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The $r^{p}$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications

Georgios Moschidis

TL;DR

The paper broadens the r^p-weighted energy method of Dafermos–Rodnianski to a wide class of asymptotically flat spacetimes, including Kerr exhaustively and radiating spacetimes, establishing a robust decay framework. By constructing an r^p-energy hierarchy and combining it with integrated local energy decay and Morawetz-type estimates, the authors obtain bar{t}^{-1} decay and, under additional vector-field and geometric assumptions, improved bar{t}^{-d/2} decay, along with well-defined Friedlander radiation fields at I^+. The results apply to the exterior of black holes and moving obstacles, and yield concrete decay rates for solutions and their derivatives as well as radiationfields, with implications for stability questions in general relativity. The work also demonstrates the versatility of the r^p method by deriving applications to Kerr spacetimes, radiating Minkowski perturbations, and dynamical black-hole spacetimes, providing a unified approach to decay in non-stationary, radiating geometries.

Abstract

In [M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in XVIth International Congress on Mathematical Physics, Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arXiv:0910.4957], Dafermos and Rodnianski presented a novel approach to establish uniform decay rates for solutions $φ$ to the scalar wave equation $\square_{g}φ=0$ on Minkowski, Schwarzschild and other asymptotically flat backgrounds. This paper generalises the methods and results of the above paper to a broad class of asymptotically flat spacetimes $(\mathcal{M},g)$, including Kerr spacetimes in the full subextremal range $|a|<M$, but also radiating spacetimes with no exact symmetries in general dimension $d+1$, $d\ge3$. As a soft corollary, it is shown that the Friedlander radiation field for $φ$ is well defined on future null infinity. Moreover, polynomial decay rates are established for $φ$, provided that an integrated local energy decay statement (possibly with a finite loss of derivatives) holds and the near region of $(\mathcal{M},g)$ satisfies some mild geometric conditions. The latter conditions allow for $(\mathcal{M},g)$ to be the exterior of a black hole spacetime with a non-degenerate event horizon (having possibly complicated topology) or the exterior of a compact moving obstacle in an ambient globally hyperbolic spacetime satisfying suitable geometric conditions.

The $r^{p}$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications

TL;DR

The paper broadens the r^p-weighted energy method of Dafermos–Rodnianski to a wide class of asymptotically flat spacetimes, including Kerr exhaustively and radiating spacetimes, establishing a robust decay framework. By constructing an r^p-energy hierarchy and combining it with integrated local energy decay and Morawetz-type estimates, the authors obtain bar{t}^{-1} decay and, under additional vector-field and geometric assumptions, improved bar{t}^{-d/2} decay, along with well-defined Friedlander radiation fields at I^+. The results apply to the exterior of black holes and moving obstacles, and yield concrete decay rates for solutions and their derivatives as well as radiationfields, with implications for stability questions in general relativity. The work also demonstrates the versatility of the r^p method by deriving applications to Kerr spacetimes, radiating Minkowski perturbations, and dynamical black-hole spacetimes, providing a unified approach to decay in non-stationary, radiating geometries.

Abstract

In [M. Dafermos and I. Rodnianski, A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, in XVIth International Congress on Mathematical Physics, Pavel Exner ed., Prague 2009 pp. 421-433, 2009, arXiv:0910.4957], Dafermos and Rodnianski presented a novel approach to establish uniform decay rates for solutions to the scalar wave equation on Minkowski, Schwarzschild and other asymptotically flat backgrounds. This paper generalises the methods and results of the above paper to a broad class of asymptotically flat spacetimes , including Kerr spacetimes in the full subextremal range , but also radiating spacetimes with no exact symmetries in general dimension , . As a soft corollary, it is shown that the Friedlander radiation field for is well defined on future null infinity. Moreover, polynomial decay rates are established for , provided that an integrated local energy decay statement (possibly with a finite loss of derivatives) holds and the near region of satisfies some mild geometric conditions. The latter conditions allow for to be the exterior of a black hole spacetime with a non-degenerate event horizon (having possibly complicated topology) or the exterior of a compact moving obstacle in an ambient globally hyperbolic spacetime satisfying suitable geometric conditions.

Paper Structure

This paper contains 44 sections, 27 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Comparison of the two approaches
  3. The "old" approach
  4. The $r^{p}$-weighted energy method
  5. Statement of the main results
  6. The $r^{p}$-weighted energy hierarchy in dimensions $d\ge3$
  7. A $\bar{t}^{-1}$ polynomial decay estimate for solutions to the wave equation $\square\text{ \fontencoding{LGR} f}=0$
  8. An improved $\bar{t}^{-\frac{d}{2}}$ polynomial decay estimate for solutions to the wave equation $\square\text{ \fontencoding{LGR} f}=0$ in dimensions $d\ge3$
  9. Applications of the $r^{p}$-weighted energy method
  10. The Friedlander radiation field for solutions to the wave equation $\square\text{ \fontencoding{LGR} f}=0$
  11. Logarithmic hyperboloidal energy decay for solutions to the wave equation $\square\text{ \fontencoding{LGR} f}=0$ on a general class of stationary asymptotically flat spacetimes
  12. Polynomial decay for solutions to the wave equation $\square\text{ \fontencoding{LGR} f}=0$ on the exterior of subextremal Kerr spacetimes for $|a|<M$
  13. Improved polynomial decay on radiating uniformly small perturbations of Minkowski spacetime
  14. Improved polynomial decay on dynamical, radiating black hole spacetimes
  15. Outline of the paper and technical comments
  16. ...and 29 more sections

Key Result

Theorem 1

(Dafermos-Rodnianski DafRod7, specialised here to Schwarzschild) On Schwarzschild exterior spacetime $(\mathcal{M}_{Sch},g_{M})$, the following statements hold for any solution $\text{ \fontencoding{LGR} f}$ to the wave equation (eq:WaveEquation): 1. An $r^{p}$-weighted energy hierarchy of the form for $p\in[0,2]$ holds, where $\mathcal{D}_{\text{ \fontencoding{LGR} t}_{1}}^{\text{ \fontencodin

Theorems & Definitions (59)

  • Theorem
  • Remark
  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Theorem 1.3
  • Remark
  • Remark
  • Theorem 1.4
  • Remark
  • ...and 49 more