The general relativistic thin disc evolution equation

TL;DR

This paper extends the classical thin disc accretion theory into Kerr geometry, deriving a relativistic diffusion equation that accounts for black hole effects and singularities, aiding black hole accretion studies.

Contribution

It develops a relativistic thin disc evolution equation in Kerr spacetime, incorporating singularities and boundary conditions for stable mode analysis.

Findings

A diffusion-like equation with a singularity at the angular momentum zero gradient.

Stable external modes exist with a vanishing stress boundary condition.

Provides a new tool for numerical and phenomenological accretion disc studies.

Abstract

In the classical theory of thin disc accretion discs, the constraints of mass and angular momentum conservation lead to a diffusion-like equation for the turbulent evolution of the surface density. Here, we revisit this problem, extending the Newtonian analysis to the regime of Kerr geometry relevant to black holes. A diffusion-like equation once again emerges, but now with a singularity at the radius at which the effective angular momentum gradient passes through zero. The equation may be analysed using a combination of WKB, local techniques, and matched asymptotic expansions. It is shown that imposing the boundary condition of a vanishing stress tensor (more precisely the radial-azimuthal component thereof) allows smooth stable modes to exist external to the angular momentum singularity, the innermost stable circular orbit, while smoothly vanishing inside this location. The extension of the disc diffusion equation to the domain of general relativity introduces a new tool for numerical and phenomenolgical studies of accretion discs, and may prove to be a useful technique for understanding black hole X-ray transients.