Equilibrium configurations of fluids and their stability in higher dimensions

TL;DR

The paper analyzes equilibrium configurations and stability of fluids in higher spatial dimensions under two cohesive mechanisms: surface tension and Newtonian self-gravity. It develops a unified framework using virial theorems, surface-energy tensors, and perturbation theory to examine rotating drops, toroidal shapes, and cylinder/cylinder-like instabilities, extending classic 3D results to arbitrary D. A central thread is the proposed universality between fluid configurations and black-object dynamics, illustrated by analogies to Rayleigh-Plateau versus Gregory-Laflamme instabilities and to Dyson-Chandrasekhar-Fermi modes, as well as by connecting MacLaurin/Jacobi sequences to higher-dimensional analogs and black rings. The work highlights both the utility and the limitations of Newtonian fluid models for capturing horizon-like physics, and it outlines future directions for numerical construction of higher-dimensional bifurcation diagrams and deeper correspondence with general relativity."

Abstract

We study equilibrium shapes, stability and possible bifurcation diagrams of fluids in higher dimensions, held together by either surface tension or self-gravity. We consider the equilibrium shape and stability problem of self-gravitating spheroids, establishing the formalism to generalize the MacLaurin sequence to higher dimensions. We show that such simple models, of interest on their own, also provide accurate descriptions of their general relativistic relatives with event horizons. The examples worked out here hint at some model-independent dynamics, and thus at some universality: smooth objects seem always to be well described by both ``replicas'' (either self-gravity or surface tension). As an example, we exhibit an instability afflicting self-gravitating (Newtonian) fluid cylinders. This instability is the exact analogue, within Newtonian gravity, of the Gregory-Laflamme instability in general relativity. Another example considered is a self-gravitating Newtonian torus made of a homogeneous incompressible fluid. We recover the features of the black ring in general relativity.

Figures (11)

• Figure 1: The bifurcation and stability diagram for rotating liquid drops with surface tension (self-gravity is neglected). Shapes are stable along solid branches, unstable along broken ones. The plot depicts dimensionless angular velocity as function of dimensionless angular momentum. The dimensionless angular velocity is $\Omega=\sqrt{\frac{\rho \hat{\Omega}\hat{R}}{8T}}$, where all hatted quantities are dimensionfull, $\hat{R}$ is the radius of a sphere with the same volume as that of the drop, $\rho$ and $T$ are the density and surface tension of the liquid. Likewise, the dimensionless angular momentum J is defined as $J=\hat{J}/4T \hat{R}^2\left (8T\hat{R}^2 \rho\right )$. Although not shown in the plot, the axisymmetric curve bends back to lower angular velocities after it has passed the four-lobed family neutral point. Note also that the maximum allowed angular momentum for stable shapes is denoted by $J_{II}$ in the diagram. Taken from brownscriven.
• Figure 2: This plot shows a sequence of members of the two-lobed family of drop shapes for several values of angular velocity. The two-lobed drops seem to approach a limiting surface consisting of two spheres touching each other in one point as $\Omega$ approaches 0. The first shape, with $\Omega=0.5599$, corresponds to the first (two-lobed) bifurcation point in Fig. \ref{['fig:brown']}. For $\Omega=0.2885$ for instance, the figure is peanut shaped, and we can situate the corresponding family in Fig. \ref{['fig:brown']}. Taken from phdheine.
• Figure 3: The bifurcation diagram for rotating self-gravitating, incompressible homogeneous stars. Here we show the angular velocity as function of the angular momentum for several possible families. The data and notation are those of Eriguchi and Hachisu eriguchi. The line $A-A'$ is the MacLaurin sequence, $B-B'$ the Jacobi sequence, $C-C'$ triangle sequence, $D-D'$ the square sequence, $E-E'$ ammonite sequence, $F-F'$ the one-ring sequence, $G-G'$ the two-ring sequence, $H-H'$ the pear-shaped sequence, $I-I'$ the dumb-bell sequence, $J-J'$ the Darwin sequence and finally $K-K'$ is the two point-mass sequence. Note the similarities with Fig. \ref{['fig:brown']}. See eriguchi for further details.
• Figure 4: Typical (axi-symmetric) equilibrium figures of rotating drops for case I (one non-vanishing angular momentum), in $D=4$. The (blue) dashed line corresponds to a spherical surface with $\Sigma_a=0$. The (magenta) dotted line corresponds to $\Sigma_a=0.46$ and the (black) solid line corresponds to the limit $\Sigma_a=0.86$. In all three cases the equatorial radius is unity.
• Figure 5: Typical equilibrum figures of rotating drops. The (blue) dashed line corresponds to a spherical surface with $\Sigma_b=0$. The (magenta) dotted line corresponds to $\Sigma_b=2$ and the (black) solid line corresponds to the limit $\Sigma_b=2.329$. In all three cases the equatorial radius was chosen to be unity.