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Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces

Edgar Gasperín, Mariem Magdy, Filipe C. Mena

TL;DR

The paper analyzes spin-0 field propagation on $n$-dimensional Minkowski space near spatial and null infinity using Friedrich's cylinder to obtain formal asymptotic expansions. It derives the conformal wave equation in the cylinder framework, decomposes solutions into Legendre-type modes, and establishes regularity conditions that determine when asymptotic charges are well-defined. The main results show an infinite family of well-defined charges in even dimensions, while odd higher dimensions admit at most a single trivial or no non-trivial charges, depending on the angular mode. Overall, the approach clarifies how initial data control the asymptotic structure and conserved quantities at the critical sets where null and spatial infinity meet, within a formal, polyhomogeneous setting.

Abstract

We use conformal geometry methods and the construction of Friedrich's cylinder at spatial infinity to study the propagation of spin-$0$ fields (solutions to the wave equation) on $n$-dimensional Minkowski spacetimes in a neighbourhood of spatial and null infinity. We obtain formal solutions written in terms of series expansions close to spatial and null infinity and use them to compute non-trivial asymptotic spin-$0$ charges. It is shown that if one considers the most general initial data within the class considered in this paper, the expansion is poly-homogeneous and hence of restricted regularity at null infinity. Furthermore, we derive the conditions on the initial data needed to obtain regular solutions and well-defined limits for the asymptotic charges at the critical sets where null infinity and spatial infinity meet. In even dimensions, we find that there are infinitely many well-defined asymptotic charges at the critical sets, while for odd higher dimensions there exists no non-trivial asymptotic charges that remain regular at the critical sets.

Asymptotics of spin-0 fields and conserved charges on n-dimensional Minkowski spaces

TL;DR

The paper analyzes spin-0 field propagation on -dimensional Minkowski space near spatial and null infinity using Friedrich's cylinder to obtain formal asymptotic expansions. It derives the conformal wave equation in the cylinder framework, decomposes solutions into Legendre-type modes, and establishes regularity conditions that determine when asymptotic charges are well-defined. The main results show an infinite family of well-defined charges in even dimensions, while odd higher dimensions admit at most a single trivial or no non-trivial charges, depending on the angular mode. Overall, the approach clarifies how initial data control the asymptotic structure and conserved quantities at the critical sets where null and spatial infinity meet, within a formal, polyhomogeneous setting.

Abstract

We use conformal geometry methods and the construction of Friedrich's cylinder at spatial infinity to study the propagation of spin- fields (solutions to the wave equation) on -dimensional Minkowski spacetimes in a neighbourhood of spatial and null infinity. We obtain formal solutions written in terms of series expansions close to spatial and null infinity and use them to compute non-trivial asymptotic spin- charges. It is shown that if one considers the most general initial data within the class considered in this paper, the expansion is poly-homogeneous and hence of restricted regularity at null infinity. Furthermore, we derive the conditions on the initial data needed to obtain regular solutions and well-defined limits for the asymptotic charges at the critical sets where null infinity and spatial infinity meet. In even dimensions, we find that there are infinitely many well-defined asymptotic charges at the critical sets, while for odd higher dimensions there exists no non-trivial asymptotic charges that remain regular at the critical sets.
Paper Structure (10 sections, 4 theorems, 61 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conformal compactification and the Friedrich cylinder
  3. Preliminaries
  4. Friedrich's cylinder on the $n$-dimensional Minkowski space
  5. Asymptotic solutions to the wave equation in $n$-dimensions
  6. Solutions for $\nu \in \mathbb{Z}$
  7. The subcase $\nu\ge p$
  8. Solutions for $\nu \notin \mathbb{Z}$
  9. Asymptotic charges for the spin-$0$ field
  10. Solutions to the Legendre's differential equation

Key Result

Lemma 1

Given initial data as in Eq. initial-data at $\tau=0$, the solution to Eq. ODE-general-a-coefficient for any $p \geq 0$ is given by where $P^{p}_{\nu}(\tau)$ is the associated Legendre polynomial of order $\nu$, $Q^{p}_{\nu}(\tau)$ is the associated Legendre function of the second kind of order $\nu$ and The constants $C_{p;\ell,{\bm m}}$ and $D_{p;\ell,{\bm m}}$ are given by

Figures (1)

  • Figure 1: A schematic representation of the conformal transformation map $\Phi: \hat{{\bm \eta}} \to {\bm \eta}$. In particular, the diagram illustrates the blow-up of $i^{0} \cong \mathbb{S}^{n-2}$ to Friedrich's cylinder $\mathcal{I} \cong \mathbb{R} \times \mathbb{S}^{n-2}$ that connects $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • ...and 1 more