ASYMPTOTIC SILENCE: THE END OF SPACE-TIME

The great Albert Einstein imagined himself "riding on the back of a ray of light." Einstein realized that traveling at the speed of light, the laws of electromagnetism displayed an uncomfortable asymmetry. As he explained years later, this is one of the thought experiments that led him to discover the theory of relativity. More than a century later, a modern version of this thought experiment seems to peek out from deep within the "seams" of space-time. Recently, theoretical physicists have discovered a phenomenon that appears to be repeated in many of the candidate theories to describe quantum gravity: string theory, loop quantum gravity, causal triangle dynamics, asymptotically safe quantum gravity, quantum foam models...

This phenomenon is called "asymptotic silence" because of its similarity to what is believed to happen very close to a cosmic singularity: as if we were traveling "on the back of a ray of light," the light cones associated with our trajectory are compressed so much that they become a straight line, and adjacent points in space become causally disconnected, so that they cannot "hear" each other. This phenomenon implies the end of classical space-time as we know it. In this article, we will explain how this phenomenon arises in our modern approaches to quantum gravity and how, if confirmed, it would mean the discovery of a fundamental quantum property of space-time.

The concept of space-time dimension

If we were on a strange planet whose geometry is unknown, how could we experimentally determine its spatial dimension?

One option would be to draw a straight line on its surface, then draw another line perpendicular to it, and repeat this process. If we can only draw three perpendicular lines, the space is three-dimensional. However, this is only valid in plane or Euclidean geometries; if the space is curved, this method is invalid, and we would have to use geodesics. Can we find a more general definition?

The first question is: What do we understand by the dimension of space-time? Certain mathematical theorems, such as Cantor's work demonstrating that a segment and a square of unit area have the same number of points, require a more appropriate definition of the concept of dimension. On the other hand, a straight line may appear one-dimensional if we observe it from a distance, but if we zoom in closely enough, we will see that it has a thickness, so it is not actually one-dimensional. This tells us that the concept of dimension is not absolute but depends on system variables such as energy or the distance at which we make the observations.

In string theory, for example, the natural scale is the Planck scale. At this energy, strings "feel" additional dimensions. This tells us that in quantum gravity, the concept of dimension is even more diffuse. Therefore, we need appropriate "markers" that allow us to estimate the dimension of a quantum system based on the variables or observables we use in each case. Some of the most commonly used "markers" in fundamental physics are the following:

Hausdorff Dimension or Fractal Dimension: It is calculated by filling the space-time of unknown dimension with balls of radius R. By measuring how the volume V of these balls evolves as we vary the energy scale of the system, the dimension of space-time can be determined through the formula V= r D where r is the radius of the ball and D the dimension of space (for example, in a three-dimensional space the volume scales as the cube of the radius).

Spectral dimension: To estimate the dimension of a space-time, we can define a random walk process. This process measures the time it takes to complete a closed random path in space-time. The central idea is that the larger the space-time dimension, the more degrees of freedom will affect this process, and the longer it will take to complete the path.

Thermodynamic dimension: Certain thermodynamic quantities, such as the free energy or internal energy of a system, depend directly on the space-time dimension. By measuring how these quantities evolve, we can estimate the space-time dimension.

Green's functions: Ehrenfest showed that the Newtonian gravitational potential in d-dimensional spacetime varies as r -(d-3) . Based on this fact, the following generalization can be made:

Where the function sigma(x,x') is half the square of the geodesic distance between x and x'. This is an indicator of the space-time dimension.

Scale dimension:

Physical fields have a natural scale D that describes how the field changes when we change the scale of the lengths and masses of the system under consideration. For example, a scalar field has a term in its action of the form:

Where the action is integrated over D dimensions of space-time. For the action to be invariant under changes in scale, the field must scale like L (-X)

where X=d-2/2. The dimension d is the so-called scale dimension.

Recently physicists have found something surprising: by calculating the evolution of these markers in different approximations of quantum gravity for smaller and smaller scales we find that as we approach the Planck scale the number of effective dimensions is reduced, specifically the effective dimension

of space-time is reduced from four to two! This seems to indicate that the fundamental quantum structure of space-time is a two-dimensional structure!

Before explaining how physicists have discovered this fascinating phenomenon

In the different approximations of quantum gravity we will see how the phenomenon of "asymptotic silence" occurs in cosmology when reaching a singularity in space-time.

Asymptotic silence in a space-time singularity

Describing what happens near a singularity like those found inside black holes is impossible without a definitive theory of quantum gravity. However, by assuming certain reasonable approximations, we can try to understand what happens in the vicinity of these "cosmic monsters." Physicists believe that a quantum theory of gravity must satisfy the so-called Wheeler-DeWit equation:

Where lp is the Planck length whose value is:

The Wheeler-DeWit equation contains information about the metric at all scales. To see what happens at small scales near a singularity, we must analyze the behavior of the equation as the gravitational coupling G tends to infinity, that is, as lp tends to infinity. We can also make lp tend to infinity by making c tend to 0. As lp tends to infinity, the only term in the Wheeler-DeWit equation that has spatial derivatives disappears. This means that the light cones close in on themselves and transform into straight lines. This implies that the spatial points become decoupled, and we have a two-dimensional system: that of a point evolving in time (a "world line"). In this way, adjacent points become causally disconnected and cannot "hear" each other. This phenomenon is therefore called asymptotic silence.

