Angular Momentum and Killing Potentials

TL;DR

This paper investigates the limitations of the Penrose-Goldberg superpotential in computing angular momentum, showing it only works asymptotically for Schwarzschild and Kerr solutions due to constraints on Killing potentials.

Contribution

It proves that Killing potentials satisfying Penrose's equation do not exist at finite radius for Schwarzschild and Kerr, restricting PG superpotential's use to asymptotic regions.

Findings

Killing potentials do not satisfy Penrose's equation at finite r.

PG superpotential is only valid asymptotically for angular momentum.

Results apply specifically to Schwarzschild and Kerr solutions.

Abstract

When the Penrose-Goldberg (PG) superpotential is used to compute the angular momentum of an axial symmetry, the Killing potential $Q_{(\varphi )}^{\mu\nu}$ for that symmetry is needed. Killing potentials used in the PG superpotential must satisfy Penrose's equation. It is proved for the Schwarzschild and Kerr solutions that the Penrose equation does not admit a $%Q_{(\varphi )}^{\mu\nu}$ at finite r and therefore the PG superpotential can only be used to compute angular momentum asymptotically.