Complex Line Bundles Over Two-Sphere and Black Hole Physics

TLDR: In this paper, the authors discuss a long-standing problem of the global topological non-trivial properties of the four-dimensional space-times underlying black hole physics and observe that the standard space-time topology of the ℝ2×S2 form admits topologically inequivalent configurations of a complex scalar field on black hole.

Abstract: We discuss a long-standing problem of the global topological non-trivial properties of the four-dimensional space-times underlying black hole physics and observe that the standard space-time topology of the ℝ2×S2 form for black hole physics admits topologically inequivalent configurations of a complex scalar field on black hole by virtue of the availability of non-trivial complex line bundles over S2. Each configuration can be labeled by its Chern number n∈ℤ. For the Schwarzschild black hole we formulate an appropriate wave equation for these configurations in massless case and describe its solutions as a first step to study quantum effects for the above configurations within the framework of black hole physics.