The effect of negative-energy shells on Schwarzschild geometry and their Penrose diagrams

This work demonstrates some novel procedures for joining multiple solutions of the Einstein field equation in a way which creates a new analytical tool for studying them. Joining known solutions along shells ensures that we have solutions off the boundary. In 1985 Dray and 't Hooft showed how to use use spherically symmetric shells of massless matter to join spacetime regions with Schwarzschild geometry. We demonstrate here how to use these solutions to draw information-rich Penrose diagrams usable for investigating the characteristics of the solutions. In particular, we show that the boundaries proposed by Dray and 't Hooft between Schwarzschild geometries of equal mass are best described using a shift that changes the shape of the resulting Penrose diagram. The drawing of diagrams with shifts of unequal mass lead to a shift with a more involved structure. The shift that results from the presence of a negative energy shell, in both equal and unequal mass cases, allows for travel between the heretofore causally separated exterior regions of the Schwarzschild geometry. Close attention to these shifted diagrams gives us a powerful tool, enabling experimentation in joining various solutions of the Einstein field equation.