Metrics on End-Periodic Manifolds as Models for Dark Matter

Christopher L Duston

doi:10.3390/universe8030167

Abstract

In this paper we will detail an approach to generate metrics and matter models on end-periodic manifolds, which are used extensively in the study of the exotic smooth structures of R4. After an overview of the technique, we will present two specific examples, discuss the associated matter models by solving the Einstein equations, and determine the physical viability by examining the energy conditions. We compare the resulting model directly with existing models of matter distributions in extragalactic systems, to highlight the viability of utilizing exotic smooth structures to understand the existence and distribution of dark matter.

Highlights

There is no reason that exotic smooth structure could not play a role in the study of the early universe ([22,23], for example), but the signal would presumably be cleaner at the galactic scale because the only physics involved there is that of the gravitational field, and only at relatively low energies
This paper is organized as follows: Section 2 will briefly present the theoretical and mathematical background needed to understand the metric generation procedure on end periodic manifolds, and Section 3 will give some context for how this approach fits into the larger question, “what is the dynamical source of exotic smoothness structures?” We will present two specific examples of this approach, using a Kruskal black hole (Section 4) and an embedded conformal surface (Section 5) as building blocks
We will not attempt to solve the field equations for this metric here, but draw some inspiration from (2): our formulation of the Z-transformation produces a conformal transformation on the R2 part of the spacetime. These types of metrics may have interesting properties in their own right, but we will instead use this as inspiration for another choice of our building block metric

Summary

Introduction

End-Periodic Manifolds as Models for Dark Matter. Universe 2022, 8, 167. There is no reason that exotic smooth structure could not play a role in the study of the early universe ([22,23], for example), but the signal would presumably be cleaner at the galactic scale because the only physics involved there is that of the gravitational field, and only at relatively low energies. We will point out the various and interesting similarities and differences in this approach compared with the one we are taking Beyond this introduction, this paper is organized as follows: Section 2 will briefly present the theoretical and mathematical background needed to understand the metric generation procedure on end periodic manifolds, and Section 3 will give some context for how this approach fits into the larger question, “what is the dynamical source of exotic smoothness structures?” We will present two specific examples of this approach, using a Kruskal black hole (Section 4) and an embedded conformal surface (Section 5) as building blocks.

Metrics on End-Periodic Manifolds

Perspective Taken in This Paper

The Exotic Kruskal Metric

A Building Block with a Locally Embedded Conformal Surface

A Self-Consistent Matter Model for M

Summary