Mining the geodesic equation for scattering data

The geodesic equation encodes test-particle dynamics at arbitrary gravitational coupling, hence retaining all orders in the post-Minkowskian (PM) expansion. Here we explore what geodesic motion can tell us about dynamical scattering in the presence of perturbatively small effects such as tidal distortion and higher derivative corrections to general relativity. We derive an algebraic map between the perturbed geodesic equation and the leading PM scattering amplitude at arbitrary mass ratio. As examples, we compute formulas for amplitudes and isotropic gauge Hamiltonians for certain infinite classes of tidal operators such as electric or magnetic Weyl to any power, and for higher derivative corrections to gravitationally interacting bodies with or without electric charge. Finally, we present a general method for calculating closed-form expressions for amplitudes and isotropic gauge Hamiltonians in the test-particle limit at all orders in the PM expansion

Effective Field Theory for Extreme Mass Ratios

We derive an effective field theory describing a pair of gravitationally interacting point particles in an expansion in their mass ratio, also known as the self-force (SF) expansion. The 0SF dynamics are trivially obtained to all orders in Newton's constant by the geodesic motion of the light body in a Schwarzschild background encoding the gravitational field of the heavy body. The corrections at 1SF and higher are generated by perturbations about this configuration -- that is, the geodesic deviation of the light body and the fluctuation graviton -- but crucially supplemented by an operator describing the recoil of the heavy body as it interacts with the smaller companion. Using this formalism we compute new results at third post-Minkowskian order for the conservative dynamics of a system of gravitationally interacting massive particles coupled to a set of additional scalar and vector fields.Comment: 9 pages, 1 figur

QCD Meets Gravity 2023

The metric and corresponding geodesic equation describing test-particle dynamics in a background encode gravitational data to all orders in the post-Minkowskian (PM) expansion and effectively resum certain infinite classes of flat-space Feynman diagrams. In the context of the connection between the bound gravitational two-body problem and the relativistic scattering of massive particles interacting via gravity, I will describe how this property can be leveraged to obtain information about post-Minkowskian conservative dynamics as a systematic expansion in the mass ratio of the interacting bodies, also known as the self-force (SF) expansion. At the leading PM order for arbitrary mass ratios and all PM orders in the test-particle limit, data such as isotropic gauge Hamiltonians and scattering amplitudes can be algebraically extracted for a non-spinning black hole binary system and small perturbations away from it. Higher SF dynamics can be determined using an effective field theory (EFT) setup that accounts for the recoil of the heavy body due to effects from the light body through “recoil operators”. This EFT formalism for extreme mass ratios will be discussed

Scattering and Gravitational Effective Field Theory

Advances in the methodologies developed in quantum field theory and in the scattering amplitudes program have led to their application to questions pertaining to the classical physics of gravitationally interacting binary systems. The perturbative and relativistic nature of the quantum field theoretic setup is perfectly suited for obtaining results in an expansion in the gravitational constant, also known as the post-Minkowskian (PM) expansion. However, there are several practical scenarios where the gravitational waves produced by the inspiral or interaction of two massive bodies arise from dynamics in the strong field regime and the PM expansion breaks down. Extreme mass ratio inspirals, where a lighter body interacts with a much heavier black hole, are examples of such systems. In contrast, classical solutions, such as the Schwarzschild metric, and the geodesic trajectories of test bodies traversing in these nontrivial backgrounds encode information to all orders in the gravitational constant. In fact, these solutions can be viewed as the summation of certain infinite sets of Feynman diagrams from the perspective of point particle effective field theory (EFT). Alternatively, metrics and related geodesic trajectories can be seen as performing enormous simplifications of the tensor structures arising in these equivalent sets of Feynman integrals. We describe how the all order in PM information present in classical solutions can be utilized to simplify PM calculations in point particle EFT and set up a systematic framework for studying the classical dynamics of binary systems as an expansion in their mass ratio. We also delve into questions about the origin and scope of validity of color-kinematics duality and the double copy relation, which can be used to generate amplitudes of one theory from another. For example, graviton amplitudes can be obtained from gluon amplitudes. Unveiling the underlying structure that gives rise to these relations would not only deepen our understanding of the properties of these theories but could also serve in streamlining their application to computations of practical interest such as those showing up in the study of the gravitational two-body problem using field theory techniques. Specifically, we analyze a toy system in two dimensions where we find a Lagrangian-level manifestation of the duality in a classical equivalent of the nonlinear sigma model. We unpack the implications of an off-shell formulation of the color-kinematics duality and double copy in order to understand the possible wider implications for these relations in other theories.</p

Mining the geodesic equation for scattering data

The geodesic equation encodes test-particle dynamics at arbitrary gravitational coupling, hence retaining all orders in the post-Minkowskian (PM) expansion. Here we explore what geodesic motion can tell us about dynamical scattering in the presence of perturbatively small effects such as tidal distortion and higher derivative corrections to general relativity. We derive an algebraic map between the perturbed geodesic equation and the leading PM scattering amplitude at arbitrary mass ratio. As examples, we compute formulas for amplitudes and isotropic gauge Hamiltonians for certain infinite classes of tidal operators such as electric or magnetic Weyl to any power, and for higher derivative corrections to gravitationally interacting bodies with or without electric charge. Finally, we present a general method for calculating closed-form expressions for amplitudes and isotropic gauge Hamiltonians in the test-particle limit at all orders in the PM expansion

Deriving spin-1 quartic interaction vertices from closure of the Poincaré algebra

We derive the quartic interaction vertex of pure Yang–Mills theory by demanding closure of the light-cone Poincaré algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We prove that there is no correction to the spin generator, for spin one, at this order. We comment briefly on higher spin fields in this context