Communications in Mathematical Physics

, Volume 205, Issue 2, pp 249-262

First online:

Non-Existence of Black Hole Solutions¶for a Spherically Symmetric, Static Einstein–Dirac–Maxwell System

  • Felix FinsterAffiliated with{Max Planck Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany.¶E-mail:
  • , Joel SmollerAffiliated withMathematics Department, The University of Michigan, Ann Arbor, MI 48109, USA.¶E-mail:
  • , Shing-Tung YauAffiliated withMathematics Department, Harvard University, Cambridge, MA 02138, USA.¶E-mail:

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We consider for j=½, \(\)… a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field.

The Einstein–Dirac–Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner–Nordström solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a “cloud” of spin-½-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon.