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

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 characterize the distinctive magnetic field signatures through time-domain and frequency-domain analysis.} 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.

💡 Research Summary

This paper presents a systematic post-Newtonian (PN) analysis of binary charged black hole systems immersed in an external, uniform magnetic field within the framework of Einstein-Maxwell theory. The primary goal is to understand how ambient magnetic fields, ubiquitous in astrophysical environments, alter the orbital dynamics and gravitational-wave (GW) emission of charged black hole binaries.

The authors begin by constructing the theoretical framework. The total action combines the matter action for point particles (with mass and charge) and the field action encompassing both gravity and electromagnetism. A key decomposition separates the total electromagnetic field into the internal field generated by the binary’s charges and an external, uniform, and stationary magnetic field. Crucially, the gravitational back-reaction of this external field is neglected, an approximation justified by showing that the magnetic energy density is orders of magnitude smaller than the gravitational energy density for field strengths up to ~10^14 G. Using the Fokker action method within the harmonic (gravity) and Lorenz (electromagnetism) gauges, the authors derive an effective two-body Lagrangian expanded to first PN order (1PN). This Lagrangian includes the standard Newtonian and 1PN terms for an isolated charged binary (L_iso), plus an additional interaction term (L_B) describing the coupling of each charge to the external magnetic field via the minimal coupling scheme.

From this Lagrangian, the complete equations of motion are derived via the Euler-Lagrange equations, followed by an order-reduction procedure to express accelerations consistently at 1PN accuracy. The final equations incorporate Newtonian gravity, Coulomb attraction/repulsion, the magnetic Lorentz force, and relativistic corrections at 1PN order. A fundamental consequence of the external field is the explicit breaking of spatial translation symmetry, leading to the non-conservation of the system’s total linear and angular momentum. This induces a qualitative change in the orbital dynamics: initially planar orbits gradually evolve into three-dimensional trajectories, a phenomenon absent for isolated binaries.

The paper then proceeds to a numerical investigation. The equations of motion are integrated for a representative binary system, and the resulting trajectories are used to compute the emitted gravitational waveforms via the quadrupole formula. The analysis is conducted in both the time and frequency domains. The results demonstrate that a strong background magnetic field (e.g., B = 8.0 x 10^12 G) can substantially modify the orbital inspiral. This modification leaves distinctive imprints on the GW signal, including changes in the phase evolution and characteristic patterns in the frequency spectrum. These “magnetic field signatures” are quantified, and their detectability is discussed in the context of matched-filtering techniques used by GW observatories. The analysis suggests that future precision GW measurements could, in principle, simultaneously constrain the charges of the black holes and the strength of the surrounding magnetic field.

In conclusion, this work provides the first systematic 1PN treatment of binary charged black holes interacting with an external magnetic field. It establishes a foundational framework for future higher-order PN calculations, inclusion of spin effects, and studies of non-uniform fields. The findings highlight the potential of gravitational-wave astronomy not only to detect exotic objects like charged black holes but also to probe the extreme magnetized environments in which such cosmic events may occur, opening a new multi-messenger avenue in astrophysics.