A UNIVERSE INSIDE AN ETERNAL BLACK HOLE

Schwarzschild geometry was the first exact solution to Einstein's equations of general relativity. As is well known, this solution represents a static, spherical black hole. However, if we perform an appropriate change of coordinates and use the Euclidean metric of this geometry, we discover a whole new and fascinating world, a world that represents a system of two entangled spacetimes. In this article, we delve into the deepest secrets of this fascinating geometry.

Eternal black holes and Einstein-Rosen bridges

The equations of general relativity possess a symmetry called "diffeomorphism invariance." This basically means that the solutions to the equations do not change with any change in coordinates (1). The repercussions derived from this symmetry are enormous: any coordinate system, and therefore any distance measurement system, we use is equivalent to any other. All we have to do to change from one reference system to another is to adapt the metric to the new coordinate system.

The first exact solution to the equations of general relativity was found by physicist Karl Schwarzschild. This solution represents a static (eternal) spherical black hole with the following metric:

This metric presents a singularity on the horizon for r=2GM. Since the solutions do not change when changing the reference system, we can use any other coordinate system. We will use the following coordinate system below:

In this new coordinate system our previous metric becomes:

If we now calculate the metric at the horizon at r=2GM, we find that the singularity has disappeared! This tells us that this singularity is not real but an artifact derived from the choice of a specific metric. These new coordinates are called Kruskal coordinates, and if we draw the graph with the values au and v, we obtain four disconnected regions separated by an event horizon:

Zones I and II represent the exterior of the black hole while zones III and IV represent the interior of the black hole and have an asymptotically AdS geometry (this is basically the shape of a hyperboloid see for example this article ).

Since these zones are causally disconnected, the four zones can be interpreted as a system of two black holes (each with its interior and exterior) with AdS geometry. that do not interact with each other. The central region of the diagram is a region common to both black hole systems and is called the "Einstein-Rosen bridge." This "bridge" is the region where the throats of the two AdS geometries meet:

This "bridge" is also called a "wormhole" and is not traversable because it closes before any signal can pass through. This is why we say that both black holes are causally disconnected.

The time direction in Kruskal geometry

Next we will analyze the time coordinate of this eternal black hole.

The Euclidean metric of a Lorentzian metric such as the Schwarzschild metric is obtained by allowing the time coordinate to take complex values. Although this may seem strange, it is a fairly widespread technique called "Wick rotation" and is mathematically permissible under certain special conditions, such as the one at hand. The Euclidean metric of a 3D black hole can be written, equivalently, in the following forms:

Where tau and alpha are the coordinates of the boundary of AdS space and B is the temperature. Performing an analytical continuation for z other than 0 and doing:

z=-vyz=u we obtain the Kruskal metric in Euclidean coordinates:

Where u = t + x and v = tx. This is where we come to the key point: this metric is not static but depends on time. The time coordinate does not change if we make the following change:

Analyzing the time trajectories of this symmetry, we find that in Zone I time flows "forward" (positive sign) , in Zone II it flows "backward" (negative sign) , and in Zones III and IV time flows toward the singularity. Therefore, the time coordinate has a mirror symmetry with respect to the t=0 axis. The next question would be: What happens at the point t=0? Can we calculate the metric at this point? The answer is yes: if we take the Euclidean metric equivalent to the Kruskal metric and analyze the geometry right at z=0, we obtain the following section:

This geometry looks quite familiar: It is the construction of the Hartle-Hawking wavefunction! At time t=0 there exists a Euclidean geometry corresponding to the Hartle-Hawking no-boundary state. This initial geometry gives rise to the usual Lorentzian Universe corresponding to one half of the eternal Kruskal black hole. If we perform the same construction on the other half and glue both Universes together at time t=0 we can consider the point z=0 as the creation instant of two entangled Universes. In this way this eternal black hole can be interpreted as a spacetime with two entangled Universes connected by a non-traversable wormhole produced at the instant of its creation.

The original Hartle-Hawking non-boundary function implies the creation of a single Universe where the energy of matter is exactly equal but opposite in sign to the energy of gravity. Therefore the total energy of the Universe is zero. However, subsequent studies indicate that these two energies may not exactly cancel out. In this case, the cancellation of the total energy would occur through the creation of two entangled Universes whose combined total energy cancels out. Time in both Universes flows in opposite directions but an observer in either Universe would measure that both Universes are either expanding or contracting (2).

