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Non-conformal Hydrodynamics in Einstein-dilaton Theory

In the Einestein-dilaton theory with a Liouville potential parameterized by $η$, 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 $0 \le η< 2$. 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 $η$. 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 $η$. Interestingly, the vector fluctuation with the dilaton coupling shows that the DC conductivity increases with temperature, a feature commonly found in electrolytes.