First-principle derivation of stable first-order generic-frame relativistic dissipative hydrodynamic equations from kinetic theory by renormalization-group method

We derive first-order relativistic dissipative hydrodynamic equations from the rela-tivistic Boltzmann equation by the renormalization-group (RG) method. We introduce a macroscopic-frame vector, which does not necessarily coincide with the flow velocity, to specify the local rest frame on which the macroscopic dynamics is described. The five hydrodynamic modes are naturally identified with the same number of zero modes of the linearized collision operator, i.e., the collision invariants. After defining the inner product in the function space spanned by the distribution function, the higher-order terms, which give rise to the dissipative effects, are constructed so that they are precisely orthogonal to the zero modes in terms of the inner product: Here, no ansatzs, such as the so-called conditions of fit used in the standard methods in an ad hoc way, are necessary. We elucidate that the Burnett term does not affect the hydrodynamic equations owing to the very nature of the hydrodynamic modes as the zero modes. Then, applying the RG equation, we obtain the hydrodynamic equation in a generic frame specified by the macroscopic-frame vector, as the coarse-grained and covariant equation. Our generic hydrodynamic equation reduces to hydrodynamic equations in various local rest frames, including the energy and particl