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

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 $ η/s = 1/(4πc_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)}(μ) $ and $ c^{(2)}(μ) $ 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.

Figures (1)

• Figure 1: Integrand of the sum rule (\ref{['sum']}) for $d=4$. Solid (blue): conformal field theory; dashed (orange): free massive Dirac field (as an example, the mass is set equal to $m=1/\rho$). The vanishing integral ensures the isotropy sum rule. Quantities are made dimensionless using $\rho$.