Rotating magnetic solution in three dimensional Einstein gravity

TL;DR

This paper extends the three-dimensional Einstein gravity magnetic solutions to include rotation, conserved quantities, and new interpretations, revealing bounds on physical parameters and connecting magnetic sources to electric charge systems.

Contribution

It introduces rotating magnetic solutions with conserved quantities and bounds, and offers a novel interpretation of the magnetic source in three-dimensional Einstein gravity.

Findings

Both static and rotating solutions have negative mass.

Upper bounds exist for magnetic field strength and angular momentum.

Rotating magnetic solutions reduce to BTZ when magnetic source vanishes.

Abstract

We obtain the magnetic counterpart of the BTZ solution, i.e., the rotating spacetime of a point source generating a magnetic field in three dimensional Einstein gravity with a negative cosmological constant. The static (non-rotating) magnetic solution was found by Clement, by Hirschmann and Welch and by Cataldo and Salgado. This paper is an extension of their work in order to include (i) angular momentum, (ii) the definition of conserved quantities (this is possible since spacetime is asymptotically anti-de Sitter), (iii) upper bounds for the conserved quantities themselves, and (iv) a new interpretation for the magnetic field source. We show that both the static and rotating magnetic solutions have negative mass and that there is an upper bound for the intensity of the magnetic field source and for the value of the angular momentum. The magnetic field source can be interpreted not as a vortex but as being composed by a system of two symmetric and superposed electric charges, one of the electric charges is at rest and the other is spinning. The rotating magnetic solution reduces to the rotating uncharged BTZ solution when the magnetic field source vanishes.