The Shape of the Universe

2003-04-01 00:00:00

Ancient people, considering it very important to determine Earth'shape, derived two important clues from the night skies. According to Aristotle (384-322 bce), these were lunar eclipses and the North Star. Lunar eclipses occur when the sun, Earth, and the moon line up in such a way that Earth temporarily blocks the sun's light from reaching the moon while its circular shadow gradually crosses the moon's face. The North Star appears lower in the sky the further south we go: at the Equator it lies directly on the horizon, at a latitude of 45 it is 45 above the horizon; and at the North Pole it is directly overhead. However, it is not visible south of the Equator.(1) As both of these indicate a spherical Earth, the scholars of that time discarded the idea of a flat Earth.

The more challenging question was how to determine Earth's size. Eratosthenes of Alexandria (third century bce) had a simple yet brilliant idea: insert a gnomon (a vertical stick) into a level piece of ground. This enabled him to determine noon's exact time (when the shadow was the shortest). It was also used as a compass, for in the Northern Hemisphere the gnomon's shadow points north.

But can such a simple device determine Earth's size? Aswan, located about 500 miles south of Alexandria, sits on the Tropic of Cancer. So, at noon of June 21 (the summer solstice), a gnomon inserted there has no shadow. By doing just that in Alexandria, Eratosthenes found that the angle was 1/50 of a circle's circumference (i.e., 2p/50). In other words, the angle at Earth's center corresponding to the arc between Aswan and Alexandria on Earth's surface is 1/50 of a circle's circumference. Since the distance between Alexandria and Aswan is 500 miles, Earth's circumference should be 25,000 miles, which is its actual circumference.(2) Thus, Earth's size and shape was pretty well established over 2,000 years ago.

This knowledge was lost to Europe when the ancient civilizations crumbled. But Islamic civilization and culture, which was rising at roughly the same time as the West was declining, produced scholars and scientists who translated and refined quite a bit of this ancient knowledge. For example, in 1424 al-Kashi used Archimedes' method of computing to determine its values to 16 decimal places. Ulug Beg compiled the greatest star catalog known at that time. During al-Ma'mun reign (813-833), al-Khwarizmi measured one degree of latitude on Earth's surface and obtained the result of 57 miles. This means that Earth's circumference is 360x57 = 20,520 miles.(3) Thus, in the ninth century, Muslim scientists knew that Earth was spherical and had a good idea of its size. Most Europeans at that time, believed that Earth was flat and the universe impenetrable.

The Qur'an describes Earth's geographical shape and change in that shape: Do they not see how We gradually shrink the land from its outlying borders? Is it then they who will be victors? (21:44).(4) The reference to shrinking could relate to the now-known fact that Earth is compressed at the poles.

At a time when people generally believed that Earth was flat and stationary, the Qur'an explicitly and implicitly revealed that it is round. More unexpectedly still, it also says that its precise shape is more like an ostrich egg than a sphere: After than He shaped Earth like an egg, whence He caused to spring forth the water thereof, and the pasture thereof (79: 30-32).

The verb daha' means "to shape like an egg," and its derived noun da'hia is still used to mean "an egg." As this may have appeared incorrect to pre-modern scientists, some interpreters misunderstood the word's meaning as "stretched out," perhaps fearing that its literal meaning would only confuse people. Modern scientific instruments recently established that Earth is shaped more like an egg than a perfect sphere, and that there is a slight flattening around the poles and a slight curving around the Equator.

An enduring Western myth is that Columbus had to overcome a pervasive belief that he would sail off the edge of a flat Earth by sailing west to Asia. This myth stems in part from compressing the past and conflating the early Middle Ages, when Europe's belief in a flat Earth was widespread, with the late Middle Ages, when Europe's knowledge had caught up with and partially surpassed that of ancient Greece and medieval Islam.

During the Renaissance, Europe came into contact with "lost" knowledge by translating Greek and Arabic works. One important book was Ptolemy's Geography, which accepts Earth's spherical shape. Geography once more became available in the original Greek, which was not widely known in the thirteenth century. This book was translated into Latin in the late fifteenth century and became widely known. Columbus owned a copy printed in 1479.

By the time of Columbus, the idea of a spherical Earth was widely accepted in theory. Columbus believed this and wanted to sail west to the eastern shores of Asia. Earth's size was the real issue. Ptolemy's estimate was as much as 20% too low. Also, he vastly overestimated Asia's size. The resulting map depicted an Earth with oceans between Europe's western tip and Asia's eastern tip, which was well within range of the provisions that ships of that time could carry. Columbus' estimate of the distance to Asia was wrong, as was his assumption that there was no land between Europe and Asia. Fortunately for him, these two "wrongs" made a "right," with all of its attendant fame and glory.

