Motivated by conceptual problems in quantum theories of gravity, the gravitational eikonal approach, inspired by its electromagnetic predecessor, has been successfully applied to the transplanckian energy collisions of elementary particles and strings since the late eighties, and to string-brane collisions in the past decade. After the direct detection of gravitational waves from black-hole mergers, most of the attention has shifted towards adapting these methods to the physics of black-hole encounters. For such systems, the eikonal exponentiation provides an amplitude-based approach to calculate classical gravitational observables, thus complementing more traditional analytic methods such as the Post-Newtonian expansion, the worldline formalism, or the Effective-One-Body approach. In this review we summarize the main ideas and techniques behind the gravitational eikonal formalism. We discuss how it can be applied in various different physical setups involving particles, strings and branes and then we mainly concentrate on the most recent developments, focusing on massive scalars minimally coupled to gravity, for which we aim at being as self-contained and comprehensive as possible.

We calculate the spin-orbit corrections to the loss of angular momentum in a two-body scattering at third post-Minkowskian order, O(G³), from scattering amplitudes using the eikonal operator. These results include effects linear in spin, are valid for generic spin orientations, and are presented in a manifestly Poincaré covariant way. We include both radiative losses, by means of the leading-order gravitational waveform, and static losses, by means of the appropriate −i0 prescription in the leading soft graviton theorem, finding agreement with known results in the post-Newtonian limit.

We calculate the tidal corrections to the loss of angular momentum in a two-body collision at leading post-Minkowskian order from an amplitude-based approach. The eikonal operator allows us to efficiently combine elastic and inelastic amplitudes, and captures both the contributions due to genuine gravitationalwave emissions and those due to the static gravitational field. We calculate the former by harnessing powerful collider-physics techniques such as reverse unitarity, thereby reducing them to cut two-loop integrals, and cross check the result by performing an independent calculation in the post-Newtonian limit. For the latter, we can employ the results of P. Di Vecchia et al. [Angular momentum of zero-frequency gravitons, J. High Energy Phys. 08 (2022) 172.], where static-field effects were calculated for generic gravitational scattering events using the leading soft graviton theorem.

We propose two possible eikonal operators encoding the effects of classical radiation as coherent states of gravitons and show how to compute from them different classical observables. In the first proposal, only genuinely propagating gravitons are included, while, in the second, zero-frequency modes are added in order to recover the effects of a static gravitational field. We first calculate the radiated energy-momentum and the change in each particle's momentum, or impulse, to 3PM order finding agreement with the literature. We then calculate the angular momentum of the gravitational field after the collision. In order to do so, we adapt the method of reverse unitarity to the presence of derivatives in the operators describing the angular momentum and reproduce the result of [1] obtained by resumming the small-velocity expansion. As a new application, we derive also the variation in each particle's angular momentum up to 3PM: calculating separately field and particle contributions allows us to check the balance laws explicitly. We also show how the eikonal operator encodes the linear-response formula of Bini-Damour by deriving the linear radiation-reaction contribution to the transverse impulse at 4PM.

We calculate the inelastic 2 → 3 one-loop amplitude for the scattering of two point-like, spinless objects with generic masses involving the additional emission of a single graviton. We focus on the near-forward, or classical, limit. Our results include the leading and subleading orders in the soft-region expansion, which captures all non-analytic contributions in the transferred momentum and in the graviton's frequency. This allows us to check the first constraint arising from the inelastic exponentiation put forward in refs. [1-3], and to calculate the 2 → 3 one-loop matrix element of the N-operator, linked to the S-matrix by S = e^iN, showing that it is real, classical and free of infrared divergences. We discuss how our results feature in the calculation of the O(G³⁾) corrections to the asymptotic waveform.

We construct a basis of conformal primary wavefunctions (CPWs) for p-form fields in any dimension, calculating their scalar products and exhibiting the change of basis between conventional plane wave and CPW mode expansions. We also perform the analysis of the associated shadow transforms. For each family of p-form CPWs, we observe the existence of pure gauge wavefunctions of conformal dimension increment = p, while shadow p-forms of this weight are only pure gauge in the critical spacetime dimension value D = 2p + 2. We then provide a systematic technique to obtain the large -r asymptotic limit near .Q based on the method of regions, which naturally takes into account the presence of both ordinary and contact terms on the celestial sphere. In D = 4, this allows us to reformulate in a conformal primary language the links between scalars and dual two-forms.

We study the renormalized action and the renormalized presymplectic potential for Maxwell fields on Anti de Sitter backgrounds of any dimensions. We then use these results to explicitly derive finite boundary charges for angle-dependent asymptotic symmetries. We consider both Poincaré and Bondi coordinates, the former allowing us to control the systematics for arbitrary D, the latter being better suited for a smooth flat limit.

By following closely Weinberg’s soft theorem, which captures the 1/ω pole contribution to the amplitude for soft graviton emissions (ω < Λ) on top of an arbitrary background hard process, we calculate the expectation value of the graviton’s angular momentum operator for arbitrary collisions dressed with soft radiation. We find that the result becomes independent of the cutoff Λ on the graviton’s frequency, effectively localizing at ω = 0. In this way, our result captures the contribution to the angular momentum that comes from the zero-frequency modes. Like the soft theorem, our formula has an exact dependence on the kinematics of the hard particles and is only a function of their momenta. As an example, we discuss in some detail the case of the 2 → 2 scattering of spinless particles in General Relativity and N = 8 supergravity.

Observables related to the real part of the gravitational eikonal, such as the deflection angle and time delay, have been found so far to have a smooth post-Minkowskian (PM) expansion whose validity extends from the non-relativistic to the most extreme ultra-relativistic (UR) regime, which smoothly connects with massless particle collisions. To describe gravitational radiation, the eikonal phase has to be promoted to a unitary operator for which we motivate a proposal and start discussing properties in the soft-radiation limit. A convergent PM expansion is found to only hold below an UR bound (discussed in the GR literature in the seventies) above which a different expansion is instead needed implying, in general, some non-analyticity in Newton's constant. In this extreme UR regime soft radiative observables receive contributions only from gravitons and are therefore universal. This generalises the pattern discussed in [1] beyond the elastic case.

The aim of this article is to explore the interplay between the eikonal resummation in impact-parameter space and the exponentiation of infrared divergences in momentum space for gravity amplitudes describing collisions of massive objects. The eikonal governs the classical dynamics relevant to the two-body problem, and its infrared properties are directly linked to the zero-frequency limit of the gravitational wave emission spectrum and to radiation-reaction effects. Combining eikonal and infrared exponentiations it is possible to derive these properties at a given loop order starting from lower-loop data. This is illustrated explicitly in N = 8 supergravity and in general relativity by deriving the divergent part of the two-loop eikonal from tree-level and one-loop elastic amplitudes.