2023/12/29 by Yakov Shlapentokh-Rothman, Shlapentokh-Rothman, Yakov, Maxime Van de Moortel +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2401.00048
openalex publication_date 2023/12/29 · openalex created_date 2024/01/03 · openalex updated_date 2026/07/28
We consider the massive scalar field equation \Box_gRN ϕ= m2 ϕ on any subextremal Reissner--Nordström exterior metric gRN. We prove that solutions with localized initial data decay pointwise-in-time at the polynomial rate t-(5)/(6)+δ in any spatially compact region (including the event horizon), for some small δ≤ (1)/(23) . Moreover, assuming the validity of the Exponent Pair Conjecture on exponential sums in Number Theory, our result implies that decay upper bounds hold at the rate t-(5)/(6)+ε, for any arbitrarily small ε>0. In our previous work, we proved that each fixed angular mode decays at the exact rate t-(5)/(6), thus the upper bound t-(5)/(6)+ε is sharp, up to a tε loss. Without the restriction to a fixed angular mode, the solution turns out to have an unbounded Fourier transform due to discrete frequencies associated to quasimodes, and caused by the occurrence of stable timelike trapping. Our analysis nonetheless shows that inverse-polynomial asymptotics in t still hold after summing over all angular modes.