Hawking radiation and the ER=EPR conjecture

There are two possible interpretations of the Einstein-Rosen (ER) bridge of the eternal Schwarzschild black hole. In the first interpretation, the bridge "connects" two asymptotically AdS geometries of two different spacetimes:

The second interpretation implies a "connection" between two asymptotically AdS geometries of the same spacetime :

As we have emphasized, this connection does not imply violations of causality because

It is not possible to transmit information across the bridge.

The famous Bell experiment implies that if we create two entangled particles and measure the spin of one of them, we automatically know that the spin of the other particle must have the opposite value. This is true even if the particles are separated by light years! This experiment was originally proposed by physicists Einstein, Podolsky, and Rosen; in their honor, this phenomenon is called the EPR experiment.

If we analyze the properties we have seen of the Einstein-Rosen (ER) bridge and the properties of the Einstein-Podolsen-Rosen (EPR) experiment with entangled particles, we observe striking similarities. These similarities, together with other theoretical evidence, led Juan Maldacena and Leonard Susskind (two of the most important theoretical physicists today) to propose the well-known ER=EPR conjecture : every entangled particle is connected to its partner through an Einstein-Rosen (micro)bridge. Although this conjecture seems extremely exotic and speculative, there are quite solid theoretical indications that seem to support it.

If this conjecture is true, the Hawking radiation particles would be "connected" to their companion particles inside the black hole by a wormhole:

This would have important consequences for the famous black hole information paradox and would support recent discoveries indicating that wormholes are responsible for information escaping from the black hole.

Finally, in the next section we will discover an incredible phenomenon: although it is not possible to send signals across the ER bridge, we can connect the interior of both black holes. It would actually be possible to establish communication, but this information cannot be seen by the rest of the Universe: the information will be hidden forever inside the black hole (3).

Connecting the interior of two eternal black holes

Suppose we create a collection of N entangled particles, separate them several light-years apart, and collapse them to form two mini-black holes. Following the principles of quantum mechanics and general relativity , these two black holes will be entangled.

Let's imagine that somewhere in our immense Universe a highly advanced civilization manages to achieve the technology necessary to compress matter to the Schwarzschild radius, thus being able to manufacture macroscopic black holes. Suppose this civilization manufactures a pair of identical black holes inside a very intense magnetic field. Precise theoretical studies of this phenomenon (4) indicate that these two black holes will be entangled. Next, an astronaut named Bob transports one of the black holes in an ultra-fast rocket to a region located 50 light years away. When he arrives at his destination, Bob wants to communicate with Alice, who has stayed on Earth, but each message sent from her position will take 50 years to reach Earth, which is an unbearable wait. Then Bob has a surprising idea: if they both jump into the black hole, they will be able to meet inside and communicate. This is because the interior of both entangled black holes is common to both exterior regions and is causally accessible from the outside, as can be seen in the Penrose conformal diagram:

If Alice and Bob jump at the same time once the black hole is formed, both will be able to meet inside.

However, both must jump into their respective black holes shortly after the hole forms; if they wait too long, they will both reach the singularity before meeting:

If both wait too long, once the hole formation time has started, the meeting will no longer be possible since they will first collide with the singularity.

This incredible phenomenon seems to support the famous work of physicist Raamsdonk in which he describes how entanglement connects space-time itself: if we increase the entanglement between two regions the space-time between them gets closer, if we decrease it the space-time separates until it becomes completely disconnected (see this article ).

Of course, this experiment is not recommended, since no one will ever know what the two astronauts discussed inside the black hole, and the singularity will ultimately mark the tragic fate of anyone who enters the hole.

Grades:

(1) Coordinate changes must respect the topology of the original space-time

(2) See for example this article

(3) This will be valid for an eternal black hole, i.e., without evaporation.

(4) See for example: D. Gar nkle and A. Strominger, Semiclassical Wheeler wormhole production, Phys. Lett. B 256, 146

Sources:

Cool horizons for entangled black holes , Eternal black holes in Anti-de-Sitter