The Thin Shell Formalism (MTW 1973 p.551ff) is used to properly paste together different vacuum solutions to the Einstein equations. At the junction of the two solutions is a hypersurface of matter – the so-called thin shell. The thin shell formalism not only permits timelike and lightlike thin shells, it permits spacelike ones. A spacelike shell implies that when a stationary observer’s timelike world line encounters the shell (by simply moving forward in time), the observer would experience the momentary existence of a surrounding volume of matter – like momentarily finding yourself underwater. I have two questions about this:
1) How is it that the momentary appearance of a spacelike thin shell, apparently permitted by the formalism, is not a violation of energy conservation?
2) Imagine the maximally extended Schwarzschild solution with $r = R$ in the black hole sector of the solution being identified with (pasted onto) $r = R$ in the white hole sector. If $R$ is less than the radius of the event horizon, $r = R$ describes a spacelike hypersurface, which the thin shell formalism seems to have no problem with. Does this mean that this static wormhole construction (a black hole with an aperture beneath its event horizon that connects to a white hole) is perfectly valid?
This is my attempt to answer my own question. Energy conservation in GR actually means that the divergence of the stress-energy tensor is zero. This in turn means that any change in energy within any 4-volume is due to flows of energy through its bounding 3-surface. This permits the instantaneous appearance/disappearance of a thin shell of matter. The matter could have entered a given 4-volume through its boundaries and left the same way. For it to be spacelike shell existing at a single instant, its speed in doing so, however, would have to have been infinite. The only argument I have for infinite speed is that it isn’t forbidden by Special Relativity per se. Rather, it’s the transition from subluminal to superluminal speed that’s forbidden.
I found evidence that physicists are perfectly happy to consider these spacelike thin shells. Here is an example of a summary of someone’s talk at a conference on regular black holes in December 2011:
“Can construct regular black holes by filling the inner space with matter up to a certain surface and make a smooth junction, through a boundary surface, to the Schwarzschild solution as was done in (Mars CQG 1996, Magli RMP 1999, Elizalde and Hildebrandt PRD 2002, Conboy and Lake PRD 2005). The junction to Schwarzschild is made through a spacelike surface, rather than an usual timelike surface. This means the junction exists at a single instant of time.”