Relativistic Hydrodynamics around Black Holes and Horizon Adapted Coordinate Systems

TL;DR

This paper introduces horizon-adapted coordinate systems for relativistic hydrodynamics around black holes, eliminating coordinate singularities at the horizon and improving numerical simulations of accretion flows.

Contribution

It proposes a class of coordinate systems free of horizon singularities, enabling more accurate and stable numerical simulations of black hole accretion in general relativity.

Findings

Exact solutions in Eddington-Finkelstein coordinates for dust and perfect fluid accretion.

Numerical simulations show improved stability and accuracy with horizon-adapted coordinates.

Demonstrated benefits in axisymmetric and higher-dimensional accretion studies.

Abstract

Despite the fact that the Schwarzschild and Kerr solutions for the Einstein equations, when written in standard Schwarzschild and Boyer-Lindquist coordinates, present coordinate singularities, all numerical studies of accretion flows onto collapsed objects have been widely using them over the years. This approach introduces conceptual and practical complications in places where a smooth solution should be guaranteed, i.e., at the gravitational radius. In the present paper, we propose an alternative way of solving the general relativistic hydrodynamic equations in background (fixed) black hole spacetimes. We identify classes of coordinates in which the (possibly rotating) black hole metric is free of coordinate singularities at the horizon, independent of time, and admits a spacelike decomposition. In the spherically symmetric, non-rotating case, we re-derive exact solutions for dust and perfect fluid accretion in Eddington-Finkelstein coordinates, and compare with numerical hydrodynamic integrations. We perform representative axisymmetric computations. These demonstrations suggest that the use of those coordinate systems carries significant improvements over the standard approach, especially for higher dimensional studies.