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Asymptotics of scalar waves on long-range asymptotically Minkowski spaces

Dean Baskin, Andras Vasy, Jared Wunsch

TL;DR

The paper develops a comprehensive framework for the full compound asymptotics of scalar waves on long-range, non-trapping Lorentzian scattering manifolds, extending short-range results to include Bondi-mass-type mass effects. By combining b-geometry, scattering calculus, Mellin transform techniques, and a logification step, the authors prove joint polyhomogeneous expansions at both null infinity $\mathscr{I}^+$ and timelike infinity, with resonance-driven coefficients governed by a family of reduced normal operators $P_\sigma$. The long-range case introduces logarithmic terms in the radiation-field expansion, reflecting the mass term, and in Kerr-type spacetimes a nonzero leading log term is established. The work provides precise asymptotics for the Friedlander radiation field and clarifies how log terms arise from the mass corrections, with implications for Price-law-like decay and the behavior of waves in physically relevant spacetimes.

Abstract

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation. In two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the Kerr family. In these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero.

Asymptotics of scalar waves on long-range asymptotically Minkowski spaces

TL;DR

The paper develops a comprehensive framework for the full compound asymptotics of scalar waves on long-range, non-trapping Lorentzian scattering manifolds, extending short-range results to include Bondi-mass-type mass effects. By combining b-geometry, scattering calculus, Mellin transform techniques, and a logification step, the authors prove joint polyhomogeneous expansions at both null infinity and timelike infinity, with resonance-driven coefficients governed by a family of reduced normal operators . The long-range case introduces logarithmic terms in the radiation-field expansion, reflecting the mass term, and in Kerr-type spacetimes a nonzero leading log term is established. The work provides precise asymptotics for the Friedlander radiation field and clarifies how log terms arise from the mass corrections, with implications for Price-law-like decay and the behavior of waves in physically relevant spacetimes.

Abstract

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation. In two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the Kerr family. In these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero.

Paper Structure

This paper contains 18 sections, 30 theorems, 261 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Sketch of proof
  4. Basics of b- and scattering-geometry
  5. b-geometry
  6. Scattering geometry
  7. Long-range scattering geometry
  8. The Hamilton vector field and its radial set
  9. Propagation of b-regularity and module regularity
  10. Fredholm properties
  11. Logification
  12. The radiation field blow-up
  13. Asymptotic expansions
  14. Asymptotics at $C_+$
  15. Expansion at $\mathscr{I}^+$: the short-range case
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a non-trapping Lorentzian scattering manifold, and let with $u \in \mathcal{C}^{-\infty}(M),$$f \in \dot{\mathcal{C}}^\infty(M).$ Assume that $u$ is a forward solution. Then $u$ has a joint polyhomogeneous asymptotic expansion in $s\to\infty,\ r \to \infty$ (where $r$ and $s$ are as in equation eq:s-coord If $m=0$ then only $\alpha=0$ terms appear.

Figures (1)

  • Figure 1: A schematic view of the blow-up. The lapse function $s$ increases along $\mathscr{I}^+$ towards $C_+$. In the typical Penrose diagram of Minkowski space, $C_\pm$ are collapsed to $i_\pm$ and $C_0$ is collapsed to $i_0$.

Theorems & Definitions (61)

  • Theorem 1.1
  • Proposition 2.1: Mazzeo, Melrose
  • Proposition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 5.1
  • Lemma 5.2
  • proof
  • Proposition 5.3
  • ...and 51 more