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The Fermionic Signature Operator in the Exterior Schwarzschild Geometry

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Abstract

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integral involving the fermionic signature operator as well as a double integral which takes into account the flux of Dirac currents across the event horizon. The spectrum of the fermionic signature operator is computed. The corresponding generalized fermionic projector states are analyzed.

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Acknowledgements

We would like to thank Niky Kamran for helpful discussions. We are grateful to the referees for valuable suggestions.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Felix Finster

  2. Departamento de Geometría y Topología Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva s/n, 18071, Granada, Spain

    Christian Röken

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  1. Felix Finster
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  2. Christian Röken
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Correspondence to Felix Finster.

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Communicated by Karl-Henning Rehren.

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Supported by the DFG research grant “Dirac Waves in the Kerr Geometry: Integral Representations, Mass Oscillation Property and the Hawking Effect”.

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Finster, F., Röken, C. The Fermionic Signature Operator in the Exterior Schwarzschild Geometry. Ann. Henri Poincaré 20, 3389–3418 (2019). https://doi.org/10.1007/s00023-019-00837-9

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