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## L. Andersson## Self-gravitating elastic bodiesSelf-gravitating elastic bodies in Einstein gravity provide models for extended bodies in GR. I will discuss constructions of static and rotating bodies, multi-body configurations, as well as recent results on the initial value problem for self-gravitating elastic bodies. ## S. Babak## Perturbed geodesic motionExtreme mass ratio inspirals are among the most interesting sources of gravitational waves for the space based detectors. Those are results of capture of a stellar mass compact object by a massive black hole in the galactic nuclei. A small body can be treated as a test mass, it spirals toward the central hole loosing its orbital energy and angular momentum through gravitational radiation. We will discuss a particular method for computing gravitational waveform from such binaries. The system might not be clean, there could be an environmental influence which leaves measurable imprint in the phase of emitted gravitational waves. In addition, if mass ratio is not too extreme, we could expect to measure effect of coupling of the small body spin to the background curvature. ## L.F. Costa## Spacetime dynamics of spinning particles -- exact Gravito-Electromagnetic analogiesIn this talk the equations of motion for spinning test particles in gravitational and electromagnetic fields are compared, and exact gravito-electromagnetic analogies are seen to emerge. These analogies provide a familiar formalism to treat gravitational problems, as well as a means for a comparison of the two interactions. Fundamental differences are manifest in the symmetries and time projections of the electromagnetic and gravitational "tidal tensors". The physical consequences of the symmetries are explored comparing the following analogous setups: magnetic dipoles in the field of non-spinning/spinning charges, and gyroscopes in the Schwarzschild, Kerr, and Kerr-de Sitter spacetimes. The implications of the time-projections of the tidal tensors are illustrated by the work done on the particle in various frames; in particular, a reciprocity is found to exist: in a frame comoving with the particle, the electromagnetic (but not the gravitational) field does work on it, causing a variation of its proper mass; conversely, for "static observers", a stationary gravitomagnetic (but not a magnetic) field does work on the particle, accounting for the Hawking-Wald spin interaction energy. At the heart of the analogies is the Mathisson-Pirani spin condition, which is poorly understood even today, namely its degeneracy and the famous helical solutions it allows. I shall briefly discuss the problem of the spin supplementary condition and of the center of mass definition in General Relativity, demystifying the helical motions. They are seen to be valid and equivalent descriptions of the motion of a spinning body (only somewhat complicated); and their dynamical consistency is explained through the concept of "hidden momentum", which is analogous to the hidden momentum of electromagnetic systems. ## W.G. Dixon## The New Mechanics of Myron Mathisson (1897-1940) and its subsequent developmentIn 1937, Myron Mathisson published a paper in German in Acta Physica Polonica with the ambitious title (in translation) "A New Mechanics of Material Systems". This was the first paper to study the motion of an extended body in general relativity in a way that took into account its departure from being a point particle. It described such a body by an infinite set of multipole moments, with the expectation that a good approximation to the deviation of its motion from the geodesic path of a point particle could be determined by retaining only the first few moments. Equations of motion were then derived based on retention of only the monopole (mass) and dipole (spin) moments. There was also partial success in extending this to include the quadrupole moment. Mathisson continued working on this problem until his early death in 1940 by studying the situation in special relativity in a search for clues on how to progress further. Papapetrou rediscovered Mathisson's results in 1951 by a very different method, referring to Mathisson's 1937 paper but asserting that it only dealt with special relativity. Papapetrou's work revived interest in the problem, which was then taken up during the next 20 years by a number of authors including Tulczyjew, Taub, Madore and myself, with improvements to Papapetrou's approach and attempts to extend it. None of this work got past the quadrupole approximation and none made any significant improvement on the original results of Mathisson. There seemed to be a barrier preventing Mathisson's ambitious program being taken beyond the point that he had reached. I shall show the nature of this barrier, how it was breached and his ambitious program completed in 1974 by returning to Mathisson's original methods and essentially continuing from where he had left it upon his premature death. ## A. Eckart## Sagittarius A* at the center of the Galaxy as a laboratory for relativistic physicsThe super-massive 4 million solar mass black hole (SMBH) SgrA* shows flare emission from the millimeter to the X-ray domain. A detailed analysis of the infrared light curves allows us to address the accretion phenomenon in a statistical