Geometry of the universe

Summary: Characteristics of the universe such as size, shape, and composition have long concerned mathematicians and astronomers and over the course of history various models have been offered.

The shape of the universe and its geometry have been the topic of human interest for millennia. Researchers in scientific disciplines such as physics, astronomy, and cosmology, along with mathematicians, especially those working in geometry, are seeking to discover what shape the universe is, whether it is finite or infinite, and how many dimensions it has. Not only do researchers investigate this topic; it is also popular for philosophical debates in the media and educational settings, for example, as the theme of Mathematics Awareness Month in 2005. Generally speaking, the density of the universe determines its geometry. The shape of the universe could therefore be estimated by measuring the average density of the matter within it, assuming that all matter is evenly distributed—though there might be considered distortions caused by very dense objects with mass accumulated locally, such as galaxies. This assumption is well justified by cosmological observations showing that, while the universe appears to be weakly inhomogeneous and anisotropic locally, on average it is homogeneous and isotropic. Therefore, all considerations about the geometry of the universe have to be seen from two perspectives: the local geometry that is related to the observable universe and the global geometry related to the universe as a whole, where also that is included which humans have yet to be able to measure in the early twenty-first century.

Measurements are closely related to the origins of geometry, a discipline flourishing more than 5000 years ago from the early stages of the human civilization in ancient Egypt and later in ancient Greece, and are practical and necessary in connection to the geodetic measurements of Earth. Later, developed as a theoretical abstract branch of mathematics, geometry offered mathematical background for the description of geometric abstract spaces with more dimensions, which cannot be visualized in the three-dimensional spaces, but can be used as models in modern physical and cosmological theories describing the possible form, structure, and principal laws of the universe.

From the History

For thousands of years, people believed that the universe revolved around Earth, and astronomers created mathematical models to explain observations in the sky. Eudoxus of Cnidus created a model containing rotating spheres centered about Earth. With this model, Aristotle was able to partially explain some of the planetary motions by rotating the spheres at different velocities, but other observations, such as differences in brightness levels, could not be resolved.

In the next century after Aristotle, Euclid of Alexandria expressed the parallel postulate. While it was not linked with models of the universe at the time, it was to eventually take on an important role in the geometry of the universe. Euclid is the author of the famous Elements, one of the earliest and most influential works in the history of mathematics, consisting of 13 books. Here, all principles of the geometric space, today called “Euclidean,” were deduced in the form of mathematically proved propositions and constructions from a small set of postulates and definitions. Postulates were not proved or demonstrated, but considered to be self-evident and true. They described all basic relations and measures between ideal geometric figures as points, lines, triangles, circles, or solids, and also numbers that were treated geometrically as line segments with various lengths. The introduced list of postulates referred to the following five groups of relations: incidence, congruence, order, continuity, and parallelism.

The fifth postulate about parallelism says: “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles lesser than the two right angles.” From the time of its publication until the late nineteenth century, this postulate, apparently different from all others and of more complicated form, attracted mathematicians, who strived to prove it as a consequence of the first four groups. New equivalent formulations of this famous parallel postulate appeared. The most familiar form is this: “Through a point not on a given straight line, at most one straight line can be drawn that never meets the given line” (see Figure 1).

All efforts to prove the fifth parallel axiom appeared to be pointless. On the contrary, different possible formulations of this special property were introduced, as the negations of Euclid’s postulate, revealing thus the existence of new kinds later called “non-Euclidean” geometries with unusual properties emerging from these formulations.

Geometric Spaces

Even in the face of overwhelming evidence, it took a long time for humanity to accept that Earth is not at the center of the universe because this revolution required an imaginative leap that surpassed problematic religious and philosophical implications. In his famous work, the Almagest, Claudius Ptolemy, a second-century philosopher, refined and improved an Earth-centered model based on the earlier work of Apollonius of Perga and Hipparchus of Rhodes. In the Ptolemaic universe, planets now moved along epicycles, which had circles attached to the spheres around Earth, and yet this model still did not completely resolve contradictions with astronomical observations. Aristarchus had suggested a heliocentric system, and in the sixteenth century, Nicolaus Copernicus gave substance to Aristarchus’s ideas by carrying out the detailed mathematical calculations. His model still utilized epicycles in order to explain the circular motion of the planets, but it placed a motionless sun close to the center of the universe. Johannes Kepler revolutionized astronomy by finally overthrowing the stranglehold of purely circular motions. His introduction of elliptical orbits together with his other two laws of planetary motion form the basis of celestial mechanics to this day. They were also critical in the formulation and verification of Sir Isaac Newton’s laws of gravity and of motion, which in turn became the basis for cosmology for the following two centuries.

