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The Spacetime Positive Mass Theorem with Multiple Time Dimensions

Sven Hirsch, Alec Payne, Yiyue Zhang

Abstract

We generalize the spacetime positive mass theorem to include multiple time dimensions. In particular, we show that the mass remains nonnegative in the sense that the energy $E$ is bounded from below by the trace norm of the linear momenta $J^1,...,J^m$. Equality in this energy inequality implies a foliation by flat submanifolds of a generalized initial data set. Moreover, under an additional umbilicity assumption, we find that the initial data set isometrically embeds into a generalized pp-wave.

The Spacetime Positive Mass Theorem with Multiple Time Dimensions

Abstract

We generalize the spacetime positive mass theorem to include multiple time dimensions. In particular, we show that the mass remains nonnegative in the sense that the energy is bounded from below by the trace norm of the linear momenta . Equality in this energy inequality implies a foliation by flat submanifolds of a generalized initial data set. Moreover, under an additional umbilicity assumption, we find that the initial data set isometrically embeds into a generalized pp-wave.
Paper Structure (11 sections, 22 theorems, 118 equations)

This paper contains 11 sections, 22 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotically flat initial data sets
  4. Spin geometry
  5. Trace Norm and Dominant Energy Condition
  6. The positive mass inequality
  7. Case of equality
  8. Multiple Killing developments
  9. Spinorial symmetries
  10. Connection formula for spinors
  11. Verification of parallel spinors

Key Result

Theorem 1.1

Witten1981DingSpacetimePMT Let $(M^n,g,k)$ be an asymptotically flat spin initial data set. Assume that the dominant energy condition holds, where $\mu$ denotes the energy density and $J$ the momentum density. Then the ADM energy $E$ and linear momentum $P$ satisfy

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 38 more