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Building upon the Noether charge formalism of Iyer and Wald, we derive a variational formula for spacetimes admitting a Killing vector field, for a generic energy-momentum distribution with compact support. Applying this general result to the particular case of a binary system of spinning compact objects moving along an exactly circular orbit, modelled using the multipolar gravitational skeleton formalism, we derive a first law of compact binary mechanics at dipolar order. We prove the equivalence of this new result with the canonical Hamiltonian first law previously derived for binary systems of spinning compact objects, for spins colinear with the orbital angular momentum. This paper paves the way to an extension of the first law of binary mechanics to the next quadrupolar order, thereby accounting for the spin-induced and tidally-induced deformability of the compact bodies., Comment: 44 pages, 3 figures

The open question of whether a black hole can become tidally deformed by an external gravitational field has profound implications for fundamental physics, astrophysics and gravitational-wave astronomy. Love tensors characterize the tidal deformability of compact objects such as astrophysical (Kerr) black holes under an external static tidal field. We prove that all Love tensors vanish identically for a Kerr black hole in the nonspinning limit or for an axisymmetric tidal perturbation. In contrast to this result, we show that Love tensors are generically nonzero for a spinning black hole. Specifically, to linear order in the Kerr black hole spin and the weak perturbing tidal field, we compute in closed form the Love tensors that couple the mass-type and current-type quadrupole moments to the electric-type and magnetic-type quadrupolar tidal fields. For a dimensionless spin ~ 0.1, the nonvanishing quadrupolar Love tensors are ~ 0.002, thus showing that black holes are particularly "rigid" compact objects., Comment: 7 pages; matches version accepted in Phys. Rev. Lett

The open question of whether a Kerr black hole can become tidally deformed or not has profound implications for fundamental physics and gravitational-wave astronomy. We consider a Kerr black hole embedded in a weak and slowly varying, but otherwise arbitrary, multipolar tidal environment. By solving the static Teukolsky equation for the gauge-invariant Weyl scalar $\psi_0$, and by reconstructing the corresponding metric perturbation in an ingoing radiation gauge, for a general harmonic index $\ell$, we compute the linear response of a Kerr black hole to the tidal field. This linear response vanishes identically for a Schwarzschild black hole and for an axisymmetric perturbation of a spinning black hole. For a nonaxisymmetric perturbation of a spinning black hole, however, the linear response does not vanish, and it contributes to the Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an application, we compute explicitly the rotational black hole tidal Love numbers that couple the induced quadrupole moments to the quadrupolar tidal fields, to linear order in the black hole spin, and we introduce the corresponding notion of tidal Love tensor. Finally, we show that those induced quadrupole moments are closely related to the well-known physical phenomenon of tidal torquing of a spinning body interacting with a tidal gravitational environment., Comment: 65 pages; matches version accepted in Phys. Rev. D

For binary systems of corotating black holes, the zeroth law of black hole mechanics states that the surface gravity is constant over each component of the horizon. Using the approximation of a conformally flat spatial metric, we compute sequences of quasi-equilibrium initial data for corotating black hole binaries with irreducible mass ratios in the range 10:1-1:1. For orbits outside the innermost stable one, the surface gravity is found to be constant on each component of the apparent horizon at the sub-percent level. We compare those numerical results to the analytical predictions from post-Newtonian theory at the fourth (4PN) order and from black hole perturbation theory to linear order in the mass ratio. We find a remarkably good agreement for all mass ratios considered, even in the strong-field regime. In particular, our findings confirm that the domain of validity of black hole perturbative calculations appears to extend well beyond the extreme mass-ratio limit., 39 pages, 10 figures; matches published version

The existence of gravitational radiation is a natural prediction of any relativistic description of the gravitational interaction. In this chapter, we focus on gravitational waves, as predicted by Einstein's general theory of relativity. First, we introduce those mathematical concepts that are necessary to properly formulate the physical theory, such as the notions of manifold, vector, tensor, metric, connection and curvature. Second, we motivate, formulate and then discuss Einstein's equation, which relates the geometry of spacetime to its matter content. Gravitational waves are later introduced as solutions of the linearized Einstein equation around flat spacetime. These waves are shown to propagate at the speed of light and to possess two polarization states. Gravitational waves can interact with matter, allowing for their direct detection by means of laser interferometers. Finally, Einstein's quadrupole formulas are derived and used to show that nonspherical compact objects moving at relativistic speeds are powerful gravitational wave sources.

