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.

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    $\begingroup$ I imagine that the "thin shell formalism" is used to make approximations to solutions of Einstein's equation in some limit. Your wormhole is then going to never be an approximate solution. But you don't need to go to all this rigamarole. Every Reissner Nordstrom or Kerr solution is already a wormhole without any approximations or modications, as is known at least since the 1960s, and probably earlier too. $\endgroup$ – Ron Maimon Mar 29 '12 at 17:46
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    $\begingroup$ Solutions obtained with the thin shell formalism are exact, not approximate. The shell of matter is the price paid for the exactness. The wormhole solution described is not interesting merely because it's a wormhole. It's interesting because one would expect that a shell of matter beneath the event horizon must necessarily contract to zero. But this doesn't seem to be the case, if it's okay -- as the formalism seems to imply -- to have a spacelike thin shell -- i.e. a shell that does not move forward in time, but merely exists at a single instant. $\endgroup$ – Belizean Mar 30 '12 at 2:38
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    $\begingroup$ I don't see how this is possible--- if you have a shell of matter at one instant, it looks like it violates local energy momentum conservation, which is a consistency condition in GR. Even if it somehow magically didn't violate energy conservation, this matter would then have to violate energy conditions in a severe way--- I don't think this is a great approach. What's wrong with the rotating or charged wormhole? $\endgroup$ – Ron Maimon Mar 30 '12 at 3:58
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    $\begingroup$ I get it now, this method glues together different known exact solution with "thin shell" of grossly unphysical matter, with negative energies impossible dispersion. You need energy conditions in order to have something that would be considered a GR solution, otherwise any manifold is a solution with some stress-energy, derived from the curvature. $\endgroup$ – Ron Maimon Mar 30 '12 at 15:01
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    $\begingroup$ All horizon-free wormholes violate the NEC, even thin-shell wormholes with timelike junctions. The non-energy-condition-violating matter that I'm referring to is that in my example of an expanding empty universe that suddenly begins to contract.[Incidentally, no one takes the Trace energy condition (violated in neutron stars) or the Strong energy condition (violated by inflation) seriously. Even the Dominant energy condition is violated in expanding/contracting cosmologies.] $\endgroup$ – Belizean Apr 1 '12 at 10:20

These constructions are not valid at all when you are working in the interior of a black hole, because the sphere on which you are doing the pasting is a spacelike sheet, which represents matter which appears at an instant, then disappears.

This is not a violation of energy conservation, because there is no energy positivity. These solutions violate energy conditions, and always must have negative energy. When you make a thin-shell that appears and disappears, it always is an equal amount of negative and positive energy that appears which separates and then annihilates again. This is not completely obvious because the conservation law is for the pseudo-stress energy, which is coordinate dependent and includes the gravitational field, but it is easy to prove that the null energy condition is violated.

To prove this, for the case of a black hole interior, you just have to note that the singularity theorem is avoided--- the interior becomes nonsingular after the wormhole pasting. So folding in the proof of the singularity theorem: the null geodesics pointing outwards from a sphere just-inside the horizon start out focusing, they are losing area, because they are going inward toward the center of the black hole. The moment they hit the reversing surface, the gluing point, they bounce out, and become unfocusing again--- the area of the light-front grows. This means that the bounce surface spreads incoming geodesics outwards, which means it has violated the null energy condition.

The negative/positive aspects of energy in cases where you introduce a momentary spacelike curvature surface are obvious from the fact that any spacetime satisfies Einstein's equations by defining the stress-energy using Einstein equations. The resulting stress energy is covariantly conserved. This means that the momentarily appearing curvature consists of positive and negative energy which can appear and annihilate quickly.

This is not completely trivial to see, because the additive energy in GR has to include gravitational energy, and is a pseudotensor. But the violations of the energy conditions are obvious from the focusing properties of the null geodesics and are coordinate independent.

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  • $\begingroup$ The NEC violation has nothing to do with the question of whether energy is conserved at spacelike junctions. If you take the same wormhole construction, but paste the black hole and white hole sectors together at a radius exterior to the horizon, you will have a timelike junction, clear energy conservation, and still have an NEC violation. Moreover, the question of energy conservation at spacelike junctions persists in the absence of any NEC violations, as in the example of an expanding universe that is shocked into contraction by the momentary appearance of spacelike matter shell. $\endgroup$ – Belizean Apr 1 '12 at 10:22
  • $\begingroup$ @Belizean: That's true, but so what? It is obvious there is negative energy produced because the stuff appears and disappears, and the NEC violation is obvious inside the black hole. Even if you just paste some other nonsingular spacetime, not a white hole, in the interior of a black hole you get a NEC violation (no singularity). But I don't know why you go through all this trouble--- a charged black hole produces a wormhole without any of this hassle, and there is no pasting or shells required, it is just an electrovac wormhole. $\endgroup$ – Ron Maimon Apr 1 '12 at 15:39
  • $\begingroup$ Thin shell wormholes are normally created with timelike spherical junctions. Spherical junctions beneath the horizon are not considered, because one expects the associated shell of matter to ineluctably contract to zero. If spacelike junctions are allowed, as they are in the Then Shell Formalism, junctions beneath the horizon are allowed. The shell need not contract to zero, contrary to expectations. The NEC violations are irrelevant. They occur for any wormhole. The point is not to create a wormhole. It is to reconcile intuition with what is allowed by a venerable GR technique. $\endgroup$ – Belizean Apr 2 '12 at 9:42
  • $\begingroup$ @Belizean: I explained it already--- there is no reconciliation, this technique is busted. It is only useful in cases where the geometry has the gross feature of an energy condition. If you want a wormhole, use a charged or rotating black hole. These are perfectly traversable. $\endgroup$ – Ron Maimon Apr 2 '12 at 12:57
  • $\begingroup$ @RonMaimon: the thin shell formalism is not 'totally busted' timelike and null shells certainly are physical, and they are used to derive physically useful things like the Oppenheimer-Snyder solution and the frame-dragging effect inside a spinning massive body. The most frequent use of spacelike thin shells is to model something like a ''sudden'' explosion in a cosmology. It's an approximation to an explosion, rather than a realistic model of a real one that would propogate subluminally, but there is physical content there in a certain approximation, and if the model isn't taken too literally $\endgroup$ – Jerry Schirmer May 25 '12 at 5:24

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