way. The analysis shows that the near-infrared flare amplitudes are dominated by a single state power law, with the low states in SgrA* are limited by confusion though the unresolved stellar background. The are several dusty objects in the close vicinity of SgrA*. The source G2/DSO is one of them. It nature is unclear. It may be similar to nearby similar stellar dusty sources or may consist predominantly of gas and dust. Particularly in this case an enhanced accretion activity onto SgrA* is expected. Here the interpretation of recent data and upcoming observations are discussed. ## G. Faye## The post-Newtonian formalism and the two-body problem in general relativityThe post-Newtonian formalism in harmonic coordinates is a powerful perturbative scheme that allows one to investigate the dynamics of "slowly"-moving systems in general relativity. In particular, it has been used in the last decade to compute the equations of motion of inspiralling compact binaries to high accuracy, with the aim of building approximate analytical waveforms for gravitational-wave-experiment data analysis. The description of the system evolution in some appropriate region of space-time relies on: (1) a systematic expansion of all relevant quantities in powers of $1/c^2$, (2) an analytical matching of the post-Newtonian metric with a multipolar post-Minkowskian solution of the Einstein equations, valid outside the matter source, (3) an effective modeling of compact bodies as point particles endowed with a multipole structure, (4) a systematic use of dimensional regularization, which can often be replaced in practice by a "Hadamard-Schwartz" prescription. This approach presents the advantage of resting on a strong theoretical basis and making the various physical effects quite explicit. It was used successfully to obtain, for instance, the equations of motions of compact binaries at 3.5PN order (i.e. neglecting $1/c^8$ contributions) including the spin-orbit interactions and to describe tidal corrections to the two-body dynamics at the next-to-next-to-leading order. ## T. Futamase## Strong-field point particle limit in Equations of Motion in GRStrong-field point particle limit is a method to incorporate strong internal gravity and finiteness of the body in the equation of motion. I will give an introductory lecture how to incorporate this limit in the post-Newtonian approximation applicable for relativistic compact binaries. In doing so we do not have any arbitrariness in 3rd post-Newtonian order. I also show how to deal with extendedness of small body in this formalism and derive explicitly some of well-known multipole-related forces. I would like to mention a possibility to apply this limit to fast-motion approximation. ## D. Giulini## Energy-momentum tensors and motion in Special RelativityThe motion of an extended structured body in Special Relativity is described by conserved quantities, like linear and angular momenta (so called Poincare charges). They can be used to define various world lines within the convex hull of the body, which may describe centres of mass, centres of inertia, and centres of spin. To set the stage for further discussions by others in the context of General Relativity, I shall describe the algebraic and geometric background behind these constructions in the simplest case of Poincare invariance. I will start from remarks concerning the mathematical habitat of global momenta and then discuss the ambiguity in defining the worldlines of centres in case of spinning bodies. ## A. Harte## Self-interaction and its effects on motionSufficiently small test bodies in general relativity fall on geodesics. There is a sense in which small self-gravitating objects also fall on geodesics. The precise meaning of this is, however, rather subtle. Objects do not generically fall on geodesics of the physical metric. Rather, they fall on geodesics associated with an appropriate effective connection. I show that under mild assumptions, all laws of motion satisfied by extended test bodies generalize essentially without change for self-interacting bodies. Dixon's multipole expansions for the force and torque acting on extended test bodies continue to hold for generic objects if the physical metric is replaced by an appropriate "effective metric". This effective metric is related to the physical one via a "nonlinear subtraction" which tends to remove all metric variations on the scale of the body. The natural momenta appearing in the laws of motion are Dixon's together with a certain renormalization due to the self-field. Similarly, all naturally-appearing higher multipole moments are renormalized versions of Dixon's definitions. The theory of extended test bodies and the theory of gravitational self-force are both special cases of the results described herein. ## F.W. Hehl## The Gravity Probe B experiment cannot sense a possible torsion of spacetime: introduction into the Poincare gauge theory of gravity and remarks on its equations of motionThe rigid fundament of the Standard Model of elementary particle physics is the Minkowski space of special relativity. Its group of motion is built from the four translations $T(4)$ and the six Lorentz rotations $SO(1,3)$. Their semidirect product is then the Poincar\'e group $P(1,3)=T(4) \semidirect SO(1,3)$. The $P(1,3)$ classifies matter according to its