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Decay of Maxwell fields on Reissner–Nordström–de Sitter black holes

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Abstract

In this paper we use Morawetz and geometric energy estimates—the so-called vector field method—to prove decay results for the Maxwell field in the static exterior region of the Reissner–Nordström–de Sitter black hole. We prove two types of decay: the first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of \(t^{-1}\) which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field. The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper.

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Notes

  1. Thus, differing from the previous methods of analysing the fundamental solution.

  2. See, for example, [2].

  3. Although the natural way of obtaining an energy–momentum tensor is by means of a Lagrangian, one can as well directly consider 2-tensors with the desired properties and which might not be derived from a Lagrangian.

  4. For electromagnetic fields represented by 2-forms on the manifold, we actually vary the (local) potential and not the field (the 2-form) itself.

  5. Depending on the sign conventions.

  6. Admitting a global Cauchy hypersurface. See [34].

  7. This is the conformal energy of the solution to the wave equation, but to avoid confusion with the conformal energy we call it a conformal charge.

  8. The conformal weight and the spin weight are, respectively, related to the way the component change when we rescale the complex vector of the tetrad by a complex constant and the conjugate vector by the conjugate constant, and when rescaling the first null vector of the tetrad by a real constant and the second by the inverse constant. More precisely, the components transform as powers of the real rescaling constant, the power being the index of the component.

  9. The details can be found in [50] for instance.

  10. A spanning set A of a vector space is a subset of the space with the property that every vector in the space can be written as a linear combination of vectors in A only.

  11. The definition of the energy given in the introduction using the Hodge star operator and the one given here are actually equal, see [57] around equation (3.10).

  12. See [50] Appendix A for a detailed calculation leading from (80) to (81).

  13. It is worth mentioning that in addition to checking these identities by hand, we had also carried out these calculations using Sage which proves to be a useful symbolic computational program for differential geometry, in both calculus and algebra.

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Acknowledgements

The results of this paper and the mentioned conformal scattering results on RNDS spacetime [49, 51], were obtained during my Ph.D. thesis [50]. I would like to thank my thesis advisor Pr. Jean-Philippe Nicolas for his indispensable guidance during the thesis.

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    Mokdad Mokdad

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Mokdad, M. Decay of Maxwell fields on Reissner–Nordström–de Sitter black holes. Lett Math Phys 110, 1961–2018 (2020). https://doi.org/10.1007/s11005-020-01273-1

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