Starting from the recently derived conservative tail-of-tail action [High precision black hole scattering: Tutti Frutti vs worldline effective field theory, arXiv:2504.20204.] we compute several dynamical observables of binary systems (Delaunay Hamiltonian, scattering angle), at the 6.5 post-Newtonian accuracy and up to the eighth post-Minkowskian order. We find perfect agreement with previous self-force results, and (when inserting a recent high-post-Newtonian order derivation of radiated angular momentum [Scattering of a point mass by a Schwarzschild black hole: Radiated energy and angular momentum, arXiv:2507.03442.]) with state-of-the-art post-Minkowskian scattering results [Emergence of Calabi–Yau manifolds in high-precision black-hole scattering, Nature (London) 641, 603 (2025).].

We compute the next-to-leading-order radiation-reaction modification to the harmonic coordinate quasi- Keplerian parametrization of the binary dynamics, the two bodies undergoing a scattering process. The solution for the radiation-reaction corrections to the orbital parameters is examined both in the time domain (exact results) and in the frequency domain (results presented in the limit of large angular momentum, i.e., as a post-Minkowskian series expansion). The knowledge of the radiation-reaction corrected orbit is a key ingredient for the calculation of the fractional 3.5 post-Newtonian corrections to the radiative losses as well as to the radiative multipole moments needed to build up the waveform at the same accuracy.

We show that the null geodesic radial action for unbound orbits in the Kerr spacetime, and consequently the scattering angle, can be resummed in terms of hypergeometric functions, extending previous results [Ivanov et al., Resummation of universal tails in gravitational waveforms, arXiv:2504.07862.]. We provide explicit expressions as series expansions in powers of the Kerr rotational parameter up to the fourth order included.We finally use the Mano-Suzuki-Takasugi formalism to prove the relation between the renormalized angular momentum and the radial action highlighted in previous works.

Explicit solution of the gravitational two-body problem at the second post-Minkowskian order

Using the multipolar post-Minkowskian formalism, we compute the frequency-domain waveform generated by the gravitational scattering of two nonspinning bodies at the fourth post-Minkowskian order (O(G4), or two-loop order), and at the fractional second post-Newtonian accuracy [O(v4/c4)]. The waveform is decomposed in spin-weighted spherical harmonics and the needed radiative multipoles, Um(ω),Vm(ω), are explicitly expressed in terms of a small number of master integrals. The basis of master integrals contains both (modified) Bessel functions, and solutions of inhomogeneous Bessel equations with Bessel-function sources. We show how to express the latter in terms of Meijer G functions. The low-frequency expansion of our results is checked against existing classical soft theorems. We also complete our previous results on the O(G2) bremsstrahlung waveform by computing the O(G3) spectral densities of radiated energy and momentum, in the rest frame of one body, at the thirtieth order in velocity.

In this paper we present an approach to compute analytical post-Minkowskian corrections to unbound two-body scattering in the self-force formalism. Our method relies on a further low-velocity (post-Newtonian) expansion of the motion. We present a general strategy valid for gravitational and nongravitational self-force, and we explicitly demonstrate our approach for a scalar charge scattering off a Schwarzschild black hole. We compare our results with recent calculations in [L. Barack, Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model, Phys. Rev. D 108, 024025 (2023)PRVDAQ2470-001010.1103/PhysRevD.108.024025], showing complete agreement where appropriate and fixing undetermined scale factors in their calculation. Our results also extend their results by including in our dissipative sector the contributions from the flux into the black hole horizon.