mass $m$ and its spin $s$. Because of the existence of fermions and half-integer spin, we have to introduce in Minkowski space orthonormal coframes (tetrads) $\vt^\a=e_i{}^\a dx^i$. The parallel transport in Minkowski space is governed by a flat and uncontorted Lorentz connection $\Gamma^{\a\b}= \Gamma_i{}^{\a\b}dx^i=-\Gamma^{\b\a}$; in Cartesian coordinates it vanishes. In order to switch on gravity in this Minkowski space, one has, generalizing the procedure of Einstein \`a la Sciama and Kibble, to gauge the Poincar\'e group, see for the original papers. Then the $\vt^\a$ become translation potentials and the $\Gamma^{\a\b}$ Lorentz potentials coupling to the energy-momentum and to the spin angular momentum of matter, respectively: $\frak{T}_\a=\d L_{\text{mat}}/\d \vt^\a$ and $\frak{S}_{\a\b}=\d L_{\text{mat}}/\d \Gamma^{\a\b}$; here $L_{\text{mat}}$ is the matter Lagrangian. The spacetime arena is now a {\it Riemann-Cartan spacetime} with the field strengths torsion $T^\a=D\vt^\a$ and curvature $R^{\a\b}=d\Gamma^{\a\b} -\Gamma^{\a\g} \wedge\Gamma^\b{}_\g $. Taking Einstein's theory of gravity as being valid to a good approximation, the described procedure seems to be the only consistent way to introduce, besides the curvature, a {\it torsion} of spacetime. With this proviso, it can be shown conclusively that in the framework of the Poincar\'e gauge theory torsion couples only to the {\it spin of elementary particles}, not, however, to the orbital angular momentum of the rotating quartz gyroscope of Gravity Probe B. Extensions of this result to metric-affine gravity (MAG) will be shortly mentioned. ## P.A. Hogan## Equations of Motion of Schwarzschild, Reissner-Nordstrom and Kerr ParticlesA technique for extracting from the appropriate field equations the relativistic motion of Schwarzschild, Reissner-Nordstrom and Kerr particles moving in external fields is motivated and illustrated. The key assumptions are that (a) the particles are isolated and (b) near the particles the wave fronts of the radiation generated by their motion are smoothly deformed spheres. No divergent integrals arise in this approach. The particles are not test particles. The formalism is used, however, to derive Papapetrou's equations of motion for a spinning test particle, neglecting spin-spin terms. ## Y. Itin## Equations of motion as derived from the field equationsThe problem of derivation of the equations of motion from the field equations is considered. Einstein's field equations have a specific analytical form: They are linear in the second order derivatives and quadratic in the first order derivatives of the field variables. We utilize this particular form and propose a novel algorithm for the derivation of the equations of motion from the field equations. It is based on the condition of the balance between the singular terms of the field equation. We apply the algorithm to a nonlinear Lorentz invariant scalar field model. We show that it results in the Newton law of attraction between the singularities of the field moved on approximately geodesic curves. The algorithm is applicable to the N-body problem of the Lorentz invariant field equations. ## B.R. Iyer## Gravitational Waves from inspiraling compact binaries in general orbits and their applications to EOMThe Multipolar Post-Minkowskian formalism discussed by G. Faye in this seminar has been applied to inspiraling compact binaries moving in general orbits and used to compute the gravitational waveform and the associated far zone fluxes of energy and angular momentum to order 3PN beyond the leading order. This allows one to generalize the classical treatment of Peters and Mathews at the leading order for the secular evolution of the elliptical orbits under gravitational radiation reaction and provides the basic inputs for the construction of high accuracy GW templates for binaries moving in quasi-elliptic orbits. These computations use the 3PN accurate generalized quasi-Keplerian representations of Memmesheimer, Gopakumar and Schaefer and are more involved due to the presence of the hereditary contributions that depend on the past history of the source. By the method of variation of constants one can go beyond the secular evolution and compute the much smaller orbital scale variations. The expressions for the fluxes also allow one to partially reconstruct the radiation reaction or odd terms in equations of motion. The standard high accuracy treatment of inspiral by PN approximants breaks down at late stages of inspiral. However, the Effective One Body approach provides a good analytical description of the late inspiral, merger and ringdown. Pade resummation of the PN waveforms is one of the crucial inputs to describe the radiation reaction in this approach. ## S.A. Klioner## High-accuracy astronomical observations in solar system and relativistic equations of motionThe tremendous progress in technology, which we witnessed during the last 35 years, has led to enormous improvements of accuracy in the disciplines of astrometry and celestial mechanics. Considering the growth of accuracy of positional observations in the course of time, we see that during the 25 years between 1988 and 2013 we expect the same gain in accuracy (4.5 orders of magnitude) as that realized during the whole previous history of astrometry, from Hipparchus till 1988 (over 2000 years). Observational techniques like Lunar and Satellite Laser Ranging, Radar and Doppler Ranging, Very Long Baseline Interferometry, high-precision atomic clocks, etc., have made it possible to probe the kinematical and dynamical properties of celestial bodies to unprecedented accuracy. Moreover, optical interferometry in space and interplanetary laser ranging are expected to boost the accuracy even further. It is clear that this accuracy require careful application of General Relativity in the modeling of those observations. In this presentation, practice-driven requirements for the relativistic equations of motion are discussed. This includes post-Newtonian equations of motion of N bodies with arbitrary multipolar structure, post-Minkowskian and post-post-Newtonian equations of light propagation in the field of N bodies of arbitrary structure, relativistic equations of rotational motion of an extended body being a part of N-body system, etc. ## B. Kol## Equations of motion in Post-Newtonian gravity from the Effective Field Theory approachWe shall describe how the effective field theory approach to GR is used to determine the equations of motion of non-relativistic (post-Newtonian) gravitating objects by using Feynman diagrams and the associated integrals together with a non-relativistic decomposition of Einstein's gravitational field to economize computation and add insight. A theory of radiation and self-force will be incorporated. ## S. Kopeikin## Equations of Motion in an Expanding UniversePost-Newtonian theory of motion of celestial bodies and propagation of light is instrumental in conducting the critical experimental tests of general relativity and in building the astronomical ephemerides of celestial bodies in the solar system with an unparalleled precision. The cornerstone of the theory is the postulate that the solar system is gravitationally isolated from the rest of the universe and the background spacetime is asymptotically flat. We extend this theoretical concept and formulates the principles of celestial dynamics of particles and light moving in gravitational field of a localized astronomical system embedded to the expanding Friedmann-Lemaître-Robertson-Walker (FLRW) universe. We formulate the precise mathematical concept of the Newtonian limit of Einstein's field equations in the conformally-flat FLRW spacetime and analyze the geodesic motion of massive particles and light in this limit. We prove that by doing conformal spacetime transformations, one can reduce the equations of motion of particles and light to the classical form of the Newtonian theory. However, the time arguments in the equations of motion of particles and light differ from each other in terms being proportional to the Hubble constant H. This leads to the important conclusion that equations of light propagation used currently by Space Navigation Centers for fitting the range and Doppler-tracking observations of celestial bodies are missing some terms of the cosmological origin that are proportional to the Hubble constant H. We also analyze the effect of the cosmological expansion on motion of electrons in atoms. We prove that the Hubble expansion does not affect the atomic frequencies and, hence, does not affect the atomic time scale used in creation of astronomical ephemerides. We derive the cosmological correction to the light travel time equation and argue that their measurement opens a fascinating opportunity to determine the local value of the Hubble constant in the solar system independently of cosmological observations. ## B. Krishnan## Status and prospects for detecting gravitational waves from compact binary inspiralsIn this talk I will discuss some of the currently used techniques and results used in gravitational wave searches for binary inspirals with ground based interferometric detectors. I will also briefly discuss the impact of waveform modeling on these searches. ## J. Mueller## Tracking the Moon to Study RelativitySince 1969, lunar laser ranging (LLR) is carried out from a few observatories on Earth to a few reflector arrays on the Moon. To enable data analysis at the mm level of accuracy, all elements of the tracking process have to be modeled at appropriate (relativistic) approximation, i.e. the orbits of the major bodies of the solar system, the rotation of Earth and Moon, the signal propagation, but also the involved reference and time systems. We will show where relativity enters the LLR analysis and how the whole measurement process is modeled, including the relevant classical (Newtonian) effects like gravity field of Earth and Moon, tidal effects, ocean loading, lunar tidal acceleration (that causes the increase of the Earth-Moon distance by about 3.8 cm/year), etc. On the other hand, LLR is one of the best tools to test General Relativity in the solar system. It allows for constraining gravitational physics parameters related to the strong equivalence principle, geodetic precession, preferred-frame effects, or the time variability of the gravitational constant. We will present recent results for the various relativistic parameters. We will also show what the limitations are, and give an outlook to future perspectives if technology and modeling are further developed. ## Y.N. Obukhov## Dynamics of matter with microstructureIn classical continuum mechanics, a