A method of incorporating general relativity in electromagnetic particle-in-cell code

An algorithm is presented that incorporates the tensor form of Maxwell’s equations in a general relativistic electromagnetic particle-in-cell code. The code simplifies to Schwartzschild space-time for a non-spinning central mass. The particle advance routine uses a fourth-order Runge-Kutta algorithm to integrate the four-velocity form of Lorentz force. The current density is calculated using the curved space-time of the metric.

💡 Research Summary

This paper presents a comprehensive algorithm for a General Relativistic electromagnetic Particle-in-Cell (GRPIC) code, designed to simulate plasma dynamics in the strong gravitational fields surrounding massive, rotating objects like Kerr black holes. The core achievement is the seamless integration of Einstein’s theory of general relativity into the established PIC framework, which traditionally operates in flat Minkowski spacetime.

The authors, Michael Watson and Ken-Ichi Nishikawa, develop their algorithm based on the tensor formulations of Maxwell’s equations and the Newton-Lorentz force law within a general metric, specifically the Kerr metric. This allows the code to naturally simplify to the Schwarzschild metric for non-spinning central masses. The computational cycle self-consistently advances electromagnetic fields and particle motions. For field updates, the components of the Maxwell field tensor are placed on a generalized Yee lattice, and their evolution equations are discretized to include Christoffel symbols, directly incorporating spacetime curvature.

A significant technical advancement is the use of a fourth-order Runge-Kutta (RK4) method for particle integration, as opposed to the standard second-order Boris push. This is crucial for maintaining accuracy and stability in regions of extreme curvature near the event horizon. The RK4 scheme is ingeniously modified to account for the time dilation experienced by each individual particle, with the proper time interval (Δτ) adjusted via the Lorentz factor (γ) at each sub-step.

The algorithm also generalizes key PIC operations for curved spacetime. The “cloud-in-cell” interpolation of fields from grid nodes to particle locations uses volume weights calculated from the metric determinant, ensuring the influence of spacetime geometry on field values. Similarly, the current deposition scheme, based on a charge-conserving method, employs area weights derived from the metric to calculate current density contributions on cell faces.

The paper discusses the challenges of simulation stability in a relativistic context, showing how classical criteria like the plasma frequency condition are modified by gravitational time dilation. To bridge the vast scale difference between gravitational (Schwarzschild radius) and plasma (skin depth) scales, the authors propose using a normalized Stoney unit system, making global simulations computationally feasible while preserving the essential physics.

As a demonstration, the paper briefly describes an application simulating jet formation from a Keplerian disk and a free-falling corona around a rapidly spinning (a=0.95) central mass. The results indicate the formation of a collimated jet structure along the rotation axis, with a velocity distribution where the fastest particles are concentrated at the jet’s core.

In summary, this work provides a complete and detailed algorithmic framework for a GRPIC code. It represents a critical tool for first-principles investigations of collisionless plasma processes—such as particle acceleration, magnetic reconnection, and radiation mechanisms—in the most extreme gravitational environments in the universe, areas where magnetohydrodynamic (MHD) approximations may be insufficient.