Exotic Black Holes?

Exotic smooth manifolds, R ×Θ S , are constructed and discussed as possible space-time models supporting the usual Kruskal presentation of the vacuum Schwarzschild metric locally, but not globally. While having the same topology as the standard Kruskal model, none of these manifolds is diffeomorphic to standard Kruskal, although under certain conditions some global smooth Lorentz-signature metric can be continued from the local Kruskal form. Consequently, it can be conjectured that such manifolds represent an infinity of physically inequivalent (non-diffeomorphic) space-time models for black holes. However, at present nothing definitive can be said about the continued satisfaction of the Einstein equations. This problem is also discussed in the original Schwarzschild (t, r) coordinates for which the exotic region is contained in a world tube along the time-axis, so that the manifold is spatially, but not temporally, asymptotically standard. In this form, it is tempting to speculate that the confined exotic region might serve as a source for some exterior solution. Certain aspects of the Cauchy problem are also discussed in terms of R Θ models which are “half-standard”, say for all t < 0, but for which t cannot be glob-