constraints on the neighboring mass distribution.

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.

Evolution of the Carter constant for inspirals into a black hole: Effect of the black hole quadrupole

We analyze the effect of gravitational radiation reaction on generic orbits around a body with an axisymmetric mass quadrupole moment $Q$ to linear order in $Q$, to the leading post-Newtonian order, and to linear order in the mass ratio. This system admits three constants of the motion in absence of radiation reaction: energy, angular momentum along the symmetry axis, and a third constant analogous to the Carter constant. We compute instantaneous and time-averaged rates of change of these three constants. For a point particle orbiting a black hole, Ryan has computed the leading order evolution of the orbit's Carter constant, which is linear in the spin. Our result, when combined with an interaction quadratic in the spin (the coupling of the black hole's spin to its own radiation reaction field), gives the next to leading order evolution. The effect of the quadrupole, like that of the linear spin term, is to circularize eccentric orbits and to drive the orbital plane towards antialignment with the symmetry axis. In addition we consider a system of two point masses where one body has a single mass multipole or current multipole of order $l$. To linear order in the mass ratio, to linear order in the multipole, and to the leading post-Newtonian order, we show that there does not exist an analog of the Carter constant for such a system (except for the cases of an $l=1$ current moment and an $l=2$ mass moment). Thus, the existence of the Carter constant in Kerr depends on interaction effects between the different multipoles. With mild additional assumptions, this result falsifies the conjecture that all vacuum, axisymmetric spacetimes possess a third constant of the motion for geodesic motion.

Spin and energy evolution equations for a wide class of extended bodies

We give a surface integral derivation of the leading-order evolution equations for the spin and energy of a relativistic body interacting with other bodies in the post-Newtonian expansion scheme. The bodies can be arbitrarily shaped and can be strongly self-gravitating. The effects of all mass and current multipoles are taken into account. As part of the computation one of the 2PN potentials parametrizing the metric is obtained. The formulae obtained here for spin and energy evolution coincide with those obtained by Damour, Soffel and Xu for the case of weakly self-gravitating bodies. By combining an Einstein–Infeld–Hoffman-type surface integral approach with multipolar expansions we extend the domain of validity of these evolution equations to a wide class of strongly self-gravitating bodies. This paper completes in a self-contained way a previous work by Racine and Flanagan on translational equations of motion for compact objects.