Can holography provide a complete description of a space-time?

[Heidi]

I read that the parameters of each these two theories can be obtained from parameters of the other theory and if one event may appear in ads5 with a given probability there is another event in the CFT side which will get the same probability.
But is there a one to one map between the two space time events?

 

[Demystifier]

But is there a one to one map between the two space time events?

Probably no. It is believed that quantum gravity is somehow nonlocal, so that physical events are not associated with spacetime points. But I think it's fair to say that nobody really understands what exactly does it mean. The bulk reconstruction problem is still unsolved.

 

[mitchell porter]

Just a comment on how the correspondence works.

In QFT in flat Minkowski space, there is an "S-matrix" describing scattering events, in which particles come from various directions, interact when they are close, and then the products of the interaction move apart in various directions.

You can perhaps visualize this by imagining particles moving towards you from various directions, interacting, and then moving away from you in various directions.

There is now a version of holographic duality for Minkowski space, called "celestial holography", in which this S-matrix is equivalent to correlation functions in a QFT that exists "on the celestial sphere", i.e. a QFT defined on a sphere that surrounds you at a great distance.

The point is that "particle approaching you from a particular direction", corresponds to an event on the distant sphere, at the point that lies in that direction. Similarly for "particle moving away from you in a particular direction".

And so to compute e.g. the probability that a particle coming from above you, collides with a particle coming from below you, and produces particles that move away from you horizontally... you would use a correlation function on the celestial sphere, with inputs at the north and south poles, and outputs at various places on the celestial equator. But the calculation would only consider processes "on" the celestial sphere, and not "in" it.

I don't know how clear that is. But if it has been understood... then AdS/CFT works in a similar way. The AdS theory has an S-matrix too, with particles entering and exiting in specific directions. These asymptotic directions correspond to points on the boundary, and the dual CFT computes these scattering probabilities via correlation functions on the boundary.

So this is the 1-to-1 map that is understood. The harder part, as @Demystifier suggests, is the "bulk reconstruction problem", i.e. saying something about the interior of AdS space, and how that maps to the boundary.

If you read the literature, you may find drawings where a point in the interior, is the tip of a wedge which ends in some finite region of the boundary. Typically, the end of the wedge consists of all those points on the boundary that are spacelike separated from the interior point, and the idea is that the bulk physics at that interior point, corresponds to some kind of weighted sum over the boundary physics throughout the wedge. (If you like Plato's cave, you could think of the wedge as the shadow cast on the boundary, by the point in the interior.)

A lot of work has been done on this and similar mappings (David Berenstein might be one author worth reading), but it's clearly still a work in progress. Nonetheless, this is the potentially profound part of the holographic principle, telling us how a spatial dimension is built up from entanglement, and similar things.

 

[Mattergauge]

Just a comment on how the correspondence works.

In QFT in flat Minkowski space, there is an "S-matrix" describing scattering events, in which particles come from various directions, interact when they are close, and then the products of the interaction move apart in various directions.

You can perhaps visualize this by imagining particles moving towards you from various directions, interacting, and then moving away from you in various directions.

There is now a version of holographic duality for Minkowski space, called "celestial holography", in which this S-matrix is equivalent to correlation functions in a QFT that exists "on the celestial sphere", i.e. a QFT defined on a sphere that surrounds you at a great distance.

The point is that "particle approaching you from a particular direction", corresponds to an event on the distant sphere, at the point that lies in that direction. Similarly for "particle moving away from you in a particular direction".

And so to compute e.g. the probability that a particle coming from above you, collides with a particle coming from below you, and produces particles that move away from you horizontally... you would use a correlation function on the celestial sphere, with inputs at the north and south poles, and outputs at various places on the celestial equator. But the calculation would only consider processes "on" the celestial sphere, and not "in" it.

I don't know how clear that is. But if it has been understood... then AdS/CFT works in a similar way. The AdS theory has an S-matrix too, with particles entering and exiting in specific directions. These asymptotic directions correspond to points on the boundary, and the dual CFT computes these scattering probabilities via correlation functions on the boundary.

So this is the 1-to-1 map that is understood. The harder part, as @Demystifier suggests, is the "bulk reconstruction problem", i.e. saying something about the interior of AdS space, and how that maps to the boundary.

