Magical wormholes, more than just science fiction

Many people must have heard of "wormholes", whether from the fantasy of traveling through time and space from a science fiction perspective, or from the academic news at the forefront of theoretical physics, but what exactly is a wormhole? How does it become a structure connecting time and space? Is it just a toy for physicists? In fact, in recent years, in the study of quantum gravity, wormholes have hidden deep meanings that we have not yet discovered.

Written by | An Yusen

A wormhole is a magical space-time structure. At the same time, research in physics has increasingly proved that wormholes are the key to connecting quantum theory and gravitational theory. This article intends to introduce the basic concept of wormholes and their role in theoretical physics from two aspects: Lorentz wormholes (including time and space) and Euclidean wormholes.

Lorentz wormhole

First, we introduce the Lorentz wormhole. The Lorentz wormhole is a wormhole structure that may exist in space-time, and it is a real physical object.

The earliest research on dragon holes was inspired by Carl Sagan's novel Contact, which was also successfully adapted into a film and television series. The film Contact directed by Robert Zemeckis was well received. In the original manuscript of the novel, the author used black holes to realize space-time tunnels. However, his friend Kip Throne expressed his concerns. As a researcher of general relativity, he knew that black holes were difficult to use as space-time tunnels. However, this aroused Kip Throne's research interest, and he later carried out a series of initial studies on dragon holes.

Wormholes and energy conditions

Time travel is an eternal interest of science fiction fans, and traversable dragon holes seem to be a good way to achieve it. Therefore, an important aspect of dragon hole research is to study its traversability. In the usual general relativity research, a material distribution is known, and then the space-time shape given by this material distribution is studied; however, in wormhole research, the goal of physicists is to achieve a specific space-time shape - so Morris and Throne considered doing the opposite, first giving restrictions on the space-time structure, and then solving the material distribution through the Einstein field equation.

The initial calculations were performed in a spherically symmetric coordinate system. They found that if a specific wormhole spacetime structure is to be satisfied, the required material distribution must violate the energy condition. In layman's terms, it is necessary to introduce strange negative energy matter [1]. This can be seen naturally through the method of geodesic sinks. Generally, in general relativity, in order to explore some properties of spacetime, some conclusions can be drawn through the changes in geodesic sinks without solving Einstein's equations. For example, if a wormhole structure is required to connect two different spacetime regions and enable traversal, then the light passing through it needs to first converge to the throat of the wormhole (that is, the narrowest part of the wormhole structure) and then emit from the throat. In general relativity, the convergence or divergence of light can be given by the expansion of a light-like geodesic sink. The equation describing it is usually called the Ray-Chaudhuri equation, which is as follows:

We can choose a line sink that satisfies both rotation and shear of 0, σ=ω=0. Based on the characteristics of the line sink passing through the hole, we know that there must be a position where dθ/dλ=0 at the throat of the hole, which implies the following equation:

According to the general theory of relativity,

This destroys the light-like energy condition, so the existence of a dragon hole requires the introduction of negative energy exotic matter into its throat.

The introduction of this exotic matter makes the construction of a hole very difficult. This kind of matter that violates the light-like energy condition is generally only allowed in quantum theory, and is usually very tiny. At the same time, if the hole is passable, we also need to consider the tidal effect of the hole on the human body. Under the condition of tidal force that the human body can tolerate, the theory predicts that the hole will be very large, and it is even more difficult to support such a huge space with exotic matter. However, perhaps as the science fiction novel "The Three-Body Problem" imagines, an infinitely developed civilization can do anything under the conditions allowed by the laws of physics and without the restrictions of technical barriers - things like building a hole are still imaginable.

Wormholes and time machines

Since a dragon hole can be seen as a shortcut between two distant points in the universe, perhaps a dragon hole can be transformed into a time machine. [2] In the discussion of the time machine, we ignore some details and only regard the dragon hole as a machine connecting two points (t, 0) and (t, L) in spacetime. The entrance of the dragon hole corresponds to (t, 0), and the exit corresponds to (t, L). If we let the exit move at a high speed relative to the entrance, then according to the clock slowing effect of special relativity (such as the twin paradox), a time difference T will be formed between the exit and the entrance; then we shorten the spatial distance L to 0 and let the exit and the entrance return to a point. Then from the entrance to the exit, the time will jump by a time of T, and this completes the operation of traveling to the past or the future. This is a simplified version of building a time machine through a dragon hole.

Compared with the hole, the time machine may arouse people's interest more, because life is always full of regrets. When people reach old age, they also have all kinds of regrets. The time machine may give people a chance to start over and make up for these regrets. Therefore, countless poignant and moving love stories can be unfolded against this background.

However, the emergence of a time machine will cause many causal problems, so most of the time, time machines are only seen as a joke, not a serious scientific research topic. Perhaps "nature hates time machines", and what physicists need to do is to find the corresponding physical principles to prove that a time machine cannot be made.

