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Asymptotics and Scattering for Massive Maxwell–Klein–Gordon Equations

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Abstract

We study the asymptotic behavior and the scattering from infinity problem for the massive Maxwell–Klein–Gordon system in the Lorenz gauge, which were previously only studied for the massless system. For a general class of initial data, in particular of nonzero charge, we derive the precise asymptotic behaviors of the solution, where we get a logarithmic phase correction for the complex Klein–Gordon field, and a combination of interior homogeneous field, radiation fields, and an exterior charge part for the gauge potentials. Moreover, we also derive a formula for the charge at infinite time, which shows that the charge is concentrated at timelike infinity, a phenomenon drastically different from the massless case. After deriving the forward asymptotics, we formulate the scattering from infinity problem by defining the correct notion of scattering data, and then solve this problem. We show that one can determine the correct charge contribution using the information at timelike infinity, which is a crucial step for us to obtain backward solutions not only for the reduced equations in the Lorenz gauge but also for the original physical system.

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Notes

  1. Throughout this paper, we raise and lower indices using the standard Minkowski metric \(m_{\mu \nu }=\textrm{diag}\{-1,1,1,1\}\), and we use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean.

  2. There are ambiguities in the terms like “asymptotic behavior" and “global dynamics". In many previous works like [26, 43, 15], such terms essentially mean global decay estimates. On the other hand, in this paper, the word “asymptotics" is reserved for the precise asymptotic expansion, i.e., the answer to the forward problem (1).

  3. We use the notation \(a\lesssim b\) to denote that there exists a constant \(C>0\) such that \(a\le Cb\).

  4. The decay rates of \(a_\pm (y)\) can be improved, provided enough regularity of the initial data, so the exact power of \((1-|y|^2)\) is not important.

  5. The timelike infinity here can be thought of as the uppermost cap in the octagonal compactification of Minkowski spacetime, which blows up the “timelike infinity" (a sphere) in the Penrose diagram. See e.g. [3].

  6. Of course, there is a smoothness problem for the solution when \(t=r\), but it does not matter in view of the strong Huygens’ principle.

  7. We remark that for the wave-Klein–Gordon model, the pair \(a_\pm (y)\) are in fact conjugate to each other. Here this is nonzero because \(\phi \) is complex.

  8. We use the Einstein summation convention. Also, when the repeated index is spatial, we define the expression to be the sum regardless of whether it is upper or lower, as the spatial part of the Minkowski metric is Euclidean.

  9. In some literatures, the notation \(D_X D_Y \phi \) could mean the X, Y-component of the tensor \(D_\mu D_\nu \phi \), but here it represents \(D_X(Y^\mu D_\mu \phi )\).

  10. One can pick any positive number \(\varepsilon >0\) instead of 2 here, and then the part where \(q<-\varepsilon \) will be exactly \(U_\mu (y)/\tau \). This is simply a consequence of strong Huygens’ principle.

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Acknowledgements

The author would like to thank his advisor, Hans Lindblad, for many helpful discussions and his encouragement in studying this problem.

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  1. Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD, 21218, USA

    Xuantao Chen

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Chen, X. Asymptotics and Scattering for Massive Maxwell–Klein–Gordon Equations. Commun. Math. Phys. 405, 133 (2024). https://doi.org/10.1007/s00220-024-05023-5

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