Dispersion relation for the linear theory of relativistic Rayleigh Taylor instability in magnetized medium revisited

TL;DR

This paper revisits the linear theory of relativistic Rayleigh-Taylor instability in magnetized media, deriving the correct dispersion relation and analyzing how relativistic effects slow down the growth rate of the instability.

Contribution

It provides a corrected analytical formulation of relativistic RTI, including the impact of Lorentz transformations and time dilation on instability growth in magnetized, relativistic flows.

Findings

Relativistic effects slow the RTI growth rate by a factor of gamma_*.

The Atwood number criterion remains unchanged in relativistic regimes.

Instability growth is significantly reduced in systems like jets of microquasars and gamma-ray bursts.

Abstract

The Rayleigh-Taylor instability (RTI) arises at the interface between two fluids of different densities, notably when a heavier fluid lies above a lighter one in an effective gravitational field. In astrophysical systems with high velocities, relativistic corrections are necessary. We investigate the linear theory of relativistic Rayleigh-Taylor instability (R-RTI) in a magnetized medium, where fluids can move parallel to the interface at relativistic velocities. We chose an "intermediate frame" where fluids on each side of the interface move in opposite directions with identical Lorentz factors gamma_*. This symmetry facilitates analytical derivations and the study of relativistic effects on the instability's dynamics. We derive the correct version of the R-RTI. We find that the instability is activated when the Atwood number A = (rho1 h1 - rho2 h2) / (rho1 h1 + rho2 h2) > 0, where rho1 and rho2 are densities measured in the rest frame of the fluids, and that this criterion does not contain relativistic corrections. The relativistic effect is mostly contained in the Lorentz transformation of the gravitational acceleration g' = g / (gamma_*)^2, which, combined with time dilation, leads to a much slower growth of instability (omega' = omega0 / gamma_*), and a slightly elongated length of the unstable patch, due to weaker g in the intermediate frame. Taking time dilation into account, when viewed in the rest frame of the medium, we expect the instability to grow at a much reduced rate. The analytical results should guide further explorations of instability in systems such as microquasars (muQSOs), active galactic nuclei (AGNs), gamma-ray bursts (GRBs), and radio pulsars (PSRs), where the apparent stability of the jet can be attributed to either the intrinsic stability (e.g. the Atwood number) or the much prolonged duration through which R-RTI can grow.