We formulate a Hamiltonian description of the orbital motion of a point particle in Kerr spacetime for generic (eccentric, inclined) orbits, which accounts for the effects of the conservative part of the gravitational self-force. This formulation relies on a description of the particle's motion as geodesic in a certain smooth effective spacetime, in terms of (generalized) action-angle variables. Clarifying the role played by the gauge freedom in the Hamiltonian dynamics, we extract the gauge-invariant information contained in the conservative self-force. We also propose a possible gauge choice for which the orbital dynamics can be described by an effective Hamiltonian, written solely in terms of the action variables. As an application of our Hamiltonian formulation in this gauge, we derive the conservative self-force correction to the orbital frequencies of Kerr innermost stable spherical (inclined or circular) orbits. This gauge choice also allows us to establish a "first law of mechanics" for black-hole-particle binary systems, at leading order beyond the test-mass approximation., Comment: 55 pages, 0 figures, matches the published version

Using the canonical Arnowitt-Deser-Misner Hamiltonian formalism, a "first law of mechanics" is established for binary systems of point masses moving along generic stable bound (eccentric) orbits. This relationship is checked to hold within the post-Newtonian approximation to general relativity, up to third (3PN) order. Several applications are discussed, including the use of gravitational self-force results to inform post-Newtonian theory and the effective one-body model for eccentric-orbit compact binaries., 26 pages; matches published version

The gravitational self-force (GSF) and post-Newtonian (PN) schemes are complementary approximation methods for modelling the dynamics of compact binary systems. Comparison of their results in an overlapping domain of validity provides a crucial test for both methods, and can be used to enhance their accuracy, e.g. via the determination of previously unknown PN parameters. Here, for the first time, we extend such comparisons to noncircular orbits---specifically, to a system of two nonspinning objects in a bound (eccentric) orbit. To enable the comparison we use a certain orbital-averaged quantity $\langle U \rangle$ that generalizes Detweiler's redshift invariant. The functional relationship $\langle U \rangle(\Omega_r,\Omega_\phi)$, where $\Omega_r$ and $\Omega_\phi$ are the frequencies of the radial and azimuthal motions, is an invariant characteristic of the conservative dynamics. We compute $\langle U \rangle(\Omega_r,\Omega_\phi)$ numerically through linear order in the mass ratio $q$, using a GSF code which is based on a frequency-domain treatment of the linearized Einstein equations in the Lorenz gauge. We also derive $\langle U \rangle(\Omega_r,\Omega_\phi)$ analytically through 3PN order, for an arbitrary $q$, using the known near-zone 3PN metric and the generalized quasi-Keplerian representation of the motion. We demonstrate that the $\mathcal{O}(q)$ piece of the analytical PN prediction is perfectly consistent with the numerical GSF results, and we use the latter to estimate yet unknown pieces of the 4PN expression at $\mathcal{O}(q)$., Comment: 44 pages, 2 figures, 4 tables

For a self-gravitating particle of mass ��in orbit around a Kerr black hole of mass M >> ��, we compute the O(��/M) shift in the frequency of the innermost stable circular equatorial orbit (ISCEO) due to the conservative piece of the gravitational self-force acting on the particle. Our treatment is based on a Hamiltonian formulation of the dynamics in terms of geodesic motion in a certain locally-defined effective smooth spacetime. We recover the same result using the so-called first law of binary black-hole mechanics. We give numerical results for the ISCEO frequency shift as a function of the black hole's spin amplitude, and compare with predictions based on the post-Newtonian approximation and the effective one-body model. Our results provide an accurate strong-field benchmark for spin effects in the general relativistic two-body problem., 5 pages, 1 table, 1 figure, matches version published in PRL. Raw data of H_int/mu are available at http://link.aps.org/supplemental/10.1103/PhysRevLett.113.161101