We revisit the quantum-amplitude-based derivation of the gravitational waveform emitted by the scattering of two spinless massive bodies at the third order in Newton's constant, h∼G+G2+G3 (one-loop level), and correspondingly update its comparison with its classically derived multipolar-post-Minkowskian counterpart. A spurious-pole-free reorganization of the one-loop five-point amplitude substantially simplifies the post-Newtonian expansion. We find complete agreement between the two results up to the fifth order in the small velocity expansion after taking into account three subtle aspects of the amplitude derivation: (1) in agreement with [A. Georgoudis et al., J. High Energy Phys. 03 (2024) 08910.1007/JHEP03(2024)089], the term quadratic in the amplitude in the observable-based formalism [D. A. Kosower et al., J. High Energy Phys. 02 (2019) 137JHEPFG1029-847910.1007/JHEP02(2019)137] generates a frame rotation by half the classical scattering angle; (2) the dimensional regularization of the infrared divergences of the amplitude introduces an additional (d-4)/(d-4) finite term; and (3) zero-frequency gravitons are found to contribute additional terms both at order h∼G1 and at order h∼G3 when including disconnected diagrams in the observable-based formalism.

We investigate the geometrical properties, spectral classification, geodesics, and causal structure of Bonnor's spacetime [W. B. Bonnor, A rotating dust cloud in general relativity, J. Phys. A 10, 1673 (1977)JPHAC50305-447010.1088/0305-4470/10/10/004], i.e., a stationary axisymmetric solution with a rotating dust as a source. This spacetime has a directional singularity at the origin of the coordinates (related to the diverging vorticity field of the fluid there), which is surrounded by a toroidal region where closed timelike curves (CTCs) are allowed, leading to chronology violations. We use the effective potential approach to provide a classification of the different kind of geodesic orbits on the symmetry plane as well as to study the helical-like motion around the symmetry axis on a cylinder with constant radius. In the former case we find that, as a general feature for positive values of the angular momentum, test particles released from a fixed space point and directed toward the singularity are repelled and scattered back as soon as they approach the CTC boundary, without reaching the central singularity. In contrast, for negative values of the angular momentum there exist conditions in the parameter space for which particles are allowed to enter the pathological region. Finally, as a more realistic mechanism, we study accelerated orbits undergoing friction forces due to the interaction with the background fluid, which may also act in order to prevent particles from approaching the CTC region.

We discuss the solution proposed by Fermi to the so called “4/3 problem” in the classical theory of the electron, a problem which puzzled the physics community for many decades before and after his contribution. Unfortunately his early resolution of the problem in 1922–1923 published in three versions in Italian and German journals (after three preliminary articles on the topic) went largely unnoticed. Even more recent texts devoted to classical electron theory still do not present his argument or acknowledge the actual content of those articles. The calculations initiated by Fermi at the time are completed here by formulating and discussing the conservation of the total 4-momentum of the accelerated electron as seen from the instantaneous rest frame in which it is momentarily at rest.

The class of Petrov type I curvature tensors is further divided into those for whichthe span of the set of distinct principal null directions has dimension four (maximallyspanning type I) or dimension three (nonmaximally spanning type I). Explicit examplesare provided for both vacuum and nonvacuum spacetimes.

A recently introduced approach to the classical gravitational dynamics of binary systems involves intricate integrals (linked to a combination of nonlocal-in-time interactions with iterated 1r-potential scattering) which have so far resisted attempts at their analytical evaluation. By using computing techniques developed for the evaluation of multiloop Feynman integrals (notably harmonic polylogarithms and Mellin transform) we show how to analytically compute all the integrals entering the nonlocal-in-time contribution to the classical scattering angle at the sixth post-Newtonian accuracy, and at the seventh order in Newton's constant, G (corresponding to six-loop graphs in the diagrammatic representation of the classical scattering angle).

