Viscosity of Causal Hydrodynamcis

The effect of dissipations in relativistic hydrodynamics is one of topical subjects in heavy-ion physics. However, the introduction of dissipation to relativistic hydrodynamics is not trivial. The simple relativistic generalization of the Navier-Stokes equation is not adequate because the propagation speed of signals exceeds the speed of light and hence causality is broken. To discuss the collective dynamics of relativistic heavy-ion collisions, we need a new hydrodynamics which is consistent causality. So far, there are several approaches to construct the theory consistent with causality. These theories are called the causal hydrodynamics. On the other hand, to solve the causal hydrodynamics, we have to know the transport coefficients; the shear viscosity coefficient, the bulk viscosity coefficient, the heat conduction coefficient and the corresponding relaxation times. Usually, it has been considered that the shear viscosity, bulk viscosity and heat conduction are given by the Green-Kubo-Nakano (GKN) formula even in the causal hydrodynamics. However, the formula is only for the relativistic Navier-Stokes equation which does not obey causality. To estimate the transport coefficients of the causal hydrodynamics, we have to develop other formula. In this work, we propose a modified GKN formula which is applicable to the causal hydrodynamics. In this formula, we can predict not only he shear viscosity, bulk viscosity and heat conduction, but also the corresponding relaxation times. Moreover, we would like to point out that the famous conjecture of the lower limitation of the shear viscosity predicted by using AdS/CFT correspondence is true only for the shear viscosity of the relativistic Navier-Stokes equation. We do not know nothing about the characteristic of the transport coefficients of the causal hydrodynamics.