Inspiralling and coalescing binary black holes are promising sources of gravitational radiation. The orbital motion and gravitational-wave emission of such system can be modelled using a variety of approximation schemes and numerical methods in general relativity: the post-Newtonian formalism, black hole perturbation theory, numerical relativity simulations, and the effective one-body model. We review recent work at the multiple interfaces of these analytical and numerical techniques, emphasizing the use of coordinate-invariant relationships to perform meaningful comparisons. Such comparisons provide independent checks of the validity of the various calculations, they inform the development of a universal, semi-analytical model of the binary dynamics and gravitational-wave emission, and they help to delineate the respective domains of validity of each approximation method. For instance, several recent comparisons suggest that perturbation theory may find applications in a broader range of physical problems than previously thought, including the radiative inspiral of intermediate mass-ratio and comparable-mass black hole binaries., Comment: 35 pages, 9 figures; invited review for IJMPD; v2: references added, matches published version

We study the general relativistic periastron advance in spinning black hole binaries on quasi-circular orbits, with spins aligned or anti-aligned with the orbital angular momentum, using numerical-relativity simulations, the post-Newtonian approximation, and black hole perturbation theory. By imposing a symmetry by exchange of the bodies' labels, we devise an improved version of the perturbative result, and use it as the leading term of a new type of expansion in powers of the symmetric mass ratio. This allows us to measure, for the first time, the gravitational self-force effect on the periastron advance of a non-spinning particle orbiting a Kerr black hole of mass M and spin S = -0.5 M^2, down to separations of order 9M. Comparing the predictions of our improved perturbative expansion with the exact results from numerical simulations of equal-mass and equal-spin binaries, we find a remarkable agreement over a wide range of spins and orbital separations., Comment: 18 pages, 12 figures; matches version to appear in Phys. Rev. D

The Hamilton-Jacobi equation for test particles in the Kerr geometry is separable. Using action-angle variables, we establish several relations between various physical quantities that characterize bound timelike geodesic orbits around a spinning black hole, including the particle's rest mass, energy, angular momentum, mean redshift and fundamental frequencies. These relations are explicitly checked to hold true in the particular case of equatorial circular orbits. An application to the gravitational wave-driven, adiabatic inspiral of extreme-mass-ratio compact binaries is briefly discussed., 7 pages; matches version to appear in Class. Quant. Grav

For a Kerr black hole perturbed by a particle on the "corotating" circular orbit (angular velocity equal to that of the unperturbed event horizon), the black hole remains in equilibrium in the sense that the perturbed event horizon is a Killing horizon of the helical Killing field. The associated surface gravity is constant over the horizon and should correspond to the physical Hawking temperature. We calculate the perturbation in surface gravity/temperature, finding it negative: the moon has a cooling effect on the black hole. We also compute the change in horizon angular frequency, which is positive, and the change in surface area/entropy, which vanishes., 8 pages, 4 figures; matches the published version

We calculate the effect of self-interaction on the "geodetic" spin precession of a compact body in a strong-field orbit around a black hole. Specifically, we consider the spin precession angle $\psi$ per radian of orbital revolution for a particle carrying mass $\mu$ and spin $s \ll (G/c) \mu^2$ in a circular orbit around a Schwarzschild black hole of mass $M \gg \mu$. We compute $\psi$ through $O(\mu/M)$ in perturbation theory, i.e, including the correction $\delta\psi$ (obtained numerically) due to the torque exerted by the conservative piece of the gravitational self-field. Comparison with a post-Newtonian (PN) expression for $\delta\psi$, derived here through 3PN order, shows good agreement but also reveals strong-field features which are not captured by the latter approximation. Our results can inform semi-analytical models of the strong-field dynamics in astrophysical binaries, important for ongoing and future gravitational-wave searches., Comment: 5 pages, 1 table, 1 figure. Minor changes to match published version

