A Note on Causal Hydrodynamics for M-Theory Branes

Abstract

The AdS/CFT duality is a powerful tool to study hydrodynamics of gauge theory plasmas (see Ref. 1) for a review). However, standard hydrodynamics (first-order formalism) has severe problems such as acausality. One can restore causality by introducing a new set of transport coefficients. The resulting theory is known as “causal hydrodynamics” or “second-order formalism”. At present, there is no unique formalism for causal hydrodynamics, but probably the most used formalism is the “Israel-Stewart theory”.2),3) The Israel-Stewart theory has been widely discussed in the context of heavy-ion collisions. Recently, a number of papers appeared which study the causal hydrodynamics of gauge theory plasmas using the AdS/CFT duality.4)−8),∗∗∗) Some of the main lessons drawn from these works are, 1. One should not take the Israel-Stewart theory too literally; it should be really considered as an effective theory. For example, a single transport coefficient τπ appears both in the “shear mode” and in the “sound mode”. However, if one naively uses the Israel-Stewart theory to interpret the AdS/CFT results, two different values arise. Reference 5) argues that the shear mode result is unreliable due to the corrections from “third-order” hydrodynamics. Namely, the shear mode case exceeds the validity of the effective theory. References 5) and 7) confirmed this for SAdS5,†) and Ref. 7) confirmed this for SAdS4 and SAdS7 as well. 2. Reference 5) proposed a Kubo-like formula for conformal theories to obtain τπ from the “tensor mode”. The formula yields the value of τπ which appears in the sound mode for SAdS5.