Fake Supersymmetry and Extremal Black Holes

TL;DR

This paper develops a fake supersymmetry approach to derive first order equations for rotating black holes in 3D gravity with scalar fields, identifying necessary constraints and confirming BPS bounds through boundary stress tensor calculations.

Contribution

It introduces a modified fake supersymmetry formalism for rotating black holes, including an additional constraint to recover full equations of motion, and computes conserved charges confirming BPS saturation.

Findings

Derived first order equations for rotating black holes in 3D gravity.

Identified an additional constraint needed for full equations of motion.

Confirmed black hole solutions saturate the BPS bound via boundary stress tensor.

Abstract

We derive the BPS type of first order differential equations for the rotating black hole solutions in the three-dimensional Einstein gravity coupled minimally with a self-interacting scalar field, using fake supersymmetry formalism. It turns out that the formalism is not complete and should be augmented by an additional equation to imply the full equations of motion. We identify this additional equation as a constraint by using an effective action method. By computing the renormalized boundary stress tensor, we obtain the mass and angular momentum of the black hole solutions of these first order equations and confirm that they saturate the BPS bound.