So far, we have given external information (i.e., lunar eclipses and the North Star) about Earth's spherical shape based upon its position in the universe. If we use this method to determine the universe's shape, we must observe it in an external manner. As this is not possible, let's reconsider the question of Earth's shape with a slight change: Can we determine Earth's shape by using measurements and observations done only on its surface, and thereby acquire intrinsic information that can inform us of the universe's shape?

Karl Gauss (1777-1855) answered this question positively by inventing "curvature," which measures a given surface's "bumpiness" at a specific point. A flat piece of paper has no bumps and so its curvature is zero. But if we look at a sphere at each point, we see some bumpiness. Gauss called such bumpiness "positive curvature." Another kind of bumpiness is "saddle-shaped." We can think of positive curvature at a point as follows: If we put a piece of flat paper on a surface at that point, the surface lies totally on one side of the paper. But in negatively curved space, this cannot happen.

To describe this concept formally (minus some technicalities), assume constant curvatures on the shapes in question. In other words, the shape is totally symmetric and every point has the same amount of bumpiness. There are several ways to describe curvature. Gauss's formulation for curvature is brilliant. But before that, let's look at his intrinsic proof for a spherical Earth. Imagine an orchard so large that any deviation from flatness is perceptible. First plant trees on the Equator every 100 kms (the approximate distance between two meridians on the Equator). Then plant another tree 100 kms (the approximate distance between two parallels) north of each tree, and do this several times. If Earth is flat, the distance between them would be same. But since the distance between the two consecutive trees (on the same parallel) decreases, Earth is spherical.

Having seen that an intuitively positive curvature implies a spherical shape, we want to follow this method to get an idea about the universe's shape. Georg Riemann (1826-66), trying to do just that, invented "curved space" and explained how to compute its curvature. We could launch six probes at equally spaced points along the Equator, and have each of them continually monitor the distance to the two adjacent probes. If space is flat, the distances at any point in its journey would equal the distance from the probe to Earth's center (an equilateral triangle). For negative curvature, the distance between probes would grow faster than the distance the probe had traveled from Earth; in positively curved space, the distance between probes would grow slower than the distance covered by the probes since leaving Earth.

There are two common misconceptions about the curvature of space. The first one is that curvature is a rather vague or qualitative concept. In reality, it is quite precise and assigns to each point in space and each direction at that point an exact number determined by the shape of the space near the specific location. The second one is that to describe curved space, one must think of it as "curving" into a fourth dimension. This can be useful in visualizing curved space for people familiar with four-dimensional Euclidean space (four-dimensional coordinate space). Unfortunately, science popularizers and science fiction writers often lace this concept with mystical overtones. This is more likely to confuse average people. In other words, measurements made in ordinary three-dimensional space may disagree with the results embodied in Euclidean geometry, for curvature measures the degree and kind of deviation from the Euclidean model.

Riemann also proposed a radically different (non-Euclidian) model for the universe: "spherical space." This would be the case if space had a constant positive curvature. Based on this, he said that the universe should be a hypersphere (a three-dimensional sphere). The usual sphere is two-dimensional and lives in three-dimensional Euclidean space. In general, n-dimensional sphere is described as in the (n+1)-dimensional Euclidean space, and the set of points whose distance from origin (the point 0) is 1.

The more intuitive way to describe hypersphere comes from the usual sphere. Starting from a point in the sphere called the South Pole, and as we go in a direction in the sphere, we see concentric circles becoming larger until we reach the Equator, after which they become smaller and we finally reach North Pole. The situation is similar in hypersphere. Start from a point in the sphere called the South Pole, and as we go in a direction in the sphere, the concentric "spheres" become larger until we reach the Equator, after which they become smaller until we reach the North Pole. We can generalize this concept for any sphere of any dimension.

Earlier philosophers speculated that the universe was infinite in extent; others (e.g., Plato, Aristotle, Newton, and Leibniz) rejected this as implausible. But the alternative seemed equally dubious: If it did not go on forever, then "like the flat Earth" it had to end somewhere. And, what was beyond that? This model solved the Euclidean paradox of the universe's "edge," for if the universe is positively curved, it can be finite in extent and still not have any "edge." In Riemann's model, every part of the universe looks just like every other part, as far as shapes and measurements go.

Qur'an 51:47-48 mentions the universe's spreading out or expansion in space: And the firmament: We constructed it with power and skill, and We are spreading it. This verse reveals that the distance between celestial bodies is increasing, which means that the universe is expanding.

The most surprising discovery of the twentieth century was made by Edwin Hubble in 1929: The universe is not static, but is in a state of rapid expansion. Based upon his observations, he stated Hubble's Law: Other galaxies are receding from us, the rate at which they recede depends upon their distance, and there is a constant ratio (the Hubble constant) between their velocity and their distance from us.

