The gravitational-wave signal from inspiralling neutron-star--neutron-star (or black-hole--neutron-star) binaries will be influenced by tidal coupling in the system. An important science goal in the gravitational-wave detection of these systems is to obtain information about the equation of state of neutron star matter via the measurement of the tidal polarizability parameters of neutron stars. To extract this piece of information will require to have accurate analytical descriptions of both the motion and the radiation of tidally interacting binaries. We improve the analytical description of the late inspiral dynamics by computing the next-to-next-to-leading order relativistic correction to the tidal interaction energy. Our calculation is based on an effective-action approach to tidal interactions, and on its transcription within the effective-one-body formalism. We find that second-order relativistic effects (quadratic in the relativistic gravitational potential $u=G(m_1 +m_2)/(c^2 r)$) significantly increase the effective tidal polarizability of neutron stars by a distance-dependent amplification factor of the form $1 + α_1 \, u + α_2 \, u^2 +...$ where, say for an equal-mass binary, $α_1=5/4=1.25$ (as previously known) and $α_2=85/14\simeq6.07143$ (as determined here for the first time). We argue that higher-order relativistic effects will lead to further amplification, and we suggest a Padé-type way of resumming them. We recommend to test our results by comparing resolution-extrapolated numerical simulations of inspiralling-binary neutron stars to their effective one body description.
Thorne elucidated that the relativistic tidal heating is the same as the Newtonian theory. Moreover, Thorne also claimed that the tidal heating is independent of how one localizes gravitational energy and is unambiguously given by a certain formula. Purdue and Favata calculated the tidal heating for different classical pseudotensors including Moller and obtained the results all matched with the Newtonian perspective. After re-examined this Moller pseudotensor, we find that there does not exist any tidal heating value. Thus we claim that the relativistic tidal heating is pseudotensor independen…
In his study of tidal stabilization of fully relativistic neutron stars Thorne showed that the fully relativistic expression for tidal heating is the same as in non-relativistic Newtonian theory. Furthermore, Thorne also noted that the tidal heating must be independent of how one localizes gravitational energy and is unambiguously given by that expression. Purdue and Favata calculated the tidal heating for a number of classical gravitational pseudotensors including that of Moller, and obtained the result that all of them produced the same (Newtonian) value. However, in a re-examination of the …
We consider the relativistic tidal effects on frequency shift of clock-comparison experiments. The relativistic formulation for frequency shift and time transfer is derived in the gravitational field of a tidal, axisymmetric, and rotating Earth. With the help of Love numbers describing the tidal response of solid Earth, we formulize the mathematical connection between tidal effects from the ground-based clock-comparison experiments and the local gravity tides from the gravimeters, which in turn provides us an approach to eliminate tidal influences on clock comparison with the local gravity tid…
With continuous advances in technology, future satellite gradiometry missions will be capable of performing precision relativistic experiments and imposing constraints on modern gravity theories. To this end, the full first-order post-Newtonian tidal tensor under inertially guided and Earth-pointing local frames along post-Newtonian orbits is worked out. The physical picture behind the "Mashhoon-Theiss anomaly" is explained at the post-Newtonian level. The relativistic precession of the local frame with respect to the sidereal frame will produce modulations of Newtonian tidal forces along cert…
Purdue and Favata calculate the tidal heating used certain classical pseudotensors. Booth and Creighton employed the quasi-local mass formalism of Brown and York to demonstrate the same subject. All of them give the result matched with the Newtonian theory. Here we present another Hamiltonian quasi-local boundary expressions and all give the same desired value. This indicates that the tidal heating is unique as Thorne predicted. Moreover, we discovered that the pseudo-tensor method and quasi-local method are fundamentally different.
