arXiv:gr-qc/9910030v1 8 Oct 1999 The Einstein-Dirac-Maxwell Equations – Black Hole Solutions 1 by F. Finster 2 , J.A. Smoller 3 , and S.-T. Yau 4 1 Introduction. We are interested in studying how different force fields interact with gravity, at a “fundamental” level, but not at the level of Quantum Field Theory. This is because there exists no theory of quantum gravity, and no understanding of “Planck-scale” physics; that is, physics at “Planck-lengths”, where the Planck-length is given by  Gh c 3  1/2 ≈ 10 −33 cm , where G is Newton’s gravitational constant, h is Planck’s constant, and c denotes the speed of light. Before discussing our results, we think that it is worthwhile to see what is the difficulty in making a theory of quantum gravity. In order to understand why a solution to the problem of reconciling gravity and quantum mechanics has been so elusive, we must consider the implications of the Heisenberg Uncertainty Principle (HUP) at small distance scales. Recall that the HUP states that “the more precisely a spatial measurement is made, the less precisely the momentum (or the energy) of the system being measured, is known”. When the spatial measurement Δx ≈ 10 −13 cm, there are large uncertainties in the energy 1 To appear in the Proceedings of the IMS Conference on Differential Equations in Mechanics, Chinese University of Hong Kong, June 1999. 2 Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. 3 University of Michigan, Ann Arbor, MI, USA; research supported in part by the NSF, Grant No. DMS-G-9501128. 4 Harvard University, Cambridge, MA, USA; research supported in part by the NSF, Grant No. 33- 585-7510-2-30. 1