Around 1830, Hungarian mathematician János Bolyai and Russian mathematician Nikolai Ivanovich Lobachevsky published their papers on non-Euclidean geometry, independently and unaware of each other—hyperbolic geometry is therefore also called Bolyai–Lobachevskian geometry. The famous mathematician Johann Karl Friedrich Gauss explored such geometry about 20 years earlier, but he never published his work. Lobachevsky developed a theory of a new geometric space, in which the fifth postulate was not true, by negating the Euclid’s postulate about the existence of a unique parallel to a given line. He stated a new, nowadays called the Lobachevsky, axiom of parallelism: “Through a point not on a given straight line, at least two different lines can be drawn that never meet the given line.” Lobachevsky based this reasoning on his own findings received from measuring distances of stars calculated from their trajectories traced on the celestial sphere because of the movements of Earth in the solar system. In his gigantic triangles, the sum of the interior angles measured less than 180 degrees. Bolyai worked out a geometric theory whereby both the Euclidean and the hyperbolic geometry were possible, depending on a special introduced parameter. Bolyai wrote in his work that it is not possible to decide whether the geometry of the physical universe is Euclidean or non-Euclidean through mathematical reasoning alone, and he regarded this to be a task for the physical sciences.

Bernhard Riemann was a German mathematician who founded a new field of geometry, later called the “Riemannian geometry,” in his famous lecture in 1854. He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in the Euclidean space. His theory of Riemannian surfaces—which can be divided into three types: hyperbolic, parabolic, and elliptic or spherical corresponding to negative, zero, or positive curvature—can be generalized by his uniformization theorem in terms of conformal geometry. Every connected Riemann surface X admits a unique complete two-dimensional real Riemannian metric with constant Gaussian curvature equal to –1, 0, or 1 inducing the same conformal structure. The surface X is then called “hyperbolic,” “parabolic,” and “elliptic,” respectively, according to its universal cover.

Later on, Riemann’s remarkable work was elaborated by German mathematician Felix Christian Klein, who established a new classification of geometric spaces based on algebraic theory of the underlying group of transformations and their invariants, which is known as the “Erlangen program” presented at the University of Erlangen in 1872. Basic properties of a specific geometry can be represented as sets of invariant properties of the space figures under a given group of transformations. This definition of geometric spaces encompassed both Euclidean and non-Euclidean geometry in a unifying theory of geometric spaces, taking into consideration not only geometric figures and the space dimension, but also specified geometric transformations and their invariants.

The development of non-Euclidean geometries was inevitably important to physics in the twentieth century. Modern geometry shows multiple strong bonds with physics, exemplified by the links between Riemannian geometry and relativity . In 1917, Albert Einstein used Bernhard Riemann’s mathematics in order to present a model for the universe that was consistent with his theory of relativity. His model was based on a finite spherical universe. Geometry, where the curvature changes locally from point to point, is the Riemannian geometry of continuous manifolds. One of the youngest physical theories, string theory, is also very geometric in flavor.

Dimensions: Shape of the Universe

There is a direct link between the geometry of the universe and its shape. The homogeneous and isotropic universe allows for a spatial geometry with a constant curvature, and three different possible types of geometric spaces can be distinguished, depending on the sign of the curvature.

If the density of the universe equals exactly the critical density, then the geometry of the universe is flat, like a plane. One has to consider a geometric space with zero curvature and Euclidean geometry as described by Euclid. As Euclid’s fifth postulate on parallelism is true, the sum of the triangle’s inner angles equals exactly 180 degrees, and light photons traveling on parallel lines never meet each other (see Figure 2).

If the density of the universe exceeds the critical density, then the geometry of space is closed and positively curved like the surface of a sphere. No parallel lines exist, and the sum of the triangle’s inner angles is more than 180 degrees. This implies that, initially, parallel photon paths converge slowly. Eventually, they cross and return back to their starting point if the universe lasts long enough (see Figure 3).

If the density of the universe is less than the critical density, then the geometry of space is open, negatively curved like the quadratic surface called “hyperbolic paraboloid.” Infinitely many parallels exist through a point to a given line and the sum of the triangle’s inner angles is less than 180 degrees. Parallel photon paths can be considered as traveling to infinity in different directions from one starting point (see Figure 4).

Global geometry describes the topology of the whole universe—the observable part and beyond. For a flat spatial geometry, any topological property may or may not be directly detectable, as the scale of all such properties is arbitrary. Probability to detect the topology of spherical and hyperbolic geometries by direct observation depends on the spatial curvature. Using the radius of curvature as a scale, a small curvature of the local geometry, with a corresponding scale greater than the observable horizon, makes the topology difficult to detect. In a hyperbolic geometry, the radius scale is unlikely to be within the observable horizon, while a spherical geometry may have a radius of curvature that can be detected.