The linear-order effects of radiation-reaction on the classical scattering of two point masses, in general relativity, are derived by a variation-of-constants method. Explicit expressions for the radiation-reaction contributions to the changes of 4-momentum during scattering are given to linear order in the radiative losses of energy, linear-momentum, and angular momentum. The polynomial dependence on the masses of the 4-momentum changes is shown to lead to nontrivial identities relating the various radiative losses. At order G3 our results lead to a streamlined classical derivation of results recently derived within a quantum approach. At order G4 we compute the needed radiative losses to next-to-next-to-leading-order in the post-Newtonian expansion, thereby reaching the absolute fourth and a half post-Newtonian level of accuracy in the 4-momentum changes. We also provide explicit expressions, at the absolute sixth post-Newtonian accuracy, for the radiation-graviton contribution to conservative O(G4) scattering. At orders G5 and G6 we derive explicit theoretical expressions for the last two hitherto undetermined parameters describing the fifth-post-Newtonian dynamics. Our results at the fifth-post-Newtonian level confirm results of [Nucl. Phys. B965, 115352 (2021)NUPBBO0550-321310.1016/j.nuclphysb.2021.115352] but exhibit some disagreements with results of [Phys. Rev. D 101, 064033 (2020)PRVDAQ2470-001010.1103/PhysRevD.101.064033].

The energy radiated (without the 1.5PN tail contribution which requires a different treatment) by a binary system of compact objects moving in a hyperboliclike orbit is computed in the frequency domain through the second post-Newtonian level as an expansion in the large-eccentricity parameter up to next-to-next-to-leading order, completing the time domain corresponding information (fully known in closed form at the second post-Newtonian of accuracy). The spectrum contains quadratic products of the modified Bessel functions of the first kind (Bessel K functions) with frequency-dependent order (and argument) already at Newtonian level, so preventing the direct evaluation of Fourier integrals. However, as the order of the Bessel functions tends to zero for large eccentricities, a large-eccentricity expansion of the spectrum allows for analytical computation beyond the lowest order.

The need for more and more accurate gravitational-wave templates requires taking into account all possible contributions to the emission of gravitational radiation from a binary system. Therefore, working within a multipolar-post-Minkowskian framework to describe the gravitational-wave field in terms of the source multipole moments, the dominant instantaneous effects should be supplemented by hereditary contributions arising from nonlinear interactions between the multipoles. The latter effects include tails and memories and are described in terms of integrals depending on the past history of the source. We compute higher-order tail (i.e., tail-of-tail, tail-squared, and memory) contributions to both energy and angular momentum fluxes and their averaged values along hyperboliclike orbits at the leading post-Newtonian approximation, using harmonic coordinates and working in the Fourier domain. Because of the increasing level of accuracy recently achieved in the determination of the scattering angle in a two-body system by several complementary approaches, the knowledge of these terms will provide useful information to compare results from different formalisms.

In Cosmology and in Fundamental Physics there is a crucial question like: where the elusive substance that we call Dark Matter is hidden in the Universe and what is it made of? that, even after 40 years from the Vera Rubin seminal discovery [1] does not have a proper answer. Actually, the more we have investigated, the more this issue has become strongly entangled with aspects that go beyond the established Quantum Physics, the Standard Model of Elementary particles and the General Relativity and related to processes like the Inflation, the accelerated expansion of the Universe and High Energy Phenomena around compact objects. Even Quantum Gravity and very exotic Dark Matter particle candidates may play a role in framing the Dark Matter mystery that seems to be accomplice of new unknown Physics. Observations and experiments have clearly indicated that the above phenomenon cannot be considered as already theoretically framed, as hoped for decades. The Special Topic to which this review belongs wants to penetrate this newly realized mystery from different angles, including that of a contamination of different fields of Physics apparently unrelated. We show with the works of this ST that this contamination is able to guide us into the required new Physics. This review wants to provide a good number of these "paths or contamination" beyond/among the three worlds above; in most of the cases, the results presented here open a direct link with the multi-scale dark matter phenomenon, enlightening some of its important aspects. Also in the remaining cases, possible interesting contacts emerges. Finally, a very complete and accurate bibliography is provided to help the reader in navigating all these issues.

The post-Minkowskian approach to gravitationally interacting binary systems (i.e., perturbation theory in G, without assuming small velocities) is extended to the computation of the dynamical effects induced by the tidal deformations of two extended bodies, such as neutron stars. Our derivation applies general properties of perturbed actions to the effective field theory description of tidally interacting bodies. We compute several tidal invariants (notably the integrated quadrupolar and octupolar actions) at the fast post-Minkowskian order. The corresponding contributions to the scattering angle are derived.

