Shear Waves, Sound Waves On A Shimmering Horizon

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.