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An Overview of Bartnik's Quasi-Local Mass

Stephen McCormick

Abstract

This article provides a concise introduction to Bartnik's quasi-local mass, and surveys a selection of results pertaining to the understanding of it. The aim is to serve as both an entry point to the topic, and a quick reference of results for those already familiar with it.

An Overview of Bartnik's Quasi-Local Mass

Abstract

This article provides a concise introduction to Bartnik's quasi-local mass, and surveys a selection of results pertaining to the understanding of it. The aim is to serve as both an entry point to the topic, and a quick reference of results for those already familiar with it.
Paper Structure (23 sections, 6 theorems, 23 equations)

This paper contains 23 sections, 6 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Asymptotically flat manifolds and the ADM mass
  3. Defining the Bartnik mass
  4. A first definition
  5. Definition from Bartnik boundary data
  6. Non-degeneracy conditions
  7. The spacetime Bartnik mass
  8. The equivalence of definitions
  9. Properties of a good quasi-local mass
  10. Stationary and static extensions
  11. Static metrics as mass minimisers
  12. Stationary metrics as mass minimisers
  13. Existence and uniqueness of static and stationary extensions
  14. Estimates and constructions of extensions
  15. Related quasi-local mass quantities
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $(M,\gamma)$ be a complete asymptotically flat manifold (without boundary) satisfying $R(\gamma)\geq0$ and $R(\gamma)\in L^1(M)$ then $\mathfrak{m}_{ADM}(M,\gamma)\geq0$ with equality if and only if $(M,\gamma)$ is isometric to Euclidean space.

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • ...and 9 more