We use the canonical Hamiltonian formalism to generalize to spinning point particles the first law of mechanics established for binary systems of non-spinning point masses moving on circular orbits [Le Tiec, Blanchet, and Whiting, Phys. Rev. D 85, 064039 (2012)]. We find that the redshift observable of each particle is related in a very simple manner to the canonical Hamiltonian and, more generally, to a class of Fokker-type Hamiltonians. Our results are valid through linear order in the spin of each particle, but hold also for quadratic couplings between the spins of different particles. The knowledge of spin effects in the Hamiltonian allows us to compute spin-orbit terms in the redshift variable through 2.5PN order, for circular orbits and spins aligned or anti-aligned with the orbital angular momentum. To describe extended bodies such as black holes, we supplement the first law for spinning point-particle binaries with some "constitutive relations" that can be used for diagnosis of spin measurements in quasi-equilibrium initial data., 16 pages; matches the published version

The phenomenology of the modified Newtonian dynamics (MOND) can be recovered from a mechanism of "gravitational polarization" of some dipolar medium playing the role of dark matter. We review a relativistic model of dipolar dark matter (DDM) within standard general relativity to describe, at some effective level, a fluid polarizable in a gravitational field. At first order in cosmological perturbation theory, this model is equivalent to the concordance cosmological scenario, or Lambda-cold dark matter (CDM) model. At second order, however, the internal energy of DDM modifies the curvature perturbation generated by CDM. This correction, which depends quadratically on the dipole, induces a new type of non-Gaussianity in the bispectrum of the curvature perturbation with respect to standard CDM. Recent observations by the Planck satellite impose stringent constraints on the primordial value of the dipole field., Comment: 9 pages; to appear in the proceedings of the conference HTGRG (Quy Nhon, Vietnam, 2013)

First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are explored for PN spacetimes. As one such consequence, a simple relation between the PN binding energy of the binary system and Detweiler's redshift observable is established. Through it, we are able to determine with high precision the numerical values of some previously unknown high order PN coefficients in the circular-orbit binding energy. Finally, we propose new gauge invariant notions for the energy and angular momentum of a particle in a binary system., 43 pages, 3 figures; matches the published version

Using the main result of a companion paper, in which the binding energy of a circular-orbit non-spinning compact binary system is computed at leading-order beyond the test-particle approximation, the exact expression of the effective-one-body (EOB) metric component g^eff_tt is obtained through first order in the mass ratio. Combining these results with the recent gravitational self-force calculation of the periastron advance for circular orbits in the Schwarzschild geometry, the EOB metric component g^eff_rr is also determined at linear order in the mass ratio. These results assume that the mapping between the real and effective Hamiltonians at the second and third post-Newtonian (PN) orders holds at all PN orders. Our findings also confirm the advantage of resumming the PN dynamics around the test-particle limit if the goal is to obtain a flexible model that can smoothly connect the test-mass and equal-mass limits., Comment: 11 pages, 2 figures; appendix generalized to include the logarithmic contributions in the post-Newtonian Hamiltonian. Results unchanged. Matches version to be published in Phys. Rev. D

The curved spacetime geometry of a system of two point masses moving on a circular orbit has a helical symmetry. We show how Kepler's third law for circular motion, and its generalization in post-Newtonian theory, can be recovered from a simple, covariant condition on the norm of the associated helical Killing vector field. This unusual derivation can be used to illustrate some concepts of prime importance in a general relativity course, including those of Killing field, covariance, coordinate dependence, and gravitational redshift., Comment: 11 pages, 3 figures; minor changes and text improvements; matches version to appear in Class. Quant. Grav