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 defo…
In his study of tidal stabilization of fully relativistic neutron stars Thorne showed that the fully relativistic expression for tidal heating is the same as in non-relativistic Newtonian theory. Furthermore, Thorne also noted that this tidal heating must be independent of how one localizes gravitational energy and is unambiguously given by that expression. Favata calculated the tidal heating for a number of classical gravitational pseudotensors including that of M$ø$ller, and obtained the result that all of them produced the same (Newtonian) value. After a re-examination of the calculation us…
We examine the tidal deformation of a nonrotating compact body (material body or black hole) in general relativity. The body's exterior metric is calculated in a simultaneous expansion in powers of the ratio between the distance to the body and three distinct length scales: the radius of curvature of the external spacetime in which the body is inserted, the scale of spatial inhomogeneity of the curvature, and the scale of temporal variation. The metric is valid in the body's immediate neighborhood, which excludes the external matter responsible for the tidal environment. The body's tidal respo…
We construct the external metric of a slowly rotating, tidally deformed material body in general relativity. The tidal forces acting on the body are assumed to be weak and to vary slowly with time, and the metric is obtained as a perturbation of a background metric that describes the external geometry of an isolated, slowly rotating body. The tidal environment is generic and characterized by two symmetric-tracefree tidal moments E_{ab} and B_{ab}, and the body is characterized by its mass M, its radius R, and a dimensionless angular-momentum vector χ^a << 1. The perturbation accounts for all c…
In 1998 Thorne claimed that all pseudotensors give the same tidal work as the Newtonian theory. In 1999, Purdue used the Landau-Lifshitz pseudotensor to calculate the tidal heating and the result matched with the Newtonian gravity. Soon after in 2001, Favata employed the same method to examine the Einstein, Bergmann-Thomson and Møller pseudotensors, all of them give the same result as Purdue did. Inspired by the work of Purdue and Favata, for the completeness, here we manipulate the tidal work for Papapetrou, Weinberg and Goldberg pseudotensors. We obtained the same tidal work as Purdue achiev…
The metric outside a compact body deformed by a quadrupolar tidal field is universal up to its Love numbers, constants which encode the tidal response's dependence on the body's internal structure. For a non-rotating body, the deformed external geometry is characterized by the familiar gravitational Love numbers $K_2^{\text{el}}$ and $K_2^{\text{mag}}$. For a slowly rotating body, these must be supplemented by rotational-tidal Love numbers, which measure the response to couplings between the body's spin and the external tidal field. By integrating the interior field equations, I find that the …
Tidal interactions have a significant influence on the late dynamics of compact binary systems, which constitute the prime targets of the upcoming network of gravitational-wave detectors. We refine the theoretical description of tidal interactions (hitherto known only to the second post-Newtonian level) by extending our recently developed analytic self-force formalism, for extreme mass-ratio binary systems, to the computation of several tidal invariants. Specifically, we compute, to linear order in the mass ratio and to the 7.5$^{\rm th}$ post-Newtonian order, the following tidal invariants: t…
When a self-gravitating body (e.g., a neutron star or black hole) interacts with an external tidal field (e.g., that of a binary companion), the interaction can do work on the body, changing its mass-energy. The details of this "tidal heating" are analyzed using the Landau-Lifshitz pseudotensor and the local asymptotic rest frame of the body. It is shown that the work done on the body is gauge-invariant, while the body-tidal-field interaction energy contained within the body's local asymptotic rest frame is gauge dependent. This is analogous to Newtonian theory, where the interaction energy is…
An extended body orbiting a compact object undergoes tidal deformations by the background gravitational field. Tidal invariants built up with the Riemann tensor and their derivatives evaluated along the world line of the body are essential tools to investigate both geometrical and physical properties of the tidal interaction. For example, one can determine the tidal potential in the neighborhood of the body by constructing a body-fixed frame, which requires Fermi-type coordinates attached to the body itself, the latter being in turn related to the spacetime metric and curvature along the consi…
To detect the gravitational-wave (GW) signal from binary neutron stars and extract information about the equation of state of matter at nuclear density, it is necessary to match the signal with a bank of accurate templates. We present the two longest (to date) general-relativistic simulations of equal-mass binary neutron stars with different compactnesses, C=0.12 and C=0.14, and compare them with a tidal extension of the effective-one-body (EOB)model. The typical numerical phasing errors over the $\simeq 22$ GW cycles are $Δϕ\simeq \pm 0.24$ rad. By calibrating only one parameter (representing…
Within the framework of general relativity, we investigate the tidal acceleration of astrophysical jets relative to the central collapsed configuration ("Kerr source"). We neglect electromagnetic forces throughout. The rest frame of the Kerr source is locally defined via the set of hypothetical static observers in the spacetime exterior to the source. Relative to such a fiducial observer fixed on the rotation axis of the Kerr source, jet particles are tidally accelerated to almost the speed of light if their outflow speed is above a certain threshold, given roughly by one half of the Newtonian…
The LIGO collaboration recently reported the first gravitational-wave constraints on the tidal deformability of neutron stars. I discuss an inherent ambiguity in the notion of relativistic tidal deformability that, while too small to affect the present measurement, may become important in the future. I propose a new way to understand the ambiguity and discuss future prospects for reliably linking observed gravitational waveforms to compact object microphysics.
