Replicas of entangled black holes: the solution to the information paradox

In 1997, physicist John Preskill bet the great Stephen Hawking and the renowned physicist Kip Thorn that information was not destroyed in black holes. Imagining a universe with "sinks" where information was lost forever seemed inconceivable and contradicted several fundamental laws of physics. However, Hawking's calculation showing that information was destroyed in black holes seemed impeccable, and no one had found a single error in his approach. Many physicists believe that the solution to this paradox will bring with it a new way of understanding space-time at its fundamental level. Almost five decades after the paradox was first posed, we are finally beginning to glimpse the solution, and as many predicted, this solution may entail a paradigm shift. In this article, we will analyze the recent and exciting work that is beginning to create a fascinating new vision of the fundamental dynamics of space-time.

Holography and Ryu-Takayanagi surfaces

As we saw in this article, the new proposal to solve the famous information paradox in black holes involves concepts of holography and so-called Ryu-Takayanagi surfaces. In this section, we will briefly review these concepts.

An AdS (Anti-deSitter) space-time such as that inside a black hole (BH) can be represented in conformal coordinates as a cylinder in which the vertical axis represents time and the horizontal axis the AdS space. By the holographic principle, we obtain that the information of the AdS geometry of the BH must be encoded in a quantum system (CFT) at the asymptotic edge, i.e., at the edge of the cylinder:

If we make a section of the cylinder at a given time we obtain:

Let's now consider two CFTs: the CFTan corresponding to the states inside the AN, and the CFTrad corresponding to the Hawking radiation states. The so-called Ryu-Takayanagi (RT) surface is proportional to the entanglement entropy between the two systems and is defined as the extremal surface of both systems with the lowest entropy. Therefore, the RT surface represents a "junction zone" between the two CFTs.

Once the AN begins to evaporate the entropy of the AN begins to decrease while the entropy associated with Hawking radiation begins to increase.

The Page time is the instant at which the black hole's entropy equals the entropy of Hawking radiation. Therefore, before the Page time, the entropy of the tanCF is greater than that of the tradCF, and vice versa.

The RT surface is the minimum surface of both systems, that is, the area where the entropy is lowest. Before the page time the area of lowest entropy is any empty surface outside the AN, therefore, in this period there is no RT surface (this surface is zero) and both CFTs are "disconnected", that is, they are independent.

However, in Page time, something extraordinary happens: the region of least entanglement is now inside the black hole. Therefore, the RT surface is no longer zero. To understand what's happening, we must plot the RT surface in the CFTan. The RT surface associated with two boundary points is described as the minimum area covered by a path in AdS space between those points that contains the boundary between A and B. The minimum area in AdS space is the shortest path (the geodesic) between A and B:

Now, an observer within AdS space who could access the information of the AN CFT would find that this CFT no longer contains the information of the RT surface region. Therefore, the red line behaves like a holographic boundary surrounding an "island." The information of this "island" no longer belongs to the black hole CFT but to the external Hawking radiation CFT:

This means that from time p to 1, the RT surface of the CFTrad is inside the NA, and therefore, the Hawking radiation contains information from the interior of the black hole. The information has escaped from the black hole! But how is this possible? How does information actually escape from the NA?

A fundamental part of this whole scheme is missing: the formula for RT can be obtained independently using a method called "black hole replicas." In fact, these replicas can be considered the very origin of this formula. As we will see in the next section, replicas may provide the ultimate solution to the information paradox.

Replicas of black holes and wormholes

What would happen if, instead of considering a single black hole, I considered a set of n disconnected black holes? At first glance, this seems very strange, but entropy is a statistical quantity, meaning it only makes sense for a large number of measurements. Therefore, instead of measuring the same black hole n times, I consider the measurement of n identical black holes. This method is called black hole replicas. To perform this calculation, we must use the most widely used mathematical tool in quantum gravity approximations: the gravitational path integral:

Essentially, this expression means summing all possible geometries that begin in state A and end in state B, weighted by the value of their action. In general, this sum involves infinite terms and is extremely difficult to compute. One way to solve it is to find the terms in the sum that contribute the most and discard those whose contribution is negligible.

