$\mathtt{Entity}$ -- Hardware-agnostic Particle-in-Cell Code for Plasma Astrophysics. II: General Relativistic Module

TL;DR

This work develops a general-relativistic particle-in-cell module within the open-source Entity framework to simulate collisionless, relativistic plasmas around black holes. It implements a $3+1$ GR PIC scheme in Kerr–Schild coordinates with a coordinate-agnostic, performance-portable design, including a tetrad-based Boris pusher and a curved-spacetime field solver on a Yee grid. The authors validate the GRPIC module with tests in vacuum (Wald solution), axisymmetric magnetospheres, and Blandford–Znajek monopole configurations, demonstrating stable energy and charge conservation and agreement with analytic solutions. The GR module enables axisymmetric 2.5D explorations of particle acceleration and jet/corona dynamics in strong gravity on GPUs and CPUs, with planned 3D extensions using cubed-sphere grids.

Abstract

Black hole environments often host plasmas that are fully collisionless or contain intrinsically collisionless regions, including relativistic jets and coronae where particle energization is ubiquitous. Capturing the physics of these systems requires numerical methods capable of modeling relativistic, magnetized, collisionless plasmas in strong gravitational fields. In this work, we introduce the general-relativistic module for the Entity -- the first open-source, coordinate-agnostic performance-portable particle-in-cell code. The code enables fast axisymmetric simulations of collisionless plasmas around black holes on any modern high-performance computing architecture (both GPUs and CPUs).

Figures (6)

• Figure 1: Left: locations of $B^i$ and $D^i$ components for which boundary conditions on both of the axes are defined. The solid (no arrow) lines for $B^2_{(\mathtt{i+1/2},0)}$, $B^2_{(\mathtt{i+1/2},\mathtt{N_\theta})}$, as well as solid circles for $D^3_{(i,0)}$, and $D^3_{(i,\mathtt{N_\theta})}$ indicate that these components are set to zero by symmetry. The other components are either reflected exactly ($r$-components) or reflected with the opposite sign ($\theta$-components). Right: Location of the components of electromagnetic fields used for each of the four main routines. The deduced components for the cell $(\mathtt{i_{1}=0}, \mathtt{i_2})$ are shown with gray. All the magnetic and electric field components used in the respective routines are shown in blue and orange, respectively.
• Figure 2: Test particles' orbits (top row) and their corresponding relative energy errors (bottom).
• Figure 3: Conservation of Wald solution in vacuum, from the initial state (left side of each panel) to $t=250~r_g$ (right side of each panel). The top row shows the magnetic field components $B^i$, while the bottom row shows the electric field components $D^i$. In the first panel, black contours trace the magnetic field lines and we show a zoom-in close to the BH. In all panels, the event horizon is indicated by a thin, round black outline.
• Figure 4: Panels (a-c): Snapshots of the radial component of the electric field $D^r$ (in color) at three different times ($t = 1\,r_g$, $10\,r_g$, and $20\,r_g$). The dotted semicircle indicates the reference radius $r = 10\,r_g$. The blue and red markers denote the two particles (electron and positron). Panel (d): Time evolution of the angular integral $\int D^r\,d\Omega$ at $r = 10\,r_g$, with vertical dashed lines marking the snapshot times.
• Figure 5: Snapshot of the plasma-filled magnetosphere around a rapidly rotating black hole ($a = 0.95$) at $t = 90\,r_g$. The color scales show $H_\phi$ normalized to the fiducial $B_0$ (left), and the plasma number density normalized to the fiducial density $n_0$ (right). White curves indicate magnetic field lines. The black circle marks the event horizon, and the vertical solid line denotes the rotation axis. Continuous pair injection maintains densities at or above the Goldreich–Julian level, resulting in a filled magnetosphere consistent with the high-supply regime of Parfrey2019PhRvL.