Rotating charged fluids: Theorems and results for Weyl-type systems

Abstract

We perform a systematic study of rotating charged fluids, and extend several well known theorems regarding static Weyl-type systems which were recently compiled by Lemos and Zanchin [Phys. Rev. D 80, 024010 (2009)] to rotating and axisymmetric systems. Static Weyl-type systems are composed by static charged fluid configurations obeying the Newton-Maxwell or the Einstein-Maxwell systems of equations in which the electric potential $ϕ$ and the timelike metric potential $g_{tt}\equiv - W^ 2$ satisfy the Weyl hypothesis, i.e., $W=W(ϕ)$. In the present analysis, both the Newton-Maxwell and Einstein-Maxwell theories that describe non-relativistic and relativistic systems, respectively, are used to perform a detailed analysis of the general properties of rotating charged fluids rotating charged dust as well as rotating charged fluids with pressure in four-dimensional spacetimes. In comparison to the static (nonrotating) systems, two additional potentials, a metric potential related to rotation and an electromagnetic potential related to the magnetic field, come into play for rotating systems. In each case, constraints between the fluid quantities and the metric and electromagnetic potentials are identified in order to generalize the theorems holding for static charged systems to rotating charged systems. New theorems regarding equilibrium configurations with differential rotation in both the Newtonian and the relativistic theories are stated and proved. For rigidly rotating charged fluids in the Einstein-Maxwell theory, a new ansatz involving the gradient of the metric potentials and the gradient of the electromagnetic potentials is considered in order to prove new theorems. Such an ansatz leads to new constraints between the fluid quantities and field potentials, so implying new equations of state for the charged fluids.