Extremal black holes, attractor equations, and harmonic functions

TL;DR

This paper reviews a method for constructing multi-centered black hole solutions using harmonic functions and directly deriving attractor equations from second order field equations, bypassing traditional reliance on Killing spinors.

Contribution

It introduces a novel approach to black hole solution construction that avoids the use of Killing spinor equations, focusing instead on solving second order field equations with harmonic functions.

Findings

Black hole solutions can be constructed via harmonic functions without Killing spinors.

Attractor equations are derived directly from second order field equations.

The method simplifies the analysis of multi-centered black holes.

Abstract

We review the construction of multi-centered black hole solutions through dimensional reduction over time. This method does not rely on Killing spinor equations or gradient flow equations, but on solving the second order field equations in terms of harmonic functions. The black hole attractor equations are obtained directly from the field equations.