  Following the recent success of anisotropic hydrodynamics we propose a new,
general prescription for the hydrodynamics expansion around an anisotropic
background. The anisotropic distribution is fixing exactly the complete
energy-momentum tensor, just like the effective temperature is fixing the
proper energy density in the ordinary expansion around local equilibrium. This
means that momen- tum anisotropies are already included at the leading order,
allowing for large pressure anisotropies without the need of a next to leading
order treatment. The first moment of the Boltzmann equation (local
four-momentum conservation) provides the time evolution of the proper energy
density and the four velocity. Differently from previous prescriptions, the
dynamic equations for the pressure corrections are not derived from the zeroth
or second moment of the Boltzmann equation, but they are taken directly from
the exact evolution given by the Boltzmann equation. We check the effec-
tiveness of this new approach by matching with the exact solution of the
Boltzmann equation in the Bjorken limit with the collisional kernel treated in
relaxation time approximation, finding an unprecedented agreement.