material medium consists of structureless points. In the early 20th century, the Cosserat brothers proposed a generalization of this simple picture, in which the material body or fluid is formed by particles whose microscopic properties contribute to the macroscopic dynamics of the medium. These more complex continuous mechanical models are known under different names, such as the theories of multipolar, micromorphic, or oriented media. Important particular cases of continua with microstructure is represented by the spinning fluids and bodies, ferromagnetic materials, cracked media, liquid crystals, superfluids, and granular media. We discuss the Lagrangian theories of material media with microstructure in flat and curved spacetime. The symmetries, conservation laws and the equations of motion are derived for the models of fluids, particles and extended bodies with microstructure. Their possible applications as detectors of the geometrical structure of spacetime are reviewed. ## R. F. O'Connell## General Relativistic Two-Body Problem, including Spin (Rotation) EffectsSince the electromagnetic interaction between two electrons, with inclusion of relativistic and spin effects, is most rigorously calculated by one-photon exchange, we obtain the gravitational interaction between two electrons by using one-graviton exchange and then generalizing to the classical case of two arbitrary masses by letting $(1/2) \hbar \sigma^{1,2} \rightarrow S^{1,2}$, where $S^{1,2}$ are the classical spin angular momenta. In essence, we adopted the universality of the gravitational interaction, which was verified by immediate derivation of the one-body Lense-Thirring results. Next, we obtained the generalization to the two-body case. Our subsequent procedure made use of the fact the Runge-Lenz vector $A$ is a constant in the non-relativistic case and that $A L=0$ , where $L$ is the angular momentum. Our spin precession results readily followed and our orbital precession results showed that $A$ and precessed at the same rate, so that $A L$ remained zero. We also pointed out that the spin contributions to the orbital results are non-observable (as distinct from the spin precession results which involve just angles), due to a special relativistic effect which states that a spinning body has a finite size so that there are various choices of its centre-of-mass or, equivalently, various choices of the spin supplementary condition, which leads to various choices for the position coordinate (analogous to the famous Foldy-Wouthuysen transformation of the coordinate in Dirac's theory). Finally, we discuss the verification of our two-body results by Breton et al. ## V. Perlick## On the self-force in electrodynamics and in gravityIn the first part of this talk I'll discuss the electromagnetic self-force of a point charge in classical electrodynamics. In particular, I'll recall that the problems associated with a point charge in standard Maxwell theory are related to the fact that the field energy in a finite ball around a point charge is infinite. I'll then discuss the question of whether these problems can be remedied in modified theories of electrodynamics. I'll concentrate on two such modified theories: The non-linear Born-Infeld theory and the higher-order Podolsky theory. In the second part of the talk I'll then discuss what can be learned about the gravitational self-force of point masses from the electrodynamic analogue. ## R. Plyatsko## Mathisson-Papapetrou equations as a source of knowledge on highly relativistic spinning particle motions in the gravitational fieldThe Mathisson-Papapetrou (MP) equations enable to describe possible motions of a classical (nonquantum) spinning test particle in the gravitational field according to general relativity. It is of importance that the MP equations can be used for investigations of the spinning particle motions with any velocity relative to the source of the gravitational field, up to the speed of light, similarly as the geodesic equations for a spinless particle. By many investigations it is known that if the particle's velocity is not too high the influence of the spin on its trajectory in the gravitational field is very small. From the second half of the 1970s in the focus of our studies is the properties of the highly relativistic spinning particle motions in the Schwarzschild and Kerr backgrounds. It is shown that as a result of the highly relativistic spin-gravity coupling the trajectory of a spinning particle differs significantly from the corresponding trajectory of a spinless particle. Dependently on the correlation of signs of the particle's spin and orbital velocity this coupling creates an essential additional attractive or repulsive action. In this sense one can conclude that antigravity in the natural way is present in gravity. Note that from the point of view of an observer comoving with a particle the spin-gravity action is proportional to the second power of the relativistic Lorentz factor as calculated by the particle's velocity relative to Schwarzschild's mass. We also discuss adequate choosing the supplementary condition for the MP equations; the connection between the MP equations and the general relativistic Dirac equation; the possibilities of registration of the highly relativistic spin-gravity effects in some astrophysical