According to the "no-hair" conjecture, a Kerr black hole (BH) is completely described by its mass and spin. In particular, the spin-induced quadrupole moment of a Kerr BH with mass $m$ and dimensionless spin $χ$ can be written as $Q=-κ\,m^3χ^2$, where $κ_{\rm BH}=1$. Thus by measuring the spin-induced quadrupole parameter $κ$, we can test the binary black hole nature of compact binaries and distinguish them from binaries comprised of other exotic compact objects, as proposed in [N. V. Krishnendu et al., PRL 119, 091101 (2017)]. Here, we present a Bayesian framework to carry out this test where we measure the symmetric combination of individual spin-induced quadrupole moment parameters fixing the anti-symmetric combination to be zero. The analysis is restricted to the inspiral part of the signal as the spin-induced deformations are not modeled in the post-inspiral regime. We perform detailed simulations to investigate the applicability of this method for compact binaries of different masses and spins and also explore various degeneracies in the parameter space which can affect this test. We then apply this method to the gravitational wave events, GW151226 and GW170608 detected during the first and second observing runs of Advanced LIGO and Advanced Virgo detectors. We find the two events to be consistent with binary black hole mergers in general relativity. By combining information from several more of such events in future, this method can be used to set constraints on the black hole nature of the population of compact binaries that are detected by the Advanced LIGO and Advanced Virgo detectors.
We propose a novel method to test the binary black hole (BBH) nature of compact binaries detectable by gravitational wave (GW) interferometers and hence constrain the parameter space of other exotic compact objects. The spirit of the test lies in the "no-hair" conjecture for black holes where all properties of a black hole are characterised by the mass and the spin of the black hole. The method relies on observationally measuring the quadrupole moments of the compact binary constituents induced due to their spins. If the compact object is a Kerr black hole (BH), its quadrupole moment is expres…
Testing the Kerr hypothesis beyond the quadrupole with GW241011
Indian Institute of Technology Madras（印度马德拉斯理工学院）；University of Birmingham（伯明翰大学）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）；Chennai Mathematical Institute（金奈数学研究所）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
We describe a new numerical algorithm for ray tracing in the external spacetimes of spinning compact objects characterized by arbitrary quadrupole moments. Such spacetimes describe non-Kerr vacuum solutions that can be used to test the no-hair theorem in conjunction with observations of accreting black holes. They are also appropriate for neutron stars with spin frequencies in the 300-600 Hz range, which are typical of the bursting sources in low-mass X-ray binaries. We use our algorithm to show that allowing for the quadrupole moment of the spacetime to take arbitrary values leads to observab…
We consider tidal coupling in a binary stellar system to first-post-Newtonian order. We derive the orbital equations of motion for bodies with spins and mass quadrupole moments and show that they conserve the total linear momentum of the binary. We note that spin-orbit coupling must be included in a 1PN treatment of tidal interactions in order to maintain consistency (except in the special case of adiabatically induced quadrupoles); inclusion of 1PN quadrupolar tidal effects while omitting spin effects would lead to a failure of momentum conservation for generic evolution of the quadrupoles. W…
Utilizing the tools of tendex and vortex, we study the highly dynamic plunge and merger phases of several $π$-symmetric binary black hole coalescences. In particular, we observe a decline of the strength of the current quadrupole moment as compared to that of the mass quadrupole moment during the merger phase, contrary to a naive estimate according to the dependence of these moments on the separation between the black holes. We further show that this decline of the current quadrupole moment is achieved through the remnants of the two individual spins becoming nearly aligned or anti-aligned wit…
Black holes in General Relativity are known as Kerr black holes and are characterized solely by two parameters, the mass $M$ and the spin $J$. All the higher multipole moments of the gravitational field are functions of these two parameters. For instance, the quadrupole moment is $Q=-J^2/M$, which implies that a measurement of $M$, $J$, and $Q$ for black hole candidates would allow one to test whether these objects are really black holes as described by General Relativity. While future gravitational-wave experiments will be able to test the Kerr nature of these objects with very high accuracy,…
We analytically determine the quasinormal mode (QNM) frequencies of a black hole with quadrupole moment in the eikonal limit using the light-ring method. The generalized black holes that are discussed in this work possess arbitrary quadrupole and higher mass moments in addition to mass and angular momentum. Static collapsed configurations with mass and quadrupole moment are treated in detail and the QNM frequencies associated with two such configurations are evaluated to linear order in the quadrupole moment. Furthermore, we touch upon the treatment of rotating systems. In particular, the gene…
We consider rapidly rotating black holes and neutron stars in dilatonic Einstein-Gauss-Bonnet (EGBd) theory and determine their quadrupole moments, which receive a contribution from the dilaton. The quadrupole moment of EGBd black holes can be considerably larger than the Kerr value. For neutron stars, the universality property of the $\hat I$-$\hat Q$ relation between the scaled moment of inertia and the scaled quadrupole moment appears to extend to EGBd theory.