The general relativistic (Mercury-type) periastron advance is calculated here for the first time with exquisite precision in full general relativity. We use accurate numerical relativity simulations of spinless black hole binaries with mass ratios 1/8 < m1/m2 < 1 and compare with the predictions of several analytic approximation schemes. We find the effective-one-body model to be remarkably accurate, and, surprisingly, so also the predictions of self-force theory [replacing m1/m2 --> m1m2/(m1+m2)^2]. Our results can inform a universal analytic model of the two-body dynamics, crucial for ongoing and future gravitational-wave searches., 5 pages, 3 figures; v2: new abstract, updated figures correcting minor error, added references, minor changes to match published version; to appear in Phys. Rev. Lett

The orbital motion of inspiralling and coalescing black hole binaries can be investigated using a variety of approximation schemes and numerical methods within general relativity: post-Newtonian expansions, black hole perturbation theory, numerical relativity, and the effective-one-body formalism. We review two recent comparisons of the predictions from these various techniques. Both comparisons rely on the calculation of a coordinate invariant relation, in the case of non-spinning binary black holes on quasi-circular orbits. All methods are shown to agree very well in their common domain of validity., Comment: 4 pages, 2 figures; to appear in the Proceedings of the "46th Rencontres de Moriond & GPhyS Colloquium on Gravitational Waves and Experimental Gravity", La Thuile (Italy), March 2011

We continue a previous work on the comparison between the post-Newtonian (PN) approximation and the gravitational self-force (SF) analysis of circular orbits in a Schwarzschild background. We show that the numerical SF data contain physical information corresponding to extremely high PN approximations. We find that knowing analytically determined appropriate PN parameters helps tremendously in allowing the numerical data to be used to obtain higher order PN coefficients. Using standard PN theory we compute analytically the leading 4PN and the next-to-leading 5PN logarithmic terms in the conservative part of the dynamics of a compact binary system. The numerical perturbative SF results support well the analytic PN calculations through first order in the mass ratio, and are used to accurately measure the 4PN and 5PN non-logarithmic coefficients in a particular gauge invariant observable. Furthermore we are able to give estimates of higher order contributions up to the 7PN level. We also confirm with high precision the value of the 3PN coefficient. This interplay between PN and SF efforts is important for the synthesis of template waveforms of extreme mass ratio inspirals to be analysed by the space-based gravitational wave instrument LISA. Our work will also have an impact on efforts that combine numerical results in a quantitative analytical framework so as to generate complete inspiral waveforms for the ground-based detection of gravitational waves by instruments such as LIGO and Virgo., Comment: 32 pages, 2 figures; to appear in Phys. Rev. D

The ringdown phase of a black hole formed from the merger of two orbiting black holes is described by means of the close-limit (CL) approximation starting from second-post-Newtonian (2PN) initial conditions. The 2PN metric of point-particle binaries is formally expanded in CL form and identified with that of a perturbed Schwarzschild black hole. The multipolar coefficients describing the even-parity (polar) and odd-parity (axial) components of the linear perturbation consistently satisfy the 2PN-accurate perturbative field equations. We use these coefficients to build initial conditions for the Regge-Wheeler and Zerilli wave equations, which we then evolve numerically. The ringdown waveform is obtained in two cases: head-on collision with zero-angular momentum, composed only of even modes, and circular orbits, for which both even and odd modes contribute. In a separate work, this formalism is applied to the study of the gravitational recoil produced during the ringdown phase of coalescing binary black holes., 34 pages, 6 figures; to appear in Class. Quant. Grav

The problem of a compact binary system whose components move on circular orbits is addressed using two different approximation techniques in general relativity. The post-Newtonian (PN) approximation involves an expansion in powers of v/c<, 36 pages, 3 figures; matches the published version

In previous work [L. Blanchet and A. Le Tiec, Phys. Rev. D 78, 024031 (2008)], a model of dark matter and dark energy based on the concept of gravitational polarization was investigated. This model was shown to recover the concordance cosmological scenario (Lambda-CDM) at cosmological scales, and the phenomenology of the modified Newtonian dynamics (MOND) at galactic scales. In this article we prove that the model can be formulated with a simple and physically meaningful matter action in general relativity. We also provide alternative derivations of the main results of the model, and some details on the variation of the action., Comment: 11 pages, 2 figures; minor stylistic corrections, added references, added appendix; to appear in Phys. Rev. D

The gravitational recoil or "kick" of a black hole formed from the merger of two orbiting black holes, and caused by the anisotropic emission of gravitational radiation, is an astrophysically important phenomenon. We combine (i) an earlier calculation, using post-Newtonian theory, of the kick velocity accumulated up to the merger of two non-spinning black holes, (ii) a "close-limit approximation" calculation of the radiation emitted during the ringdown phase, and based on a solution of the Regge-Wheeler and Zerilli equations using initial data accurate to second post-Newtonian order. We prove that ringdown radiation produces a significant "anti-kick". Adding the contributions due to inspiral, merger and ringdown phases, our results for the net kick velocity agree with those from numerical relativity to 10-15 percent over a wide range of mass ratios, with a maximum velocity of 180 km/s at a mass ratio of 0.38., Comment: 9 pages, 5 figures; to appear in Class. Quant. Grav

After reviewing the modified Newtonian dynamics (MOND) proposal, we advocate that the associated phenomenology may actually not result from a modification of Newtonian gravity, but from a mechanism of "gravitational polarization" of some dipolar medium playing the role of dark matter. We then build a relativistic model within standard general relativity to describe (at some phenomenological level) the dipolar dark matter polarizable in a gravitational field. The model naturally involves a cosmological constant, and is shown to reduce to the concordance cosmological scenario (Lambda-CDM) at early cosmological times. From the mechanism of gravitational polarization, we recover the phenomenology of MOND in a typical galaxy at low redshift. Furthermore, we show that the cosmological constant Lambda scales like a0^2, where a0 is the constant MOND acceleration scale, in good agreement with observations., 8 pages; to appear in the Proceedings of the "43rd Rencontres de Moriond", La Thuile (Italy), March 2008, edited by J. Dumarchez, Y. Giraud-Heraud and J. Tran Thanh Van

A model of dark matter and dark energy based on the concept of gravitational polarization is investigated. We propose an action in standard general relativity for describing, at some effective or phenomenological level, the dynamics of a dipolar medium, i.e. one endowed with a dipole moment vector, and polarizable in a gravitational field. Using first-order cosmological perturbations, we show that the dipolar fluid is undistinguishable from standard dark energy (a cosmological constant Lambda) plus standard dark matter (a pressureless perfect fluid), and therefore benefits from the successes of the Lambda-CDM (Lambda-cold dark matter) scenario at cosmological scales. Invoking an argument of "weak clusterisation" of the mass distribution of dipole moments, we find that the dipolar dark matter reproduces the phenomenology of the modified Newtonian dynamics (MOND) at galactic scales. The dipolar medium action naturally contains a cosmological constant, and we show that if the model is to come from some fundamental underlying physics, the cosmological constant Lambda should be of the order of a0^2/c^4, where a0 denotes the MOND constant acceleration scale, in good agreement with observations., 38 pages, 4 figures; to appear in Phys. Rev. D

In previous work [L. Blanchet and A. Le Tiec, Phys. Rev. D 80, 023524 (2009)], motivated by the phenomenology of dark matter at galactic scales, a model of dipolar dark matter (DDM) was introduced. At linear order in cosmological perturbations, the dynamics of the DDM was shown to be identical to that of standard cold dark matter (CDM). In this paper, the DDM model is investigated at second order in cosmological perturbation theory. We find that the internal energy of the DDM fluid modifies the curvature perturbation generated by CDM with a term quadratic in the dipole field. This correction induces a new type of non-Gaussianity in the bispectrum of the curvature perturbation with respect to standard CDM. Leaving unspecified the primordial amplitude of the dipole field, which could in principle be determined by a more fundamental description of DDM, we find that, in contrast with usual models of primordial non-Gaussianities, the non-Gaussianity induced by DDM increases with time after the radiation-matter equality on super-Hubble scales. This distinctive feature of the DDM model, as compared with standard CDM, could thus provide a specific signature in the CMB and large-scale structure probes of non-Gaussianity., 20 pages; v2: extended discussion in Sec. V D, added references; matches published version