There are three primary methods to measure curvature: luminosity, scale length, and density. Luminosity requires an observer to fix some standard source of light, such as the brightest quasars, and follow them out to high red shifts. Scale length requires determination and usage of some standard size, which can be the size of the largest galaxies. Density is a number of galaxies in a chosen box as a function of distance. Recently, all these methods have been inconclusive because the size and number of observable galaxies and their brightness are changing with time in unpredictable ways. As of 2011, the cosmological measurements were consistent with the model of a flat universe, based on data from sources such as NASA’s Wilkinson Microwave Anisotropy Probe (WMAP). NASA has declared the universe to be flat within a 2% margin of error.

Two following investigations are decisive in the study of the global geometry of the universe:

• Whether the universe is a compact space or it is infinite in extent
• Whether the topology of the universe is simply or nonsimply connected

Both of these topological properties depend on the mass distribution and, therefore, on the total strength of gravitation within the universe. However, each of them implies a different history and future development of the universe:

1. If the universe is a space with negative curvature, there is insufficient mass to cause the universe to cease expansion. Therefore, the universe has no boundaries, and it will continue expanding forever, ending in a Heat Death. This model of the universe is presented as an “open universe.”

2. If the universe is a space with zero curvature, there is exactly enough mass to stop its expansion, but this will take an infinite amount of time. In this case, the universe has also no bounds and will expand forever; but after an infinite amount of time, the rate of its expansion will be gradually approaching zero. This is a “Euclidean flat universe” model.

3. If the universe is a space with positive curvature, there is more than enough mass to stop its expansion. The universe is not infinite, but it is endless. The present expansion of the universe might eventually stop and turn into a contraction, and the universe will start collapsing on itself. This model is called a “closed universe.”

Scientists still do not know which of these three scenarios of the future of the universe could be correct, as they have not yet been able to determine exactly how much mass is in the universe.

If the three-dimensional manifold of a spatial section of the universe is compact, then the universe has a definable volume, as on a sphere. If the geometry of the universe is not compact, then the universe is infinite in extent with no definable volume, such as the Euclidean plane. Therefore, if the spatial geometry is spherical, then its topology is compact, while for a flat or a hyperbolic spatial geometry the topology can be either compact or infinite.

Particle physics, quantum field theory, and cosmological theories led to a revolution in thought and new paradigms of subatomic matter that require the existence of a so-called hyperspace, which is an ultimate universe of many dimensions. In an ongoing quest for a synthesis of quantum mechanics and relativity physics into a superstring theory of universe unifying four fundamental forces (gravity, electromagnetism, and the strong and weak nuclear forces), the idea of a Theory of Everything has been born. This unified field theory, as it is understood in the early twenty-first century, does not preclude any of such hypotheses as, for instance, the existence of superstrings, black holes, wormholes, other parallel universes, and time travel ideas. Modern physics still needs a more powerful mathematical theory and topology of the 10-dimensional space in order to understand completely our expanding and evolving cosmos. The theory of hyperspace introduced by American mathematician Michio Kaku may be the leading candidate for the Theory of Everything, for which Albert Einstein spent the last years of his life searching.

When, in 1990, scientists sent the Hubble Space Telescope into space, they did not expect to find that the expansion of the universe was speeding up, nor did they realize the existence of the black matter and the dark energy that became the dominant force in the universe, recently accelerating its expansion. The James Webb Space Telescope, NASA’s next orbiting observatory and the successor to the Hubble Space Telescope, is scheduled to be launched in 2014 to distant orbits. This infrared telescope detecting infrared radiation will be capable of seeing wavelengths of light difficult to observe from Earth, thus opening new horizons of the visible universe. It is hard to imagine and predict what discoveries and answers to the mysteries of the universe scientists will gain using its observations in the future.

Bibliography

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Frank, Adam and Erika Larsen. “Is the Universe Actually Made of Math?” Discover Magazine (July 2008).

Kaku, Michio. Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps and the Tenth Dimension. New York: Oxford University Press, 1994.

Rucker, Rudy. Spaceland. New York: Tom Doherty Associates, 2002.

Stewart, Ian. Flatterland. New York: Perseus Publishing, 2001.

Weeks, Jeff. The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds. New York: Dekker, 1985.

Yau, Shing-Tung and Steve Nadis. The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions. New York: Basic Books, 2010.