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Construction of Global Solutions to the Linearized Field Equations for Causal Variational Principles

Felix Finster, Margarita Kraus

TL;DR

The work addresses the problem of obtaining global solutions to the linearized field equations for causal variational principles without relying on shielding conditions or Exhaustion by lens-shaped regions. It introduces a gluing scheme that stitches local lens-shaped-region weak solutions into a global solution by applying future-boundary cutoffs and managing induced inhomogeneities that propagate toward future infinity, complemented by an inductive construction. The authors develop causal Green's operators $S^\wedge$, $S^\vee$ and the causal fundamental solution $G$, and establish an exact sequence that links test jets, the linearized operator $\Delta$, and spatially compact solutions, thereby providing a robust global framework. They also propose a preliminary causal-cone structure by defining forward–in–time influence via global weak solutions, aiming to relate these notions to classical closed-cone causality in smooth settings. Overall, the approach yields a simpler, more flexible path to global solutions and clarifies the causal structure of spacetime in the context of causal variational principles.

Abstract

We give a novel construction of global solutions to the linearized field equations for causal variational principles. The method is to glue together local solutions supported in lens-shaped regions. As applications, causal Green's operators and cone structures are introduced.

Construction of Global Solutions to the Linearized Field Equations for Causal Variational Principles

TL;DR

The work addresses the problem of obtaining global solutions to the linearized field equations for causal variational principles without relying on shielding conditions or Exhaustion by lens-shaped regions. It introduces a gluing scheme that stitches local lens-shaped-region weak solutions into a global solution by applying future-boundary cutoffs and managing induced inhomogeneities that propagate toward future infinity, complemented by an inductive construction. The authors develop causal Green's operators , and the causal fundamental solution , and establish an exact sequence that links test jets, the linearized operator , and spatially compact solutions, thereby providing a robust global framework. They also propose a preliminary causal-cone structure by defining forward–in–time influence via global weak solutions, aiming to relate these notions to classical closed-cone causality in smooth settings. Overall, the approach yields a simpler, more flexible path to global solutions and clarifies the causal structure of spacetime in the context of causal variational principles.

Abstract

We give a novel construction of global solutions to the linearized field equations for causal variational principles. The method is to glue together local solutions supported in lens-shaped regions. As applications, causal Green's operators and cone structures are introduced.
Paper Structure (12 sections, 4 theorems, 78 equations, 3 figures)

This paper contains 12 sections, 4 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Causal Variational Principles in the Non-Compact Setting
  4. The Euler-Lagrange Equations and Jet Spaces
  5. The Linearized Field Equations
  6. Surface Layer Integrals
  7. Weak Solutions in Lens-Shaped Regions
  8. Construction of Global Solutions
  9. Local Solutions with Cutoff in the Future
  10. Gluing Local Solutions
  11. Construction of Causal Green's Operators
  12. Causal Cone Structures

Key Result

Theorem 2.8

(existence) Assume that $L$ is a lens-shaped region inside $U$ with foliation $(\eta_t)_{t \in I}$ with $I=[{t_{\min}},{t_{\max}}]$. Then for every $\mathfrak{w} \in L^2(L)$ there is a weak solution $\mathfrak{v} \in L^2(L)$ of the Cauchy problem weak. This solution is bounded by

Figures (3)

  • Figure 1: A local foliation.
  • Figure 2: The sets $W$ and $Z$ of a lens-shaped region $L$.
  • Figure 3: Future-related sets.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 3.1
  • Theorem 3.2
  • ...and 8 more