The grand challenges of contemporary fundamental physics-dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem-all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on 'Black holes, Gravitational waves and Fundamental Physics'., SCOPUS: re.j, info:eu-repo/semantics/published

The Wirst part of the thesis focuses on the relativistic, two-body problem in the context of general relativity. More precisely, we derive a variational identity known as the ``First Law of Mechanics" that relates physical parameters of a binary system of compact objects, such as its total energy and angular momentum, to the characteristics of the objects themselves, such as their masses and spins. Our derivation is based on the gravitational skeleton formalism for compact objects at quadrupolar order, combined with an extended version of a general variational identity established for spacetimes endowed with a helical isometry describing circular orbits. We also propose a review of the various multipolar skeleton models and the different types of First Laws that exist in the literature, and discuss applications and physical implications of our results in the context of gravitational wave astronomy. The second part of the thesis deals with a classical problem of potential theory, in Newtonian gravity. In particular, we continue the exploration of isochrone potentials, introduced in the Wifties by Michel Hénon. These potentials are deWined by the property that any test particle orbits within it with a radial period that is independent of its angular momentum. After a complete classiWication of the isochrone potentials using nothing but euclidean geometry, we explore the dynamics in these potentials by classifying their orbits, providing analytical solutions to the equation of motion. Using a Hamiltonian treatment, we also derive action-angle coordinates for the isochrone problem, providing new insight of several well known result of classical celestial mechanics to all isochrone orbits, such as the Kepler equation, Bertrand's theorem and Kepler's laws of motion, and a generalization of these to all isochrone orbits. Finally, we compute the Birkhoff normal form of the corresponding Hamiltonian for a generic potential, and derive the fundamental theorem of isochrony from the inspection of the Birkhoff invariants of the system; La première partie de cette thèse se place dans le contexte du problème à deux corps en relativité générale. Nous établissons une identité variationnelle appelée ``Première Loi de la Mécanique'' qui fait le lien entre différents paramètres physiques caractérisant un système binaire d'objets compacts comme son énergie de liaison et son moment cinétique total, aux propriétés des corps, comme leur masse ou leur rotation propre. Notre résultat se base sur le modèle du ``squelette gravitationnel'' poussé à l'ordre quadrupolaire, combiné avec une généralisation d'une identité variationnelle établie pour des espaces-temps dotés d'une isométrie helicoı̈dale pour décrire des orbites circulaires. Nous présentons également un tour d'horizon des modèles de squelettes gravitationels et des différentes premières lois existant dans la littérature, et discutons des applications et de l'impact de la première loi à l'ordre quadrupolaire dans le contexte de l'astronomie gravitationnelle. La seconde partie de la thèse concerne un problème de gravitation Newtonienne, dans lequel il est question d'une classe particulière de potentiels gravitationels appelés ``potentiels isochrones''. Ceux-ci, introduits dans les années 1950 par Michel Hénon, sont déWini par la propriété que toute particule test y orbite avec une période radiale indépendante de son moment cinétique. Après avoir établi une classiWication complète de cette classe de potentiels, nous explorons la dynamique qu'ils génèrent et donnons une solution analytique des équations du mouvement exclusivement à l'aide d'outils géométriques. Puis, nous proposons une revisite du problème sous l'angle de la mécanique Hamiltonienne, permettant ainsi de voir avec un angle nouveaux certains résultats de mécanique classique, comme l'équation de Kepler, le théorème de Bertrand et les lois de Kepler, et de généraliser ceux-ci à tout orbite isochrone. EnWin, en calculant le forme normale de Birkhoff du système, nous donnons une démonstration purement mécanique du théorème fondamental de l'isochronie, basée sur l'étude des invariants de Birkhoff du système.