Chapter 1: The Intriguing Concept of Fewer Dimensions

The classic science fiction tale "Flatland" by Edwin Abbott comes alive in modern physics. The story is narrated by a two-dimensional character known as "A Square," who discovers he is unwittingly situated within a three-dimensional universe referred to as Spaceland. This concept resonates with physicists who have been contemplating whether our universe is merely a fragment of a larger four- or even ten-dimensional reality. Imagine the possibilities: accessing a locked safe by reaching through an additional dimension, or seeing the intricate details of a human body laid bare before us. Such ideas have tantalizing implications about the fundamental unity of nature.

However, recent developments in dimensional theory suggest a surprising twist: space might actually have fewer dimensions than we perceive. This notion stems from the complexities surrounding gravity, one of the four fundamental forces of nature. As two objects draw closer, their gravitational attraction intensifies. This behavior is also observed in other forces like electromagnetism and the weak force. Yet, gravity exhibits unique characteristics at tiny scales, where its strength escalates dramatically, influenced by the quantum uncertainty principle.

When two bodies approach one another closely, the energy minimum dictated by quantum physics begins to rise inversely with distance, resulting in a striking increase in gravitational attraction. While other forces exhibit minor variations, gravity escalates to match them at an incredibly small scale known as the Planck scale, approximately a few trillionths of a trillionth of a trillionth of a meter. At this scale, gravity doesn't become more potent; instead, it equalizes with the other forces, obscuring the differences between various scales.

Interestingly, standard gravitational theory posits that gravity's strength could become so immense at this scale that matter would collapse into black holes. Since gravity reflects the curvature of spacetime, this overwhelming intensity hints that spacetime, as we perceive it, cannot be a fundamental characteristic of reality. Some alternative structure must exist, or gravity must be curtailed at small scales to maintain the essence of spacetime—perhaps both scenarios are valid. Regardless, the current theory appears incomplete.

The quest to understand the microscopic structure of spacetime is encapsulated in the challenge of quantum gravity, a conundrum that has puzzled physicists since Einstein's time. Gather three theorists, and you'll encounter a multitude of theories: a grid-like arena of discrete cells, a foamy geometry, a collection of vibrating strings, or a network of interwoven loops.

Astonishingly, many of these theories share a commonality: at the most minuscule scales, space appears to shed all but one dimension, effectively controlling gravity. In fact, a one-dimensional space is the only scenario where gravitational force does not escalate as distance decreases.

How Three Spatial Dimensions Affect the Size of Atoms in Our Universe

This concept of dimensional reduction raises intriguing questions. Is this transition merely a step toward the ultimate disappearance of spacetime, or does it preserve it as a fundamental aspect of nature? Frank Saueressig, a theoretical physicist at Radboud University in the Netherlands, suggests that "two dimensions—one for space and one for time—seem to be the natural dimensions where gravity prefers to exist."

Chapter 2: The Tangible Implications of Dimensions

The dimensionality of space has profound implications: it dictates how the size of objects, the speed of events, and the strength of forces change with scaling. In our familiar three-dimensional space, doubling the radius of a sphere results in an eightfold increase in volume. In four-dimensional space, the increase is 16 times, while in two dimensions, it's only four.

If you were unaware of the number of dimensions in space, you could deduce it from how volumes scale. A deviation from the expected scaling would suggest a variation in dimensionality, which is what many physicists suspect may be occurring with gravity.

The inspiration for this line of thought arose from computer simulations of Planck-scale spacetime shapes conducted in the mid-2000s. Renate Loll and her team at Radboud developed an algorithm called causal dynamical triangulations, infusing their simulated spacetimes with quantum properties. Their findings revealed that observable space indeed possesses three dimensions, as we experience it. However, if gravity's strength were to fluctuate, space could transition into a structure with fewer dimensions or become crumpled into a denser form, akin to a freezing or boiling process.

