Exotic Black Holes?

Abstract

Exotic smooth manifolds, ${\bf R^2\times_ΘS^2}$, are constructed and discussed as possible space-time models supporting the usual Kruskal presentation of the vacuum Schwarzschild metric locally, but {\em 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 ${\bf R^4_\Θ}$\ models which are ``half-standard&#39;&#39;, say for all $t<0,$ but for which $t$ cannot be globally smooth.