The article reviews the current status of a theoretical approach to the problem of the emission of gravitational waves by isolated systems in the context of general relativity. Part A of the article deals with general post-Newtonian sources. The exterior field of the source is investigated by means of a combination of analytic post-Minkowskian and multipolar approximations. The physical observables in the far-zone of the source are described by a specific set of radiative multipole moments. By matching the exterior solution to the metric of the postNewtonian source in the near-zone we obtain the explicit expressions of the source multipole moments. The relationships between the radiative and source moments involve many nonlinear multipole interactions, among them those associated with the tails (and tails-of-tails) of gravitational waves. Part B of the article is devoted to the application to compact binary systems. We present the equations of binary motion, and the associated Lagrangian and Hamiltonian, at the third post-Newtonian (3PN) order beyond the Newtonian acceleration. The gravitational-wave energy flux, taking consistently into account the relativistic corrections in the binary moments as well as the various tail eects, is derived through 3.5PN order with respect to the quadrupole formalism. The binary’s orbital phase, whose prior knowledge is crucial for searching and analyzing the signals from inspiralling compact binaries, is deduced from an energy balance argument.

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Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.

Using effective field theory (EFT) methods we present a Lagrangian formalism which describes the dynamics of nonrelativistic extended objects coupled to gravity. The formalism is relevant to understanding the gravitational radiation power spectra emitted by binary star systems, an important class of candidate signals for gravitational wave observatories such as LIGO or VIRGO. The EFT allows for a clean separation of the three relevant scales: ${r}_{s}$, the size of the compact objects, $r$, the orbital radius, and $r/v$, the wavelength of the physical radiation (where the velocity $v$ is the expansion parameter). In the EFT, radiation is systematically included in the $v$ expansion without the need to separate integrals into near zones and radiation zones. Using the EFT, we show that the renormalization of ultraviolet divergences which arise at ${v}^{6}$ in post-Newtonian (PN) calculations requires the presence of two nonminimal worldline gravitational couplings linear in the Ricci curvature. However, these operators can be removed by a redefinition of the metric tensor, so that the divergences arising at ${v}^{6}$ have no physically observable effect. Because in the EFT finite size features are encoded in the coefficients of nonminimal couplings, this implies a simple proof of the decoupling of internal structure for spinless objects to at least order ${v}^{6}$. Neglecting absorptive effects, we find that the power counting rules of the EFT indicate that the next set of short distance operators, which are quadratic in the curvature and are associated with tidal deformations, does not play a role until order ${v}^{10}$. These operators, which encapsulate finite size properties of the sources, have coefficients that can be fixed by a matching calculation. By including the most general set of such operators, the EFT allows one to work within a point-particle theory to arbitrary orders in $v$.

Effective field theory of gravity for extended objects

We present the amplitude for classical scattering of gravitationally interacting massive scalars at third post-Minkowskian order. Our approach harnesses powerful tools from the modern amplitudes program such as generalized unitarity and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. Adapting methods for integration and matching from effective field theory, we extract the conservative Hamiltonian for compact spinless binaries at third post-Minkowskian order. The resulting Hamiltonian is in complete agreement with corresponding terms in state-of-the-art expressions at fourth post-Newtonian order as well as the probe limit at all orders in velocity. We also derive the scattering angle at third post-Minkowskian order and find agreement with known results.

Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order

We describe a systematic framework for computing the conservative potential of a compact binary system using modern tools from scattering amplitudes and effective field theory. Our approach combines methods for integration and matching adapted from effective field theory, generalized unitarity, and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. With these methods we derive the third post-Minkowskian correction to the conservative two-body Hamiltonian for spinless black holes. We describe in some detail various checks of our integration methods and the resulting Hamiltonian.

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Black Hole Binary Dynamics from the Double Copy and Effective Theory

The Theory of Space, Time and Gravitation. By V. Fock. Pp. 411. 100s. 1959. (Pergamon Press)

Using a fast-motion approximation method we obtain the second-order gravitational field and equations of motion for two pointlike objects in algebraically closed form. A regularization procedure is used which is shown to guarantee the consistency of the approximation scheme. The equations of motion are then transformed within the framework of relativistic predictive mechanics into a system of ordinary differential equations.

Poincaré-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity

Historia de la edición en España, 1836-1936

Historia de la edición en España (1939-1975)

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Lectura y lectores en el Madrid del siglo XIX

We define the Hamilton Form (H. F.) in the past (resp. future) of a Predictive Poincaré Invariant System (P. I. S.). Under rather general assumptions we prove that the H. F. in the past (resp. future), when they exist, they are unique. We define a Conservative System as a P. I. S. for which the H. F.’s in the past and in the future exist and coincide. Conservative Systems are, we believe, the closest generalization of Hamiltonian non relativistic systems. Using perturbation theory we prove, at the first order, the existence of H. F.’s in the past and in the future for short range P. I. S. And with a slight change in the definition, the existence ofnot unique, H. F.’s for long range P. I. S. The paper ends with some applications of the general Theory of H. F.’s and Conservative Systems, including the calculation of the Energy-Momentum and the Intrinsic Angular Momentum for scalar and vector interactions.

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Jesús Antonio Martínez Martín

Papers

Poincaré-invariant gravitational field and equations of motion of two pointlike objects: The postlinear approximation of general relativity

Historia de la edición en España, 1836-1936

Historia de la edición en España (1939-1975)

Lectura y lectores en el Madrid del siglo XIX

Formes hamiltoniennes et systèmes conservatifs