  We calculate the shear viscosity of field theories with gravity duals using
Kubo-formula by calculating the Green function of dual transverse gravitons and
confirm that the value of the shear viscosity is fully determined by the
effective coupling of transverse gravitons on the horizon. We calculate the
effective coupling of transverse gravitons for Einstein and Gauss-Bonnet
gravities coupled with matter fields, respectively. Then we apply the resulting
formula to the case of AdS Gauss-Bonnet gravity with $F^4$ term corrections of
Maxwell field and discuss the effect of $F^4$ terms on the ratio of the shear
viscosity to entropy density.
