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Reissner-Nordström Black Holes at second post-Minkowskian order from Scattering Amplitudes

Allan Alonzo-Artiles, Manfred Kraus

Abstract

We employ one-loop scattering amplitudes in Einstein-Maxwell theory to compute the classical Hamiltonian of a binary system of two charged, non-spinning compact objects. The Hamiltonian is valid to all orders in velocity and up to second post-Minkowskian order (2PM), i.e. $\mathcal{O}(G^2)$. The classical interaction potential is extracted via matching to a non-relativistic classical effective field theory. We also provide the scattering angle at 2PM order. We perform several cross checks on our results and find full agreement with existing results in the literature. Finally, we also briefly discuss a comparison for the scattering angle, the binding energy and the periastron shift of a bound system up to the second post-Newtonian order.

Reissner-Nordström Black Holes at second post-Minkowskian order from Scattering Amplitudes

Abstract

We employ one-loop scattering amplitudes in Einstein-Maxwell theory to compute the classical Hamiltonian of a binary system of two charged, non-spinning compact objects. The Hamiltonian is valid to all orders in velocity and up to second post-Minkowskian order (2PM), i.e. . The classical interaction potential is extracted via matching to a non-relativistic classical effective field theory. We also provide the scattering angle at 2PM order. We perform several cross checks on our results and find full agreement with existing results in the literature. Finally, we also briefly discuss a comparison for the scattering angle, the binding energy and the periastron shift of a bound system up to the second post-Newtonian order.
Paper Structure (11 sections, 82 equations, 2 figures)

This paper contains 11 sections, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Einstein-Maxwell Field Theory
  3. The static Reissner-Nordström black hole solution
  4. Weak Field Expansion of Einstein-Maxwell theory
  5. Scattering Amplitudes and their Classical Limit
  6. Effective Field Theory
  7. Classical Hamiltonian and Observables
  8. Point charge in a Reissner-Nordström background
  9. Scattering Angle
  10. Post-Newtonian expansion and observables for bound systems
  11. Conclusions

Figures (2)

  • Figure 1: Representative diagrams for the various gauge-invariant building block of the one-loop scattering amplitude in Einstein-Maxwell theory.
  • Figure 2: The tree-level and one-loop EFT diagram necessary for the matching. The loop momentum is given by $\ell^\mu = (\omega,\boldsymbol{\ell})$ and therefore $\boldsymbol{k} = \boldsymbol{p}+\boldsymbol{\ell}$.