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 tides data. Moreover, we develop a method of the perturbed Kepler orbit to determine relativistic effects of clock comparison for space missions, which allows more precise calculations comparing to the conventional method of unperturbed Kepler orbit. With this perturbed method, it can give the perturbation of relativistic effects due to the orbital changes under the influences of tidal forces, Earth's oblateness etc. In addition, as the applications of our results, we simulate tidal effects in frequency shift for the clock comparison on the ground and also give some estimates for TianQin mission and GPS.
A cyclic nature of quantum mechanical clock is discussed as ``quantization of time." Quantum mechanical clock is seen to be equivalent to the relativistic classical clock.
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 …
How does one measure the gravitational field? We give explicit answers to this fundamental question and show how all components of the curvature tensor, which represents the gravitational field in Einstein's theory of General Relativity, can be obtained by means of two different methods. The first method relies on the measuring the accelerations of a suitably prepared set of test bodies relative to the observer. The second methods utilizes a set of suitably prepared clocks. The methods discussed here form the basis of relativistic (clock) gradiometry and are of direct operational relevance for…
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…
General Relativistic Chronometry with Clocks on Ground and in Space
Center of Applied Space Technology and Microgravity (ZARM), University of Bremen（不莱梅大学应用空间技术与微重力中心（ZARM））；Gauss-Olbers-Center, University of Bremen（不莱梅大学高斯-奥尔伯斯中心）；Institute for Theoretical Physics, University of Bremen（不莱梅大学理论物理研究所）
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 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 …
This work closes certain gaps in the literature on material reference systems in general relativity. It is shown that perfect fluids are a special case of DeWitt's relativistic elastic media and that the velocity--potential formalism for perfect fluids can be interpreted as describing a perfect fluid coupled to a fleet of clocks. A Hamiltonian analysis of the elastic media with clocks is carried out and the constraints that arise when the system is coupled to gravity are studied. When the Hamiltonian constraint is resolved with respect to the clock momentum, the resulting true Hamiltonian is f…
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…
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…
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…
A review is given of the implications of the absence of an intrinsic notion of instantaneous 3-space, so that a clock synchronization convention has to be introduced, for relativistic theories.
Lecture at BOSS2011 on relativistic metrology, on clock synchronization, relativistic dynamics and non-inertial frames in Minkowski spacetime, on relativistic atomic physics, on ADM canonical tetrad gravity in asymptotically Minkowskian spacetimes, on the York canonical basis identifying the inertial (gauge) and tidal degrees of freedom of the gravitational field, on the Post-Minkowskian linearization in 3-orthogonal gauges, on the Post-Newtonian limit of matter Hamilton equations, on the possibility to interpret dark matter as a relativistic inertial effect connected with relativistic metrolo…
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…
Relativistic Modeling for Solid Earth Tide Estimation via Space-to-Ground Clock Comparison
National Gravitation Laboratory, MOE Key Laboratory of Fundamental Physical Quantities Measurement, and School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院，引力中心，基础物理量测量教育部重点实验室）；MOE Key Laboratory of TianQin Mission, TianQin Research Center for Gravitational Physics & School of Physics and Astronomy, Frontiers Science Center for TianQin, CNSA Research Center for Gravitational Waves, Sun Yat-sen University (Zhuhai Campus)（中山大学（珠海校区）物理与天文学院，天琴引力物理研究中心，天琴前沿科学中心，国家航天局引力波研究中心）；Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学技术大学物理科学学院，微观尺度物质科学国家研究中心）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
The conflict between quantum theory and the theory of relativity is exemplified in their treatment of time. We examine the ways in which their conceptions differ, and describe a semiclassical clock model combining elements of both theories. The results obtained with this clock model in flat spacetime are reviewed, and the problem of generalizing the model to curved spacetime is discussed, before briefly describing an experimental setup which could be used to test of the model. Taking an operationalist view, where time is that which is measured by a clock, we discuss the conclusions that can be…
We perform a protocol for multipartite quantum clock synchronization under the influence of Unruh thermal noise. The clocks consisting of Unruh-DeWitt detectors when one of detectors accelerated is obtained. To estimate the time difference between the clocks, we calculate the time probability and analyze how the probability is influenced by the Unruh thermal noise and other factors. It is shown that both relativistic motion and interaction between the atom and the external scalar field affect the choice of optimal number of excited atoms in the initial state, which leads to a higher clock adju…
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.
