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Linking Past and Future Null Infinity in Three Dimensions

Stefan Prohazka, Jakob Salzer, Friedrich Schöller

TL;DR

This work analyzes three-dimensional Einstein gravity in asymptotically flat spacetimes to relate the independent BMS symmetries at future and past null infinity. By examining the asymptotic phase space and imposing a symmetry-based linking, the authors derive a unique antipodal matching between I+ and I− that preserves energy and enforces relations between mass and angular momentum aspects: Θ^+(φ) = Θ^−(φ+π) and Ξ^+(φ) = Ξ^−(φ+π). This matching implies an infinite set of conserved charges Q^+_{T,Y} = Q^-_{Ṫ,Ŷ} for corresponding boundary functions related by a π-shift, effectively coupling the two null boundaries under a single global BMS symmetry. The construction extends to scenarios with matter and has potential implications for flat-space holography, where Θ and Ξ may be interpreted as boundary stress-tensor components and the entanglement structure across I± becomes accessible through the linked boundary theories.

Abstract

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

Linking Past and Future Null Infinity in Three Dimensions

TL;DR

This work analyzes three-dimensional Einstein gravity in asymptotically flat spacetimes to relate the independent BMS symmetries at future and past null infinity. By examining the asymptotic phase space and imposing a symmetry-based linking, the authors derive a unique antipodal matching between I+ and I− that preserves energy and enforces relations between mass and angular momentum aspects: Θ^+(φ) = Θ^−(φ+π) and Ξ^+(φ) = Ξ^−(φ+π). This matching implies an infinite set of conserved charges Q^+_{T,Y} = Q^-_{Ṫ,Ŷ} for corresponding boundary functions related by a π-shift, effectively coupling the two null boundaries under a single global BMS symmetry. The construction extends to scenarios with matter and has potential implications for flat-space holography, where Θ and Ξ may be interpreted as boundary stress-tensor components and the entanglement structure across I± becomes accessible through the linked boundary theories.

Abstract

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

Paper Structure

This paper contains 7 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Asymptotically Flat Spacetimes
  3. Phase Space and Validity of the Mapping
  4. Linking Past and Future Null Infinity
  5. Adding Matter
  6. Discussion
  7. Coordinate Transformations

Figures (3)

  • Figure 1: The phase space of the spacetimes given in equation \ref{['eq:constrep']}. The cross at $M=-1, J=0$ is Minkowski space. The snake line indicates that the linking between past and future null infinity appears nonsensical at $M \ge 0, J = 0$. The energy of a spacetime with angular excess is not bounded from below when acted upon by BMS transformations.
  • Figure 2: Penrose diagrams for spacetimes with $M < 0$ (except $M=-1, J=0$ where there is no singularity) as well as spacetimes with $M = 0, J \neq 0$ (left) and flat space cosmologies (right).
  • Figure 3: Penrose diagram of a constant $Y$ slice of Minkowski space. The snake lines indicate where causal singularities develop when taking the quotient to obtain flat space cosmologies. The gray and the dotted regions mark different coordinate patches.