The super-massive objects at the center of many galaxies are commonly thought to be black holes. In 4-dimensional general relativity, a black hole is completely specified by its mass $M$ and by its spin angular momentum $J$. All the higher multipole moments of the gravitational field depend in a very specific way on these two parameters. For instance, the mass quadrupole moment is $Q = - J^2/M$. If we can estimate $M$, $J$, and $Q$ for the super-massive objects in galactic nuclei, we over-constrain the theory and we can test the black hole hypothesis. While there are many works studying how th…
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 (UV) 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 …
A rotating star's oblateness creates a deformation in the gravitational field outside the star, which is measured by the quadrupole-moment tensor. We consider the effect of the quadrupole moment on the orbital motion and rate of inspiral of a compact binary system, composed of neutron stars and/or black holes. We find that in the case of circular orbits, the quadrupole-monopole interaction affects the relation between orbital radius and angular velocity, and also the rate of inspiral, by a quantity of order (v/c)^4, where v is the orbital velocity and c the speed of light.
Recently, spacetimes described by metrics with three parameters (mass, rotation and small quadrupole moment) was found, and in this paper, null geodesics for these metrics are calculated and visualized. Light scattering, as well as the role that the quadrupole moment plays in deforming these kinds of spacetimes are studied. This is a new application of a symbolic--numeric program that we previously used to study the Bonnor metric.
The trajectories of test particles moving in the gravitational field of a non-spherically symmetric mass distribution become affected by the presence of multipole moments. In the case of hyperbolic trajectories, the quadrupole moment of an oblate mass induces a displacement of the trajectory towards the mass source, an effect that can be interpreted as an additional acceleration directed towards the source. Although this additional acceleration is not constant, we perform a general relativistic analysis in order to evaluate the possibility of explaining Pioneer's anomalous acceleration by mean…
In this contribution, we calculate the light deflection, perihelium shift, time delay and gravitational redshift using an approximate metric that contains the Kerr metric and an approximaction of the Erez-Rosen spacetime. The results were obtained directly using Mathematica. The results agree with the ones presented in the literature, but they are extended until second order terms of mass, angular momentum and mass quadrupole.
The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) d…
A metric representing a slowly rotating object with quadrupole moment is obtained using a perturbation method to include rotation into the weak limit of the Erez-Rosen metric. This metric is intended to tackle relativistic astrometry and gravitational lensing problems in which a quadrupole moment has to be taken into account.
We study weak gravitational lensing around a compact object with arbitrary quadrupole moment in the presence of plasma. The studied compact objects are considered to be spherically symmetric. The additional parameter $ε$ regulating the quadrupole moment in the metric alters the deflection angle of light rays along with the plasma parameters. In the vacuum, the number of images due to the presence of the parameter $ε$ increases and causes the increase of the magnification of the image source. The effects of uniform and nonuniform plasma on gravitational lensing around the compact object are als…
We study geometrically thick perfect-fluid tori with constant specific angular momentum, so-called "Polish doughnuts", orbiting deformed compact objects with a quadrupole moment. More specifically, we consider two different asymptotically flat, static and axisymmetric vacuum solutions to Einstein's field equation with a non-zero quadrupole moment, the q-metric and the Erez-Rosen spacetime. It is our main goal to find features of Polish doughnuts in these two spacetimes which qualitatively distinguish them from Polish doughnuts in the Schwarzschild spacetime. As a main result we find that, for …
We study the effect of non-quadrupolar modes in the detection and parameter estimation of gravitational waves (GWs) from non-spinning black-hole binaries. We evaluate the loss of signal-to-noise ratio and the systematic errors in the estimated parameters when one uses a quadrupole-mode template family to detect GW signals with all the relevant modes, for target signals with total masses $20 M_\odot \leq M \leq 250 M_\odot$ and mass ratios $1 \leq q \leq 18$. Target signals are constructed by matching numerical-relativity simulations describing the late inspiral, merger and ringdown of the bina…
In this paper we use the second order formalism of Hartle to study slowly and rigidly rotating stars with focus on the quadrupole moment of the object. The second order field equations for the interior fluid are solved numerically for different classes of possible equations of state and these solutions are then matched to a vacuum solution that includes the general asymptotically flat axisymmetric metric to second order, using the Darmois-Israel procedure. For these solutions we find that the quadrupole moment differs from that of the Kerr metric, as has also been found for some equations of s…
Extracting Energy from a Non-Kerr Rotating Spacetime with an Anomalous Quadrupole Moment via Magnetic Reconnection
School of Material Science and Engineering, Chongqing Jiaotong University（重庆交通大学材料科学与工程学院）；College of Physics and Optoelectronic Engineering, Chongqing Normal University（重庆师范大学物理与光电工程学院）
An exact solution of Einstein's equations representing the static gravitational field of a quasi-spherical source endowed with both mass and mass quadrupole moment is considered. It belongs to the Weyl class of solutions and reduces to the Schwarzschild solution when the quadrupole moment vanishes. The geometric properties of timelike circular orbits (including geodesics) in this spacetime are investigated. Moreover, a comparison between geodesic motion in the spacetime of a quasi-spherical source and non-geodesic motion of an extended body also endowed with both mass and mass quadrupole momen…
