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A Positive Quasilocal Mass for Causal Variational Principles

Felix Finster, Niky Kamran

TL;DR

The paper develops a coherent positivity program for causal variational principles by deriving a new positive nonlinear surface layer inequality under a volume constraint and then removing the constraint via asymptotic alignment. It introduces a positive mass theorem without volume constraint, a notion of equivariant mass, and a nonnegative quasilocal mass that bounds the total mass, along with a synthetic notion of scalar curvature. The framework is illustrated in ultrastatic and Schwarzschild spacetimes, with connections to the ADM and Brown–York masses and a rigorous link to the underlying geometry through a linearized regime. The results significantly weaken regularity assumptions and provide robust mass concepts that apply to non-smooth and non-manifold settings, while aligning with familiar gravitational notions in classical limits.

Abstract

A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alignment. Moreover, a positive quasilocal mass and a synthetic definition of scalar curvature are introduced in the setting of causal variational principles. Our notions and results are illustrated by the explicit examples of causal fermion systems constructed in ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the correspondence to the ADM mass and similarities to the Brown-York mass are worked out.

A Positive Quasilocal Mass for Causal Variational Principles

TL;DR

The paper develops a coherent positivity program for causal variational principles by deriving a new positive nonlinear surface layer inequality under a volume constraint and then removing the constraint via asymptotic alignment. It introduces a positive mass theorem without volume constraint, a notion of equivariant mass, and a nonnegative quasilocal mass that bounds the total mass, along with a synthetic notion of scalar curvature. The framework is illustrated in ultrastatic and Schwarzschild spacetimes, with connections to the ADM and Brown–York masses and a rigorous link to the underlying geometry through a linearized regime. The results significantly weaken regularity assumptions and provide robust mass concepts that apply to non-smooth and non-manifold settings, while aligning with familiar gravitational notions in classical limits.

Abstract

A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alignment. Moreover, a positive quasilocal mass and a synthetic definition of scalar curvature are introduced in the setting of causal variational principles. Our notions and results are illustrated by the explicit examples of causal fermion systems constructed in ultrastatic spacetimes and the Schwarzschild spacetime. In these examples, the correspondence to the ADM mass and similarities to the Brown-York mass are worked out.
Paper Structure (34 sections, 33 theorems, 291 equations, 3 figures)

This paper contains 34 sections, 33 theorems, 291 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Causal Variational Principles in the Non-Compact Setting
  4. The Restricted Euler-Lagrange Equations and Jets
  5. The Linearized Field Equations and Inner Solutions
  6. The Total Mass as a Surface Layer Integral
  7. A Positive Nonlinear Surface Layer Integral
  8. A Positivity Argument under a Volume Constraint
  9. Generalizations of the Positive Nonlinear Surface Layer Integral
  10. A New Proof of the Positive Mass Theorem with Volume Constraint
  11. A Positive Mass Theorem without Volume Constraint
  12. Rewriting Surface Layer Integrals as Surface Integrals
  13. Asymptotic Alignment
  14. Local Alignments and the Local Volume Condition
  15. Statement and Proof of the Positive Mass Theorem
  16. ...and 19 more sections

Key Result

Theorem 3.1

(Positivity argument under volume constraint Let $\mu$ be a minimizer with respect to variations of finite volume and $\tilde{\mu}$ a measure on ${\mathscr{G}}$. Moreover, let $\Omega \subset N:= \mathop{\mathrm{supp}}\nolimits \mu$ and $\tilde{\Omega} \subset \tilde{N} := \mathop{\mathrm{supp}}\nol

Figures (3)

  • Figure 1: The measure $\hat{\mu}$.
  • Figure 2: The measure $\hat{\mu}$.
  • Figure 3: Isometric transformation of $\tilde{\Omega}$.

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • Definition 3.2
  • Theorem 3.3
  • proof : Proof of Theorem \ref{['thm2']}.
  • Remark 3.4
  • ...and 67 more