processes. ## A. Pound## Motion of small bodies in curved spacetimes: An introduction to gravitational self-forceThe gravitational self-force describes the effect of a small body's gravitational field on its own motion. Incorporating this effect with high accuracy is essential in modelling binary inspirals with extreme mass ratios, which will be an important source of gravitational waves for future detectors. In recent years, asymptotic approximation schemes have been developed to describe the motion of a small compact body and the metric perturbation it produces to any order in perturbation theory. The schemes are based on rigorous methods of matched asymptotic expansions, in which one accounts for the finite size of the small body. I describe the foundations of these schemes and two ways of representing a body's motion within them: a perturbative description in terms of small deviations from a reference geodesic, valid on short timescales; and a self-consistent description in terms of a worldline that obeys a self-accelerated equation of motion, potentially valid on any timescale. I then discuss recent results at second order in perturbation theory, which have extended the first-order result that, up to couplings of the body's multipole moments to the external background curvature, a small compact body moves on a geodesic of a certain smooth metric satisfying the vacuum Einstein equation. These second-order results promise to be the first approximation that is sufficiently accurate to extract system parameters from detected waveforms. ## D. Puetzfeld## Multipolar approximation schemes for extended test bodiesMultipolar approximation schemes aim for a simplified description of extended test bodies in given gravitational fields. Within such schemes test bodies are characterized by a set of multipole moments, and their motion is described by taking only a finite number of moments, e.g. mass and spin, into account. Several schemes have been proposed to provide a multipolar description of test bodies in General Relativity. In this talk I will give a brief overview, starting with the work of Mathisson (1937), and highlight the differences between some approaches in the literature. The completion of Mathisson's program will be discussed by W.G. Dixon in this seminar. ## G. Schaefer## Higher order post-Newtonian dynamics of compact binary systems in Hamiltonian formThe Hamiltonian approach to general relativity by Arnowitt, Deser, and Misner (ADM) is used to derive and discuss the higher oder post-Newtonian dynamics and motion of compact binary systems also including the spins of the components. Explicit analytic Hamiltonians will be presented for the conservative and dissipative dynamics, the latter resulting from gravitational radiation damping. Explicit analytic expressions for the orbital motion and emitted gravitational wave forms will be given. ## O. Semerak## On motion and notion of pole-dipole particlesMotion of spinning particles in relativistic space-times is usually treated within pole-dipole approximation. Inclusion of spin may lead to interesting departures from geodesic motion, though its values occurring in astrophysical circumstances are rarely large enough to produce noticeable effects. In order to have a numerical idea, we show some results of integration of Mathisson-Papapetrou-Dixon equations in a Kerr space-time. A considerable part of the world-tube of pole-dipole approach comes from discussion of the "spin supplementary condition" which has to be added in order to close the system of MPD equations. Generically, this condition reads $S_{ab}V^b=0$, where $V^a$ is some time-like vector, but different choices of $V^a$ lead to different properties of the MPD system: they yield different world-lines, with only some of them fixing the world-line uniquely, and they simplify the system in different specific cases. In order to compare the effect of the spin conditions which have been favoured in the literature, we found their relationships and illustrated the difference on trajectories of pole-dipole particles in black-hole fields. However, in a highly non-homogeneous field, such as close to a black hole, the applicability of pole-dipole approximation itself (actually of the multipole expansion in general) starts to be the main issue. In order to reveal this limitation, we introduced the "minimal world-tube" of a particle, determined by history of the most compact body consistent with a given size of spin. The figures obtained prove that in strong-curvature regions the MPD model tends to become problematic; in any case, a mass quadrupole of the body should be incorporated then as well. ## A.J. Silenko## Motion of Spinless Particles in Gravitational FieldsInteraction of spinless particles with gravitational fields is defined by the covariant Klein-Gordon-Fock equation. The generalized Case-Foldy-Feshbach-Villars transformation applicable for both massive and massless particles presents the initial equation in a Hamiltonian form. The consequent exact Foldy-Wouthuysen transformation allows obtaining the even form of the Hamiltonian convenient for a derivation of equations of motion. Quantum mechanical and semiclassical equations of motion can be found for a wide class of gravitation fields and noninertial frames. We obtain the classical limit of the Hamiltonian and the