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Peeling on Kerr Spacetime: Linear and Semi-linear Scalar Fields

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Abstract

We study the peeling on Kerr spacetime for fields satisfying conformally invariant linear and semi-linear scalar wave equations. We follow the approach initiated by Mason and Nicolas (J Inst Math Jussieu 8(1):179–208, 2009; J Geom Phys 62(4):867–889, 2012. arXiv:1101.4333) for the Schwarzschild metric, based on a Penrose compactification and energy estimates. This approach provides a definition of the peeling at all orders in terms of Sobolev regularity near \({{\mathscr {I}}}\) instead of \({{\mathcal {C}}}^k\) regularity at \({{\mathscr {I}}}\), allowing to characterise completely and without loss the classes of initial data ensuring a certain order of peeling at \({{\mathscr {I}}}\). This paper extends the construction to the Kerr metric, confirms the validity and optimality of the flat spacetime model (in the sense that the same regularity and fall-off assumptions on the data guarantee the peeling behaviour in flat spacetime and on the Kerr metric) and does so for the first time for a nonlinear equation. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics.

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Notes

  1. Larger than the one obtained using the embedding in the Einstein cylinder.

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Acknowledgements

This research, based on a chapter of Pham Truong Xuan’s PhD thesis, was partly supported by the ANR funding ANR-12-BS01-012-01.

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Authors and Affiliations

  1. LMBA UMR CNRS 6205 Department of Mathematics, University of Brest, 6 Avenue Victor Le Gorgeu, 29200, Brest, France

    Jean-Philippe Nicolas

  2. Department of Mathematics Faculty of Information Technology, Thuyloi university, Khoa Cong nghe Thong tin Bo mon Toan Dai hoc Thuy loi 175 Tay Son Dong Da, Hanoi, Vietnam

    Truong Xuan Pham

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  1. Jean-Philippe Nicolas
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  2. Truong Xuan Pham
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Correspondence to Jean-Philippe Nicolas.

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Communicated by Jan Derezinski.

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Nicolas, JP., Pham, T.X. Peeling on Kerr Spacetime: Linear and Semi-linear Scalar Fields. Ann. Henri Poincaré 20, 3419–3470 (2019). https://doi.org/10.1007/s00023-019-00832-0

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