Fundamentals Page 3

  Conversely, to the extent that GPS works, its success reinforces our confidence in all the underlying assumptions, including the assumption that Euclidean geometry describes, with good accuracy, the reality of spatial geometry on earthly scales. And so far, GPS has worked flawlessly.

  More generally, science builds. The most advanced, adventurous experiments and technologies rely on tangled webs of underlying theories. When those adventurous applications hold up, they increase our confidence in their supporting webs. The fact that fundamental understanding forms a tangled, mutually reinforcing web of ideas will be a recurring theme in what follows.

  Before concluding this prelude, I must add a qualification. When we come to consider space on gigantic cosmic scales, as we’re about to do, or with exquisite precision, or in the vicinity of black holes, Euclidean geometry stops matching reality. Albert Einstein, in his special and general relativity theories (in 1905 and 1915, respectively), exposed its inadequacies theoretically and suggested how to get beyond them. Since then, his theoretical ideas have been confirmed in many experiments.

  Einstein taught us, in special relativity, that when we claim to measure “distance” we must consider carefully what it is we’re measuring and how we are measuring it. Real measurements take time, and things can move in time. What we can actually measure is separations between events. Events are located in both space and time. The geometry of events must be constructed within that larger framework: space-time, not just space. In general relativity, we learn further that the geometry of space-time can be warped by the influence of matter, or by waves of distortion that travel through it. (More on this in chapters 4 and 8.)

  Within the more comprehensive frameworks of space-time and general relativity, Euclidean geometry serves as an approximation to more accurate theories. It is accurate enough for use in the many practical applications mentioned above. Surveyors, architects, and designers of space missions use Euclidean geometry because they can get away with it, and it eases their work. The more comprehensive theories, while more accurate, are much more complicated to use.

  The fact that Euclidean geometry fails to provide a complete model of reality does not detract from its mathematical consistency nor invalidate its many successes. But it does confirm the wisdom of Gauss’s fact-checking, radically conservative approach. The relationship between geometry and reality is a question for Nature to settle.

  Surveying the Universe

  Having taken the measure of nearby space, we can proceed to survey the cosmos. The primary tools in this endeavor are various kinds of telescopes. Besides the familiar telescopes that employ visible light, astronomers use telescopes that gather “light” from many other parts of the electromagnetic spectrum, including radio waves, microwaves, infrared, ultraviolet, x-rays, and gamma rays. There are also more exotic eyes on the sky, not based on electromagnetic radiation, notably including a very recent addition, gravitational wave detectors. I’ll say more about those in later chapters.

  Let me begin by highlighting the amazingly simple conclusions of this survey. Then I’ll review how astronomers reached them. That is more complicated—though, given the context, still amazingly simple.

  The most fundamental conclusion is that we find the same kind of material everywhere. Furthermore, we observe that the same laws apply everywhere.

  Second, we find that matter is organized into a hierarchy of structures. Everywhere we look, we can recognize stars. They tend to cluster into galaxies, which commonly contain anywhere from a few million to billions of stars. Our own star, the Sun, has a retinue of planets and moons (and also comets, asteroids, the beautiful “rings” of Saturn, and other debris). Jupiter, the largest planet, has about one-thousandth the weight of the Sun, while Earth has about three-millionths the weight of the Sun. Despite their modest share of mass, planets and their moons should be especially dear to our hearts. We live on one, of course, and there are reasons to suspect that others might support new forms of life—if not in our solar system, then elsewhere. Astronomers have long suspected that other stars might have planets, but it is only recently that they’ve developed the technical strength to detect them. By now, hundreds of extrasolar planets have been discovered, and new discoveries keep flooding in.

  Third, we find that all this stuff is sprinkled nearly uniformly throughout space. We find roughly the same density of galaxies in all directions, and at all distances.

  Later we will refine and supplement these three fundamental conclusions, notably to bring in the big bang, “dark matter,” and “dark energy.” But their central message endures: One finds the same sorts of substances, organized in the same sorts of ways, spread uniformly over the visible universe, in vast abundance.

  By now you may be wondering how astronomers arrive at such far-reaching conclusions. Let’s have a closer look, while filling in concrete values of the sizes and distances.

  It is not immediately obvious how to measure the distance to very distant objects. Obviously, you can’t lay down rulers, stretch tape measures into the sky, or monitor time-stamped radio transmissions. Instead, astronomers use a bootstrap technique, called the cosmic distance ladder. Each rung of the ladder takes us to larger distances. We use our understanding at one rung to prepare us for the next.

  We can start by surveying distances in the immediate neighborhood of Earth. Using similar techniques to GPS—that is, bouncing light (or radio signals) around, and measuring transit times—we can determine distances on Earth, and distances from Earth to other objects in the solar system. There are several other ways to do this, including some ingenious, though not very accurate, methods invented by the ancient Greeks. For present purposes, it is enough to note that all of these methods give consistent results.