Wormholes and quantum entanglement

In 1997, Maldacena’s original AdS/CFT paper shocked the theoretical physics community, and since then, more and more scholars have begun to study the holographic properties of gravity. [3] Later, based on the conclusions of Maldacena’s 2001 paper [4], Raamsdonk first discovered through simple arguments that there is an essential connection between spider holes and quantum entanglement, namely the ER=EPR conjecture. [5] (The name ER=EPR was formally proposed by Susskind and Maldacena in 2013 to solve the firewall problem of black holes [6].) ER refers to the Einstein-Rosen bridge, which is the region connecting two black holes and can be regarded as the predecessor of spider hole research. However, it is not traversable. Any attempt to cross the Einstein-Rosen bridge will inevitably fall into the black hole singularity. EPR refers to quantum entanglement.

Einstein-Rosen Bridge | Image source: arXiv: 2110.14958

Let us briefly introduce this view. In 2001, Maldacena's research found that the thermal field doublet TFD in quantum field theory

Corresponding to a corresponding AdS Schwab's black hole, its Penrose diagram is consistent with the Penrose diagram of the maximum analytical extension of the Schwab's black hole. Of course, if you look at a certain spatial cross-section of the Penrose diagram, it can be understood as two black holes connected by the middle of the hole structure.

Correspondence between thermal field doublet and Schweizer black hole | Image source: arXiv: 1005.3035

People have found that this thermal field doublet state is an entangled state, and adjusting the temperature (that is, β here) corresponds to adjusting the entanglement on the left and right sides. When the temperature is very low, the entangled state above will become a direct product state without entanglement; when the temperature is very high, it will become a maximum entangled state. Studies have found that as the temperature changes from high to low, the throat in the middle of the wormhole structure will gradually narrow until it is disconnected. Therefore, we find that from the perspective of boundary theory, the operation of reducing entanglement corresponds to reducing the size of the connecting hole between the two black holes. Therefore, this implies that quantum entanglement and hole have a profound connection, and even that they are essentially the same thing.

The shape of the hole gradually narrows as the temperature decreases. Image source: arXiv: 1005.3035

The ER=EPR conjecture suggests that the origin of space and time may come from quantum entanglement. The metric that usually describes quantum entanglement is entanglement entropy, but the growth time of the ER bridge will greatly exceed the thermal equilibrium time (and after thermal equilibrium, the entanglement entropy will tend to a constant value), so the concept of entropy seems difficult to describe the change in the volume of the ER bridge. Based on this, physicists have proposed a physical quantity that may have different properties from entropy and is associated with the volume of the hole, namely computational complexity. Its physical meaning is to specify a series of operation gates, the minimum number of operation gates required to prepare from an initial state to the final state.

At the same time, it is interesting that although the Einstein-Rosen bridge mentioned above is not traversable, we can construct a corresponding model to realize this traversable hole, that is, introduce an operation called double trace deformation at the boundary, and introduce the following operator perturbation

This operation is equivalent to introducing a negative energy flow into the background space-time. Its energy will become very large near the black hole horizon due to gravitational blueshift, so it will cause a large reaction on the background, thereby affecting the position of the horizon and causing the black hole horizon to shrink inward. Therefore, the photons emitted from one boundary and originally falling into the singularity will run outside the horizon and reach another boundary again. That is, the traversability of the dragon hole is realized.

According to the idea of ​​ER=EPR, this process is equivalent to the gravitational version of quantum teleportation, while the double trace deformation is similar to the classical channel. In quantum teleportation, it seems that the quantum bit is reconstructed in another place through quantum entanglement; but under the image of gravity, it has a completely new understanding, that is, it is transmitted through a hole connecting two places [7].

Physical image of a traversable hole | Image source: arXiv: 1704.05333

Euclidean wormhole

The above introduces the conditions and corresponding physics required for a spacetime hole to be a possible physical object. However, in recent years of quantum gravity research, a new hole structure has aroused more interest, namely the Euclidean wormhole.

Before introducing what is a Euclidean wormhole, let us first introduce the Euclidean operation that is often performed in theoretical physics research. By analyzing the similarities between the path integral in quantum field theory and the partition function in statistical physics, we find that if we perform the following Wick rotation operation t=iτ on time (for more information about Wick rotation, see "Temperature and the Mysterious Imaginary Time | The Door to Wonders"), that is, make the time coordinate imaginary, we can equate the problems of quantum field theory with those of statistical physics, and the resulting Euclidean path integral. In the Euclidean path integral, there is no time direction, and it can be regarded as physics on a certain time plane. (Of course, we can also combine the Euclidean path integral with the Lorentz path integral.)

The Euclidean path integral is an extremely effective tool for studying many theoretical physics problems. Later, we will introduce that when using the Euclidean path integral to specifically calculate the fine entropy of black hole Hawking radiation, a wormhole structure that has never been discovered before will appear. This wormhole structure can help us understand many difficult problems, such as the information loss problem of black holes.

