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| Fig. 1: Visualization of AdS3/CFT2 space. (Source: G. Anikeeva) |
One of the main paradoxes while developing a theory of quantum gravity is how to resolve the tension between the information loss that we expect in black holes with the understanding that no information is lost in a quantum theory. This is the setup where information goes into the black hole at finite time, the black hole evaporates at a large time, and we check if we can recover the information after the black hole disappears. The calculations done by Hawking showed that this is not possible. [1]
The holographic principle, and AdS/CFT duality in particular, has emerged recently as the most promising theory of quantum gravity. It allows us to map a quantum gravitational picture into a pure quantum field theory without gravity and vice versa. This makes preparation and analysis simpler, as we have more options to work with.
We present a review of Akers et al. that provides two holographic models of black hole evaporation. [2] While superficially similar, one of these models predicts information loss, while the other does not. They attempt to clarify why recent models of black holes seem to conserve information, in apparent contradiction to Hawking's famous calculations. In particular, they keep the relation between the bulk geometry and the boundary explicit.
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| Fig. 2: Examples of wormhole and multi-boundary wormhole. |
AdS/CFT duality, and more generally the gauge/gravity duality, is the idea that for every theory of gravity, there is a dual quantum field theory "in the boundary", and most concepts on one have a mapping in the other one. [3] In it's most concrete iteration, it shows that we can obtain a conformal field theory (CFT) dual to the Anti-de Sitter (AdS) space. In particular, we have a duality between operators in the CFT and AdS. One way to construct this operator correspondence is by taking the large radius limit of AdS.
Thus we have quantum gravity in asymptotically AdS space in the center (bulk) and a quantum field theory without gravity in the boundary. We will be working with AdS3/CFT2, as visualized in Fig. 1, which means that our theory will have two space dimensions plus one time dimension in the bulk, and one space dimension plus one time dimension in the boundary. [4] A nice characteristic of this model is that gravitational waves do not exist, simplifying a lot of the needed results.
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| Fig. 3: Page curve of the information conservation case. [11] |
In AdS space there are two kinds of stationary (Schwarschild) black holes. We have small black holes, that is black holes with radius smaller than the AdS scale L = LAdS, which are thermodynamically unstable. They evaporate but do so in an extreme way. [5] The techniques in the original calculations require slow evolution of the black hole. This makes small black holes not suitable for the experiment, since we need it to be parallel to the calculations proposed by Hawking. [1] To set up the models we need slow dynamics, since the techniques rely on this. Additionally, this instability means they have not been studied as much.
We also have large black holes, that is black holes with radius bigger than the AdS scale, which are stable. These which reabsorb radiation faster than they evaporate, and reach a thermal equilibrium. Thus they do not disappear at late times. This stability makes their connection to CFT simpler.
We want to be working with these latter black holes. To induce evaporation, we will couple the CFT with an external system, so that radiation does not bounce back. More precisely, we will be connecting a main CFT additional CFTs to create stable wormholes between them, shifting the thermal equilibrium. We will be manually driving evaporation forward through each coupling step.
A large black hole in AdS at temperature T has as a dual in the CFT the thermal density state [6]
| ρTDS = | ∑ i |
pi |Ψi⟩⟨Ψi| ∝ | e−βEi |Ψi⟩⟨Ψi| |
Here β−1 = kBT and kB is the Boltzmann constant, and |Ψi⟩ is the eigenstate of the CFT at energy Ei.
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| Fig. 4: Evolution of the information conservation case. Note the multi-boundary wormhole. |
Wormholes can also be viewed through the lens of AdS/CFT. In particular, Schwarzschild wormhole in the AdS/CFT view are understood to be duals of the thermofield double state. That is, given two identical CFTs, we can construct the entangled state [7]
| |TFD⟩ = | ∑ i |
e−βEi |Ψi⟩0 |Ψ i⟩1 |
where the reduced density matrix on each side is given by the thermal density state. This backs up the interpretation of the state as a wormhole. This interpretation is the core of the conjecture ER = EPR. [8]
One of the tools we use to construct one of the two models are multi-boundary wormholes. [9] The extra mouths of the wormhole, from the point of view of these models, will be created from black hole evaporation. In the CFT picture, this is a multi-CFT generalization of the thermofield double. Note that the above describes a wormhole where each mouth of the wormhole is of the same radius. The corresponding expression for more general geometries is more complicated. Invoking the duality will give us great leverage here. Fig. 2 shows us a visualization of wormholes.
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| Fig. 5: Page curve of the information loss case. [11 |
Note that by creating wormholes between the CFTs, we will be placing black holes in the incoming CFTs. The wormholes will be prepared in such a way that the black holes in the new CFTs are exactly the size needed to be stable.
We present two alternative models for the information paradox in black holes. One of our models will preserve information and the other will lose it. Recall that the relevant metric to measure is the entropy of the state outside the black hole. In both cases this state will start pure. We are interested in the entropy of the state outside the black hole at late times. In the information conservation case this will be approximately zero at late times, while in the information loss case the late-time entropy will be high at late times.
As mentioned before, we will be manually driving evaporation forward by coupling the initial black hole with an additional CFT through a worm hole. In one model, the worm hole will be a multi-bundary shared wormhole, and in the other there will be independent worm holes.
We avoid direct entropy calculation, and instead strive to get a result purely from the geometry. [10] For this, we work directly within AdS3/CFT2. As mentioned before, the lack of gravitational waves simplifies the results. In particular, quantum gravitational corrections go away, and we find that the entropy is the length of the minimum cut separating our main black hole from the rest.
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| Fig. 6: Evolution of the information loss case. Note that the wormholes are disjoint. |
In the information conservation, at each time step the black hole evaporation process produces an additional boundary in a wormhole. The possible cuts are "entrances," where the wormhole has the opening towards the main black hole, and the alternative is the union of the additional boundaries. Through time, the minimum between these two first increases, then decreases and reaches 0 again. We interpret this as information conservation at late times. The first steps of this process are shown in Fig. 4. Through the entropy-energy relation and conservation of energy, we can calculate the length of the entrance in the main CFT, shown in the picture. We can see the plot of the entropy versus time in Fig. 3.
In the information loss model, at each time step the black hole evaporation produces an additional copy of the thermofield double state. The relevant cut is the union of the cross-section of each thermofield double. The total area is constant through time. We interpret this as the traditional information loss computation. The first steps of this process are shown in Fig. 6. Through the entropy-energy relation and conservation of energy, we can calculate the length of the entrance in the main CFT, shown in the picture. Note that the lengths are the same as the other case, but the connectivity is different. We can see the plot of the entropy versus time in Fig. 5.
© Galit Anikeeva. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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