  Earth itself is a near-perfect sphere, whose radius is roughly 6,400 kilometers, or 4,000 miles. In this age of air travel, that is a readily comprehensible distance. It is roughly equal to the overland distance between New York and Stockholm, or slightly more than half the distance between New York and Shanghai.

  There is another way of stating distance, which is beautifully adapted to astronomy and cosmology, and is widely used in those subjects. Namely, to specify a distance we can specify how long it would take a light beam to travel that distance. For Earth’s radius, that computes to about one-fiftieth of a second. We say, therefore, that Earth’s radius is equal to one-fiftieth of a light-second.

  At higher rungs in the cosmic distance ladder it becomes more practical to measure distances in light-years, rather than light-seconds. To get started with that, and for comparison purposes, let me record now that Earth’s radius is roughly one-billionth of one light-year. Keep that tiny number in mind as we expand our survey of the world. It will soon encompass whole light-years, and then hundreds, millions, and finally billions of them.

  Our next milestone length is the distance from Earth to our Sun. That distance is about 150 million kilometers, or 94 million miles. It is also 8 light-minutes, or about 15 millionths of one light-year.

  Notably, the distance from Earth to the Sun is about 24,000 times Earth’s radius. That startlingly large number emphasizes that even within the solar system, all of Earth, let alone a single human, really is “swallowed like a speck.”

  If you let such things bother you, be warned that it gets much worse. Our climb up the cosmic distance ladder has barely begun.

  Knowing the size of Earth’s orbit around the Sun, we can use it to determine the distance to some relatively nearby stars directly, using Euclidean geometry. Those stars are close enough that their position in the sky changes perceptibly over the course of the year, due to Earth’s motion around the Sun. This effect is known as parallax. Our binocular vision uses parallax to gauge our distance to much nearer objects, which present different angles to our two eyes. The Hipparcos space mission, which operated from 1989 to 1993, used parallax to catalogue distances to about a hundred thousand (
relatively) nearby stars.

  The nearest star, Proxima Centauri, is a little over four light-years away. It has two nearby partners. Barnard’s Star, the next nearest independent star, is about six light-years away. Communications with (hypothetical) extraterrestrials based in either of those systems, or with their future cyborg settlers, will require an abundance of patience.

  Relative to interstellar space, our solar system is a cozy little den. The distance from our Sun to Proxima Centauri is about half a million times the distance from Earth to the Sun.

  The key technique for extending the cosmic distance ladder still further exploits the fact, mentioned earlier, that we find the same sorts of objects and materials wherever we look. If we can identify a class of objects that all have the same intrinsic brightness, we say that those objects supply a “standard candle.” If we know the distance to one realization of a standard candle, we can determine the distance to any other, simply by comparing the brightness we observe. For example, if one such source is twice as far away as another, then it will appear one-fourth as bright.

  Now, all this begs the question of how we can convince ourselves that objects seen at different faraway places would have the same brightness if we got up close. The basic idea is that we look for classes of objects that have many properties in common, hope for the best, and check for consistency. A simple example will illustrate the basic idea, and its pitfalls.

  Stars in general are much too diverse to serve as standard candles. White-hot Sirius A is about twenty-five times brighter than our Sun, while its nearby companion Sirius B, a dwarf star, is about one-fortieth as bright, even though both are— astronomically speaking—roughly equally distant from Earth. We can do much better by restricting our comparisons to stars that have the same color—or, more precisely, stars that emit the same electromagnetic spectrum.* When we compare such otherwise identical-looking stars it is reasonable to hope that the difference in their brightness arises from a difference between their distances. The physical theory of stars, which explains many of their observed features, predicts this. But how can we check? One way is to find a compact group containing many stars close to one another. The Hyades cluster, which contains many hundreds of stars, is a prime example. If stars with very similar spectra have very similar intrinsic brightness, then two such stars that are within the same cluster should appear equally bright. And that’s basically what we find.

  Professional astronomers need to take several other complications, such as the effect of interstellar dust, into account. This dust, by absorbing light, can make objects appear more distant than they are. I hope my colleagues will excuse me for gliding over that and many other technicalities, which don’t change the central idea.

  We can extend our cosmic distance ladder, and “climb” from nearby objects to the limits of the visible universe, by using a variety of standard candles. Some kinds work better for relatively nearby objects, others for very distant ones. We must also check that they yield mutually consistent results.

  The Hipparcos catalogue, mentioned earlier, gives us solid footing for our next step up the cosmic distance ladder. Having learned that similar stars have the same intrinsic brightness, we can use them to get the distance to more distant clusters, which are too far away to show observable parallax.

  In this way, we can survey our own galaxy, the Milky Way. We discover that the stars in the Milky Way define a fairly flat disc, with a bulge in the middle. And we measure that the Milky Way is about a hundred thousand light-years across.

  Cepheid variables are bright stars that pulsate. By careful study of Cepheid variables in the Magellanic Clouds,* Henrietta Leavitt (1868–1921) established that Cepheid variables that pulsate at the same rate also have the same brightness, and so provide standard candles. Cepheid variables are relatively easy to spot, because they are unusually bright as well as uniquely variable. Using Cepheid variables as their standard candles, astronomers have measured our distance to many galaxies.