quantum mechanical equations of motion which is fully consistent with corresponding equations of classical gravity. ## D. Singh## The MPD Equations in Analytic Perturbative FormThis presentation explores the recent development of an analytic perturbative approach to spinning particle dynamics in the pole-dipole approximation, based upon the Mathisson-Papapetrou-Dixon (MPD) equations of motion. The basis for this formulation is to hypothesize that the linear and spin angular momentum degrees of freedom can be expressed in terms of a power series expansion with respect to the particle's spin magnitude. In combination with the particle's kinematic degrees of freedom and supplementary spin conditions to determine the system, it is possible to solve for all relevant parameters in a completely general form and to conceivably infinite order in the expansion parameter, without prior reference to any specific space-time background. Furthermore, it can be shown that the particle's squared mass and spin magnitudes can shift due to a classical analogue of radiative corrections that arise from spin-curvature coupling. Possible applications and future directions based upon this formalism are also presented for consideration. ## J. Steinhoff## Spin and Quadrupole Contributions to the Motion of Astrophysical BinariesCompact objects in general relativity approximately move along geodesics of spacetime. Corrections to their equations of motion due to spin (dipole), quadrupole, and higher multipoles were derived, e.g., by Mathisson, Papapetrou, and Dixon. These corrections can be modeled by an extension of the point mass action. We couple such actions to the gravitational field within the canonical formalism of Arnowitt, Deser, and Misner. Based on these developments, approximate post-Newtonian Hamiltonians describing the motion of self-gravitating compact astrophysical binaries can be calculated. We discuss how Dixon's quadrupole can be related to astrophysical objects like neutron stars or black holes. This is not only important for the post-Newtonian approximation, but also in small mass ratio situations. Quadrupole effects can encode information about the internal structure of the compact objects, e.g., they allow a distinction between black holes and neutron stars, and also different equations of state. Furthermore, an inclusion of oscillation modes of the object via the quadrupole is considered. For this purpose the Newtonian case is reviewed and formulated in a novel manner. Problems for an extension to the general relativistic case and a tentative solution are discussed. ## I. Vega## Self-force: physical aspects and applicationsOver the past decade, the prospect of detecting low-frequency gravitational waves with space-based observatories has inspired a resurgence of interest in the motion of point particles in curved spacetime, centered on understanding the role played by backreacting self-force effects. Building on other talks in this meeting, I shall survey a number of physical contexts where modern self-force calculations have found relevance, ranging from the modeling of black-hole binaries to questions of cosmic censorship. ## B. Wardell## Self-force: Computational StrategiesBuilding on substantial foundational work in understanding the effect of a small body's self-field on its own motion, the past 15 years has seen the development of several strategies for explicitly computing the first order self-force on a small charge, along with the corresponding equations of motion. These approaches broadly fall into three categories: (i) mode-sum regularization, (ii) effective source approaches and (iii) Green function methods. In this talk, I will review the various approaches and give details of how each one is implemented in practice, highlighting the advantages and disadvantages in each case. I will also discuss the potential for their use in extensions to second perturbative order. ## N. Wex## Testing the motion of strongly self-gravitating bodies with radio pulsarsBefore the 1970s, precision tests for gravity theories were constrained to the weak-field environment of the Solar System. In terms of relativistic equations of motion, the Solar System gave access to the first order corrections to Newtonian dynamics, notably the well-measured anomalous precession of the Mercury orbit. Testing anything beyond the first post-Newtonian contributions was for a long time out of reach. Then the discovery of the first binary pulsar by Russell Hulse and Joseph Taylor in summer 1974, provided not only the first laboratory to test the gravitational wave damping in a binary motion, but also gave us the possibility to investigate the interaction of strongly self-gravitating bodies. To date there are a number of binary pulsars known, which can be utilized to test different aspects of relativistic dynamics. They not only allow for tests of specific gravity theories, like general relativity or scalar-tensor gravity, but also in addition to it, give generic constraints on potential deviations of gravity from general relativity in the interaction of strongly self-gravitating bodies. In my talk I will give an introduction to gravity tests with pulsars, and summarize some of the most important results. Furthermore, I will give a brief outlook into the future of this exciting field of experimental gravity. |