Post-Newtonian Dynamics of Radiating Charges: Canonical Formulation and Binary Inspiral Laws

Technical Summary: Post-Newtonian Dynamics of Radiating Charges

Problem Statement
The paper addresses the need for an explicit, directly implementable electromagnetic analogue of the Post-Newtonian (PN) canonical Hamiltonian framework widely used in gravitational wave physics. While the dynamics of neutral compact binaries in General Relativity (GR) are well-described by PN expansions augmented with radiation reaction (e.g., via the Effective-One-Body approach), a unified canonical dissipative framework for charged many-body systems has been lacking. Specifically, the authors aim to construct a phase-space system for charged binaries that incorporates the conservative Darwin interaction (1PN) and the leading-order dissipative radiation reaction (1.5PN) derived from the Landau-Lifshitz reduction of the Lorentz-Dirac equation. The work seeks to derive analytic inspiral laws for both circular and eccentric orbits and to extend these results to charged compact binaries in Einstein-Maxwell theory, identifying the transition scale between electromagnetic dipole dominance and gravitational quadrupole dominance.

Methodology
The authors employ a perturbative expansion in powers of $v/c$ (the post-Newtonian ordering) to construct the dynamics:

1. Order Reduction: Starting from the covariant Lorentz-Dirac (LD) equation, which admits unphysical runaway and preacceleration solutions, the authors implement the Landau-Lifshitz (LL) order-reduction procedure. This yields a causal, second-order equation of motion where the self-force is treated perturbatively.
2. Canonical Formulation:
   • Conservative Sector: The system utilizes the Darwin Hamiltonian, truncated at 1PN order ($O(c^{-2})$), which includes relativistic corrections to the Coulomb interaction.
   • Dissipative Sector: The 1.5PN radiation-reaction force ($O(c^{-3})$) is derived from the near-zone limit of the LL equation. Crucially, this force is expressed entirely in terms of canonical variables (positions and momenta) as a double sum over the $N$ bodies, ensuring compatibility with the Hamiltonian formalism.
3. Binary Specialization: The $N$-body system is specialized to $N=2$. The authors derive the explicit relative acceleration, demonstrating that the dipole radiation reaction is proportional to the charge-to-mass asymmetry ($q_1/m_1 - q_2/m_2$).
4. Analytic Derivations:
   • Purely Electromagnetic Context: Analytic inspiral laws are derived for circular and eccentric binaries, including 1PN conservative corrections.
   • Einstein-Maxwell Context: The framework is extended to relativistic charged binaries by combining the known 2PN ADM-type conservative Hamiltonian (from Placidi et al.) with the leading 1.5PN dipole dissipation.
5. Numerical Validation: Direct numerical integrations of the full canonical equations of motion are performed for charge-neutral and multi-charge systems to validate the analytic inspiral laws, circularization rates, and energy loss.

Key Contributions and Results

• Unified Phase-Space System: The paper presents a closed 1PN+1.5PN canonical phase-space system. In the absence of dissipation, the system strictly conserves the Darwin Hamiltonian. With dissipation, it exhibits monotonic energy loss, secular inspiral, and robust circularization.
• Dipole Radiation Reaction Structure: For binaries, the radiation-reaction force is shown to vanish identically when the charge-to-mass ratios are equal ($q_1/m_1 = q_2/m_2$), recovering the result that a constant dipole moment produces no radiation. The force scales with the asymmetry parameter $\Delta = m_1 q_2 - m_2 q_1$.
• Inspiral Laws and Scaling:
  • Chirp Scaling: In the dipole-dominated regime, the frequency evolution follows $\dot{\Omega} \propto \Omega^3$, contrasting with the gravitational quadrupole scaling of $\dot{\Omega} \propto \Omega^{11/3}$.
  • Phase Evolution: The orbital phase scales as $\Phi \propto \Omega^{-1}$ (dipole) versus $\Phi \propto \Omega^{-5/3}$ (quadrupole).
  • Eccentricity: For eccentric orbits, the authors derive a closed-form relation between semi-major axis and eccentricity, $a(e) \propto e^{4/3}/(1-e^2)$, and an explicit expression for the time to circularization.
• Dipole-Quadrupole Crossover: In Einstein-Maxwell theory, the authors identify a gauge-invariant crossover frequency ($f_{cross}$) where electromagnetic dipole losses transition to gravitational quadrupole dominance. This scale depends solely on the charge-to-mass asymmetry:
  $$f_{cross} \simeq 2.2 \times 10^3 \text{ Hz} \left(\frac{M_\odot}{M}\right) \frac{|\eta_2 - \eta_1|^3}{1 - \eta_1 \eta_2}$$
  where $\eta_A$ is the dimensionless charge-to-mass ratio.
• Numerical Verification: Simulations confirm that the system exhibits "eccentric bursts" in the Hamiltonian evolution during the circularization of initially eccentric orbits. The numerical results for the invariant $I = a(1-e^2)/e^{4/3}$ match the analytic predictions to sub-percent accuracy.

Significance and Claims
The paper claims to provide the first explicit, directly implementable electromagnetic analogue of the PN canonical paradigm for dissipative long-range dynamics. Its significance lies in:

1. Structural Analogy: It demonstrates that the structural features of gravitational radiation reaction (monotonic energy loss, secular inspiral, circularization) have a direct electromagnetic counterpart when formulated via Landau-Lifshitz reduction in a canonical framework.
2. Analytic Framework for Charged Binaries: It offers a complete set of analytic inspiral laws (circular and eccentric) and gauge-invariant energy-frequency relations for charged binaries in Einstein-Maxwell theory through 2PN conservative order.
3. Observational Diagnostics: The derived scaling laws ($\dot{\Omega} \propto \Omega^3$ and $\Psi(f) \propto f^{-1}$) provide distinct observational signatures for gravitational wave detectors. The paper notes that for dipole effects to enter the LIGO/Virgo band, the charge-to-mass asymmetry must be of order unity, implying near-extremal charges or hidden-sector interactions.
4. Universality: The work suggests a potential universality in dissipative long-range dynamics, noting that the order-reduced equations admit reductions to Painlevé transcendents in certain limits, a feature previously conjectured for binary black hole coalescence.

The authors explicitly limit their scope to leading dipole dissipation and low-PN conservative dynamics, noting that higher-order terms and EOB resummations are natural extensions for future work. They do not claim to have detected such systems but provide the theoretical tools necessary to constrain hidden-sector charges using current and future gravitational wave data.