We will call psi(i) the states inside the black hole, and i(R) the states of a hypothetical auxiliary system outside where the Hawking radiation emitted by the black hole is stored. Thus, the combined state of the two systems will be:

The so-called density matrix allows us to measure the degree of entanglement between two quantum systems. The density matrix corresponding to the previous system is:

The states psi(i) and psi(j) are gravitational amplitudes. There is a formulation called "JT gravity" where calculations in a four-dimensional gravitational system can be reduced by holography and dimension reduction methods.

to a two-dimensional CFT . In this formulation, the gravitational states psi(i) and psi(j) correspond to the initial and final conditions at the holographic boundary. This can be represented schematically as:

The black line represents the CFT boundary, and the arrow represents the time evolution from state j to state i. To solve the gravitational path integral, we must sum up all possible paths between "j" and "i" through the portion of AdS spacetime (the "bulk") corresponding to this CFT boundary. Since this sum is weighted, there are terms that contribute significantly and others that are practically negligible. The term that contributes the most to the sum is the one corresponding to the two-dimensional area (the minimum area) contained by a geodesic between je and i :

This is the dominant term that Hawking included in his calculations and represents an isolated non-entangled CFT in which all states are orthogonal.

To calculate the entanglement entropy between the black hole and the emitted Hawking radiation we consider two copies of our CFT, one representing the CFT of the interior of the black hole and the other representing the CFT of the emitted radiation:

Next, we pose the key question: In two dimensions, how many ways are there to holographically "fill" the two boundaries with a complementary AdS space-time? The answer is not just one way, as Hawking considered, but two different ways:

The first form on the left is equivalent to the "standard" configuration Hawking included in his calculations and corresponds to two disconnected CFTs with a topology equivalent to that of two disks. The second form corresponds to two connected CFTs with a single-disk topology, whose geometry is equivalent to that of two spacetimes connected by a wormhole.

Next we ask ourselves: What is the consequence of including the contribution of the second configuration in the calculations?

In our JT gravity model, we can calculate the total entropy by considering a magnitude that measures the degree of entanglement. Thus, we obtain that the total contribution to entropy of the two previous configurations is:

Where k represents the number of entangled microstates and Z 1 -Z 2 are the path integrals corresponding to each of the two configurations. The numerator of the middle quotient contains the contributions from both configurations (the denominator factor is just a normalization factor). The first value kZ 1 2 corresponds to the first configuration without entanglement since we have a single loop in k and two identical and independent configurations Z 1 . The second value k 2Z 2 corresponds to the second configuration with a wormhole since we have two loops in k and a single geometric configuration Z 2 . The sums Z 1 and Z 2 depend on the topology, their contributions are proportional to the following factor:

Where So is the action in "JT gravity" and X is a factor called the "Euler characteristic." In the case of a disk, this factor equals 1. Therefore, we have:

And we can conclude that:

This expression represents our key conclusion: in the early stages of evaporation when k is small, the first term representing the first configuration dominates, but when the black hole is already old and k is large, the second term representing the second configuration dominates. This tells us that after the Page time, the black hole information starts to escape because the system The predominant feature in the gravitational path integral is that of a configuration of two space-times connected by a wormhole.

As we saw in this article , entanglement is the "glue" of space-time. When entanglement increases between two disconnected space-times, they begin to connect and "merge" to form a single geometry. This may be the key to solving the information paradox!

Intertwined AN replicas: Towards a paradigm shift?

The solution to the information paradox presented in this article, although not yet definitive, is the most promising of all those proposed. The interpretation of this result is that the information paradox can only be resolved if we consider a sum of geometries containing identical black holes. More specifically, what we measure is a weighted average of a sum of geometries. As entanglement entropy increases, disconnected geometries become connected through a wormhole. In fact, it is the formation of a wormhole that allows the appearance of an island inside the black hole in Page time and therefore allows information to escape from the black hole.

Finally, a personal opinion: from a conceptual point of view, the picture we have described in this article is much closer to the interpretation of quantum mechanics as a Feynmann sum of paths, or to a vision similar to Everett's many worlds, or even to the Multiverse, than to more "orthodox" interpretations such as the Copenhagen interpretation. Therefore, from this perspective, we can affirm that the resolution of the information paradox in black holes is very likely to represent a paradigm shift in our overall view of the Universe we inhabit.

Sources:

Entanglement wedge reconstruction and the information paradox , Replica wormholes and the black hole interior