The tidal interaction of a (rotating or nonrotating) black hole with nearby bodies produces changes in its mass, angular momentum, and surface area. Similarly, tidal forces acting on a Newtonian, viscous body do work on the body, change its angular momentum, and part of the transferred gravitational energy is dissipated into heat. The equations that describe the rate of change of the black-hole mass, angular momentum, and surface area as a result of the tidal interaction are compared with the equations that describe how the tidal forces do work, torque, and produce heat in the Newtonian body. …
High-order effective-one-body tidal interactions and gravitational scattering
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学理论物理研究所）；Dipartimento di Fisica e Geologia, Università di Perugia, I.N.F.N. Sezione di Perugia（佩鲁贾大学物理与地质学系，意大利国家核物理学会佩鲁贾分部）；Institut des Hautes Etudes Scientifiques（高等科学研究所）
We point out novel consequences of general relativity involving tidal dynamics of ultrarelativistic relative motion. Specifically, we use the generalized Jacobi equation and its extension to study the force-free dynamics of relativistic flows near a massive rotating source. We show that along the rotation axis of the gravitational source, relativistic tidal effects strongly decelerate an initially ultrarelativistic flow with respect to the ambient medium, contrary to Newtonian expectations. Moreover, an initially ultrarelativistic flow perpendicular to the axis of rotation is strongly accelera…
The late part of the gravitational wave signal of binary neutron star inspirals can in principle yield crucial information on the nuclear equation of state via its dependence on relativistic tidal parameters. In the hope of analytically describing the gravitational wave phasing during the late inspiral (essentially up to contact) we propose an extension of the effective one body (EOB) formalism which includes tidal effects. We compare the prediction of this tidal-EOB formalism to recently computed nonconformally flat quasi-equilibrium circular sequences of binary neutron star systems. Our anal…
The deformation of a nonrotating body resulting from the application of a tidal field is measured by two sets of Love numbers associated with the gravitoelectric and gravitomagnetic pieces of the tidal field, respectively. The gravitomagnetic Love numbers were previously computed for fluid bodies, under the assumption that the fluid is in a strict hydrostatic equilibrium that requires the complete absence of internal motions. A more realistic configuration, however, is an irrotational state that establishes, in the course of time, internal motions driven by the gravitomagnetic interaction. We …
We study the various linear responses of neutron stars to external relativistic tidal fields. We focus on three different tidal responses, associated to three different tidal coefficients: (i) a gravito-electric-type coefficient Gμ_\ell=[length]^{2\ell+1} measuring the \ell^{th}-order mass multipolar moment GM_{a_1... a_\ell} induced in a star by an external \ell^{th}-order gravito-electric tidal field G_{a_1... a_\ell}; (ii) a gravito-magnetic-type coefficient Gσ_\ell=[length]^{2\ell+1} measuring the \ell^{th} spin multipole moment G S_{a_1... a_\ell} induced in a star by an external \ell^{th…
We study the tidal effects of a Kerr black hole on a neutron star in black hole-neutron star binary systems using a semi-analytical approach which describes the neutron star as a deformable ellipsoid. Relativistic effects on the neutron star self-gravity are taken into account by employing a scalar potential resulting from relativistic stellar structure equations. We calculate quasi-equilibrium sequences of black hole-neutron star binaries, and the critical orbital separation at which the star is disrupted by the black hole tidal field: the latter quantity is of particular interest because whe…
Extreme-mass-ratio inspirals (EMRIs) and intermediate-mass-ratio inspirals (IMRIs)--binaries in which a stellar-mass object spirals into a massive black hole or other massive, compact body--are important sources of gravitational waves for LISA and LIGO, respectively. Thorne has speculated that the waves from EMRIs and IMRIs encode, in principle, all the details of (i) the central body's spacetime geometry (metric), (ii) the tidal coupling (energy and angular momentum exchange) between the central body and orbiting object, and (iii) the evolving orbital elements. Fintan Ryan has given a first p…
Relativistic tidal equations are formulated with respect to the rest frame of a central gravitational source and their solutions are studied. The existence of certain relativistic critical tidal currents are thereby elucidated. Specifically, observers that are spatially at rest in the exterior Kerr spacetime are considered in detail; in effect, these fiducial observers define the rest frame of the Kerr source. The general tidal equations for the free motion of test particles are worked out with respect to the Kerr background. The analytic solutions of these equations are investigated and the e…
We demonstrate that a slowly rotating compact body subjected to a stationary tidal field undergoes a dynamical response, in which the fluid variables and the interior metric vary on the time scale of the rotation period. This dynamical response requires the tidal field to have a gravitomagnetic component generated by external mass currents; the response to a gravitoelectric tidal field is stationary. We confirm that in a calculation carried out to first order in the body's rotation, the exterior geometry bears no trace of this internal dynamics; it remains stationary in spite of the time-depen…