This law's most dramatic consequence is what it tells about how we got to where we are now. If distances between galaxies increase as we look toward the future, they must decrease as we go back in time. Each ring of galaxies must have been closer to us in the past; the further away (or back in time) we go, the closer they would have been, and the faster they appear to be moving toward us.

Hubbel's evidence was limited to a few relatively nearby galaxies. Over the years, however, thousands of observations extended and refined the measurements, and confirmed the general correctness of the velocity-distance relation. Current best estimates are that those galaxies are a billion light-years away (a light-year is roughly 6 trillion miles). Assuming that light always travels at the same speed, those galaxies must have been 1/20 of a light-year closer to us each year in the past. To have ended up a billion light years away, they must have started at exactly the same point as we did "the Big Bang" some 20 billion years ago.

Let's start by using concentric rings of galaxies at intervals of a billion light-years. Then there are 20 rings, because five rings from us represents galaxies 5 billion light-years distant from us. To see them, we need to see their light that has been traveling for 5 billion years. Thus, we now see their position 5 billion years ago. As there was nothing 20 billion years ago, the outmost ring must the twentieth ring. This might sound paradoxical, as the circles of galaxies seem to grow larger as they move further away from us. However, the paradox is only apparent. Assuming Earth is in the South Pole and that the rings are a sphere's latitudes, the rings become larger by the Equator and then become smaller until, in the twentieth ring, we reach the North Pole. This time, the rings are spheres and thus fit in the hypersphere. So Hubble's Law supports our model of hypersphere for the universe.

But how can an expanding universe fit into our picture? In the sphere, the whole surface is expanding, just like inflating a balloon. So the distance from us (at Earth) to the Big Bang is increasing in all directions. In other words, any two points in the universe recede from each other, just as any two points on the balloon recede from each other during inflation.

So far, we have considered the universe's shape at a fixed time. But, in physics, it is useful to consider space and time together. After Einstein's brilliant publications about special relativity, Hermann Minkowski (1864-1909) proposed a very useful four-dimensional space-time model as the fabric of the physical universe. In a global picture, each fixed time represents a thin slice of space-time. Like an onion, each layer (assuming there are infinite very thin layers) corresponds to the universe at different fixed times. Given that each fixed time is a hypersphere, the layers are hyperspheres. According to Hubble's Law, the hypersphere becomes larger as time passes. Just like an onion, the inner layers are smaller and the outer layers are larger.

In an ordinary onion, the layers are usually spheres; in the universe, the layers are hyperspheres. Assuming that the outer-most slice represents the universe at this time, the space-time "so far" is a four-dimensional onion, with layers of the universe at different times. The center of this "onion" corresponds to the Big Bang. We receive the picture of space-time until "this time." Now, let mathematics predict the future of the space-time, just as Newton's laws allowed a detailed description of the solar system's future course. As time evolves, the universe expands, distances between galaxies grow, and gravity weakens. Thus, the space-time curvature diminishes and successive hyperspheres grow at a slower rate.

Two possibilities emerge: The hyperspheres continue to grow indefinitely, although at an ever-decreasing rate, or reach a maximum size and start to contract, in exactly the same fashion as the parallels of latitude on Earth: starting at the North Pole, growing until they reach their maximum size at the Equator, and then begin contracting toward the South Pole. If the universe contracts, distances between galaxies would decrease, gravity and curvature would increase, and the successive hyperspheres would shrink ever faster, eventually contracting into a single point: the "Big Crunch."

We could then draw a map of the universe as a succession of hyperspheres growing in size for during the first half of its life and contracting during the second half. All space-time would then form a kind of super-hypersphere a four-dimensional object known as a "four-dimensional sphere."

At a fixed time, the universe should be a hypersphere. When the time dimension is added, space-time should be super-hypersphere, with a "Big Bang" like the South Pole, each fixed time of the universe corresponding to the parallels, and finally a "Big Crunch" corresponding to the North Pole in our space-time model.

Almost everybody has heard that time is the fourth dimension. Even though this concept is easy to imagine, people find it hard to understand because of its mystification by science popularizers. We live in three-dimensional space. This means that I can parameterize the universe such that I can describe any point in it by using just three letters (a, b, c).

- All of these statements are only approximately true. They would be exactly true if the North Star was precisely overhead at the North Pole, instead of being off center by about 1.
- The real estimate might not be 25,000 miles, as we do not know the exact correspondence between ancient and current measurements. However, this was a very good estimate for that time.
- Despite the potential errors mentioned in footnote 2, this also was a good estimate for that time.

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