Membrane paradigm

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In black hole theory, the black hole membrane paradigm is a useful "toy model" method or "engineering approach" for visualising and calculating the effects predicted by quantum mechanics for the exterior physics of black holes, without using quantum-mechanical principles or calculations. It models a black hole as a thin classically-radiating surface (or membrane) at or vanishingly close to the black hole's event horizon.

The results of the membrane paradigm are generally considered to be "safe".

An extension to the membrane idea, the holographic principle, has been suggested as a possible solution to the black hole information paradox. As of 2004, this later work is still under intense review, and there continue to be strong disagreements between respected researchers in the field as to the correct way of interpreting and presenting these sorts of extended arguments.

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Electrical resistance

Thorne (1994) relates that this approach to studying black holes was prompted by the realisation by Hanni, Ruffini, Wald and Cohen in the early 1970's that since an electrically-charged pellet dropped into a black hole should still appear to a distant outsider to be remaining just outside the critical r=2M radius, if its image persists, its electrical fieldlines ought to persist too, and ought to point to the location of the "frozen" image (1994, pp.406). If the black hole rotates, and the image of the pellet is pulled around, the associated electrical fieldlines ought to be pulled around with it to create basic "electrical dynamo" effects (see: dynamo theory).

Further calculations yielded properties for a black hole such as apparent electrical resistance (pp.408). Since these fieldline properties seemed to be exhibited down to the event horizon, and general relativity insisted that no dynamic exterior interactions could extend through the horizon, it was considered convenient to invent a surface at the horizon that these electrical properties could be said to belong to.

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Hawking radiation

After being introduced to model the theoretical electrical characteristics of the horizon, the "membrane" approach was then pressed into service to model the Hawking radiation effect predicted by quantum mechanics.

In the coordinate system of a distant stationary observer, Hawking radiation tends to be described as a quantum-mechanical particle-pair production effect (involving "virtual" particles), but for stationary observers hovering nearer to the hole, the effect is supposed to look like a purely conventional radiation effect involving "real" particles. In the "membrane paradigm", the black hole is described as it should be seen by an array of these stationary, suspended noninertial observers, and since their shared coordinate system ends at r=2M (because an observer cannot legally hover at or below the event horizon under general relativity), this conventional-looking radiation is described as being emitted by an arbitrarily-thin shell of "hot" material at or just above the critical r=2M radius, where this coordinate system fails.

As in the "electrical" case, the membrane paradigm is useful because these effects should appear all the way down to the event horizon, but are not allowed by GR to be coming through the horizon – blaming them on a hypothetical thin radiating membrane at the horizon allows them to be modelled classically without explicitly contradicting general relativity's prediction that the r=2M surface is inescapable.

In 1986, Kip S. Thorne, R. H. Price and D. H. Macdonald published an anthology of papers by various authors that examined this idea: "Black Holes: The membrane paradigm".

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"Observerspace" arguments

Although matter falling into a black hole is considered to fall through the horizon, a distant outside observer watching an infalling object should see the increase in gravitational redshift in the object's signals making it appear to slow as it approaches the horizon, with the apparent speed tending towards zero at r=2M — the object always appears to be reducing the distance between itself and the horizon, but never quite reaching the surface.

For a structure seen to be approaching the horizon, the difference in gravitational redshift between its upper and lower parts makes the deepest part of the object appear to be falling more slowly, and makes the object seem to be compacting to zero thickness as it approaches the event horizon. The combination of effects gives a description in which matter that has already passed through the event horizon is seen by distant outsiders as an apparent arbitrarily-thin film of material constantly converging on r=2M.

Treating direct observation as literal reality does not always give a complete description of a situation, but useful insights can still be gained by asking how the physics of these objects might operate if all the material of the black hole really was in the form of a thin film of material at the event horizon.

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Holographic arguments

Leonard Susskind has since taken this argument further, in the context of the black hole information paradox.

If we define a volume of space and draw a boundary surface around it, then the interaction of the entire "interior" physics of the region with its surrounding environment can be said to take place at (or through) this boundary. The external interactions of the volume are then the interactions of the surface, and if all information regarding the internal physics has some point passed through the surface, we can in theory construct a consistent description of the surface physics that explains everything that an external observer sees happening inside the volume, without the volume actually existing.

If a two-dimensional surface is capable of reproducing the exterior physics of a three-dimensional volume (the holographic principle), then it should be possible to represent the physics of how a black hole interacts with the outside universe by creating a suitable description of a bounding region around it … in this case, the obvious surface to use is the surface at or directly above the event horizon.

This has been claimed as a possible solution to the black hole information paradox: if quantum mechanics says that black holes radiate, and general relativity says that no information passes outward through the horizon (leaving us to wonder where information encoded in the HR came from, and where the infallen radiation went to), "holographic" arguments can claim that the information never really fell into the hole but was imprinted on its horizon, and that interactions within this special surface then conditioned the pattern of outgoing Hawking radiation leaving the region, so that information carried by escaping Hawking radiation originates in material that fell into the black hole.

In this description, matter and energy appear to fall into the hole, and to be ejected from the hole, but can be described as "really" being absorbed and ejected from the horizon surface.

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The holographic principle in action

For the sake of argument, suppose that a spaceship somehow violates GR rules by descending through a gravitational event horizon and then escaping (perhaps by jettisoning most of its mass, or activating a hypothetical warp drive). This might be possible under a pre-GR dark star model, but not with a black hole under general relativity.

The holographic principle can take this "GR-illegal" behaviour and make it seem reasonable: it can be argued that the spaceship's information is absorbed as it passes inward through the horizon, and Susskind's horizon then mechanistically convolves and combines the spaceship's information with all the rest of the information embedded in the surface describing the black hole's supposed interior. The spaceship's entry causes a ripple of information through the surface that converges at a new point where the surface ejects an "extrapolated" version of the same information, and uses the Hawking radiaion effect to spit out a suspiciously similar looking spaceship.

This "new" spaceship has the same basic dimensions as the original, it has the same number of crew, its chronometers show a suitably advanced time, and its databanks hold what appears to be convincing video recordings of the ship entering and successfully escaping the black hole horizon. The ship perhaps also shows some charring and damage that appears to correspond to events shown in the video recordings, and the ship's crew claim to have a memory of approaching the horizon, passing through, and exiting again.

In this situation, the holographic principle allows us to agree that physics at a black hole can appear to operate according to a different set of classical laws to general relativity, without forcing us to explicitly disagree with GR's assertion that no information "really" passes outward through r=2M. One could agree that information seems to pass in and out of the black hole, and that all our measurements could indicate that this is what happens, but still claim that general relativity has not irretrievably broken down.

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References

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