Gravitational-wave astronomy with coalescing binary neutron star sources requires the availability of gravitational waveforms with tidal effects accurate up to merger. This article presents an improved version of \TEOBResum, a nonspinning effective-one-body (EOB) waveform model with enhanced analytical information in its tidal sector. The tidal potential governing the conservative dynamics employs resummed expressions based on post-Newtonian (PN) and gravitational self-force (GSF) information. In particular, we compute a GSF-resummed expression for the leading-order octupolar gravitoelectric t…
It was recently revealed that a rotating compact body responds dynamically when it is subjected to a gravitomagnetic tidal field, even when this field is idealized as time-independent. The dynamical response is characterized by time-changing internal currents, and it was suspected to originate from zero-frequency g-modes and r-modes driven by the tidal forces. In this paper we provide additional insights into the phenomenon by examining the tidal response of a rotating body within the framework of post-Newtonian gravity. This approach allows us to develop an intuitive picture for the phenomeno…
We consider a model of the state evolution of relativistic vector bosons, which includes both the dynamical equations for the particle four-velocity and the equations for the polarization four-vector evolution in the field of a nonlinear plane gravitational wave. In addition to the gravitational minimal coupling, tidal forces linear in curvature tensor are suggested to drive the particle state evolution. The exact solutions of the evolutionary equations are obtained. Birefringence and tidal deviations from the geodesic motion are discussed.
We consider tidal encounters between a white dwarf and an intermediate mass black hole. Both weak encounters and those at the threshold of disruption are modeled. The numerical code combines mesh-based hydrodynamics, a spectral method solution of the self-gravity, and a general relativistic Fermi normal coordinate system that follows the star and debris. Fermi normal coordinates provide an expansion of the black hole tidal field that includes quadrupole and higher multipole moments and relativistic corrections. We compute the mass loss from the white dwarf that occurs in weak tidal encounters.…
We investigate the tidal deformability of a superfluid neutron star. We calculate the equilibrium structure in the general relativistic two-fluid formalism with entrainment effect where we take neutron superfluid as one fluid and the other fluid is comprised of protons and electrons, making it a charge neutral fluid. We use a relativistic mean field model for the equation of state of matter where the interaction between baryons is mediated by the exchange $σ$, $ω$ and $ρ$ mesons. Then, we study the linear, static $l=2$ perturbation on the star to compute the electric-type Love number following…
We propose an approximate approach for studying the relativistic regime of stellar tidal disruptions by rotating massive black holes. It combines an exact relativistic description of the hydrodynamical evolution of a test fluid in a fixed curved spacetime with a Newtonian treatment of the fluid's self-gravity. Explicit expressions for the equations of motion are derived for Kerr spacetime using two different coordinate systems. We implement the new methodology within an existing Newtonian Smoothed Particle Hydrodynamics code and show that including the additional physics involves very little e…
Relativistic Tidal Dissipation and the Gravitational-wave Signal of a White Dwarf Orbiting an Intermediate-Mass Black Hole
Peking University（北京大学）；Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；Beijing Institute of Mathematical Sciences and Applications（北京数学与应用理科研究所）
Nonlinear Relativistic Tidal Response of Neutron Stars
Dipartimento di Fisica, “Sapienza” Università di Roma & Sezione INFN Roma1（罗马大学物理系与INFN罗马第一分部）；Department of Physics, Center for Theoretical Physics, Columbia University（哥伦比亚大学理论物理中心物理系）；Université Paris Cité, CNRS, Astroparticule et Cosmologie（巴黎西岱大学、法国国家科学研究中心粒子物理与宇宙学）；Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique（巴黎萨克雷大学、法国国家科学研究中心、法国原子能和替代能源委员会理论物理研究所）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
We study further a general relativistic mechanism for the acquisition of tidal energy by free test particles near a gravitationally collapsed configuration. Specifically, we investigate the solutions of timelike geodesic equation in a Fermi normal coordinate system established about the world line of an accelerated observer that remains spatially at rest in the exterior Schwarzschild spacetime. Such static observers in effect define the rest frame of the collapsed source. The gain in tidal energy is due to local spacetime curvature. Previous work in this direction in connection with astrophysi…
It is well known that, when an external general relativistic (electric-type) tidal field E(t) interacts with the evolving quadrupole moment I(t) of an isolated body, the tidal field does work on the body (``tidal work'') -- i.e., it transfers energy to the body -- at a rate given by the same formula as in Newtonian theory: dW/dt = -1/2 E dI/dt. Thorne has posed the following question: In view of the fact that the gravitational interaction energy between the tidal field and the body is ambiguous by an amount of order E(t)I(t), is the tidal work also ambiguous by this amount, and therefore is th…