Dimensional reduction in string theory

The first sign of dimensionality reduction as we approach the Planck scale was found in string theory. When a string gas is heated above a critical temperature called the Hagedorn temperature, the system undergoes a phase transition. In 1988, physicists Atickk and Witten found that above this temperature the number of degrees of freedom decreases dramatically and the free energy evolves as F/V = T 2 . This means that above the Hagedorn temperature , the thermodynamic dimension changes from four to two!

In the words of the great physicist-mathematician Edward Witten: "Above the critical point, a mysterious system appears that behaves like a two-dimensional field theory."

On the other hand, it is known that the scattering amplitude in the interaction between two strings falls exponentially with energy, which also points to an effective reduction of the dimensions of space-time at very high energies.

Asymptotic silence in causal triangle dynamics

Despite the evidence found in string theory, it wasn't until 2005 that the scientific community's interest in the phenomenon of dimensionality reduction really took off. The quantum gravity approximation known as causal triangle dynamics is based on the fact that any continuous surface or volume (Riemann surfaces) can be approximated with the necessary precision using triangles (for surfaces) or tetrahedra (for volumes). This fact is used in the so-called Regge calculus, which can capture the dynamics of general relativity using triangles or tetrahedra.

A two-dimensional surface can be completely covered using triangles. The more triangles we use, the greater the precision obtained. By calculating the total angle around a point where these triangles join, we can measure the curvature: if this angle is 2PI, the surface is flat; if it is different, the space is curved.

With these basic components, a simulation of space-time can be performed using the gravitational path integral, that is, summing over all possible geometries between two time intervals. Similar to the Feynman calculus in quantum mechanics, the Regge calculus is performed by summing the number of triangles (of a given fixed length) traversed when considering all possible paths (geometries).

Example of simulation of space-time in causal triangle dynamics: from an initial instant (central area) space-time expands simulating a De-Sitter space-time (an expanding space-time with continuous positive curvature) like our real Universe.

In 2005, physicists Ambjørn, Jurkiewicz, and Loll used a new and improved model based on causal triangle dynamics and found something completely unexpected. The model not only predicted a four-dimensional expanding Universe like ours on large scales, but also when we zoom in to very small scales

The spectral dimension falls to two dimensions!

The horizontal axis represents the scale (distance), and the vertical axis represents the spectral dimension. As we approach the Planck scale, the dimension drops from four to two.

These models have been independently verified by other groups of physicists. The importance of this result lies in the fact that this model is based on geometric principles that are more general and less speculative than those used in other approaches and that are thought to be fulfilled by any theory of quantum gravity.

Asymptotic silence in loop quantum gravity

In loop quantum gravity, length and area can only take discrete values. The average area can be written as:

Where l is the length quantum and lp is the Planck length. For large values of l this average area scales with an effective dimension corresponding to four, but when l is small the area scales with two-dimensional effective dimension!

Other studies on how the effective dimension varies with distance depend heavily on the details of the model and are therefore inconclusive, so more precise simulations will be necessary.

Asymptotic silence in asymptotically safe quantum gravity

If we analyze general relativity as a conventional field theory, we find that the theory is non-renormalizable: its effective action has infinitely many terms that are impossible to control. However, if the theory had an ultraviolet fixed point at very high energies with a finite number of directions (directions of evolution of the renormalization flux), then the theory would depend only on these finite degrees of freedom, and the theory would be renormalizable, valid at all scales, and "asymptotically safe." No one yet knows whether general relativity is asymptotically safe, but there are indications that point in that direction. If the answer is positive, then the ultraviolet fixed-point operators necessarily acquire a two-dimensional scaling dimension, since the Einstein-Hilbert action is only scale-invariant in two dimensions, and at the fixed point the theory is scale-invariant. This would be another indication of asymptotic silence arising from a completely different approach.

Dimensional reduction in "quantum foam" models

In 1955, physicist John Wheeler proposed that quantum fluctuations in spacetime near the Planck scale would produce a diffuse and highly variable spacetime. The famous term "quantum foam" coined by Wheeler to describe this structure is still active in modern approaches. After Wheeler's original proposal, quantum foam models based on "stochastic gravity" emerged. One such model, proposed by Crane and Smolin, describes a quantum foam based on virtual black holes. If the distribution of these black holes is scale-invariant and sufficiently dense, the spacetime outside the black holes has a scale dimension equal to 4 - X , where the value of X depends on the details of the black hole distribution. Although the value of X is not known, this calculation again points to a dimensional reduction to very small scales.

Conclusions

It's still too early to draw definitive conclusions; future studies will have to confirm or refute the existence of this "strange two-dimensional entity" near the Planck scale. If its existence is confirmed, we would be faced with a fascinating discovery: What is the nature of this two-dimensional system? Have we discovered the fundamental components of space-time? Is there a connection between this system and the fundamental elements of our theories of quantum gravity (strings, loops, etc.)?

Fundamental physics is undoubtedly uncovering a fascinating new world, a world where space-time emerges from certain fundamental components.

Like explorers of a new Universe, physicists are beginning to glimpse the deepest secrets of space-time. The new vision of the Universe we inhabit that these new discoveries present to us is so different from the current one that it will take time to assimilate it and fully understand its radical consequences.

The quantum revolution and the relativistic revolution will be far surpassed by the new revolution of quantum gravity.

Sources:

Dimension and Dimensional Reduction in Quantum Gravity