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DGLA Actions: An Application in GR

Ryan Grady

TL;DR

This note develops a framework where differential graded Lie algebras (dglas) and their equivariant actions encode symmetry in field theories, extending the BV/BFV perspective to capture symmetries up to homotopy. It defines dgla actions, including how a dgla can act on another dgla or on a classical field theory, and presents an explicit equivariant action $S^ rak{g}$ that augments Palatini–Cartan gravity with Killing-symmetry data in a neighborhood of infinity. By enforcing a Killing algebra on a region outside a compact set, the approach reduces the spacetime geometry to highly constrained forms: the full Poincaré algebra forces a Minkowski metric on the exterior (and, under the dominant energy condition, global flatness via the Positive Mass Theorem), while spherical symmetry with $ olinebreak olinebreak olinebreak olinebreak olinebreak 0$ and $ olinebreak a0$ yields Schwarzschild in the exterior by Birkhoff’s theorem. The framework also supports gluing constructions to interpolate between different exterior geometries (e.g., Minkowski inside, Schwarzschild outside) and suggests pathways to more general asymptotic or AdS/dS geometries through topology changes, thereby linking dgla symmetry with classic GR rigidity results and spacetime assembly techniques.

Abstract

This note serves two purposes: 1) define actions by differential graded Lie algebras, and 2) apply such differential graded Lie symmetry in general relativity (GR) to constrain the spacetime geometry on a neighborhood of infinity.

DGLA Actions: An Application in GR

TL;DR

This note develops a framework where differential graded Lie algebras (dglas) and their equivariant actions encode symmetry in field theories, extending the BV/BFV perspective to capture symmetries up to homotopy. It defines dgla actions, including how a dgla can act on another dgla or on a classical field theory, and presents an explicit equivariant action that augments Palatini–Cartan gravity with Killing-symmetry data in a neighborhood of infinity. By enforcing a Killing algebra on a region outside a compact set, the approach reduces the spacetime geometry to highly constrained forms: the full Poincaré algebra forces a Minkowski metric on the exterior (and, under the dominant energy condition, global flatness via the Positive Mass Theorem), while spherical symmetry with and yields Schwarzschild in the exterior by Birkhoff’s theorem. The framework also supports gluing constructions to interpolate between different exterior geometries (e.g., Minkowski inside, Schwarzschild outside) and suggests pathways to more general asymptotic or AdS/dS geometries through topology changes, thereby linking dgla symmetry with classic GR rigidity results and spacetime assembly techniques.

Abstract

This note serves two purposes: 1) define actions by differential graded Lie algebras, and 2) apply such differential graded Lie symmetry in general relativity (GR) to constrain the spacetime geometry on a neighborhood of infinity.

Paper Structure

This paper contains 11 sections, 1 theorem, 14 equations.

Table of Contents

  1. Introduction
  2. DGLAs and Actions
  3. Actions on Classical Field Theories
  4. Application: Geometry on a Neighborhood of Infinity via an Equivariant Action
  5. Enforcing a Killing Algebra
  6. The Poincaré Algebra
  7. Spherical Symmetry
  8. Fundamental Concepts/Results
  9. Mass as a Conserved Quantity
  10. Positive Mass Theorem
  11. Birkhoff and Israel's Theorems

Key Result

Theorem A.1

Let $(M,g)$ be an asymptotically flat spacetime of dimension $n$ (which satisfies the dominant energy condition). If $3 \le n \le 7$ or $M$ is spin, then $E \ge \lvert P \rvert$ where $(E,P)$ is the ADM energy-momentum vector of $(M,g)$. Moreover, as the ADM mass is given by $m = \sqrt{E^2-\lvert P

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem A.1: Positive Mass Theorem