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On the Scattering of Waves inside Charged Spherically Symmetric Black Holes

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Abstract

In this paper, we show that there is a breakdown of scattering between the event horizon (or the Cauchy horizon) and an intermediate Cauchy hypersurface in the dynamic interior of a Reissner–Nordström-like black hole. More precisely, we show that the trace operators and their analytic counterparts, the inverse wave operators, do not have bounded inverses, even though these operators themselves are bounded. This result holds for the natural energy given by the energy–momentum tensor of the wave equation using the timelike vector field of the Regge–Wheeler variable, which asymptotically becomes normal to the horizons. The behaviour of solutions at low spatial frequencies and their behaviour at high angular momenta are the only obstructions causing this breakdown of scattering. The breakdown follows from an analysis of a \(1+1\)-dimensional wave equation with exponentially decaying potential which we treat for general potentials, and we show that the breakdown is generic.

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Notes

  1. Thus, the boundedness of the scattering operator from the horizon to the horizon is not easily provable via the method of “concatenating” intermediate operators, as is usually done in the exterior.

  2. This vector field is timelike in the interior of the black hole and asymptotically becomes normal to the horizons.

  3. E.g. scalar waves. We also expect electromagnetic fields to exhibit similar behaviours to scalar waves.

  4. See, e.g. [12].

  5. A causal vector is future-oriented if its inner product with the vector field defining the time-orientation is positive.

  6. \(\Re (z)\) and \(\Im (z)\) are, respectively, the real and the imaginary parts of a complex number z.

  7. The assumption that \(\varphi \) is even was added for convenience so that its Fourier transform becomes real, but this is not indispensable.

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Acknowledgements

The authors would like to thank D. Häfner and J.-P. Nicolas for previous valuable discussions on the subject. On behalf of M. Mokdad, the IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002).

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Authors and Affiliations

  1. University of Burgundy, Dijon, France

    Mokdad Mokdad

  2. ETH Zürich, Zurich, Switzerland

    Rajai Nasser

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  1. Mokdad Mokdad
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  2. Rajai Nasser
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Correspondence to Mokdad Mokdad.

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Communicated by Mihalis Dafermos.

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Mokdad, M., Nasser, R. On the Scattering of Waves inside Charged Spherically Symmetric Black Holes. Ann. Henri Poincaré 23, 3191–3220 (2022). https://doi.org/10.1007/s00023-022-01176-y

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