If you read the literature, you may find drawings where a point in the interior, is the tip of a wedge which ends in some finite region of the boundary. Typically, the end of the wedge consists of all those points on the boundary that are spacelike separated from the interior point, and the idea is that the bulk physics at that interior point, corresponds to some kind of weighted sum over the boundary physics throughout the wedge. (If you like Plato's cave, you could think of the wedge as the shadow cast on the boundary, by the point in the interior.)

A lot of work has been done on this and similar mappings (David Berenstein might be one author worth reading), but it's clearly still a work in progress. Nonetheless, this is the potentially profound part of the holographic principle, telling us how a spatial dimension is built up from entanglement, and similar things.

Aren't virtual quantum fields on or just above the horizon of a black hole, entangled with the interior of the hole, to ensure unitarity, to conserve information, and as such, Hawing radiation can take away this information? So, if the hole were formed from a huge 3d bike, the information would be different than if another material form collapsed, say a huge collection of books.

The difference is though, that the stuff on the horizon is a consequence of the matter inside, while in the ADS5/CFT4 the stuff on the surface is primary, and directing the stuff inside. But how can the velocity of a particle on the inside be changed by stuff on the outside? Which of the two descriptions is the real one? When the inside stuff is entangled with the surface, which gives rise to what? What if another surface is considered? Can the inside be part of the surface of another inside?

 

[mitchell porter]

Aren't virtual quantum fields on or just above the horizon of a black hole, entangled with the interior of the hole, to ensure unitarity, to conserve information, and as such, Hawing radiation can take away this information? So, if the hole were formed from a huge 3d bike, the information would be different than if another material form collapsed, say a huge collection of books.

The difference is though, that the stuff on the horizon is a consequence of the matter inside, while in the ADS5/CFT4 the stuff on the surface is primary, and directing the stuff inside. But how can the velocity of a particle on the inside be changed by stuff on the outside? Which of the two descriptions is the real one? When the inside stuff is entangled with the surface, which gives rise to what? What if another surface is considered? Can the inside be part of the surface of another inside?

In a holographic duality like AdS/CFT, it's not the case that "the stuff on the surface is... directing the stuff inside". AdS and CFT are supposed to be two equally valid descriptions of the same thing.

The clearest examples of holography also involve a holographically compressed description of an entire space-time. AdS/CFT, celestial holography for Minkowski space, possibly dS/CFT too.

The event horizon of a black hole is a surface within a space-time, and is not per se holographic. It's just a division of space-time into two parts, and you can have entanglement across that division.

I think it was Susskind who proposed something called a "stretched membrane" model of a black hole. This would be a hot surface just above the horizon, with temperature equal to the Hawking temperature. This is an example of a holographically compressed description of part of a space-time; the black hole within the stretched membrane.

There are other examples of this. There are modifications of AdS/CFT, in which a truncated CFT describes just part of a dual AdS space. It may be possible to take an arbitrary region of a space-time with gravity, and describe a QFT on the boundary of that region, that is holographically equivalent to what goes on inside. Or perhaps it can only be done for certain special submanifolds of space-time. Relevant work may come from Jacobson, Bousso, maybe Tom Banks.

Regarding whether the exact physical state of a black hole depends on the details of what went into it... In Hawking's original model, he accepted the conclusion of classical general relativity, that the only properties of the black hole are mass, spin, charge. That model implies that quantum information (along with coarse macroscopic information, like bike vs books) is lost.

The majority opinion now is that black holes have some kind of microstructure that preserves quantum information, and that correlations in the Hawking radiation still carry this information too. But there isn't a consensus about how this works. The fuzzball model of black holes says that the black hole isn't compressed down to almost a point, it's actually some kind of extended structure all the way out to the event horizon. Strominger and Pasterski say that the black hole has previously unnoticed symmetries ("soft hair") and this is where the correlations are stored. String theory has described various types of black hole as bundles of branes or tangles of string. The idea of multiple wormholes behind the horizon leading to other places, perhaps baby universes, has arisen in the study of the "Page curve" describing the intensity of Hawking radiation over time, but it's unclear whether these wormholes are real, or just a fictitious duplication of the black hole, for the sake of a calculation (this method is called the "replica trick").