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Communications in Mathematical Physics

, Volume 86, Issue 1, pp 69–86 | Cite as

The general relativistic hydrogen atom

  • Jeffrey M. Cohen
  • Robert T. Powers
Article

Abstract

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (−mc2, +mc2). It is shown that, for certain boundary conditions, neutrino bound states exist.

Keywords

Black Hole Hydrogen Atom Angular Momentum Free Electron Real Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Brill, D. R., Cohen, J. M.: Cartan frames and the general relativistic Dirac Equation. J. Math. Phys.7, 238 (1966)Google Scholar
  2. 2.
    Chernoff, P. R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal.12, 401–414 (1973)Google Scholar
  3. 3.
    Cohen, J. M., Kegeles, L. S.: Electromagnetic fields in curved spaces: A constructive procedure. Phys. Rev.D10, 1070–1084 (1974)Google Scholar
  4. 4.
    Dunford, N., Schwartz, J.: Linear operators, Part II. New York: Interscience Publishers 1963Google Scholar
  5. 5.
    Kato, T.: Perturbation theory for linear operators, New York: Springer 1966Google Scholar
  6. 6.
    Lichnerowitz, A.: Bull. Soc. Math. France92, 11 (1964)Google Scholar
  7. 7.
    Rose, M. E.: Relativistic electron theory, New York: John Wiley and Sons 1961Google Scholar
  8. 8.
    Schroedinger, E.: Commun. Pontif. Acad. Sci.2, 321 (1938)Google Scholar
  9. 9.
    Schiff, L.: Quantum mechanics New York: McGraw-Hill 1949Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Jeffrey M. Cohen
    • 1
  • Robert T. Powers
    • 2
  1. 1.Department of PhysicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Mathematics DepartmentUniversity of PennsylvaniaPhiladelphiaUSA

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