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Dynamics of Geometric Invariants in the Asymptotically Hyperboloidal Setting: Energy and Linear Momentum

Anna Sancassani, Saradha Senthil Velu

TL;DR

This work analyzes how Michel charges for energy and linear momentum evolve for asymptotically hyperboloidal initial data under Einstein dynamics, using hyperboloidal foliations that reach future null infinity without conformal compactifications. The authors derive explicit expressions for E and P^i as Michel charges and prove that E-P chargeability is preserved along the evolution with a carefully chosen time function, yielding Bondi-like energy-momentum loss relations in a spacelike setting. They extend the framework to generalized hyperboloidal backgrounds and generalized observers, showing the charges remain well defined and background-independent within a natural class. The results connect spacelike initial-data charges to radiation at infinity, offering a robust alternative to null-formalism approaches and laying groundwork for comparisons with Hamiltonian charges and extensions to angular momentum, center of mass, and matter fields.

Abstract

We investigate the evolution of geometric invariants, as defined by Michel \cite{Michel}, in the context of asymptotically hyperboloidal initial data sets. Our focus lies on the charges of energy and linear momentum, and we study their behavior under the Einstein evolution equations. We construct foliations describing the evolution of asymptotically hyperboloidal initial data sets using hyperboloidal time function. We define E-P chargeability as a property of the initial data set, and we show that it is preserved under the evolution for our choice of time function. This ensures that the charges are well-defined along the evolution, which is crucial for our approach. Along such foliations, we recover the same energy-loss and linear momentum-loss formulae as those derived by Bondi, Sachs, and Metzner \cite{Bondi-vanderBurg-Metzner} while operating under weaker asymptotic assumptions. Our approach is distinct from previous work as we do not utilize conformal compactifications and work directly at the level of the initial data set.

Dynamics of Geometric Invariants in the Asymptotically Hyperboloidal Setting: Energy and Linear Momentum

TL;DR

This work analyzes how Michel charges for energy and linear momentum evolve for asymptotically hyperboloidal initial data under Einstein dynamics, using hyperboloidal foliations that reach future null infinity without conformal compactifications. The authors derive explicit expressions for E and P^i as Michel charges and prove that E-P chargeability is preserved along the evolution with a carefully chosen time function, yielding Bondi-like energy-momentum loss relations in a spacelike setting. They extend the framework to generalized hyperboloidal backgrounds and generalized observers, showing the charges remain well defined and background-independent within a natural class. The results connect spacelike initial-data charges to radiation at infinity, offering a robust alternative to null-formalism approaches and laying groundwork for comparisons with Hamiltonian charges and extensions to angular momentum, center of mass, and matter fields.

Abstract

We investigate the evolution of geometric invariants, as defined by Michel \cite{Michel}, in the context of asymptotically hyperboloidal initial data sets. Our focus lies on the charges of energy and linear momentum, and we study their behavior under the Einstein evolution equations. We construct foliations describing the evolution of asymptotically hyperboloidal initial data sets using hyperboloidal time function. We define E-P chargeability as a property of the initial data set, and we show that it is preserved under the evolution for our choice of time function. This ensures that the charges are well-defined along the evolution, which is crucial for our approach. Along such foliations, we recover the same energy-loss and linear momentum-loss formulae as those derived by Bondi, Sachs, and Metzner \cite{Bondi-vanderBurg-Metzner} while operating under weaker asymptotic assumptions. Our approach is distinct from previous work as we do not utilize conformal compactifications and work directly at the level of the initial data set.

Paper Structure

This paper contains 11 sections, 6 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotically Hyperboloidal Initial Data Sets
  4. Energy and linear momentum on asymptotically hyperboloidal initial data
  5. Choice of Evolution
  6. Evolution of Michel Charges on Asymptotically Hyperboloidal Foliations
  7. Generalized observers
  8. Discussion
  9. Christoffel Symbols of the Standard Hyperboloid $(\mathbb{H}^{3}, b)$
  10. Christoffel Symbols of Asymptotically Hyperboloidal Initial Data Sets
  11. Evolution of hyperboloidal initial data

Key Result

Theorem 1

Let $(M,g,K,\rho,J)$ be an asymptotically hyperboloidal initial data set in the sense of Definition Def::AHyperboloidal. The energy $E$ and linear momentum $P^{i}$, for $i=1,2,3$ of the initial data are well-defined and are given by the following formulae: where $x^{i}$ denote the first spherical harmonics on the unit sphere $\mathbb{S}^2$.

Figures (1)

  • Figure 1: Foliation of Minkowski spacetime obtained by evolving the standard hyperboloid $(\mathbb{H}^3,b,b)$ with the standard Killing time $\partial_\tau=\partial_t$. The red lines represent radiation, hitting only some of the future hyperboloids, as can be seen in the first picture. The second picture shows the foliation of $\mathscr{I}^+$ produced by the hyperboloidal foliation. There are many functions $h(r)$ that satisfy \ref{['choice_H']}, and we refer the interested reader to AnilZenginolu2025 for some explicit examples of height functions. See AnilDiagrams for a guide to drawing Penrose diagrams.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Example 1
  • Remark 3
  • Definition 3: Michel Charges
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 15 more