  Galaxies are distributed irregularly, so there is no unique value of the distance between them. Still, we can identify a typical distance between a galaxy and its nearest large neighbor. That intergalactic distance turns out to be a few hundred thousand light-years. Unlike the situation for stars or planets, which almost invariably are separated from their neighbors by distances many times their own size, the typical separation between galaxies is not vastly larger than the galaxies themselves.

  There are several other useful standard candles, and many more interesting details of structure, within the realm of galaxies. Those riches of astronomy add depth to the picture I’ve sketched so far and reinforce its basic messages. But since my goal is to convey fundamentals, rather than to provide encyclopedic coverage, let us proceed to the farthest frontiers without further ado.

  The Cosmic Horizon

  In his pioneering studies of distant galaxies, using Cepheid variables as his primary tool, Edwin Hubble (1889–1953) discovered something fundamentally new, and rich in consequences. He observed that the patterns of starlight that distant galaxies emit—their spectra—are shifted toward longer wavelengths, compared with the light patterns of closer galaxies. This is called a redshift. The reason for that name is that if you systematically expand the wavelengths in a rainbow’s light, then the colors of its stripes will change. Colors on the blue side will shift toward colors on the red side. This effect continues beyond what is humanly visible: A “new” blue stripe will appear where ultraviolet was before, and the red stripe will fade out into infrared.

  Hubble’s redshift observations have a compelling interpretation, which revolutionized our picture of the universe. The interpretation relies on a simple but striking effect, first described by Christian Doppler in 1842. Doppler pointed out that if a source of waves is moving away from us, then successive peaks in the wave pattern it emits will come from farther away, so that the waves will arrive stretched out. In other words, the observed waves will be shifted toward longer wavelengths than they would have had were the source stationary. The straightforward interpretation of Hubble’s redshifts, therefore, is that they indicate the galaxies are moving away from us.

  Hubble discovered a strikingly simple pattern within the redshifts he observed: The farther the galaxy, the larger the redshift. In more detail, he observed that the size of the redshift is proportional to the distance. This means that the distant galaxies are moving away with speeds proportional to their distance.

  If we imagine reversing the galaxies’ motions to reconstruct the past, then that proportionality acquires dramatic new meaning. It means that in the reversed flow, the more distant galaxies will be moving toward us more rapidly, covering the distance in just such a way that everything comes together at the same time. Thus, we are led to suspect that in the past all the matter in the universe was packed together much more tightly than it is today. Switching back to the original direction of time, it looks like a cosmic explosion.

  Might the universe have emerged with a bang? When the Jesuit priest Georges Lemaître first proposed that interpretation of Hubble’s observations, his “big bang” was a bold and beautiful idea, but the evidence for it was skimpy, and it lacked a firm basis in physics.* (Lemaître himself spoke of “the primeval atom” or “the cosmic egg.” The less poetic name “big bang” came later.) But subsequent research has given us a much better understanding of matter in extreme conditions. Today, the evidence for the big bang concept is overwhelming. In chapter 6, we’ll discuss cosmic history much more deeply, and review that evidence.

  Here, to round off our survey of the cosmos, we’ll use the big bang picture to define the limit and extent of the visible universe. Running the movie of cosmic history backward in our minds, we found the galaxies all coming together to meet at a definite time. When did it happen? To calculate how long ago, we simply divide the distance a galaxy must travel by the speed at which it’s moving. (Since a galaxy’s speed is proport
ional to its distance, according to Hubble’s observations, we’ll find the same result consistently whichever galaxy we choose.) Doing that, we estimate that galaxies were all smushed together about 20 billion years ago. More accurate calculations, which include how the velocities change over time due to gravity, yield a somewhat smaller result. Today’s best estimate is that 13.8 billion years have elapsed since the big bang.

  When we look out to objects in the distant cosmos, we are looking at their past. Because light travels at a finite speed, the light from a distant object that we receive today had to be emitted long ago. When we look back 13.8 billion years or so, all the way to the big bang, we reach the limits of our vision. We get “blinded by the light.” The initial cosmic explosion was so bright that we can’t see beyond it. (At least, no one knows how.)

  And because we can’t see beyond a certain time, so, too, we can’t see beyond a certain distance—namely, the distance that light can travel in the limited time available. However large the universe “really” is, the presently visible universe is finite.

  How large is it? Here is where the idea of measuring distance in terms of light-years really shines. Since the limiting time is 13.8 billion years, the limiting distance is . . . 13.8 billion light-years! To bring the immensity home, let us recall here that Earth’s radius is about one-billionth of one light-year.

  With that wild contrast, our survey of cosmic size is complete. The world is large. There’s plenty of room for humans to thrive in, and plenty left over for us to admire from a distance.

  INNER PLENTY: WHAT WE KNOW AND HOW WE KNOW IT