In 1999, Sean Hayward proposed a framework for general dynamic wormholes, unifying them with black holes. Both are generically defined locally by outer trapping horizons, temporal for wormholes and spatial or null for black and white holes. Thus, wormhole horizons are two-way traversible, while black-hole and white-hole horizons are only one-way traversible. It follows from the Einstein equation that the null energy condition is violated everywhere on a generic wormhole horizon.
It is suggested that quantum inequalities constraining negative energy break down at such horizons. Wormhole dynamics can be developed as for black-hole dynamics, including a reversed second law and a first law involving a definition of wormhole surface gravity. Since the causal nature of a horizon can change, being spatial under positive energy and temporal under sufficient negative energy, black holes and wormholes are interconvertible.
In particular, if a wormhole’s negative-energy source fails, it may collapse into a black hole. Conversely, irradiating a black-hole horizon with negative energy could convert it into a wormhole horizon. This also suggests a possible final state of black-hole evaporation: a stationary wormhole. This framework proposed by Hayward allowed a fully dynamical description of the operation of a wormhole for practical transport, including the back-reaction of the transported matter on the wormhole. Lets have a closer look.
Fig.1 depicts Penrose diagrams of (i) the Schwarzschild space-time, the unique vacuum black-hole solution, which is static (or you can say black hole in simple terms); (ii) a modification in which the trapped regions have shrunk spatially so that the trapping horizons become temporal (which is dynamic wormhole); and (iii) a further modification in which the trapped regions have shrunk away completely and the trapping horizons have coalesced, yielding a static Morris-Thorne wormhole. Then (ii) is a dynamic wormhole by either the current definition or the global definition that an observer may pass from one asymptotic region to the other in either direction. This example illustrates that wormholes may contain either future or past trapped surfaces, the boundaries being either future or past trapping horizons.
The important question answered by Hayward is, static wormhole can turn into a black hole if its negative energy generator fails, but how?
Well, this can be seen as follows, fixing spherical symmetry. Suppose that a static wormhole like (iii) (given in above figure) has been constructed, necessarily maintained by negative energy, and that the negative energy subsequently weakens. Then the horizon generally bifurcates, opening up a future trapped region and forming a dynamic wormhole locally like (ii) (given in Figure). If the energy disperses smoothly and completely to leave vacuum, the solution must become Schwarzschild like (i), the horizons smoothly becoming null as in Fig.2. Thus, the wormhole has collapsed into a black hole.
Alternatively, if some positive-energy matter subsequently crosses the horizon, it becomes spatial, corresponding to a dynamic black hole. Then again, if starfleet engineers succeed in repairing the negative-energy generator, the horizon becomes temporal again and the wormhole can be restored. In this case, the wormhole has become a black hole and then a wormhole again, by any local reckoning. This makes it clear that wormhole physics cannot ignore black holes. Any actual wormhole would be in danger of becoming a black hole if the negative-energy source failed, or was overwhelmed by normal positive-energy matter. An anti-wormhole weapon could therefore be made of any sufficiently concentrated mass.
Conversely, if negative energy is appreciable, blackhole physics cannot ignore wormholes. Irradiating a black-hole horizon with negative energy would convert it into a wormhole horizon as in Fig.3. This might be a practical way to make a wormhole, though Astrophysical black holes, presumably formed by gravitational collapse, are not expected to have the right topology for global traversibility. Black holes formed by quantum fluctuations, either in the early Universe or by some hypothetical quantum engineering, may well have wormhole topology, judging by the simplest stationary solutions. This at least suggests further science fiction plots. Another such plot stems from the fact that an explorer falling into a black hole may not be trapped after all, but could escape if rescuers could beam enough negative energy into the hole quickly enough.
A theoretical example of black-hole-to-wormhole conversion already exists: black-hole evaporation. In this case, the negative-energy material is just the Hawking radiation in semi-classical theory, which is expected to cause the trapping horizons to become temporal and shrink. Locally these are future wormhole horizons. The final state of black-hole evaporation has been the subject of much discussion, to which another possibility is now suggested: an initially stationary black hole (of wormhole topology like Schwarzschild) might evaporate to leave a finally stationary wormhole. This would be possible if the particle production decreases as the trapping horizon shrinks, allowing the two horizons to approach slowly and asymptotically coalesce as in Fig.3. The existence of such a final state requires a stationary wormhole which is a self-consistent semi-classical solution, with particle production on the wormhole background providing the negative-energy matter supporting the wormhole. Such solutions do indeed exist, according to Hochberg et al.. Hochberg & Kephart also mentioned the possibility of wormholes formed by black-hole evaporation. This might also resolve the black-hole information puzzle, since the trapped region shrinks to nothing and its contents must therefore re-emerge.
A practical problem stems from the fact that the actual use of a wormhole for transport would involve mundane positive-energy matter traversing it: too much such transport would convert the wormhole into a black hole. Keeping the wormhole viable, defying its natural fate as a black hole, would require additional negative energy to balance the transported matter. Previous work has been unable to deal with this consistently, instead ignoring the back-reaction of the transported matter on the wormhole. The framework given by Hayward allows a consistent description of wormhole operation, sketched as follows.
During operation, an initially stationary wormhole horizon would bifurcate, opening up a trapped region, but could be closed up to a stationary state again by a careful balance of positive and negative energy, as in Fig.4. The temporary wormhole region could be either future or past trapped, depending on whether the positive-energy matter is sent in before or after the compensating negative energy, respectively. After operation, the wormhole would be respectively smaller or larger. Consequently an alternating process is suggested, sandwiching the transported matter between bursts of negative energy, so that the wormhole is kept tight when not in use and dilated as necessary.
If the negative-energy source fails during the dynamical stage, the wormhole would become locally a black hole or white hole, respectively. In the latter case, the white-hole horizons would presumably collapse towards each other, meet and pass through each other as in the Schwarzschild space-time, leaving a black hole again. Either way, wormhole experiments would risk accidental creation of black holes, subsequently littering the universe.
So, guys for today we have to stop here, we will continue this in next part of this article.
Featured image: Artist’s conception of a wormhole+blackhole. (Image: © Shutterstock)
Reference: Sean Hayward, “Dynamic Wormholes”, International Journal of Modern Physics D, Vol. 08, No. 03, pp. 373-382 (1999). https://www.worldscientific.com/doi/abs/10.1142/S0218271899000286 https://doi.org/10.1142/S0218271899000286
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