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…
Quantum mechanics and general relativity have been each successfully tested in numerous experiments. However, the regime where both theories are jointly required to explain physical phenomena remains untested by laboratory experiments, and is also not fully understood by theory. This contribution reviews recent ideas for a new type of experiments: quantum interference of "clocks", which aim to test novel quantum effects that arise from time dilation. "Clock" interference experiments could be realised with atoms or photons in near future laboratory experiments.
If a clock, mathematically modeled by a parametrized timelike curve in a general-relativistic spacetime, is given, the radar method assigns a time and a distance to every event which is sufficiently close to the clock. Several geometric aspects of this method are reviewed and their physical interpretation is discussed.
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…
The special relativistic test theory of Mansouri and Sexl is sketched. Theories based on different clock synchronisations are found to be equivalent to special relativity, as regards experimental results. The conventionality of clock synchronisation is shown not to hold, by means of an example, in a simple accelerated system and through the principle of equivalence in gravitational fields, especially when the metric is not static. Experimental implications on very precise clock synchronisation on earth are discussed.
The theory of relativity associates a proper time with each moving object via its world line. In quantum theory however, such well-defined trajectories are forbidden. After introducing a general characterisation of quantum clocks, we demonstrate that, in the weak-field, low-velocity limit, all "good" quantum clocks experience time dilation as dictated by general relativity when their state of motion is classical (i.e. Gaussian). For nonclassical states of motion, on the other hand, we find that quantum interference effects may give rise to a significant discrepancy between the proper time and …
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…
The theory of relativistic {\em location systems} is sketched. An interesting class of these systems is that of relativistic {\em positioning systems,} which consists in sets of four clocks broadcasting their proper time. Among them, the more important ones are the {\em auto-located positioning systems,} in which every clock broadcasts not only its proper time but the proper times that it receives from the other three. At this level, no reference to any exterior system (the Earth surface, for example) and no synchronization are needed. Some properties are presented. In the SYPOR project, such …
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…
We derive exact expressions for the relativistic redshift between an Earth-bound observer, that is meant to model a standard clock on the Earth's surface, and various (geodesic) observers in the Schwarzschild spacetime. We assume that the observers exchange radial light signals to compare the frequencies of standard clocks, which they transport along their respective worldlines. We calculate the redshift between an Earth-bound clock and static observers, observers in radial free fall, on circular geodesics, and on arbitrary bound quasi-elliptical orbits. For the latter case, we consider as exa…
In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of $2π$. For two clocks on counterrotating equatorial circular orbits around th…
Clock synchronization is the backbone of applications such as high-accuracy satellite navigation, geolocation, space-based interferometry, and cryptographic communication systems. The high accuracy of synchronization needed over satellite-to-ground and satellite-to-satellite distances requires the use of general relativistic concepts. The role of geometrical optics and antenna phase center approximations are discussed in high accuracy work. The clock synchronization problem is explored from a general relativistic point of view, with emphasis on the local measurement process and the use of the …
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.
Although time is one of the fundamental notions in physics, it does not have a unique description. In quantum theory time is a parameter ordering the succession of the probability amplitudes of a quantum system, while according to relativity theory each system experiences in general a different proper time, depending on the system's world line, due to time to time dilation. It is therefore of fundamental interest to test the notion of time in the regime where both quantum and relativistic effects play a role, for example, when different amplitudes of a single quantum clock experience different…
Clock synchronization between the ground and satellites is a fundamental issue in future quantum telecommunication, navigation, and global positioning systems. Here, we propose a scheme of near-Earth orbit satellite-based quantum clock synchronization with atmospheric dispersion cancellation by taking into account the spacetime background of the Earth. Two frequency entangled pulses are employed to synchronize two clocks, one at a ground station and the other at a satellite. The time discrepancy of the two clocks is introduced into the pulses by moving mirrors and is extracted by measuring the…
Sound propagation within certain non-relativistic condensed matter models obeys a relativistic wave equation despite such systems admitting entirely non-relativistic descriptions. A natural question that arises upon consideration of this is, "do devices exist that will experience the relativity in these systems?" We describe a thought experiment in which 'acoustic observers' possess devices called sound clocks that can be connected to form chains. Careful investigation shows that appropriately constructed chains of stationary and moving sound clocks are perceived by observers on the other chai…
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…
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.…
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）
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…
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…
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 …
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…