Even when space appears reassuringly three-dimensional, deeper analysis reveals discrepancies. In a random walk through three-dimensional space, doubling your journey length would decrease the likelihood of returning to the starting point by the square root of eight. Yet simulations near the Planck scale suggest a slower decline in this probability, indicative of a space with fewer dimensions.

This contradiction signals that while volumetrically, space may be three-dimensional, its behavior during random movements suggests a one-dimensional or fractional nature. This variance implies a fundamental change in the nature of space as gravity begins to exhibit quantum traits.

Chapter 8 Homework Solutions MyMathLab - YouTube

The complexities of causal dynamical triangulations act like a black box; they take equations as input and yield numbers as output, leaving the underlying mechanisms obscured. The theory closest to elucidating this dimensional behavior is known as asymptotic safety, a theory dating back to the late 1970s. It posits that gravity possesses an inherent limit, preventing it from intensifying endlessly and leading to a maximum finite strength near the Planck scale. This idea resonates with many particle physicists because it mirrors the behavior of forces governing atomic nuclei, which tend to diminish at short distances.

According to asymptotic safety, the loss of dimensions does not result from space flattening like a crumpled can, but rather from quantum fluctuations altering the typical navigation rules of particles. A randomly moving particle might retain a memory of its trajectory, avoiding previously traversed paths—similar to the classic video game Snake, where players steer a dot while avoiding collisions with their own path. This constraint increases the likelihood of returning to the starting point, thus creating the illusion of fewer dimensions.

Asymptotic safety suggests that if gravity stabilizes at small scales, it will prevent the formation of Planck-scale black holes, allowing spacetime to remain continuous. This means one can zoom in endlessly without losing coherence. However, this zooming leads to disorientation, as the absence of distinct scales means everything appears homogeneous. Each segment resembles the whole, embodying the essence of fractal structures.

Although a one-dimensional space may seem to provide a solution for gravity, physicists remain unconvinced of the reality of dimensional shedding at small scales. The primary criterion for any theory is its internal consistency, and proponents of asymptotic safety acknowledge that they have yet to demonstrate this.

Critics argue that asymptotic safety falters when addressing the physics of black holes, where internal complexity scales with surface area. In contrast, any continuum theory, including asymptotic safety, suggests that complexity should correlate with volume. Advocates counter that these scaling rules might shift if spatial dimensions fluctuate, potentially allowing for coherence.

Even if asymptotic safety is not the definitive theory, it may serve as a stepping stone toward a potential framework where space emerges from a state devoid of spatial dimensions. "Between 'no spacetime' and 'usual spacetime,' there may well be 'weird spacetime,'" proposes Daniele Oriti of the Max Planck Institute for Gravitational Physics.

Laboratory experiments are unlikely to resolve these questions in the near future, especially since the Planck scale is far smaller than what current instruments can assess. Cosmological observations present a more promising avenue. Reuter notes that the early universe exhibited scale invariance, with density variations across all sizes showing uniform magnitude, supporting asymptotic safety and a fundamental spacetime. However, alternative theories also offer explanations for this scale invariance.

Regardless of the outcome, theories suggesting dimensional reduction may be tapping into a profound aspect of quantum gravity. "This is part of the reason it's so interesting," states Carlip. "It might unveil something deep about quantum gravity that these various approaches share. But we remain uncertain of what that is."

In the preface to the 1884 edition of "Flatland," Abbott, through A Square, lamented that we are "all alike the Slaves of our respective Dimensional prejudices." Today, physicists are beginning to shed their own biases regarding dimensions. After all, as Mr. Square poignantly notes, "one touch of Nature makes all worlds akin."

George Musser is a writer focusing on physics and cosmology. His works include "The Complete Idiot's Guide to String Theory" and "Spooky Action at a Distance" (forthcoming from Farrar, Straus and Giroux). He has served as a senior editor at Scientific American for 14 years and has received the American Institute of Physics Science Writing Award, among other accolades.