Fully covariant hydrodynamical equations for multi-species relativistic plasma in the external electromagnetic field are derived. The derived multi-fluid description takes into account binary Coulomb collisions, annihilation, and interaction with the photon background in terms of the invariant collision cross-sections. Generalized Ohm's law is derived in a manifestly covariant form. Particular attention is devoted to the relativistic electron-positron plasma. PACS numbers: 52.60.+h, 95.30.Qd,

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... lativistic plasma attracts growing interest in connection with its possibly important role in active galactic nuclei, black hole magnetospheres, early Universe, relativistic jets, and other highly energetic astrophysical objects [1]. Among these plasmas the relativistic electron-positron plasma (sometimes contaminated with other species [2]) is of particular interest because it can be produced even in laboratory conditions [3] , but mostly because of its well-established prominent role in the pulsar operation and interaction with the surrounding matter [4] . Complete kinetic description of the relativistic plasma is difficult and in many cases redundant, for example, when large scale bulk plasma motion is considered. In this case multi-fluid hydrodynamics is the most appropriate description. In the extreme case it is desirable to coarse the description to a one-fluid magnetohydrodynamics (MHD). MHD equations usually contain the particle number conservation law, the energy-momentum conservation law, and the evolution equation for the magnetic field [5] . These have to be completed by an appropriate Ohm's law. An attempt to derive such Ohm's law for weakly collisional pair plasma have been done recently [6] . However, the restrictions imposed and non-covariant consideration make this attempt not very useful. The goal of the present paper is to derive the manifestly-covariant multi-fluid hydrodynamics for collisional plasmas, and to find the covariant form of the Ohm's law. We describe each species s by its distribution function which satisfies the 4-dimensional Vlasov-Boltzmann equation