Non-conformal Hydrodynamics in Einstein-dilaton Theory

In the Einestein-dilaton theory with a Liouville potential parameterized by $\eta$, we find a Schwarzschild-type black hole solution. This black hole solution, whose asymptotic geometry is described by the warped metric, is thermodynamically stable only for $0 \le \eta < 2$. Applying the gauge/gravity duality, we find that the dual gauge theory represents a non-conformal thermal system with the equation of state depending on $\eta$. After turning on the bulk vector fluctuations with and without a dilaton coupling, we calculate the charge diffusion constant, which indicates that the life time of the quasi normal mode decreases with $\eta$. Interestingly, the vector fluctuation with the dilaton coupling shows that the DC conductivity increases with temperature, a feature commonly found in electrolytes.