The amplitude for classical gravitational scattering at third Post-Minkowskian order

N Emil J Bjerrum-Bohr,Ludovic Planté,Pierre Vanhove,Poul H Damgaard

doi:10.1007/jhep08(2021)172

Abstract

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.

Highlights

Two-body kinematics and the iterated solution to the Lippmann-Schwinger equation leads to explicit formulas for the scattering angle in terms of the coefficients of the relativistic two-body potential [11,12,13], in a sense thereby solving the problem of kinematics of the relativistic two-body problem in general relativity by means of quantum field theory and amplitude techniques
The eikonal formalism is used to derive the classical scattering angle, an interesting feature is that an in principle infinite number of super-classical terms must be computed as well in order to confirm the exponentiation of the amplitude in impact parameter space, a phenomenon that must follow from unitarity alone [31]
A second advantage of the method of ref. [32] is that it significantly reduces the number of master integrals that need to be known. This was obvious in the case of maximal supergravity and, as we shall demonstrate in this paper, this holds in Einstein gravity as well

Summary

Introduction

Two-body kinematics and the iterated solution to the Lippmann-Schwinger equation leads to explicit formulas for the scattering angle in terms of the coefficients of the relativistic two-body potential [11,12,13], in a sense thereby solving the problem of kinematics of the relativistic two-body problem in general relativity by means of quantum field theory and amplitude techniques. [14, 16, 17] have shown how certain classical terms of the two-loop scattering, that are not captured by a limitation to the so-called potential region of the loop integrals, restore a well-behaved high-energy behavior This picture has been understood directly from general relativity in terms of gravitational radiation reaction [15]. A natural language for the calculation of the scattering angle of two black holes in general relativity is the gravitational eikonal It relies on the exponentiation of appropriate terms of the S-matrix in the small-angle limit of the involved semi-classical field theory amplitudes. [32] is that it significantly reduces the number of master integrals that need to be known This was obvious in the case of maximal supergravity and, as we shall demonstrate in this paper, this holds in Einstein gravity as well.

Objectives

Results

Conclusion