A virial theorem for rotating charged perfect fluids in general relativity

We obtain an exact form of the virial theorem in general relativity, which is sufficiently general to be applied to charged, conducting, rotating perfect fluids in electromagnetic and gravitational fields. The case of infinite conductivity is of particular importance in astrophysics and we derive the relevant equations from the general results. We indicate how to calculate the post-Newtonian limits of various expressions and show that in the absence of both, the electric and magnetic fields, they lead to Chandrasekhar's post-Newtonian virial theorem in hydrodynamics. We also note that Chandrasekhar's (Newtonian) virial theorem in hydromagnetics may be derived from the Newtonian limit of the exact equations obtained. Some possible applications are pointed out. Finally, we use the exact form of the virial theorem to obtain, in co-moving coordinates, equilibrium conditions for bounded rotating charged dust.