Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes
- 1Department of Mathematics, Princeton University, Princeton, NJ, United States, Stanford University, Stanford, CA, United States
Abstract
We consider the scalar wave equation □ g φ = 0 and the linearized Einstein-scalar field system around generalized Kasner spacetimes with spatial topology ޔ D .In suitable regimes for the Kasner exponents, it is known that solutions to such equations arising from regular Cauchy data (e.g., at t = 1) have certain quantitative blow-up asymptotics near the initial time (i.e., t = 0) singularity of Kasner.For instance, solutions to the wave equation behave asThis article provides a description, and proof, of a scattering theory for the above equations, linking Cauchy data at t = 1 and suitable asymptotic data at t = 0 in Kasner.For the scalar wave equation, this means a Hilbert space isomorphism between (φ, ∂ t φ) at t = 1 and the functions (ψ ∞ , ϕ ∞ ).A curious detail is that certain quantities, e.g., ψ ∞ , feature a gain of half a derivative when compared to ∂ t φ at t = 1.The study of the linearized Einstein-scalar field system reveals further interesting phenomena, including differences between diagonal and off-diagonal components of certain tensors in the scattering theory, and that the losses of derivatives feature a sensitive dependence on the anisotropy of the background Kasner spacetime.In fact, though our result holds for the entire subcritical regime of background Kasner exponents, the number of derivatives lost and gained in the scattering theory can become unbounded as one nears the boundary of this regime.1. Introduction 1460 2. Kasner spacetimes and the ADM equations 1476 3. Scattering for the scalar wave equation 1488 4. The linearized Einstein-scalar field system around Kasner 1499 5. Linearized Einstein-scalar field: the high-frequency energy estimate 1512 6. Linearized Einstein-scalar field: the low-frequency energy estimate 1518 7. Scattering for the linearized Einstein-scalar field equations 1525 Appendix A. Fourier decomposition and DOs 1536 Appendix B. Fuchsian ODE systems 1538 Appendix C.
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BibTeX
@article{li2026scattering,
title = {{Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes}},
author = {Warren Li},
journal = {Analysis \& PDE},
year = {2026},
month = oct,
doi = {10.2140/apde.2026.19.1459},
url = {https://doi.org/10.2140/apde.2026.19.1459},
note = {CC BY},
}
RIS
TY - JOUR
TI - Scattering towards the singularity for the wave equation and the linearized Einstein-scalar field system in Kasner spacetimes
AU - Warren Li
PY - 2026
DA - 2026/10/08
JO - Analysis & PDE
DO - 10.2140/apde.2026.19.1459
UR - https://doi.org/10.2140/apde.2026.19.1459
AB - We consider the scalar wave equation □ g φ = 0 and the linearized Einstein-scalar field system around generalized Kasner spacetimes with spatial topology ޔ D .In suitable regimes for the Kasner exponents, it is known that solutions to such equations arising from regular Cauchy data (e.g., at t = 1) have certain quantitative blow-up asymptotics near the initial time (i.e., t = 0) singularity of Kasner.For instance, solutions to the wave equation behave asThis article provides a description, and proof, of a scattering theory for the above equations, linking Cauchy data at t = 1 and suitable asymptotic data at t = 0 in Kasner.For the scalar wave equation, this means a Hilbert space isomorphism between (φ, ∂ t φ) at t = 1 and the functions (ψ ∞ , ϕ ∞ ).A curious detail is that certain quantities, e.g., ψ ∞ , feature a gain of half a derivative when compared to ∂ t φ at t = 1.The study of the linearized Einstein-scalar field system reveals further interesting phenomena, including differences between diagonal and off-diagonal components of certain tensors in the scattering theory, and that the losses of derivatives feature a sensitive dependence on the anisotropy of the background Kasner spacetime.In fact, though our result holds for the entire subcritical regime of background Kasner exponents, the number of derivatives lost and gained in the scattering theory can become unbounded as one nears the boundary of this regime.1. Introduction 1460 2. Kasner spacetimes and the ADM equations 1476 3. Scattering for the scalar wave equation 1488 4. The linearized Einstein-scalar field system around Kasner 1499 5. Linearized Einstein-scalar field: the high-frequency energy estimate 1512 6. Linearized Einstein-scalar field: the low-frequency energy estimate 1518 7. Scattering for the linearized Einstein-scalar field equations 1525 Appendix A. Fourier decomposition and DOs 1536 Appendix B. Fuchsian ODE systems 1538 Appendix C.
SN - 1948-206X
SN - 2157-5045
ER -