Curvature driven diffusion, Rayleigh-Plateau, and Gregory-Laflamme

Abstract

It can be expected that the respective endpoints of the Gregory-Laflamme black brane instability and the Rayleigh-Plateau membrane instability are related because the bifurcation diagrams of the black hole-black string system and the liquid drop-liquid bridge system display many similarities. In this paper, we investigate the non-linear dynamics of the Rayleigh-Plateau instability in a range of dimensions, including the critical dimension at which the phase structure changes. We show that near the critical dimension and above, depending on a parameter in initial conditions an unstable cylinder will either pinch off or converge to an equilibrium state. The equilibrium state is apparently non-uniform but has a constant mean curvature everywhere. The results suggest that in the gravity side, near the critical dimension and above, the final state of an unstable black string (which is not too long) is a non-uniform black string. The equation of motion adopted to describe the dynamics is the surface diffusion equation, which was originally proposed to describe a grooving process of heated metal surfaces. An interesting correspondence between the diffusion dynamics and black hole (thermo)dynamics is discussed.

14 pages, 5 figures; v2: references added, typos fixed

References in corpus (25)

• Black Strings and p-Branes are Unstable
• Holography and hydrodynamics: diffusion on stretched horizons
• Critical behaviour in gravitational collapse of radiation fluid --- A renormalization group (linear perturbation) analysis ---
• On non-uniform black branes
• Fate of the Black String Instability
• The Phase Transition between Caged Black Holes and Black Strings - A Review
• Instabilities of Black Strings and Branes
• A critical dimension in the black-string phase transition
• Topology change in General Relativity and the black-hole black-string transition
• Connecting black holes and black strings
• Towards the Final Fate of an Unstable Black String
• Rayleigh-Plateau and Gregory-Laflamme instabilities of black strings
• Small Black Holes on Cylinders
• Black Rings, Boosted Strings and Gregory-Laflamme
• On Black-Brane Instability In an Arbitrary Dimension
• Convergence to a self-similar solution in general relativistic gravitational collapse
• High and Low Dimensions in The Black Hole Negative Mode
• Inhomogeneous Near-extremal Black Branes
• Equilibrium configurations of fluids and their stability in higher dimensions
• On non-uniform smeared black branes
• Multi-black hole configurations on the cylinder
• A numerical examination of an evolving black string horizon
• New stable phase of non-uniform charged black strings
• The return of the membrane paradigm? Black holes and strings in the water tap
• Liquid bridges and black strings in higher dimensions

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• Dyonic AdS black holes from magnetohydrodynamics
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• Liquid bridges and black strings in higher dimensions
• Non-linear perturbation of black branes at large $D$
• The Bernstein Conjecture, Minimal Cones, and Critical Dimensions
• One-Dimensional Approximation of Viscous Flows
• Stability of hypersurfaces with constant mean curvature trapped between two parallel hyperplanes
• D-dependence of gap between critical temperatures in one-dimensional gauge theories