The radiation reaction in compact spinning binaries on eccentric orbits due to the quadrupole-monopole interaction is studied. This contribution is of second post-Newtonian order. As result of the precession of spins the magnitude $L$ of the orbital angular momentum is not conserved. Therefore a proper characterization of the perturbed radial motion is provided by the energy $E$ and angular average $\bar{L}$. As powerful computing tools, the generalized true and eccentric anomaly parametrizations are introduced. Then the secular losses in energy and magnitude of orbital angular momentum togeth…
Measurability of Quadrupole Deviations from Kerr in Binary black hole Mergers
Shanghai Astronomical Observatory（上海天文台）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院）；International Centre for Theoretical Physics Asia-Pacific（亚太理论物理中心）
One of the most important sources for future space-borne gravitational wave detectors such as TianQin and LISA is EMRI. It happens when a stellar orgin compact object orbiting around a massive black hole(MBH) in the center of galaxies and has many benefits in the study of astrophysics and fundamental theories. One of the most important objectives is to test the no-hair theorem by measuring the quadrupole moment of the MBH. This requires us to estimate the parameters of an EMRI system accurately enough, which means we also need an accurate waveform templet for this process. Based on the fast an…
As a crucial step towards the completion of the fourth post-Newtonian (4PN) gravitational-wave generation from compact binary systems, we obtain the expressions of the so-called "canonical" multipole moments of the source in terms of the "source" and "gauge" moments. The canonical moments describe the propagation of gravitational waves outside the source's near zone, while the source and gauge moments encode explicit information about the matter source. Those two descriptions, in terms of two sets of canonical moments or in terms of six sets of source and gauge moments, are isometric. We thus …
With the aim of providing high accuracy post-Newtonian (PN) templates for the analysis of gravitational waves generated by compact binary systems, we complete the analytical derivation of the source type mass quadrupole moment of compact binaries (without spins) at the fourth PN order of general relativity. Similarly to the case of the conservative 4PN equations of motion, we show that the quadrupole moment at that order contains a non-local (in time) contribution, arising from the tail-transported interaction entering the conservative part of the dynamics. Furthermore, we investigate the infr…
In this work we present IMRPhenomTP, a time domain phenomenological model for the dominant $l=2$, $m=|2|$ modes of coalescing black hole binary systems and its extension to describe general precessing systems within the "twisting up" approximation. The underlying non-precessing model is calibrated to the new release of Numerical Relativity simulations of the SXS Collaboration and its accuracy is comparable to the state-of-the-art non-precessing dominant mode models as IMRPhenomX and SEOBNRv4. The precessing extension allows for flexibility choosing the Euler angles of the time-dependent rotati…
An exact solution of Einstein's field equations in empty space first found in 1985 by Quevedo and Mashhoon is analyzed in detail. This solution generalizes Kerr spacetime to include the case of matter with arbitrary mass quadrupole moment and is specified by three parameters, the mass $M$, the angular momentum per unit mass $a$ and the quadrupole parameter $q$. It reduces to the Kerr spacetime in the limiting case $q=0$ and to the Erez-Rosen spacetime when the specific angular momentum $a$ vanishes. The geometrical properties of such a solution are investigated. Causality violations, direction…
The regularization and renormalization of the radiative mass-type quadrupole moment of inspiralling compact binaries (without spins) is investigated at the fourth post-Newtonian (4PN) approximation of general relativity. As clear from the conservative 4PN equations of motion, a dimensional regularization has to be implemented in order to properly treat the non-linear interactions experienced by gravitational waves during their propagation toward future null infinity. By implementing such procedure, we show that the poles coming from the source moment (computed in a companion paper) are exactly…
We present highly accurate, dimensionally-reduced gravitational waveforms for binary inspirals whose components have large spin-induced quadrupole moments. The spin-induced quadrupole of a body first appears in the phase of a waveform at the early inspiral stage of the binary coalescence, making it a relatively clean probe of the internal structure of the body. However, for objects with large quadrupolar deviations from Kerr, searches using binary black hole (BBH) models would be ineffective. In order to perform a computationally-feasible search, we present two dimensionally-reduced models whi…
We recently suggested a nonlocal modification of Einstein's field equations in which Newton's constant $G$ was promoted to a covariant differential operator $G_Λ(\Box_g)$. The latter contains two independent contributions, which operate respectively in the infrared (IR) and ultraviolet (UV) energy regimes. In the light of the recent direct gravitational radiation measurements we aim to determine the UV-modified 1.5 post-Newtonian radiative quadrupole moment of a generic $n$-body system. We eventually use these initial results in the particular context of a binary system and observe that in the…
We study whether binary black hole template banks can be used to search for the gravitational waves emitted by general binary coalescences. To recover binary signals from noisy data, matched-filtering techniques are typically required. This is especially true for low-mass systems, with total mass $M \lesssim 10 \, M_\odot$, which can inspiral in the LIGO and Virgo frequency bands for thousands of cycles. In this paper, we focus on the detectability of low-mass binary systems whose individual components can have large spin-induced quadrupole moments and small compactness. The quadrupole contrib…
We consider some implications of the departure from spherical symmetry for static solutions of the vacuum Einstein's equations describing black hole mimickers. In particular, we investigate how the presence of mass quadrupole moment affects the collision energy of test particles. We show that collision processes in the vicinity of such exotic compact sources with non vanishing mass quadrupole moment may be significantly different from those in the vicinity of black holes.
