[Paper Review] Membrane paradigm of the static black bottle

Li Li, Towe Wang|arXiv (Cornell University)|Oct 2, 2017

Black Holes and Theoretical Physics3 citations

TL;DR

This paper generalizes the membrane paradigm to static black bottles in asymptotically anti-de Sitter spacetime, where the stretched horizon's normal vector has nonvanishing acceleration—a departure from the standard zero-acceleration assumption. Despite this, the membrane stress tensor remains unmodified, and the fluid continuity and Navier-Stokes equations are satisfied in the near-horizon limit, demonstrating the robustness of the membrane paradigm for non-spherically symmetric black hole solutions.

ABSTRACT

In the membrane paradigm of black holes, it is usually assumed that the normal vector of the stretched horizon has a vanishing acceleration. This assumption breaks down for black bottles, a class of solutions discovered recently in the asymptotically anti-de Sitter spacetime. In this paper, the membrane paradigm is generalized to the stretched horizon with a nonvanishing acceleration of normal vector, and then it is applied to the static black bottle. In this example, the membrane stress tensor and the fluid quantities are similar to those of black holes, while the fluid continuity equation and the Navier-Stokes equation are well satisfied in the near-horizon limit.

Motivation & Objective

• To extend the membrane paradigm to static black bottles in asymptotically AdS spacetime, where the stretched horizon's normal vector has nonvanishing acceleration.
• To investigate whether the standard membrane paradigm—based on zero-acceleration of the normal vector—remains valid for such solutions.
• To verify that the fluid equations (continuity and Navier-Stokes) are still satisfied in the near-horizon limit despite the non-zero acceleration.
• To demonstrate the robustness of the membrane stress tensor under non-zero normal vector acceleration.
• To establish a hydrodynamic description of the stretched horizon for black bottles, analogous to black holes and black branes.

Proposed method

• Reformulate the static black bottle solution in spherical coordinates to facilitate analysis of the stretched horizon geometry.
• Perform a $2+1+1$ spacetime split by defining the timelike generator $u^a$ and spacelike normal $n^a$ of the stretched horizon.
• Compute the extrinsic curvature $K_{ab}$ and the induced metric $h_{ab}$ on the membrane to derive fluid quantities such as energy density $\rho$, pressure $p$, shear $\sigma^{AB}$, and expansion $\theta$.
• Introduce Eddington-Finkelstein-like coordinates to evaluate the surface gravity and analyze the limit where the stretched horizon approaches the true event horizon.
• Use the action-based method with a surface term to derive the membrane stress tensor and verify its independence from the normal vector's acceleration.
• Check the fluid equations—continuity and Navier-Stokes—explicitly in the near-horizon limit using the derived fluid quantities.

Experimental results

Research questions

• RQ1 Can the membrane paradigm be consistently applied to static black bottles, which violate the zero-acceleration assumption of the normal vector on the stretched horizon?
• RQ2 Does the membrane stress tensor remain unchanged when the normal vector of the stretched horizon has nonvanishing acceleration?
• RQ3 Are the fluid continuity and Navier-Stokes equations still satisfied in the near-horizon limit for the static black bottle?
• RQ4 How do the fluid quantities (density, pressure, shear, expansion) on the membrane compare to those of standard black holes or black branes?
• RQ5 What is the role of the Plebański-Demiański-like coordinate system in simplifying the membrane paradigm for black bottles?

Key findings

• The membrane stress tensor in Einstein gravity is not corrected by the non-zero acceleration of the normal vector, as shown through a general variation of the action with $a^b \neq 0$.
• The fluid quantities on the membrane—energy density $\rho = \frac{1}{8\pi G L(x-y)} Q(y)$, pressure $p = -\frac{1}{32\pi G L} \left[ \frac{2}{x-y} Q(y) + \partial_y Q(y) \right]$, shear $\sigma^{AB} = 0$, expansion $\theta = -\frac{1}{L(x-y)} Q(y)$—are well-defined and analogous to those in standard black hole membranes.
• The fluid continuity equation and Navier-Stokes equation are satisfied in the near-horizon limit ($\alpha \to 0$), confirming hydrodynamic consistency.
• The surface gravity is given by $g_{\mathcal{H}} = -\frac{1}{4L} \left[ \frac{2}{x-y} Q(y) + \partial_y Q(y) \right]$, which matches the fluid equation source term.
• The shear viscosity $\eta = \frac{1}{16\pi G}$ and bulk viscosity $\zeta = -\frac{1}{16\pi G}$ are constant and analogous to those in standard membrane paradigms.
• The results are reproducible in Plebański-Demiański-like coordinates, confirming the robustness of the formulation across coordinate systems.