# Axially-Symmetric Exact Solutions for Flagpole Fermions with Gravity

## TL;DR

This paper derives exact solutions for flagpole spinor fields, specifically Majorana spinors, interacting with gravity under axial symmetry, expanding the understanding of spinor-gravity systems beyond Dirac and Weyl fields.

## Contribution

It provides the first known exact solutions for flagpole (Majorana) spinor fields coupled with gravity in an axially symmetric setting.

## Key findings

- Exact solutions for flagpole spinor-gravity systems are obtained.
- The solutions extend previous work on Dirac and Weyl fields.
- The results enhance understanding of spinor fields in gravitational contexts.

## Abstract

The Lounesto classification splits spinors in six classes: I, II, III are those for which at least one among scalar and pseudo-scalar bi-linear spinor quantities is non-zero, its spinors are called regular, and among them we find the usual Dirac spinor. IV, V, VI are those for which the scalar and pseudo-scalar bi-linear spinor quantities are identically zero, its spinors are called singular, and they are split in further sub-classes: IV has no further restrictions, its spinors are called flag-dipole; V is the one for which the spin axial-vector vanishes, its spinors are called flagpole, and among them we find the Majorana spinor; VI is the one for which the momentum antisymmetric-tensor vanishes, its spinors are called dipole, and among them we find the Weyl spinor. In the quest for exact solutions of fully-coupled systems of spinor fields in their own gravity, we have already given examples in the case of Dirac fields and Weyl fields but never in the case of Majorana or more generally flagpole spinor fields. Flagpole spinor fields in interaction with their own gravitational field, in the case of axial symmetry, will be considered. Exact solutions of the field equations will be given.