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

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 the relativistic Rayleigh-Taylor instability (R-RTI) in a magnetized medium, where fluids can move with relativistic velocities. We chose an “intermediate frame” in which the fluids on each side of the interface move in opposite directions with identical Lorentz factors γ ⁎ and derive the new dispersion relation of the R-RTI. This symmetry facilitates analytical derivations and the study of relativistic effects on the dynamics of instabilities. We find that the instability is activated when the Atwood number A = ( ρ 1 h 1 − ρ 2 h 2 ) / ( ρ 1 h 1 + ρ 2 h 2 ) > 0 , where ρ 1 and ρ 2 are densities measured in the rest of the fluids. The relativistic effect is mostly contained in the Lorentz transformation of the effective acceleration g ′ = g γ ⁎ − 2 , which, combined with time dilation, leads to a much slower growth of instability ( ω ′ = γ ⁎ − 1 ω 0 ), 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 ( μ QSOs), 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.