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Shear Waves, Sound Waves On A Shimmering Horizon

Omid Saremi

TL;DR

The paper addresses the need for a conserved horizon stress tensor within the membrane paradigm to properly describe shear dynamics on the stretched horizon. It adopts the Balasubramanian-Kraus holographic stress tensor and proves the relevant constitutive relations by solving the near-horizon Einstein equations, enabling a direct readout of the shear viscosity. The shear viscosity is obtained consistently in two channels—sound and shear—and agrees with the general membrane-paradigm formula. This work strengthens the link between holography and horizon fluid dynamics and suggests directions to extend to additional transport coefficients and to explore implications for Schwarzschild backgrounds.

Abstract

In the context of the so called ``membrane paradigm'' of black holes/branes, it has been known for sometime that the dynamics of small fluctuations on the stretched horizon can be viewed as corresponding to diffusion of a conserved charge in simple fluids. To study shear waves in this context properly, one must define a conserved stress tensor living on the stretched horizon. Then one is required to show that such a stress tensor satisfies the corresponding constitutive relations. These steps are missing in a previous treatment of the shear perturbations by Kovtun, Starinets and Son. In this note, we fill the gap by prescribing the stress tensor on the stretched horizon to be the Brown and York (or Balasubramanian-Kraus (BK) in the AdS/CFT context) holographic stress tensor. We are then able to show that such a conserved stress tensor satisfies the required constitutive relation on the stretched horizon using Einstein equations. We read off the shear viscosity from the constitutive relations in two different channels, shear and sound. We find an expression for the shear viscosity in both channels which are equal, as expected. Our expression is in agreement with a previous membrane paradigm formula reported by Kovtun, Starinets and Son.

Shear Waves, Sound Waves On A Shimmering Horizon

TL;DR

The paper addresses the need for a conserved horizon stress tensor within the membrane paradigm to properly describe shear dynamics on the stretched horizon. It adopts the Balasubramanian-Kraus holographic stress tensor and proves the relevant constitutive relations by solving the near-horizon Einstein equations, enabling a direct readout of the shear viscosity. The shear viscosity is obtained consistently in two channels—sound and shear—and agrees with the general membrane-paradigm formula. This work strengthens the link between holography and horizon fluid dynamics and suggests directions to extend to additional transport coefficients and to explore implications for Schwarzschild backgrounds.

Abstract

In the context of the so called ``membrane paradigm'' of black holes/branes, it has been known for sometime that the dynamics of small fluctuations on the stretched horizon can be viewed as corresponding to diffusion of a conserved charge in simple fluids. To study shear waves in this context properly, one must define a conserved stress tensor living on the stretched horizon. Then one is required to show that such a stress tensor satisfies the corresponding constitutive relations. These steps are missing in a previous treatment of the shear perturbations by Kovtun, Starinets and Son. In this note, we fill the gap by prescribing the stress tensor on the stretched horizon to be the Brown and York (or Balasubramanian-Kraus (BK) in the AdS/CFT context) holographic stress tensor. We are then able to show that such a conserved stress tensor satisfies the required constitutive relation on the stretched horizon using Einstein equations. We read off the shear viscosity from the constitutive relations in two different channels, shear and sound. We find an expression for the shear viscosity in both channels which are equal, as expected. Our expression is in agreement with a previous membrane paradigm formula reported by Kovtun, Starinets and Son.

Paper Structure

This paper contains 10 sections, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Balasubramanian And Kraus Holographic Stress Tensor Prescription
  3. Conservation Of Energy and Momentum For The BK Stress Tensor
  4. Constitutive Relations For The Stress Tensor
  5. Background And Notations
  6. Constitutive Relations On The Stretched Horizon
  7. Sound Channel
  8. Shear Channel
  9. Discussion
  10. Acknowledgement

Figures (1)

  • Figure 1: Manifold $M$ with $n^{\mu}$ being the spacelike normal vector to the boundary. The timelike vector U is orthogonal to the spatial part of the boundary i.e., ${}^{D-2}\textrm{${M}$}$.