# GRay2: A General Purpose Geodesic Integrator for Kerr Spacetimes

## TL;DR

GRay2 is a new, efficient geodesic integrator for Kerr spacetimes that uses Cartesian Kerr-Schild coordinates and GPU acceleration to improve accuracy and performance near black holes.

## Contribution

It introduces a general-purpose geodesic integrator that overcomes coordinate singularities and leverages GPU computing for faster, more accurate simulations in Kerr spacetimes.

## Key findings

- Outperforms Boyer-Lindquist coordinate-based methods.
- Achieves high accuracy in geodesic calculations.
- Runs efficiently on CPUs and GPUs.

## Abstract

Fast and accurate integration of geodesics in Kerr spacetimes is an important tool in modeling the orbits of stars and the transport of radiation in the vicinities of black holes. Most existing integration algorithms employ Boyer-Lindquist coordinates, which have coordinate singularities at the event horizon and along the poles. Handling the singularities requires special numerical treatment in these regions, often slows down the calculations, and may lead to inaccurate geodesics. We present here a new general-purpose geodesic integrator, GRay2, that overcomes these issues by employing the Cartesian form of Kerr-Schild coordinates. By performing particular mathematical manipulations of the geodesic equations and several optimizations, we develop an implementation of the Cartesian Kerr-Schild coordinates that outperforms calculations that use the seemingly simpler equations in Boyer-Lindquist coordinates. We also employ the OpenCL framework, which allows GRay2 to run on multi-core CPUs as well as on a wide range of GPU hardware accelerators, making the algorithm more versatile. We report numerous convergence tests and benchmark results for GRay2 for both time-like (particle) and null (photon) geodesics.