Charged Black-Hole Binary Evolution at Second Post-Newtonian Order

TL;DR

This work extends the PN description of binary black holes to include electric charge at $2\mathrm{PN}$ order within an EFT framework. It delivers a complete conservative Lagrangian in harmonic coordinates, an ADM-type Hamiltonian with center-of-mass transformations, and gauge-invariant observables (binding energy, periastron advance, scattering angle), plus the leading $1.5\mathrm{PN}$ dissipative dipolar effects. The results reproduce known neutral GR limits, agree with recent post-Minkowskian Einstein–Maxwell findings, and establish a solid foundation for incorporating charge-physics into GW waveform models. The analysis sets the stage for future refinements including energy flux, extremal-charge regimes, and spin/finite-size effects relevant for gravitational-wave data analysis.

Abstract

We study the dynamics of electrically charged black-hole binaries and their gravitational-wave emission during the inspiral phase. Within the post-Newtonian framework, we derive the conservative and dissipative dynamics up to second order (2PN), combining Effective Field Theory and classical methods. We compute the NNLO conservative Lagrangian, LO dissipative effects in harmonic and Lorenz gauges, and provide the equations of motion, center-of-mass transformations, and the Lagrangian/Hamiltonian in ADM-type coordinates. We also obtain gauge-invariant expressions for the binding energy, periastron advance in quasi-circular orbits, and the scattering angle in unbound orbits. Our results extend previous analyses and are fully consistent with recent post-Minkowskian findings.

Figures (2)

• Figure 1: The 35 Feynman diagrams contributing up to second PN order (2PN) in the conservative sector. The first row shows the 2 diagrams first contributing at 0PN order, the second row represents the 6 diagrams first contributing at 1PN order, whereas the last three rows represent the 27 diagrams first contributing at 2PN order. In this figure we employ the generic worldline diagram representation: its connection to the explicit worldline diagram representation is reported in Fig. \ref{['fig:diagrams_equivalence']}. The worldline and bulk interaction vertices represented in each diagram are understood to include also higher PN order terms, whenever such terms are present in the action: therefore, we show each diagram only at the first PN order to which it contributes, although it may contribute at higher orders as well. Propagators of scalar fields are represented with dashed lines, in blue for $\phi$ and light blue for $\phi_{\rm EM}$, propagators of vector fields are represented with solid lines, in red for $\mathrm{A}_i$ and in yellow for $\mathrm{A}_{{\rm EM}, i}$, propagators of the $\sigma_{ij}$ tensor field are represented with green dotted lines. The annotation $n{\rm X}$ over a propagator indicates it carries $n$ retardation propagator insertions. The number in square brackets on the lower left of each diagram indicates the positive multiplicity of the corresponding diagram. This number must be divided by the inverse multiplicity to obtain the total symmetry factor. The inverse multiplicity is given by the product of the number of permutations of identical legs in each vertex and the number of permutations of identical vertices.
• Figure 2: Relation between different representations of PN diagrams (left side), and their subsequent connection to multi-loop diagrams (right side). Focusing first on the left side of the figure, the left-hand side of the equality shows the generic worldline diagram representation (employed in Fig. \ref{['fig:diagrams_PN']}), where the worldline vertices are represented as empty circles, and their Feynman rules understand the summation over the worldine indices $\sum_{A=1,2}$. The right-hand side of the equality shows the explicit worldline diagram representation, with worldlines already specified to either $A = 1$ or $A = 2$, and depicted explicitly as horizontal thick lines, even though they are not propagating. As shown in the figure, these two representations are equivalent. More specifically, each generic worldline diagram represents an equivalence class of possibly several explicit worldline diagrams, obtained by fixing the worldline labels. Incidentally, the symmetry factors reported in Fig. \ref{['fig:diagrams_PN']} refer to the generic worldline diagrams, whereas the explicit worldline diagrams belonging to the same equivalence class may carry different symmetry factors each. Focusing now on the right part of the figure (adapted from Foffa:2016rguMandal:2022ntyMandal:2022ufbMandal:2023hqaMandal:2023lgy), we depict the connection between each PN diagram, here with explicit worldlines, and multi-loop diagrams. In particular, each of the PN diagrams, after solving the worldline algebra, can be mapped to (the Fourier transform of) multi-loop diagrams, in particular exactly to two-point loop diagrams arising in a $d$-dimensional Euclidean quantum field theory with a single massless scalar field. The gray-colored area represents arbitrarily complicated loop structures.