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Next-to-leading order gravitational spin1-spin2 coupling with Kaluza-Klein reduction

Michele Levi

TL;DR

This work applies a Kaluza-Klein reduction over the time dimension within an effective field theory framework to compute the next-to-leading order gravitational spin1-spin2 interaction for a binary of spinning compact objects. The approach shows that the reduced KK action in the stationary limit suffices to describe the NLO dynamics and that the gravitomagnetic vector field principally mediates the interaction, yielding results that match the ADM Hamiltonian. By implementing both the Newton-Wigner and covariant spin supplementary conditions, the authors derive a consistent NLO spin1-spin2 Hamiltonian and resolve prior discrepancies between EFT and ADM formalisms through canonical-like transformations. The study demonstrates the equivalence of EFT action and ADM Hamiltonian formalisms for this problem and highlights substantial simplifications in diagrammatic calculations due to the KK reduction. Overall, the methodology offers a clearer physical picture of spin-mediated gravity and a practical route to higher-order PN corrections.

Abstract

We use the recently proposed Kaluza-Klein (KK) reduction over the time dimension, within an effective field theory (EFT) approach, to calculate the next to leading order (NLO) gravitational spin1-spin2 interaction between two spinning compact objects. It is shown here that to NLO in the spin1-spin2 interaction, the reduced KK action within the stationary approximation is sufficient to describe the gravitational interaction, and that it simplifies calculation substantially. We also find here that the gravito-magnetic vector field defined within the KK decomposition of the metric mostly dominates the mediation of the interaction. Our results coincide with those calculated in the ADM Hamiltonian formalism, and we provide another explanation for the discrepancy with the result previously derived within the EFT approach, thus demonstrating clearly the equivalence of the ADM Hamiltonian formalism and the EFT action approach.

Next-to-leading order gravitational spin1-spin2 coupling with Kaluza-Klein reduction

TL;DR

This work applies a Kaluza-Klein reduction over the time dimension within an effective field theory framework to compute the next-to-leading order gravitational spin1-spin2 interaction for a binary of spinning compact objects. The approach shows that the reduced KK action in the stationary limit suffices to describe the NLO dynamics and that the gravitomagnetic vector field principally mediates the interaction, yielding results that match the ADM Hamiltonian. By implementing both the Newton-Wigner and covariant spin supplementary conditions, the authors derive a consistent NLO spin1-spin2 Hamiltonian and resolve prior discrepancies between EFT and ADM formalisms through canonical-like transformations. The study demonstrates the equivalence of EFT action and ADM Hamiltonian formalisms for this problem and highlights substantial simplifications in diagrammatic calculations due to the KK reduction. Overall, the methodology offers a clearer physical picture of spin-mediated gravity and a practical route to higher-order PN corrections.

Abstract

We use the recently proposed Kaluza-Klein (KK) reduction over the time dimension, within an effective field theory (EFT) approach, to calculate the next to leading order (NLO) gravitational spin1-spin2 interaction between two spinning compact objects. It is shown here that to NLO in the spin1-spin2 interaction, the reduced KK action within the stationary approximation is sufficient to describe the gravitational interaction, and that it simplifies calculation substantially. We also find here that the gravito-magnetic vector field defined within the KK decomposition of the metric mostly dominates the mediation of the interaction. Our results coincide with those calculated in the ADM Hamiltonian formalism, and we provide another explanation for the discrepancy with the result previously derived within the EFT approach, thus demonstrating clearly the equivalence of the ADM Hamiltonian formalism and the EFT action approach.

Paper Structure

This paper contains 7 sections, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Spinning objects in an EFT approach with Kaluza-Klein reduction
  3. Next-to-leading order spin1-spin2 potential
  4. Implementation of NW SSC at the action level
  5. Implementation of the covariant SSC at the Hamiltonian level
  6. Summary of results
  7. Conclusions

Figures (3)

  • Figure 1: Feynman diagram of the leading order spin1-spin2 interaction. The thick solid lines represent the time evolution of the point particles worldlines. The blobs represent spin insertions on the worldline. The dashed line represents the $A_i$ gravitomagnetic vector field propagator.
  • Figure 2: Next-to-leading order spin1-spin2 interaction Feynman diagrams of one-graviton exchange. The solid line represents the scalar field propagator. The double line represents the 2-tensor field propagator. The cross vertex corresponds to an insertion of the graviton kinetic term. (b) should be included together with its mirror image.
  • Figure 3: Feynman diagrams of nonlinear spin1-spin2 interaction in next-to-leading order. The black blobs represent mass vertices on the worldline. These diagrams should be included together with their mirror images.