Inspiraling binary charged black holes in an external magnetic field: Application of post-Newtonian dynamics in Einstein-Maxwell theory

TL;DR

This work develops a first-order post-Newtonian framework for binary charged black holes immersed in an external, uniform magnetic field within Einstein-Maxwell theory. The authors construct the action, separate the external field as a fixed background, and derive the 1PN equations of motion via a Fokker-action approach, enabling numerical integration of orbits and computation of gravitational-wave signals. They demonstrate that the background magnetic field breaks translational symmetry, induces a secular center-of-mass drift, and drives the orbit to three dimensions, with 1PN corrections providing measurable refinements to the trajectories and waveform polarizations. The study shows that magnetized charged binaries yield distinctive gravitational-wave imprints, offering a new avenue to probe black-hole charge and ambient magnetic environments with current and future detectors, and lays the groundwork for extensions to more realistic magnetic-field geometries and magnetohydrodynamic effects.

Abstract

We present a systematic post-Newtonian treatment of binary charged black holes immersed in external magnetic fields within the framework of Einstein-Maxwell theory. By incorporating a uniform external magnetic field into the two-body Lagrangian expanded to first post-Newtonian order, we derive the complete equations of motion that capture both gravitational and electromagnetic interactions. The magnetic Lorentz force fundamentally alters the orbital dynamics, breaking the conservation of linear and angular momentum and inducing transitions from planar to three-dimensional trajectories. Through numerical integration of these equations, we compute the resulting gravitational waveforms and quantify the magnetic field imprints using matched filtering techniques. Our results demonstrate that strong background magnetic fields can substantially modify the orbital evolution and leave distinctive signatures in the gravitational wave signals. These findings provide a promising avenue for detecting charged black holes and probing magnetic field environments through gravitational wave observations.

Figures (9)

• Figure 1: The schematic diagram of the evolution of the orbits. The left panel represent the case with $q_1/m_1=q_2/m_2$ and the right is for $q_1/m_1\neq q_2/m_2$. $\vec{V}$ is the velocity of CoM which is not conserved.
• Figure 2: Cartesian components of $\bm J$ for different charge-to-mass ratios at Newtonian order. Physical parameters: $m_1=15M_{\odot}$, $q_1=\sqrt{G}\nu_{1}m_1$ with $\nu_1=0.72$; $m_2=20M_{\odot}$, $q_2=\sqrt{G}\nu_{2}m_2$ with $\nu_2=0.72$ (left panel) and $\nu_2=0.32$ (right panel).
• Figure 3: Three components of the Newtonian-order total linear momentum evolution. Physical parameters: $m_1=15M_{\odot}$, $q_1=\sqrt{G}\nu_{1}m_1$ with $\nu_1=0.52$; $m_2=20M_{\odot}$, $q_2=\sqrt{G}\nu_{2}m_2$ with $\nu_2=-0.48$. The magnetic field strength is $\bm B=5.0\times 10^{12}\,\mathrm{G}$. The linear momenta are rescaled by the total mass $M$.
• Figure 4: Comparison of trajectories with (right) and without (left) magnetic field at Newtonian order. Physical parameters: $m_1=15M_{\odot}$, $q_1=\sqrt{G}\nu_{1}m_1$ with $\nu_1=0.52$; $m_2=20M_{\odot}$, $q_2=\sqrt{G}\nu_{2}m_2$ with $\nu_2=-0.48$.
• Figure 5: Comparison of trajectories with (right) and without (left) magnetic field at 1PN order. Physical parameters identical to Figure \ref{['fig:0PN trajectory comparison']}.