Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hawking-type singularity theorems for worldvolume energy inequalities

Melanie Graf, Eleni-Alexandra Kontou, Argam Ohanyan, Yasmin Schinnerl

TL;DR

The paper develops Hawking-type singularity theorems using worldvolume energy inequalities instead of pointwise energy conditions, addressing quantum violations of classical bounds. It builds a geometric framework with forward/backward area and d'Alembert comparisons, a segment-type inequality, and translates worldvolume Ricci bounds into line-integral estimates that drive geodesic focusing. The authors apply the theory to the non-minimally coupled scalar field (Einstein–Klein–Gordon) to derive a strong energy inequality and demonstrate past timelike geodesic incompleteness in a cosmological setting, with Higgs-scale fields becoming admissible under the worldvolume approach. The work highlights advantages of worldvolume bounds, such as mass-independence, and outlines future extensions to QEIs and the null case, opening avenues for semiclassical singularity results beyond pointwise SEC limitations.

Abstract

The classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a - potentially very negative - global timelike Ricci curvature bound, a Hawking type singularity theorem is proven. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.

Hawking-type singularity theorems for worldvolume energy inequalities

TL;DR

The paper develops Hawking-type singularity theorems using worldvolume energy inequalities instead of pointwise energy conditions, addressing quantum violations of classical bounds. It builds a geometric framework with forward/backward area and d'Alembert comparisons, a segment-type inequality, and translates worldvolume Ricci bounds into line-integral estimates that drive geodesic focusing. The authors apply the theory to the non-minimally coupled scalar field (Einstein–Klein–Gordon) to derive a strong energy inequality and demonstrate past timelike geodesic incompleteness in a cosmological setting, with Higgs-scale fields becoming admissible under the worldvolume approach. The work highlights advantages of worldvolume bounds, such as mass-independence, and outlines future extensions to QEIs and the null case, opening avenues for semiclassical singularity results beyond pointwise SEC limitations.

Abstract

The classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a - potentially very negative - global timelike Ricci curvature bound, a Hawking type singularity theorem is proven. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.
Paper Structure (11 sections, 13 theorems, 123 equations, 1 table)

This paper contains 11 sections, 13 theorems, 123 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Geometric background
  4. Forward and backward comparison results
  5. The singularity theorems
  6. The strong energy inequality
  7. The bound on the effective energy density
  8. The curvature bound
  9. A singularity theorem for the Einstein-Klein-Gordon theory
  10. Application
  11. Discussion

Key Result

Lemma 2.3

The future cut function $c_{\Sigma}^+$ is lower semicontinuous and satisfies $c_{\Sigma}^+(\mathbf{x}) > 0$ for all $\mathbf{x} \in \Sigma$. Moreover, $\mathop{\mathrm{Cut}}\nolimits^+(\Sigma) \subseteq M$ is a set of measure zero with respect to the volume measure.

Theorems & Definitions (37)

  • Definition 2.1: Future normal exponential map
  • Definition 2.2: Future cut function and future cut points
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 27 more