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Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity

Dejan Gajic, Maxime Van de Moortel

TL;DR

The paper resolves sharp late-time tail behavior for scale-invariant wave equations with inverse-square potentials in Minkowski space by tying global decay to local dynamics on the AdS$_2\times\mathbb{S}^2$ geometry. The authors develop a novel geometric approach that uses a conformal embedding, twisted derivatives, and $\alpha$-twisted Sobolev spaces to convert the problem into a local well-posedness and elliptic-estimate framework on AdS$_2\times\mathbb{S}^2$, then translate back to obtain precise leading-order tails with exponents $p_\ell$. They prove existence/uniqueness of weak solutions, establish pointwise and $L^2$ tail bounds, and show generic nonzero tail amplitudes for generic data, including refined results for higher angular harmonics $\ell$. The results illuminate the connection between decay along timelike routes and radiation at null infinity (rate-doubling) and have implications for charged scalar fields in black hole spacetimes and related Maxwell–Klein–Gordon models. Overall, the work provides a rigorous, geometry-driven mechanism for capturing late-time tails in scale-invariant wave equations and extends understanding of near-horizon/null-infinity dynamics in gravitational contexts.

Abstract

We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential. We present a novel, geometric, physical-space approach for determining late-time asymptotics, based around embedding Minkowski spacetime conformally into the spacetime $AdS_2 \times \mathbb{S}^2$ (with $AdS_2$ the two-dimensional anti de-Sitter spacetime) to turn a global late-time asymptotics problem into a local existence problem for the wave equation in $AdS_2 \times \mathbb{S}^2$. Our approach is inspired by the treatment of the near-horizon geometry of extremal black holes in the physics literature. We moreover apply our method to another scale-invariant model: the (complex-valued) charged wave equation on Minkowski spacetime in the presence of a static electric field, which can be viewed as a simplification of the charged Maxwell-Klein-Gordon equations on a black hole spacetime.

Late-time tails for scale-invariant wave equations with a potential and the near-horizon geometry of null infinity

TL;DR

The paper resolves sharp late-time tail behavior for scale-invariant wave equations with inverse-square potentials in Minkowski space by tying global decay to local dynamics on the AdS geometry. The authors develop a novel geometric approach that uses a conformal embedding, twisted derivatives, and -twisted Sobolev spaces to convert the problem into a local well-posedness and elliptic-estimate framework on AdS, then translate back to obtain precise leading-order tails with exponents . They prove existence/uniqueness of weak solutions, establish pointwise and tail bounds, and show generic nonzero tail amplitudes for generic data, including refined results for higher angular harmonics . The results illuminate the connection between decay along timelike routes and radiation at null infinity (rate-doubling) and have implications for charged scalar fields in black hole spacetimes and related Maxwell–Klein–Gordon models. Overall, the work provides a rigorous, geometry-driven mechanism for capturing late-time tails in scale-invariant wave equations and extends understanding of near-horizon/null-infinity dynamics in gravitational contexts.

Abstract

We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such equations are scale-invariant and we show their solutions decay in time at a rate determined by the coefficient in the inverse-square potential. We present a novel, geometric, physical-space approach for determining late-time asymptotics, based around embedding Minkowski spacetime conformally into the spacetime (with the two-dimensional anti de-Sitter spacetime) to turn a global late-time asymptotics problem into a local existence problem for the wave equation in . Our approach is inspired by the treatment of the near-horizon geometry of extremal black holes in the physics literature. We moreover apply our method to another scale-invariant model: the (complex-valued) charged wave equation on Minkowski spacetime in the presence of a static electric field, which can be viewed as a simplification of the charged Maxwell-Klein-Gordon equations on a black hole spacetime.
Paper Structure (35 sections, 15 theorems, 178 equations, 2 figures)

This paper contains 35 sections, 15 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rate doubling and the conformal structure of null infinity.
  3. The restriction in the parameters range for $a$ and $qe$.
  4. Gauge choice and resemblances between \ref{['ISP']} and \ref{['CSF']}.
  5. Strategy to obtain decay-in-time.
  6. Outline of the introduction
  7. Precise statement of the main results
  8. Set-up of the initial data
  9. Weighted Sobolev spaces.
  10. The main theorems.
  11. Charged scalar field equations on a black hole
  12. Wave equations with (scale-invariant) potentials
  13. Late-time tails on black hole spacetimes
  14. Key ideas of the proof
  15. (A) Global $\to$ local: embedding Minkowski into a local patch of $AdS_2\times \mathbb{S}^2$
  16. ...and 20 more sections

Key Result

Theorem 1

Assume that $|q_0e| \in (0,\frac{1}{2})$ for CSF and $a>-\frac{1}{4}$ for ISP and introduce the following exponents for any $\ell\in \mathbb{N} \cup\{0\}$: Let $\phi$ be the unique solution of ISP or CSF with suitably regular and decaying initial data, such that the projection of $\phi$ onto spherical harmonics of degree $\geq \ell$ satisfies additionally the boundary conditions $\phi_{\geq \ell}

Figures (2)

  • Figure 1: A 2-dimensional representation of the Minkowski region $(\mathcal{M}=\mathbb{R}^{3+1}- \Gamma,m)$ embedded in $AdS_2\times \mathbb{S}^2$.
  • Figure 2: A 2-dimensional representation of two copies of Minkowski spacetime embedded in $AdS_2\times \mathbb{S}^2$.

Theorems & Definitions (43)

  • Theorem : Rough version
  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem I
  • Theorem II
  • Conjecture 1.1
  • Remark 1.2
  • Remark 1.3
  • ...and 33 more