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Conformal Scattering of Maxwell Potentials

Jean-Philippe Nicolas, Grigalius Taujanskas

TL;DR

This work constructs a complete conformal scattering framework for finite-energy Maxwell potentials on a broad class of curved, asymptotically flat spacetimes with nonzero ADM mass. It introduces a Lorenz-like gauge to render Maxwell's potential equations hyperbolic up to null infinity and uses Hörmander's Goursat problem (via Bar–Wafo) to relate data on scri to interior fields, yielding a bijective scattering map between initial data on a Cauchy surface and data at null infinity. In Minkowski space, a Morawetz-based formulation provides a strictly stronger data space, while in CSCD curved spacetimes the authors establish two-way energy estimates, construct gauges near scri and i0, and prove the invertibility of the scattering operator, thereby generalizing conformal scattering to nonstationary backgrounds. The results give precise decay rates and reveal potential connections to electromagnetic memory and future Yang–Mills generalizations, highlighting the role of potentials and gauges in scattering. Overall, the paper delivers a rigorous, gauge-consistent, conformal scattering theory for Maxwell potentials with two-scale (scri and interior) data frameworks and two complementary formulations.

Abstract

We construct a complete conformal scattering theory for finite energy Maxwell potentials on a class of curved, asymptotically flat spacetimes with prescribed smoothness of null infinity and a non-zero ADM mass. In order to define the full set of scattering data, we construct a Lorenz-like gauge which makes the field equations hyperbolic and non-singular up to null infinity, and reduces to an intrinsically solvable ODE on null infinity. We develop a method to solve the characteristic Cauchy problem from this scattering data based on a theorem of Hörmander. In the case of Minkowski space, we further investigate an alternative formulation of the scattering theory by using the Morawetz vector field instead of the usual timelike Killing vector field.

Conformal Scattering of Maxwell Potentials

TL;DR

This work constructs a complete conformal scattering framework for finite-energy Maxwell potentials on a broad class of curved, asymptotically flat spacetimes with nonzero ADM mass. It introduces a Lorenz-like gauge to render Maxwell's potential equations hyperbolic up to null infinity and uses Hörmander's Goursat problem (via Bar–Wafo) to relate data on scri to interior fields, yielding a bijective scattering map between initial data on a Cauchy surface and data at null infinity. In Minkowski space, a Morawetz-based formulation provides a strictly stronger data space, while in CSCD curved spacetimes the authors establish two-way energy estimates, construct gauges near scri and i0, and prove the invertibility of the scattering operator, thereby generalizing conformal scattering to nonstationary backgrounds. The results give precise decay rates and reveal potential connections to electromagnetic memory and future Yang–Mills generalizations, highlighting the role of potentials and gauges in scattering. Overall, the paper delivers a rigorous, gauge-consistent, conformal scattering theory for Maxwell potentials with two-scale (scri and interior) data frameworks and two complementary formulations.

Abstract

We construct a complete conformal scattering theory for finite energy Maxwell potentials on a class of curved, asymptotically flat spacetimes with prescribed smoothness of null infinity and a non-zero ADM mass. In order to define the full set of scattering data, we construct a Lorenz-like gauge which makes the field equations hyperbolic and non-singular up to null infinity, and reduces to an intrinsically solvable ODE on null infinity. We develop a method to solve the characteristic Cauchy problem from this scattering data based on a theorem of Hörmander. In the case of Minkowski space, we further investigate an alternative formulation of the scattering theory by using the Morawetz vector field instead of the usual timelike Killing vector field.
Paper Structure (36 sections, 17 theorems, 255 equations, 8 figures)

This paper contains 36 sections, 17 theorems, 255 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Conventions and Notation
  4. Background Spacetimes
  5. Maxwell Fields and Potentials
  6. Maxwell Components
  7. Minkowski Space
  8. Partial Conformal Compactifications
  9. A Priori Energy Estimates
  10. Conformal invariance of energies
  11. Field Equations and Gauge Fixing
  12. Gauge reduction on scri
  13. The Scattering Construction
  14. Spaces of initial and scattering data
  15. Construction of Trace Operators
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\Sigma$ be the standard initial Cauchy surface in Minkowski space. A finite energy solution to Maxwell's equations on Minkowski space admits the Coulomb, temporal, and Lorenz gauges simultaneously, and there exist bounded, invertible linear operators corresponding to the future/past Cauchy development of Maxwell's equations, which map finite energy Maxwell potential initial data on $\Sigma$

Figures (8)

  • Figure 1: The Penrose diagram of Minkowski space showing surfaces of constant $r$ and surfaces of constant $t$.
  • Figure 2: The asymptotically null foliation $\{ \Sigma_t \}_t$ of $\hat{\mathcalboondox{M}}$.
  • Figure 3: The foliation $\{\hat{\Sigma}_\tau \}_\tau$ of $\hat{\mathcalboondox{M}}$ whose leaves are transverse to $\mathscr{I}^+$.
  • Figure 4: We choose a conformal factor $\Omega$ which is equal to $R$ near $\mathscr{I}^+$ away from $i^+$, and smoothly brings $i^+$ to a finite distance.
  • Figure 5: For smooth compactly supported data the solution is smooth and compactly supported in $\hat{\mathcalboondox{M}}$.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Theorem : Scattering theory on Minkowski space
  • Theorem : Scattering theory on CSCD spacetimes
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 41 more