We investigate the interior Einstein's equations in the case of a static, axially symmetric, perfect fluid source. We present a particular line element that is specially suitable for the investigation of this type of interior gravitational fields. Assuming that the deviation from spherically symmetry is small, we linearize the corresponding field equations and find several classes of vacuum and perfect fluid solutions. We find physically meaninful spacetimes by imposing appropriate matching conditions.
We have studied numerically the shadows of a non-Kerr rotating compact object with quadrupole mass moment, which belongs to Manko-Novikov family. The non-integrable photon motion caused by quadrupole mass moment affects sharply the shadow of the compact object. As the deviation parameter related to quadrupole mass moment is negative, the shadow of compact object is prolate and there are two disconnected main shadows with eyebrows located symmetrically on both sides of the equatorial plane. As the deviation parameter is positive, the shadow becomes oblate and the main shadow is joined together …
We present the Ernst potential and the line element of an exact solution of Einstein's vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This ope…
We investigate the general geometric properties of the surface of infinite red-shift corresponding to the event horizon of the static and axisymmetric solution of the Einstein vacuum equations that only possesses mass $M$ and quadrupole moment $Q$. The deformation of the Schwarzschild surface $r=2M$ produced by the quadrupole moment is shown, and the range of values of this multipole moment is specified, which preserves a regular, closed, continuous and differentiable surface. Some thermodynamic consequences and speculations ensuing from our results are discussed.
The "no-hair" theorem can, in principle, be tested at the center of the Milky Way by measuring the spin and the quadrupole moment of Sgr A$^*$ with the orbital precession of S-stars, measured over their full periods. Contrary to the original method, we show why it is possible to test the no-hair theorem using observations from only a single star, by measuring precession angles over a half-orbit. There are observational and theoretical reasons to expect S-stars to spin rapidly, and we have quantified the effect of stellar spin, via spin-curvature coupling (the leading-order manifestation of the…
The innermost stable circular orbit equation of a test particle is obtained for an approximate Kerr-like spacetime with quadrupole moment. We derived the effective potential for the radial coordinate by the Euler-Lagrange method. This equation can be employed to measure the mass quadrupole by observational means, because from this equation a quadratic polynomial for the quadrupole moment can be found. As expected, the limiting cases of this equation are found to be the known cases of Kerr and Schwarzschild.
We analyze in detail the spin precession equations in binary black hole systems, when the tidal torque on a Kerr black hole due to quadrupole-monopole coupling is taken into account. We show that completing the precession equations with this term reveals the existence of a conserved quantity at 2PN order when averaging over orbital motion. This quantity allows one to solve the (orbit-averaged) precession equations exactly in the case of equal masses and arbitrary spins, neglecting radiation reaction. For unequal masses, an exact solution does not exist in closed form, but we are still able to …
The aim of this paper is to dynamically constrain the quadrupole mass moment J_2 of the main-sequence HD 209458 star. The adopted method is the confrontation between the measured orbital period of its transiting planet Osiris and a model of it. Osiris is assumed to move along an equatorial and circular orbit. Our estimate, for given values of the stellar mass M and radius R and by assuming the validity of general relativity, is J_2=(3.5 +/- 385.1) 10^-5. Previous fiducial evaluations based on indirect, spectroscopic measurements yielded J_2\sim 10^-6: such a value is compatible with our result…
