Causality, the Kovtun-Son-Starinets bound, and a novel sum rule for spectral densities

TL;DR

This paper establishes a universal relation between shear viscosity, entropy density, and sound speed near horizons, links causality to viscosity bounds, and introduces a new spectral sum rule relevant for extreme acceleration media.

Contribution

It reveals a universal viscosity relation involving sound speed, connects causality with the Kovtun-Son-Starinets bound, and derives a novel spectral sum rule for thermal radiation in Rindler space.

Findings

The shear viscosity to entropy density ratio is universal and related to sound speed.

The bulk viscosity ratio saturates a known holographic bound.

A new sum rule for spectral densities is validated for conformal and free fields.

Abstract

We directly show that the local ratio of the shear viscosity to the entropy density for Unruh radiation at a finite distance from the horizon is universal and satisfies the relation $ \eta/s = 1/(4\pi c_s^2) $, which involves the speed of sound $ c_s $. Since $ c_s^2 \leq 1 $ by causality, this establishes the close connection between the famous Kovtun-Son-Starinets bound and causality. Moreover, we show that the ratio of bulk to shear viscosity saturates another well-known bound for the bulk viscosity, predicted within holographic approach. We also show that the condition of isotropy of thermal radiation in the Rindler space leads to a novel sum rule relating the $ c^{(0)}(\mu) $ and $ c^{(2)}(\mu) $ spectral densities, and we explicitly demonstrate its validity for conformal field theory and free massive Dirac fields in any number of dimensions. The sum rule provides the validity of Pascal law and bears some similarity with Burkhardt-Cottingham sum rule for spin-dependent parton distributions. Our result suggests a new perspective on dissipative transport phenomena in media undergoing extreme acceleration, such as quark-gluon plasma created in relativistic heavy-ion collisions.