The Dirac and Rarita-Schwinger equations on scalar flat metrics of Taub-NUT type

TL;DR

The paper constructs a scalar-flat Taub-NUT type metric with $f(r)=\sqrt{ \frac{r^2-N^2}{h} }$ and $h=r^2+C_1 r+C_2$, showing the total mass can be negative via $E=-4NC_1$ and identifying geometric completeness and singularity criteria. It establishes that the spaces of twistor and parallel spinors are complex $2$-dimensional for the relevant $f(r)$, enabling the construction of $L^2$ harmonic spinors and $L^2$ Rarita-Schwinger fields using parallel spinors and harmonic Maxwell data. By separating the Dirac and Rarita-Schwinger equations into angular and radial components, the authors obtain explicit angular solutions and closed-form or special-function radial solutions (notably in terms of Kummer functions ${}_1F_1$) under various parameter regimes. The results provide analytic spinor-field solutions on noncompact, scalar-flat gravitational instanton backgrounds and illuminate how negative mass configurations interact with Dirac/RS dynamics, with implications for spinorial approaches to energy theorems in such geometries.

Abstract

We construct a scalar flat metric of Taub-NUT type whose total mass can be negative. The standard Taub-NUT metric and its negative NUT charge counterpart serve as particular examples, for which the complex 2-dimensional space of parallel spinors gives rise to $L^2$ harmonic spinors and Rarita-Schwinger fields. For the scalar flat Taub-NUT type metric, we study the Dirac and Rarita-Schwinger equations by separating them into angular and radial equations, and obtain explicit solutions in certain special cases.

Theorems & Definitions (14)

• Proposition 2.1
• Proposition 3.1
• Theorem 4.1
• Remark 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 5.1
• Theorem 5.2
• Theorem 5.3
• Theorem 6.1