We compute the first-order self-force contribution to Detweiler's redshift invariant for extended bodies endowed with both dipolar and quadrupolar structure (with spin-induced quadrupole moment) moving along circular orbits on a Schwarzschild background. Our analysis includes effects which arc second order in spin, generalizing previous results for purely spinning particles. The perturbing body is assumed to move on the equatorial plane, the associated spin vector being orthogonal to it. The metric perturbations are obtained by using a standard gravitational self-force approach in a radiation gauge. Our results are accurate through the 6.5 post-Newtonian order, and arc shown to reproduce the corresponding post-Newtonian expression for the same quantity computed by using the available Hamiltonian from an effective field theory approach for the dynamics of spinning binaries.

Using the new methodology introduced in a recent paper [D. Bini, T. Damour, and A. Geralico, Phys. Rev. Lett. 123, 231104 (2019)], we present the details of the computation of the conservative dynamics of gravitationally interacting binary systems at the fifth post-Newtonian (5PN) level, together with its extension at the fifth-and-a-half post-Newtonian level. We present also the sixth post-Newtonian (6PN) contribution to the third-post-Minkowskian (3PM) dynamics. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, gravitational self-force, effective one-body, and Delaunay averaging. We determine the full functional structure of the 5PN Hamiltonian (which involves 95 nonzero numerical coefficients), except for two undetermined coefficients proportional to the cube of the symmetric mass ratio, and to the fifth and sixth power of the gravitational constant, G. We present not only the 5PN-accurate, 3PM contribution to the scattering angle but also its 6PN-accurate generalization. Both results agree with the corresponding truncations of the recent 3PM result of Bern et al. [Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019)]. We also compute the 5PN-accurate, fourth-post-Minkowskian (4PM) contribution to the scattering angle, including its nonlocal contribution, thereby offering checks for future 4PM calculations. We point out a remarkable hidden simplicity of the gauge-invariant functional relation between the radial action and the effective-one-body energy and angular momentum.

Using a recently introduced method [D. Bini, T. Damour, and A. Geralico, Phys. Rev. Lett. 123, 231104 (2019)], which splits the conservative dynamics of gravitationally interacting binary systems into a nonlocal-in-time part and a local-in-time one, we compute the local part of the dynamics at the sixth post-Newtonian (6PN) accuracy. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, effective-field-theory, gravitational self-force, effective one-body, and Delaunay averaging. The full functional structure of the local 6PN Hamiltonian (which involves 151 numerical coefficients) is derived, but contains four undetermined numerical coefficients. Our 6PN-accurate results are complete at orders G(3) and G(4), and the derived O(G(3)) scattering angle agrees, within our 6PN accuracy, with the computation of [Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019)]. All our results are expressed in several different gauge-invariant ways. We highlight, and make a crucial use of, several aspects of the hidden simplicity of the mass-ratio dependence of the two-body dynamics.

We complete our previous derivation, at the sixth post-Newtonian (6PN) accuracy, of the local-in-time dynamics of a gravitationally interacting two-body system by giving two gauge-invariant characterizations of its complementary nonlocal-in-time dynamics. On the one hand, we compute the nonlocal part of the scattering angle for hyberboliclike motions; and, on the other hand, we compute the nonlocal part of the averaged (Delaunay) Hamiltonian for ellipticlike motions. The former is computed as a large-angular-momentum expansion (given here to next-to-next-to-leading order), while the latter is given as a small-eccentricity expansion (given here to the tenth order). We note the appearance of zeta(3) in the nonlocal part of the scattering angle. The averaged Hamiltonian for ellipticlike motions then yields two more gauge-invariant observables: the energy and the periastron precession as functions of orbital frequencies. We point out the existence of a hidden simplicity in the mass-ratio dependence of the gravitational-wave energy loss of a two-body system. We include a Supplemental Material that gives the explicit analytic form of a scattering integral which we could only evaluate numerically.