Massive wave propagation near null infinity

Summary

The paper discusses the propagation of massive waves near null infinity in asymptotically Minkowski spacetime. It introduces the octagonal compactification O of Minkowski spacetime, which refines both the radial and Penrose compactifications. The author uses a combination of microlocal analysis and asymptotic expansions to study the behavior of solutions to the Klein-Gordon equation near null infinity.

Highlights

• The octagonal compactification O is a refinement of the radial and Penrose compactifications of Minkowski spacetime.
• The Klein-Gordon equation is studied near null infinity using microlocal analysis and asymptotic expansions.
• The author uses the de,sc-calculus, a symbolic pseudodifferential calculus, to study the Klein-Gordon equation.
• The de,sc-calculus is used to prove propagation estimates and radial point estimates for the Klein-Gordon equation.
• The author studies the classical dynamics on the de,sc-phase space near null infinity.
• The Hamiltonian flow of the d'Alembertian is studied near null infinity, and the radial sets are identified.
• The author proves that the solution to the Klein-Gordon equation has a full asymptotic expansion at infinity.

Key Insights

• The octagonal compactification O is a useful tool for studying the behavior of solutions to the Klein-Gordon equation near null infinity. It provides a more detailed understanding of the asymptotic behavior of solutions than the radial or Penrose compactifications alone.
• The de,sc-calculus is a powerful tool for studying the Klein-Gordon equation near null infinity. It allows for the proof of propagation estimates and radial point estimates, which are essential for understanding the behavior of solutions to the equation.
• The classical dynamics on the de,sc-phase space near null infinity are complex and involve the study of the Hamiltonian flow of the d'Alembertian. The radial sets play a crucial role in understanding the behavior of solutions to the Klein-Gordon equation.
• The solution to the Klein-Gordon equation has a full asymptotic expansion at infinity, which can be obtained using the de,sc-calculus and microlocal analysis. This expansion provides a detailed understanding of the behavior of solutions to the equation near null infinity.
• The author's results have implications for the study of more general spacetimes, such as asymptotically flat spacetimes. The methods used in the paper can be applied to study the behavior of solutions to the Klein-Gordon equation in these spacetimes.
• The paper provides a new perspective on the study of the Klein-Gordon equation near null infinity. It highlights the importance of using a combination of microlocal analysis and asymptotic expansions to understand the behavior of solutions to the equation.
• The author's results can be used to study the behavior of solutions to the Klein-Gordon equation in more general spacetimes, such as black hole spacetimes. The methods used in the paper can be applied to study the behavior of solutions to the equation in these spacetimes.

Mindmap

Citation

Sussman, E. (2023). Massive wave propagation near null infinity (Version 2). arXiv. https://doi.org/10.48550/ARXIV.2305.01119