Relativistic figures of equilibrium in the Wald magnetosphere

TL;DR

This paper extends Wald's magnetosphere solution to include a self-gravitating, rotating charged fluid with specific equations of state, deriving integrable equations and providing numerical solutions.

Contribution

It demonstrates compatibility of Wald's solution with a rotating charged fluid and derives modified Einstein-Euler equations for specific cases.

Findings

Equations can be integrated for fluids with constant density or polytropic EOS.

Numerical solutions are obtained using modified pseudospectral code.

The system resembles standard Einstein-Euler equations with specific modifications.

Abstract

We consider a self-gravitating, rigidly rotating charged perfect fluid immersed in the Wald magnetosphere, constructed out of two linearly independent Killing vectors present in stationary and axially-symmetric spacetimes. We show that in non-vacuum spacetimes, Wald's solution can be compatible with the electric current associated with a rotating charged perfect fluid characterized by the vanishing electric conductivity. We prove that for rigidly rotating fluids with a constant energy density or described by the polytropic equation of state, the resulting equations expressing the conservation of the energy-momentum tensor can be integrated. Consequently, the system can be described by nearly standard Einstein-Euler equations known from the theory of general-relativistic rotating fluids, with modifications introduced in the Euler-Bernoulli equation. Numerical solutions of the Einstein-Euler equations are provided for these two cases by introducing suitable modifications in the pseudospectral code by Ansorg, Kleinw\"{a}chter, and Meinel.