Copy wormholes and information loss

The black hole information problem is the most profound contradiction between quantum mechanics and general relativity in the black hole space-time. Considering the collapse of pure matter into a black hole and then radiation, we can see a non-normal evolution from pure state to mixed state, but it is not allowed by quantum mechanics. The black hole information problem, as a hen that lays golden eggs, has inspired physicists' endless creativity.

Recently, inspired by the holographic entanglement entropy, people have discovered a way to calculate the exact entropy of Hawking radiation in gravity, which is called the island formula. (See "The Mystery of the Black Hole Information Paradox, Has Hawking's Last Question Been Solved?") The exact entropy obtained by this calculation magically satisfies the Page curve and further satisfies the positivity of quantum mechanics. We know that the RT formula of the holographic entanglement entropy was initially a semi-hypothetical work, but later it was precisely proved by the gravitational path integral. Can the island formula obtained here be proved by the gravitational path integral? If so, then which parts of the gravitational path integral should it come from?

First, we introduce how to calculate entanglement entropy in field theory. It can be calculated through a method called replica trick, which is to copy the system under study n times, perform calculations, and finally perform analytical extension. The formula is as follows:

The first equal sign above is the definition of entanglement entropy, and the second equal sign is derived by applying L’Hôpital’s rule. This operation is usually called the replica trick.

Because the physical meaning of the path integral describes the probability amplitude from the initial state to the final state.

, so the Euclidean path integral can be used to define the wave function and further define the density matrix. Under this Euclidean path integral, the calculation of the entanglement entropy above can be transformed into the calculation of the partition function on the copy manifold, which is the last step of the equation above.

Based on the above ideas, if we use the Euclidean path integral to represent the density matrix of Hawking radiation, we can accurately calculate its entropy (i.e., the partition function):

All possible copy manifold configurations need to be considered. Consider radiation and black holes as a whole forming a pure state

, because what is calculated is the entropy of Hawking radiation, the black hole part needs to be traced.

The calculation of entropy only requires that the radiation density matrix be connected sequentially from beginning to end as the boundary to form a replica structure, but its geometric interior cannot actually be restricted. Therefore, all possible internal configurations, including some connected configurations, need to be considered when calculating Zn.

A simple diagram: the left side is the boundary condition formed by the radiation density matrix (the solid line represents the black hole boundary after the trace is found, and the dotted line represents the radiation), and the right side represents the gravitational configuration required for the calculation. The first figure is a non-connected configuration, and the second figure represents a connected copy hole configuration. Image source: arXiv: 1911.11977

When the connected configuration is not considered, the entropy can be obtained in accordance with Hawking's initial calculation, which violates the correctness; and when the connected configuration (usually called the copy hole) is considered, the entropy behavior consistent with the correctness expectation is obtained. (Considering the fully connected configuration, we can get the results of the island formula in the later period, but the contribution of the real copy hole will be richer.) The meaning of this connected configuration is very similar to that of a hole, both of which connect different gravitational regions through a connected structure (except that the different regions here are obtained by doing the replica trick on a system), but the physics corresponding to it is very different from that of the Lorentz-type hole, and its specific physical meaning still needs more understanding and clarification.

The characteristics of the copy hole. From the picture we can see that the black holes on each boundary surface are connected together. Image source: arXiv: 1911.12333

The calculation of replica holes is complicated. Only the simplest model can consider all possible configurations of replica holes and analytically sum them up to obtain the most accurate radiation entropy [8]. However, physicists can already prove the correctness of the previously obtained island formula (at least in 2 dimensions) by replica holes. The emergence of replica holes has injected new vitality into the study of black hole information problems. Many problems have been re-discussed and studied, such as the correspondence problem of the gravitational ensemble [9], the global symmetry problem in quantum gravity, and the remnant after black hole radiation [10].

Perhaps the really interesting things have just begun. I hope that future research on dragon holes will bring us more surprises.

References

[1] MS Morris and KS Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56 (1988) 395.

[2] MS Morris, KS Thorne and U. Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Phys. Rev. Lett. 61 (1988) 1446.

[3] JM Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231[hep-th/9711200].

[4] JM Maldacena, Eternal black holes in anti-de Sitter, JHEP 04(2003) 021 [hep-th/0106112].

[5] MVRaamsdonk, Building up spacetime with quantum entanglement, Gen.Rel.Grav(2010) 2323-2329

[6]J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [1306.0533].

[7] P. Gao, DL Jafferis and AC Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [1608.05687].

[8] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [1911.12333].

[9] G. Penington, SH Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, 1911.11977.

[10]PS Hsin, LVIlliesiu, Z.Yang, Violation of global symmetries from replica wormholes and the fate of black hole remanants. Class.Quant.Grav.38(2021)19,194004.