Tidal deformations of general-relativistic multifluid compact stars
Institute of Astronomy and Astrophysics, Université Libre de Bruxelles（布鲁塞尔自由大学天体物理与天文学研究所）；Brussels Laboratory of the Universe（布鲁塞尔宇宙实验室）；Leuven Gravity Institute, KU Leuven（荷语鲁汶大学引力研究所）
We study how the matter dispersed when a supermassive black hole tidally disrupts a star joins an accretion flow. Combining a relativistic hydrodynamic simulation of the stellar disruption with a relativistic hydrodynamics simulation of the tidal debris motion, we track such a system until ~80% of the stellar mass bound to the black hole has settled into an accretion flow. Shocks near the stellar pericenter and also near the apocenter of the most tightly-bound debris dissipate orbital energy, but only enough to make the characteristic radius comparable to the semi-major axis of the most-bound …
The gravitomagnetic tidal Love number of a slowly rotating body was calculated previously under the assumption that the velocity perturbation created by the tidal field consists of an induction piece proportional to the vector potential, and a rotational piece that scales with $Ω$, the body's angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like $Ω^0 = 1$. The previous calculations are therefore incorrect, and the purpose of this paper is to repair the mistake. To keep the technical difficulties to a minimum, the tr…
In general relativity, relativistic gravity gradiometry involves the measurement of the relativistic tidal matrix, which is theoretically obtained from the projection of the Riemann curvature tensor onto the orthonormal tetrad frame of an observer. The observer's 4-velocity vector defines its local temporal axis and its local spatial frame is defined by a set of three orthonormal nonrotating gyro directions. The general tidal matrix for the timelike geodesics of Kerr spacetime has been calculated by Marck\cite{Marck}. We are interested in the measured components of the curvature tensor along t…
During the final stages of a neutron-star binary coalescence, stellar quasi-normal modes can become resonantly excited by tidal fields. If the strain exerted by the excited modes exceeds the extent to which the crust can respond linearly, localised crustal failures may occur. In this work, we re-examine resonant $g$-mode excitations of relativistic neutron stars in the last $\sim$ 10 seconds of an inspiral. We adopt realistic equations of state that pass constraints from GW170817, include 3rd order post-Newtonian terms for the conservation orbital motion, and employ a 2.5 post-Newtonian scheme…
Gravity gradiometry within the framework of the general theory of relativity involves the measurement of the elements of the relativistic tidal matrix, which is theoretically obtained via the projection of the spacetime curvature tensor upon the nonrotating orthonormal tetrad frame of a geodesic observer. The behavior of the measured components of the curvature tensor under Lorentz boosts is briefly described in connection with the existence of certain special tidal directions. Relativistic gravity gradiometry in the exterior gravitational field of a rotating mass is discussed and a gravitomag…
We analyze the coupling between the internal degrees of freedom of neutron stars in a close binary, and the stars' orbital motion. Our analysis is based on the method of matched asymptotic expansions and is valid to all orders in the strength of internal gravity in each star, but is perturbative in the ``tidal expansion parameter'' (stellar radius)/(orbital separation). At first order in the tidal expansion parameter, we show that the internal structure of each star is unaffected by its companion, in agreement with post-1-Newtonian results of Wiseman (gr-qc/9704018). We also show that relativi…
The Good, the Bad, and the Subtle: Relativistic mode sums for neutron-star tidal response
Princeton University（普林斯顿大学）；University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Tata Institute of Fundamental Research（塔塔基础研究所）；Montana State University（蒙大拿州立大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
There is a growing interest in investigating modified theories of gravity, primarily, with the aim of explaining the universe's accelerated expansion, which has been confirmed by several independent observations. Compact objects, like neutron stars, exhibit strong gravity effects and therefore are used to study modified gravity theories. We use the $f(R)=R+aR^2$ model, where R is the Ricci scalar and $a$ is a free parameter. This model has been studied both perturbatively and non-perturbatively. However, it was found that perturbative methods results in nonphysical solutions for the neutron st…
Unexpectedly Weak General Relativistic Effects in Strongly Relativistic Tidal Disruption Events
JILA, University of Colorado and National Institute of Standards and Technology（科罗拉多大学JILA与国家标准与技术研究院）；Department of Astrophysical and Planetary Sciences, University of Colorado（科罗拉多大学天体物理与行星科学系）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理学研究所）；Physics and Astronomy Department, Johns Hopkins University（约翰斯·霍普金斯大学物理与天文系）；Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）
