We explain how the lowest-order classical gravitational radiation produced during the inelastic scattering of two Schwarzschild black holes in General Relativity can be obtained from a tree scattering amplitude in gauge theory coupled to scalar fields. The gauge calculation is related to gravity through the double copy. We remove unwanted scalar forces which can occur in the double copy by introducing a massless scalar in the gauge theory, which is treated as a ghost in the link to gravity. We hope these methods are a step towards a direct application of the double copy at higher orders in classical perturbation theory, with the potential to greatly streamline gravity calculations for phenomenological applications.

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Inelastic Black Hole Scattering from Charged Scalar Amplitudes

We develop a general formalism for computing classical observables for relativistic scattering of spinning particles, directly from on-shell amplitudes. We then apply this formalism to minimally coupled Einstein-gravity amplitudes for the scattering of massive spin 1/2 and spin 1 particles with a massive scalar, constructed using the double copy. In doing so we reproduce recent results at first post-Minkowskian order for the scattering of spinning black holes, through quadrupolar order in the spin-multipole expansion.

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Observables and amplitudes for spinning particles and black holes

The eikonal approximation is an ideal tool to extract classical observables in gauge theory and gravity directly from scattering amplitudes. Here we consider effective theories of gravity where in addition to the Einstein-Hilbert term we include nonminimal couplings of the type ${R}^{3}$, ${R}^{4}$ and FFR. In particular, we study the scattering of gravitons and photons of frequency $\ensuremath{\omega}$ off heavy scalars of mass $m$ in the limit $m\ensuremath{\gg}\ensuremath{\omega}\ensuremath{\gg}|\stackrel{\ensuremath{\rightarrow}}{q}|$, where $\stackrel{\ensuremath{\rightarrow}}{q}$ is the momentum transfer. The presence of nonminimal couplings induces helicity-flip processes which survive the eikonal limit, thereby promoting the eikonal phase to an eikonal phase matrix. We obtain the latter from the relevant two-to-two helicity amplitudes that we compute up to one-loop order, and confirm that the leading-order terms in $\ensuremath{\omega}$ exponentiate \`a la Amati, Ciafaloni and Veneziano. From the eigenvalues of the eikonal phase matrix we then extract two physical observables, to 2PM order: the classical deflection angle and Shapiro time delay/advance. Whenever the classical expectation of helicity conservation of the massless scattered particle is violated, i.e., the eigenvalues of the eikonal matrix are nondegenerate, causality violation due to time advance is a generic possibility for small impact parameter. We show that for graviton scattering in the ${R}^{4}$ and FFR theories, time advance is circumvented if the couplings of these interactions satisfy certain positivity conditions, while it is unavoidable for graviton scattering in the ${R}^{3}$ theory and photon scattering in the FFR theory. The scattering processes we consider mimic the deflection of photons and gravitons off spinless heavy objects such as black holes.

Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity

We show that the Newman-Janis shift property of the exact Kerr solution can be interpreted in terms of a worldsheet effective action. This holds both in gravity, and for the single-copy $$ \sqrt{\mathrm{Kerr}} $$
 solution in electrodynamics. At the level of equations of motion, we show that the Newman-Janis shift holds also for the leading interactions of the Kerr black hole. These leading interactions are conveniently described using chiral classical equations of motion with the help of the spinor-helicity method familiar from scattering amplitudes.

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A worldsheet for Kerr

We study the effect of cubic and tidal interactions on the spectrum of gravitational waves emitted in the inspiral phase of the merger of two nonspinning objects. There are two independent parity-even cubic interaction terms, which we take to be ${I}_{1}={{R}^{\ensuremath{\alpha}\ensuremath{\beta}}}_{\ensuremath{\mu}\ensuremath{\nu}}{{R}^{\ensuremath{\mu}\ensuremath{\nu}}}_{\ensuremath{\rho}\ensuremath{\sigma}}{{R}^{\ensuremath{\rho}\ensuremath{\sigma}}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ and ${G}_{3}={I}_{1}\ensuremath{-}2{{{{R}^{\ensuremath{\alpha}}}_{\ensuremath{\mu}}}^{\ensuremath{\beta}}}_{\ensuremath{\nu}}{{{{R}^{\ensuremath{\mu}}}_{\ensuremath{\rho}}}^{\ensuremath{\nu}}}_{\ensuremath{\sigma}}{{{{R}^{\ensuremath{\rho}}}_{\ensuremath{\alpha}}}^{\ensuremath{\sigma}}}_{\ensuremath{\beta}}$. The latter has vanishing pure graviton amplitudes but modifies mixed scalar/graviton amplitudes which are crucial for our study. Working in an effective field theory setup, we compute the modifications to the quadrupole moment due to ${I}_{1}$, ${G}_{3}$ and tidal interactions, from which we obtain the power of gravitational waves radiated in the process to first order in the perturbations and leading order in the post-Minkowskian expansion. The ${I}_{1}$ predictions are novel, and we find that our results for ${G}_{3}$ are related to the known quadrupole corrections arising from tidal perturbations, although the physical origin of the ${G}_{3}$ coupling is unrelated to the finite-size effects underlying tidal interactions. We show this by recomputing such tidal corrections and by presenting an explicit field redefinition. In the post-Newtonian expansion our results are complete at leading order, which for the gravitational-wave flux is 5PN for ${G}_{3}$ and tidal interactions and 6PN for ${I}_{1}$. Finally, we compute the corresponding modifications to the waveforms.

From amplitudes to gravitational radiation with cubic interactions and tidal effects

We compute the scattering amplitude for classical black-hole scattering to third order in the Post-Minkowskian expansion, keeping all terms needed to derive the scattering angle to that order from the eikonal formalism. Our results confirm a conjectured relation between the real and imaginary parts of the amplitude by Di Vecchia, Heissenberg, Russo, and Veneziano, and are in agreement with a recent computation by Damour based on radiation reaction in general relativity.

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The Amplitude for Classical Gravitational Scattering at Third Post-Minkowskian Order

Illuminating Light Bending

We consider gravitational massive scalar-scalar scattering from unitarity and demonstrate how intermediate soft graviton behavior and the concept of extracting classical physics from localization of integrands on velocity cuts devise an efficient extraction scheme for computing the classical post-Minkowskian radial action perturbatively. We demonstrate the computational efficiency by deriving the scattering amplitudes in the probe regime to the fifth post-Minkowskian order in arbitrary dimensions.

L. Planté

Papers

The Amplitude for Classical Gravitational Scattering at Third Post-Minkowskian Order

Illuminating Light Bending

The Amplitude for Classical Gravitational Scattering at Third Post-Minkowskian Order

Post-Minkowskian Radial Action from Soft Limits and Velocity Cuts